Finite Group Databasebeta

← #29706555

Subgroups — #29706555

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449106C211yesmaximal, characteristic
2#21446758C211yesmaximal, characteristic, complemented
3#21446757C211yesmaximal, characteristic, complemented
4#21446435C211yesmaximal, characteristic, complemented
5#21446434C211yesmaximal, characteristic, complemented
6#21446433C211yesmaximal, characteristic, complemented
7#21446430C211yesmaximal, characteristic, complemented
8#21446113C211yesmaximal, characteristic
9#21446111C211yesmaximal, characteristic
10#21446000C211yesmaximal, characteristic, complemented
11#21446000C211yesmaximal, characteristic, complemented
12#21445997C211yesmaximal, characteristic, complemented
13#21445938C211yesmaximal, characteristic, complemented
14#21445937C211yesmaximal, characteristic, complemented
15C61×#1850C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×C22⋊D61C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839704C411yescharacteristic
31#16839703C411yescharacteristic
32#16839691C411yescharacteristic
33#16839690C411yescharacteristic
34#16839678C411yescharacteristic
35#16839677C411yescharacteristic
36#16839675C411yescharacteristic
37#16839674C411yescharacteristic
38C4×#11758198C2211yescharacteristic
39C4×#11758198C2211yescharacteristic
40C4×C22⋊Dic61C2211yescharacteristic
41#16839608C2211yescharacteristic
42#16839517C2211yescharacteristic
43#16839517C2211yescharacteristic
44#16839516C2211yescharacteristic
45#16839516C2211yescharacteristic
46#16839516C2211yescharacteristic
47#16839516C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C4×C22⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2×C411yescharacteristic
60C22×C22⋊D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—41no—
63C2×#11758189—41no—
64C2×#11758189—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C4×Dic122—41no—
70C4×Dic122—41no—
71C4×C22⋊D61—41no—
72C4×C22⋊D61—41no—
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×#11758196C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×C22⋊Dic61C2×C411yescharacteristic
80C2×#11758198C2×C411yescharacteristic
81C2×#11758198C2×C411yescharacteristic
82C2×#11758198C2×C411yescharacteristic
83C2×#11758198C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C4⋊#92418—41no—
89#14519706—41no—
90#14519706—41no—
91#14519706—41no—
92C22⋊#92418—41no—
93C22⋊#92418—41no—
94#14519689—41no—
95#14519689—41no—
96#14519689—41no—
97#14519688—41no—
98#14519687—41no—
99#14519684C2×C411yescharacteristic
100#14519684C2×C411yescharacteristic
101#14519684C2×C411yescharacteristic
102#14519684C2×C411yescharacteristic
103#14519684C2×C411yescharacteristic
104#14519684C2×C411yescharacteristic
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107#14519684C2×C411yescharacteristic
108#14519684C2×C411yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61—41no—
111C22×C8×C61—41no—
112C22×C8×C61—41no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C22⋊C4×C61C2×C411yescharacteristic
118C2×C22⋊C4×C61C2×C411yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C4⋊C8×C61—41no—
122C4⋊C8×C61—41no—
123C22⋊C8×C61—41no—
124#95×C61C2×C411yescharacteristic
125#95×C61C2×C411yescharacteristic
126C24×C61C22×C411yescharacteristic
127C23×D61C22×C411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—22no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×Dic122—41no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—44no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—41no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—22no—
179C22⋊Dic61—41no—
180C22⋊Dic61—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186C4⋊Dic61—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208#11758189—44no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61#13112yes—
216C23×C61C23×C411yescharacteristic
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C22⋊D61—44no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275Dic122—44no—
276C8×C61—41no—
277C8×C61—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#23354—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61#51811yescharacteristic
293C22×C61#52011yescharacteristic
294C22×C61#52011yescharacteristic
295C22×C61C2×#13111yescharacteristic
296C22×C61#56811yescharacteristic
297C22×C61#56811yescharacteristic
298C22×C61#56811yescharacteristic
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61H(3,𝔽4)11yescharacteristic
312C4×C61#56211yescharacteristic
313C4×C61#56211yescharacteristic
314C4×C61#56311yescharacteristic
315Dic61—41no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61—41no—
320Dic61—41no—
321Dic61—41no—
322#2398—611no—
323#1985—611no—
324#1985—611no—
325#1924—611no—
326#1924—611no—
327#1924—611no—
328#1924—611no—
329#1870—611no—
330#1869—611no—
331#1851—611no—
332#1851—611no—
333#1850—611no—
334#1850D6111yescharacteristic, complemented
335#1832—611no—
336#1832—611no—
337C2×C61—42no—
338C2×C61#237911yescharacteristic
339C2×C61#297611yescharacteristic
340C2×C61#297811yescharacteristic
341C2×C61#299211yescharacteristic
342C2×C61#299211yescharacteristic
343C2×C61#299511yescharacteristic
344C2×C61#299511yescharacteristic
345D61—81no—
346C22×M16—611no—
347C23×C8C2×D6111yescharacteristic
348C2×C4×Q8—611no—
349C2×C4×D4—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8C2×D6111yescharacteristic
355C2×C4⋊C8C2×D6111yescharacteristic
356C2×C22⋊C8—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×C22⋊C8—611no—
360C2×C22⋊C8—611no—
361C2×C22⋊C8C2×D6111yescharacteristic
362C2×#97—611no—
363C2×#97—611no—
364#400—611no—
365#400—611no—
366#397—611no—
367#397—611no—
368#394—611no—
369#394—611no—
370#393—611no—
371#393—611no—
372C4×C22⋊C4—611no—
373C4×C22⋊C4—611no—
374C4×C22⋊C4—611no—
375C4×C22⋊C4C2×D6111yescharacteristic
376#376—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387#336C2×D6111yescharacteristic
388#336C2×D6111yescharacteristic
389C61#2335411yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×D4—611no—
392C23×C4—611no—
393C23×C4C22×D6111yescharacteristic
394C2×M16—2441no—
395C2×M16—2441no—
396C2×M16—2441no—
397C22×C8—41no—
398C22×C8—41no—
399C22×C8—41no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4×Q8—2441no—
409C4×Q8—2441no—
410C4×D4—2441no—
411C4×D4—2441no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4C4×D6111yescharacteristic
425C2×C22⋊C4C4×D6111yescharacteristic
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431C4⋊C8—41no—
432C4⋊C8—41no—
433C4⋊C8—2441no—
434C4⋊C8—2441no—
435C4⋊C8—2441no—
436C4⋊C8—2441no—
437C22⋊C8—41no—
438C22⋊C8—2441no—
439C22⋊C8—2441no—
440C22⋊C8—2441no—
441C22⋊C8—2441no—
442C22⋊C8—2441no—
443#97—2441no—
444#97—2441no—
445#95—611no—
446#95—611no—
447#95—611no—
448#95—611no—
449#95—611no—
450#95—611no—
451#95—611no—
452#95—611no—
453#95—611no—
454#95—611no—
455#95C4×D6111yescharacteristic
456#95C4×D6111yescharacteristic
457C24—611no—
458C24C2×C4×D6111yescharacteristic
459C2×Q8—2441no—
460C2×Q8—2441no—
461C2×Q8—2441no—
462C2×D4—1222no—
463C2×D4—2441no—
464C2×D4—2441no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×C4×D6111yescharacteristic
487C22×C4C2×C4×D6111yescharacteristic
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—2444no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—44no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4—2441no—
520C22⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4—1222no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C22⋊C4—2441no—
529C22⋊C4—2441no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23#1451978312yes—
554C23C22×C4×D6111yescharacteristic
555Q8—2444no—
556D4—2444no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4#1451982211yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982211yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982411yescharacteristic
606C2×C4#1451982511yescharacteristic
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—2441no—
612C8—2441no—
613C8—2441no—
614C8—2441no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—2441no—
623C22—2441no—
624C22—2441no—
625C22—2441no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22#1684023411yescharacteristic
630C22#1684024111yescharacteristic
631C22#1684024311yescharacteristic
632C22C2×#1451978311yescharacteristic
633C22#1684052711yescharacteristic
634C22#1684052711yescharacteristic
635C22#1684052811yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—2441no—
642C4—2441no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4—2441no—
647C4—2441no—
648C4#1684050511yescharacteristic
649C4#1684050811yescharacteristic
650C4#1684050911yescharacteristic
651C4#1684051311yescharacteristic
652C2—42no—
653C2—4881no—
654C2#2144899211yescharacteristic
655C2#2145486111yescharacteristic
656C2#2145487811yescharacteristic
657C2#2145495211yescharacteristic
658C2#2145495511yescharacteristic
659C2#2145498511yescharacteristic
660C2#2145498711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1850C2
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1850D61
hypocenterC61#23354
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21446758C2semidirect
3. #21446757C2semidirect
4. #21446435C2semidirect
5. #21446434C2semidirect
6. #21446433C2semidirect
7. #21446430C2semidirect
10. #21446000C2semidirect
11. #21446000C2semidirect
12. #21445997C2semidirect
13. #21445938C2semidirect
14. #21445937C2semidirect
15. C61×#1850C2semidirect
334. #1850D61semidirect
389. C61#23354semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214491062
#214467582
#214467572
#214464352
#214464342
#214464332
#214464302
#214461132
#214461112
#214460002
#214459972
#214459382
#214459372
C61×#18502
#2335461