Finite Group Databasebeta

← #29701799

Subgroups — #29701799

Order 15616.

788 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839889C211yesmaximal, characteristic, complemented
2#21449087C211yesmaximal, characteristic
3#21446758C211yesmaximal, characteristic, complemented
4#21446539C211yesmaximal, characteristic, complemented
5#21446535C211yesmaximal, characteristic, complemented
6#21446534C211yesmaximal, characteristic, complemented
7#21446380C211yesmaximal, characteristic, complemented
8#21446380C211yesmaximal, characteristic, complemented
9#21446002C211yesmaximal, characteristic
10#21445997C211yesmaximal, characteristic
11#21445696C211yesmaximal, characteristic, complemented
12#21445600C211yesmaximal, characteristic
13#21445595C211yesmaximal, characteristic
14#21445585C211yesmaximal, characteristic
15C2×C61×#424C211yesmaximal, characteristic
16C2×C4×Dic122C2211yescharacteristic
17#16839889—24no—
18C2×#14519706C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519688C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839701C411yescharacteristic
31#16839698C411yescharacteristic
32#16839695C411yescharacteristic
33#16839695C411yescharacteristic
34#16839685C411yescharacteristic
35#16839685C411yescharacteristic
36#16839682C411yescharacteristic
37#16839674C411yescharacteristic
38C4×#11758196C2211yescharacteristic
39C4×#11758196C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C22⋊Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839517C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839516C2211yescharacteristic
46#16839516C2211yescharacteristic
47#16839516C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C61×#424—24no—
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×C4⋊C8×C61C2211yescharacteristic
60C2×C4⋊C8×C61C2211yescharacteristic
61C22×Dic122C2×C411yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C4×Dic122—22no—
67C4×Dic122—22no—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×C4⋊Dic61C2×C411yescharacteristic
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196C2×C411yescharacteristic
72C2×#11758196C2×C411yescharacteristic
73C2×#11758196C2×C411yescharacteristic
74C2×#11758196C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×C22⋊Dic61C2×C411yescharacteristic
80C2×C22⋊Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519706—22no—
86#14519706—22no—
87#14519706—41no—
88#14519706—41no—
89C22⋊#92418—22no—
90C22⋊#92418Q814yescomplemented
91#14519688—22no—
92#14519687—22no—
93#14519687—22no—
94#14519687—41no—
95C4×#92418—22no—
96C8×Dic61—41no—
97#14519684C2×C411yescharacteristic
98#14519684C2×C411yescharacteristic
99#14519684C2×C411yescharacteristic
100#14519684C2×C411yescharacteristic
101#14519684C2×C411yescharacteristic
102#14519684C2×C411yescharacteristic
103#14519684C2×C411yescharacteristic
104#14519684C2×C411yescharacteristic
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C2×M16×C61—22no—
109C2×M16×C61—22no—
110C2×M16×C61—41no—
111C22×C8×C61—22no—
112C22×C8×C61—22no—
113C22×C8×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118#117×C61—22no—
119#117×C61—22no—
120C2×C4⋊C4×C61C2×C411yescharacteristic
121C2×C4⋊C4×C61C2×C411yescharacteristic
122C2×C22⋊C4×C61C2×C411yescharacteristic
123C2×C22⋊C4×C61C2×C411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127C4⋊C8×C61—22no—
128C4⋊C8×C61—22no—
129C4⋊C8×C61D414yes—
130C24×C61C22×C411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C2×Q812yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C2×Dic122—22no—
152C2×Dic122—41no—
153C2×Dic122—41no—
154M16×C61—44no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
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—44no—
166C2×C8×C61C2×D412yes—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C2×D412yes—
175C42×C61#4112yes—
176C22⋊Dic61—22no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—41no—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—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×#92418C2×Q812yes—
208C2×#92418C2×Q812yes—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61C4×Q812yes—
216C23×C61C23×C411yescharacteristic
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
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×C61C4×D412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×Q812yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×Q811yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
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#13112yes—
266Dic122—44no—
267C8×C61—22no—
268C8×C61—22no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—22no—
272#92418—22no—
273#92418—41no—
274#92418—41no—
275#22961—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C2×C4×D411yescharacteristic
284C22×C61C2×C4×Q811yescharacteristic
285C22×C61#52011yescharacteristic
286C22×C61C2×#13111yescharacteristic
287C22×C61#56811yescharacteristic
288C22×C61#56811yescharacteristic
289C22×C61#56811yescharacteristic
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61#54611yescharacteristic
296C4×C61D4×Q811yescharacteristic
297C4×C61#55211yescharacteristic
298C4×C61#55411yescharacteristic
299Dic61—22no—
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305Dic61—41no—
306Dic61—41no—
307C2×#446—611no—
308C2×#424D6111yescharacteristic
309#2396—611no—
310#1985—611no—
311#1940—611no—
312#1939—611no—
313#1939—611no—
314#1917—611no—
315#1917—611no—
316#1851—611no—
317#1850—611no—
318#1771—611no—
319#1753—611no—
320#1752—611no—
321#1750—611no—
322C2×C61—42no—
323C2×C61#236011yescharacteristic
324C2×C61#293011yescharacteristic
325C2×C61#293511yescharacteristic
326C2×C61#296111yescharacteristic
327C2×C61#296211yescharacteristic
328C2×C61#299211yescharacteristic
329C2×C61#299511yescharacteristic
330C22×M16C2×D6111yescharacteristic
331C23×C8C2×D6111yescharacteristic
332C2×C4×Q8—611no—
333C2×#117C2×D6111yescharacteristic
334#446—1224no—
335#424—24no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340C2×C4⋊C8C2×D6111yescharacteristic
341C2×C4⋊C8C2×D6111yescharacteristic
342C2×C4⋊C8C2×D6111yescharacteristic
343C2×C4⋊C8C2×D6111yescharacteristic
344C2×C22⋊C8—611no—
345C2×C22⋊C8—611no—
346C2×#97—611no—
347C2×#97—611no—
348C2×#97—611no—
349C2×#97—611no—
350C2×C4×C8—611no—
351C2×C4×C8—611no—
352#400—611no—
353#399—611no—
354#398—611no—
355#398—611no—
356#396—611no—
357#396—611no—
358#395—611no—
359#393—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363C4×C22⋊C4—611no—
364C4×C22⋊C4—611no—
365#336—611no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370#336—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375C61#2296111yescharacteristic, complemented, Sylow
376C22×Q8—611no—
377C23×C4C22×D6111yescharacteristic
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—41no—
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C4×Q8—1222no—
393C4×Q8—1222no—
394#117—22no—
395#117—22no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4C4×D6111yescharacteristic
406C2×C4⋊C4C4×D6111yescharacteristic
407C2×C22⋊C4—611no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—611no—
410C2×C22⋊C4—611no—
411C2×C22⋊C4C4×D6111yescharacteristic
412C2×C22⋊C4C4×D6111yescharacteristic
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419C4⋊C8—22no—
420C4⋊C8—22no—
421C4⋊C8—22no—
422C4⋊C8—1222no—
423C4⋊C8—1222no—
424C4⋊C8—2441no—
425C4⋊C8—2441no—
426C4⋊C8C22⋊D6114yes—
427C22⋊C8—614no—
428C22⋊C8—1222no—
429#97—1222no—
430#97—1222no—
431#97—1222no—
432#97—2441no—
433C4×C8—1222no—
434C4×C8—2441no—
435#95—611no—
436#95—611no—
437#95—611no—
438#95—611no—
439#95—611no—
440#95—611no—
441#95—611no—
442#95—611no—
443#95—611no—
444#95—611no—
445C24C2×C4×D6111yescharacteristic
446C2×Q8—1222no—
447C2×Q8—2441no—
448C2×Q8—2441no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4C2×C4×D6111yescharacteristic
462C22×C4C2×C4×D6111yescharacteristic
463C22×C4C2×C4×D6111yescharacteristic
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4C2×C4×D6111yescharacteristic
466C22×C4#1175822412yes—
467C22×C4Q8×D6112yes—
468C22×C4C23×D6111yescharacteristic
469M16—44no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—44no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8C2×C22⋊D6112yes—
494C2×C8C2×C22⋊D6112yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—1222no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C42—21no—
513C42—21no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42#1175822212yes—
523C42C2×C22⋊D6112yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1451974212yes—
531C23C22×C4×D6111yescharacteristic
532Q8—2444no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—612no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4C4×C22⋊D6112yes—
574C2×C4C4×C22⋊D6112yes—
575C2×C4#1451974212yes—
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4C2×Q8×D6111yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4#1451982211yescharacteristic
581C2×C4#1451982511yescharacteristic
582C8—22no—
583C8—22no—
584C8—41no—
585C8—41no—
586C8—1222no—
587C8—1222no—
588C8—2441no—
589C8—2441no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×C4×C22⋊D6111yescharacteristic
598C22C2×#1451974211yescharacteristic
599C22#1684024311yescharacteristic
600C22#131×D6111yescharacteristic
601C22#1684052711yescharacteristic
602C22#1684052711yescharacteristic
603C22#1684052711yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—1222no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4—2441no—
614C4—2441no—
615C4—2441no—
616C4—2441no—
617C4#1684040611yescharacteristic
618C4Q8×C22⋊D6111yescharacteristic
619C4#1684044711yescharacteristic
620C4#1684046711yescharacteristic
621C2—42no—
622C2#2144882411yescharacteristic
623C2#2145452011yescharacteristic
624C2#2145456111yescharacteristic
625C2#2145475811yescharacteristic
626C2#2145476511yescharacteristic
627C2#2145495611yescharacteristic
628C2#2145498711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC2×C61×#424C2
prime radicalC24×C61C22×C4
cosocleC22×C4×C61C24
hypercenterC2×#424D61
hypocenterC61#22961
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
1. C2×#16839889C2semidirect
3. #21446758C2semidirect
4. #21446539C2semidirect
5. #21446535C2semidirect
6. #21446534C2semidirect
7. #21446380C2semidirect
8. #21446380C2semidirect
11. #21445696C2semidirect
90. C22⋊#92418Q8semidirect
375. C61#22961semidirect

Maximal subgroups

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

subgroupindex
C2×#168398892
#214490872
#214467582
#214465392
#214465352
#214465342
#214463802
#214460022
#214459972
#214456962
#214456002
#214455952
#214455852
C2×C61×#4242
#2296161