Finite Group Databasebeta

← #30617266

Subgroups — #30617266

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840813C211yesmaximal, characteristic, complemented
2C2×C4×#11758213C211yesmaximal, characteristic
3#21461993C211yesmaximal, characteristic, complemented
4#21461921C211yesmaximal, characteristic, complemented
5#21461874C211yesmaximal, characteristic, complemented
6#21461873C211yesmaximal, characteristic, complemented
7#21461767C211yesmaximal, characteristic
8#21461765C211yesmaximal, characteristic
9#21461676C211yesmaximal, characteristic
10#21461619C211yesmaximal, characteristic, complemented
11#21461618C211yesmaximal, characteristic, complemented
12#21461611C211yesmaximal, characteristic, complemented
13#21461560C211yesmaximal, characteristic
14#21461560C211yesmaximal, characteristic
15#21448357C211yesmaximal, characteristic
16C22×#11758213C2211yescharacteristic
17C22×#11758213C2211yescharacteristic
18#16840813—24no—
19C2×#14519844C2211yescharacteristic
20C2×#14519844C2211yescharacteristic
21C2×#14519844C2211yescharacteristic
22C2×#14519844C2211yescharacteristic
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4⋊#92420C2211yescharacteristic
25C2×#14519834C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C4×#11758213—24no—
28C2×#14519832C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×#14519832C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34C2×C4×#92420C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519753C411yescharacteristic
47C2×Q8×Dic61C2211yescharacteristic
48C2×C42×D61C2211yescharacteristic
49#16839660C411yescharacteristic
50#16839649C411yescharacteristic
51#16839649C411yescharacteristic
52#16839636C411yescharacteristic
53#16839636C411yescharacteristic
54#16839619C411yescharacteristic
55C4⋊C4×Dic61C2211yescharacteristic
56#16839613C2211yescharacteristic
57#16839613C2211yescharacteristic
58#16839609C2211yescharacteristic
59#16839609C2211yescharacteristic
60C61×#391C411yescharacteristic
61C2×#11758213—22no—
62C2×#11758213—22no—
63C2×#11758213—22no—
64C2×#11758213—22no—
65C2×#11758213—22no—
66C2×#11758213Q814yescomplemented
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73C22×#92420C2311yescharacteristic
74C22×#92420C2311yescharacteristic
75#14519844—22no—
76#14519844—22no—
77#14519844—22no—
78#14519844—22no—
79C4⋊#92420—22no—
80C4⋊#92420—22no—
81#14519834—22no—
82#14519834—22no—
83#14519832—22no—
84#14519832—22no—
85#14519832—22no—
86#14519832D414yes—
87C4×#92420—22no—
88C4×#92420—22no—
89C4×#92420—22no—
90C4×#92420—22no—
91C22×C4×D61C2×C411yescharacteristic
92C22×C4×D61C2×C411yescharacteristic
93C22×C4×D61C2311yescharacteristic
94#14519753—24no—
95Q8×Dic61—22no—
96Q8×Dic61—22no—
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758196C2×C411yescharacteristic
103C2×#11758196C2×C411yescharacteristic
104C2×#11758196C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C42×D61—22no—
117C42×D61—22no—
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123#14519684C2×C411yescharacteristic
124C22×Q8×C61C2×C411yescharacteristic
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C42×C61C2×C411yescharacteristic
127#95×C61C2×C411yescharacteristic
128#95×C61C2×C411yescharacteristic
129#95×C61C2×C411yescharacteristic
130#95×C61C2×C411yescharacteristic
131C23×D61C22×C411yescharacteristic
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—41no—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C2411yescharacteristic
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61C2×Q812yes—
160C2×C4×D61#4112yes—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420C2×D412yes—
180C2×#92420C2×D412yes—
181C2×#92420C2×Q812yes—
182C2×#92420C2×Q812yes—
183C2×#92420#4112yes—
184C2×#92420#4112yes—
185#11758213—28no—
186#11758213—28no—
187C4⋊C4×C61—41no—
188C42×C61—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196M1614yescomplemented
201C4×Dic61—22no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C23×C61C23×C411yescharacteristic
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61C4×Q812yes—
222Q8×C61—44no—
223C2×C4×C61—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—21no—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×Q812yes—
242C2×C4×C61C2×M1612yes—
243C2×C4×C61#13112yes—
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×Dic61C4×D412yes—
260C2×Dic61C2×M1612yes—
261C2×Dic61C2×M1612yes—
262C2×Dic61C22×D411yescharacteristic
263C2×Dic61C22×Q811yescharacteristic
264C2×Dic61C2×#4111yescharacteristic
265C2×Dic61C2×#4111yescharacteristic
266C2×Dic61C2×#4111yescharacteristic
267C2×Dic61C2×#4111yescharacteristic
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—44no—
273C4×D61—44no—
274#92420—22no—
275#92420—22no—
276#92420—22no—
277#92420—41no—
278#92420—41no—
279#92420—41no—
280#92420—41no—
281#92420C22⋊Q814yescomplemented
282#21600—611nomaximal, Sylow
283C22×C61C2×C4×D411yescharacteristic
284C22×C61C2×C4×Q811yescharacteristic
285C22×C61C4×#4111yescharacteristic
286C22×C61C22×M1611yescharacteristic
287C22×C61C2×#13111yescharacteristic
288C22×C61#56811yescharacteristic
289C22×C61#56811yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—41no—
308Dic61—41no—
309Dic61—41no—
310Dic61C2×C22⋊Q811yescharacteristic
311Dic61#52511yescharacteristic
312Dic61#52911yescharacteristic
313Dic61#53311yescharacteristic
314C2×#446—611no—
315C2×C4×M16—611no—
316C4×C22⋊Q8—611no—
317#1985—611no—
318#1940—611no—
319#1916—611no—
320#1916—611no—
321#1851—611no—
322#1851—611no—
323#1797—611no—
324#1771—611no—
325#1771—611no—
326C8×C4⋊C4—611no—
327#1753—611no—
328#1753—611no—
329C2×C61C4×C22⋊Q811yescharacteristic
330C2×C61#293311yescharacteristic
331C2×C61D4×M1611yescharacteristic
332C2×C61#296211yescharacteristic
333C2×C61Q8×M1611yescharacteristic
334C2×C61#296511yescharacteristic
335C2×C61#296711yescharacteristic
336D61—42no—
337C22×M16—611no—
338C22×M16—611no—
339C2×C22⋊Q8—611no—
340C2×C4×Q8—611no—
341C22×C42—611no—
342#446—1224no—
343C2×C4⋊C8—611no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8—611no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×C22⋊C8—611no—
350C2×C22⋊C8—611no—
351C4×M16—1224no—
352C2×#97—611no—
353C2×#97—611no—
354C2×#97—611no—
355C2×#97—611no—
356C2×C4×C8—611no—
357C2×C4×C8—611no—
358C2×C4×C8—611no—
359C2×C4×C8—611no—
360#391#692311yescharacteristic
361#389—611no—
362#387—611no—
363#387—611no—
364#385—611no—
365#385—611no—
366#380—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—611no—
371C4×C22⋊C4—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382C61#2160011yescharacteristic, complemented, Sylow
383C22×Q8C2×#692311yescharacteristic
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C2×M16—614no—
388C2×M16—1222no—
389C2×M16—1222no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22⋊Q8—1224no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4C2×#692311yescharacteristic
413C2×C22⋊C4—611no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1222no—
425C2×C42—1222no—
426C2×C42C2×#692311yescharacteristic
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435#97—614no—
436#97—1222no—
437#97—1222no—
438#97—1222no—
439C4×C8—1222no—
440C4×C8—1222no—
441C4×C8—1222no—
442C4×C8—1222no—
443#95—611no—
444#95—611no—
445#95—611no—
446#95—611no—
447#95—611no—
448#95—611no—
449#95C2×#692311yescharacteristic
450#95C2×#692311yescharacteristic
451#95C2×#692311yescharacteristic
452#95C2×#692311yescharacteristic
453C24—611no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—41no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4C22×#692311yescharacteristic
477C22×C4C22×#692311yescharacteristic
478C22×C4C22×#692311yescharacteristic
479C22×C4C22×#692311yescharacteristic
480C22×C4C22×#692311yescharacteristic
481C22×C4C22×#692311yescharacteristic
482C22×C4C22×#692311yescharacteristic
483M16—1228no—
484M16—1228no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8—612no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C4⋊C4—41no—
510C4⋊C4—614no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—2441no—
521C22⋊C4—2441no—
522C42—22no—
523C42—612no—
524C42—612no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—1222no—
534C42—1221no—
535C42—1221no—
536C23—612no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23C23×#692311yescharacteristic
544Q8—44no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1221no—
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—2444no—
591C2×C4—2444no—
592C2×C4D4×#692312yes—
593C2×C4Q8×#692312yes—
594C2×C4D61⋊M1612yes—
595C2×C4#1451986512yes—
596C8—614no—
597C8—1222no—
598C8—1222no—
599C8—1222no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C8—2441no—
604C22—1222no—
605C22—1222no—
606C22—1222no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22—2441no—
611C22C2×D4×#692311yescharacteristic
612C22C2×Q8×#692311yescharacteristic
613C22#41×#692311yescharacteristic
614C22C2×D61⋊M1611yescharacteristic
615C22C2×#1451986511yescharacteristic
616C22#1684084011yescharacteristic
617C22#1684084111yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—611no—
626C4—611no—
627C4—611no—
628C4—611no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—2441no—
633C4—2441no—
634C4—2441no—
635C2—2442no—
636C2C22⋊Q8×#692311yescharacteristic
637C2#2146255211yescharacteristic
638C2#2146255711yescharacteristic
639C2#2146261511yescharacteristic
640C2#2146262111yescharacteristic
641C2#2146262711yescharacteristic
642C2#2146263911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#391C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenter#391#6923
hypocenterC61#21600
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16840813C2semidirect
3. #21461993C2semidirect
4. #21461921C2semidirect
5. #21461874C2semidirect
6. #21461873C2semidirect
10. #21461619C2semidirect
11. #21461618C2semidirect
12. #21461611C2semidirect
66. C2×#11758213Q8semidirect
200. #11758196M16semidirect
281. #92420C22⋊Q8semidirect
382. C61#21600semidirect

Maximal subgroups

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

subgroupindex
C2×#168408132
C2×C4×#117582132
#214619932
#214619212
#214618742
#214618732
#214617672
#214617652
#214616762
#214616192
#214616182
#214616112
#214615602
#214483572
#2160061