Finite Group Databasebeta

← #27144554

Subgroups — #27144554

Order 12416.

1252 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D97C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4C22×#13095488C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260635C211yesmaximal, characteristic, complemented
8#20260615C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260613C211yesmaximal, characteristic, complemented
11#20260594C211yesmaximal, characteristic, complemented
12#20260590C211yesmaximal, characteristic, complemented
13#20260590C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#386×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C24×D97C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C22×C22⋊D97C2211yescharacteristic
21C2×D4×D97—24no—
22C2×D4×D97—24no—
23C2×D4×D97C418yescomplemented
24C23×Dic97C2211yescharacteristic, complemented
25C22×C4×D97C2211yescharacteristic
26C22×C4×D97C2211yescharacteristic, complemented
27C22×C4×D97C2211yescharacteristic, complemented
28C4×C22⋊D97—24no—
29#15690770—24no—
30C2×#13095490C2211yescharacteristic, complemented
31C2×#13095490C2211yescharacteristic, complemented
32C2×C22⋊Dic97C2211yescharacteristic, complemented
33C2×C22⋊Dic97C2211yescharacteristic, complemented
34C2×C22⋊Dic97C2211yescharacteristic, complemented
35C2×#13095488—24no—
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×#13095488C2211yescharacteristic
40C2×#13095488C2211yescharacteristic, complemented
41C2×#13095488C2211yescharacteristic, complemented
42C2×#13095488C2211yescharacteristic, complemented
43C2×#13095488C2211yescharacteristic, complemented
44C2×#13095488C2211yescharacteristic, complemented
45C2×#13095488C2211yescharacteristic, complemented
46C22⋊C4×D97—24no—
47C2×C4×Dic97C2211yescharacteristic, complemented
48C2×C4×Dic97C2211yescharacteristic, complemented
49#15690717C2211yescharacteristic, complemented
50#15690717C2211yescharacteristic, complemented
51C22×D4×C97C2211yescharacteristic
52C23×C4×C97C2211yescharacteristic, complemented
53C2×C22⋊C4×C97C2211yescharacteristic, complemented
54C2×C22⋊C4×C97C2211yescharacteristic, complemented
55C2×C22⋊C4×C97C2211yescharacteristic, complemented
56#95×C97C2211yescharacteristic, complemented
57#95×C97C2211yescharacteristic, complemented
58C23×D97—22no—
59C23×D97—22no—
60C23×D97—22no—
61C23×D97—22no—
62C23×D97—22no—
63C23×D97—22no—
64C23×D97—22no—
65C23×D97—22no—
66C23×D97—41no—
67C23×D97—41no—
68C23×D97—41no—
69C23×D97—41no—
70C23×D97C2×C414yescomplemented
71C23×D97C2×C414yescomplemented
72C23×D97C2311yescharacteristic
73C23×D97C2311yescharacteristic
74C23×D97C2311yescharacteristic
75C2×D388—22no—
76C2×D388—22no—
77C2×D388C2×C414yescomplemented
78C2×C22⋊D97—22no—
79C2×C22⋊D97—22no—
80C2×C22⋊D97C2×C414yes—
81C2×C22⋊D97C2×C414yescomplemented
82C2×C22⋊D97D414yes—
83C2×C22⋊D97D414yes—
84D4×D97—232no—
85C22×Dic97—22no—
86C22×Dic97—22no—
87C22×Dic97—41no—
88C22×Dic97C2311yescharacteristic
89C22×Dic97C2311yescharacteristic
90C22×Dic97C2311yescharacteristic
91C22×Dic97C2311yescharacteristic
92C22×Dic97C2311yescharacteristic
93C2×C4×D97—22no—
94C2×C4×D97—22no—
95C2×C4×D97—22no—
96C2×C4×D97—22no—
97C2×C4×D97—22no—
98C2×C4×D97—22no—
99C2×C4×D97—41no—
100C2×C4×D97—41no—
101C2×C4×D97C2×C414yescomplemented
102#13095490—22no—
103#13095490—22no—
104C22⋊Dic97—22no—
105C22⋊Dic97—22no—
106C22⋊Dic97—41no—
107#13095488—22no—
108#13095488—22no—
109#13095488—22no—
110#13095488—22no—
111#13095488—22no—
112#13095488—22no—
113#13095488—22no—
114#13095488—41no—
115#13095488—41no—
116#13095488D414yescomplemented
117C4×Dic97—22no—
118C4×Dic97—22no—
119C24×C97C2311yescharacteristic
120C24×C97C2311yescharacteristic
121C2×D4×C97—22no—
122C2×D4×C97—22no—
123C2×D4×C97C2×C414yescomplemented
124C22×C4×C97—22no—
125C22×C4×C97—22no—
126C22×C4×C97—41no—
127C22×C4×C97C2311yescharacteristic
128C22×C4×C97C2311yescharacteristic
129C22×C4×C97C2311yescharacteristic
130C22×C4×C97C2311yescharacteristic
131C22×C4×C97C2311yescharacteristic
132C22⋊C4×C97—22no—
133C22⋊C4×C97—22no—
134C22⋊C4×C97—41no—
135C23×C97—21no—
136C23×C97—21no—
137C23×C97—21no—
138C23×C97—21no—
139C23×C97—21no—
140C23×C97—21no—
141C23×C97—21no—
142C23×C97—21no—
143C23×C97—21no—
144C23×C97—21no—
145C23×C97C22×C412yes—
146C23×C97C22×C412yes—
147C23×C97C2×D412yes—
148C23×C97C2×D412yes—
149C23×C97C2411yescharacteristic
150C22×D97—21no—
151C22×D97—21no—
152C22×D97—21no—
153C22×D97—21no—
154C22×D97—21no—
155C22×D97—21no—
156C22×D97—21no—
157C22×D97—21no—
158C22×D97—28no—
159C22×D97—21no—
160C22×D97—21no—
161C22×D97—21no—
162C22×D97—21no—
163C22×D97—21no—
164C22×D97—21no—
165C22×D97—28no—
166C22×D97—28no—
167C22×D97—28no—
168C22×D97—44no—
169C22×D97—44no—
170C22×D97—44no—
171C22×D97—44no—
172C22×D97—44no—
173C22×D97—44no—
174C22×D97—44no—
175C22×D97—44no—
176C22×D97—44no—
177C22×D97—44no—
178C22×D97C22×C412yes—
179C22×D97C22×C412yes—
180C22×D97C22×C412yes—
181C22×D97C2×D412yes—
182C22×D97C2×D412yes—
183C22×D97C2×D412yes—
184C22×D97#4112yes—
185D4×C97—28no—
186C2×C4×C97—21no—
187C2×C4×C97—21no—
188C2×C4×C97—21no—
189C2×C4×C97—21no—
190C2×C4×C97—21no—
191C2×C4×C97—21no—
192C2×C4×C97—21no—
193C2×C4×C97—21no—
194C2×C4×C97—21no—
195C2×C4×C97—21no—
196C2×C4×C97—21no—
197C2×C4×C97—21no—
198C2×C4×C97—21no—
199C2×C4×C97—44no—
200C2×C4×C97C22×C412yes—
201C2×C4×C97C2×D412yescomplemented
202C22⋊D97—28no—
203C22⋊D97C22⋊C4116yescomplemented
204C2×Dic97—21no—
205C2×Dic97—21no—
206C2×Dic97—21no—
207C2×Dic97—21no—
208C2×Dic97—21no—
209C2×Dic97—21no—
210C2×Dic97—21no—
211C2×Dic97—21no—
212C2×Dic97—21no—
213C2×Dic97—21no—
214C2×Dic97—21no—
215C2×Dic97—44no—
216C2×Dic97C22×C412yes—
217C2×Dic97C2×D412yes—
218C2×Dic97C2×D412yes—
219C2×Dic97C2×D412yescomplemented
220D388—28no—
221C4×D97—28no—
222C4×D97—44no—
223C4×D97—44no—
224C22×C97—22no—
225C22×C97—22no—
226C22×C97—22no—
227C22×C97—22no—
228C22×C97—22no—
229C22×C97—22no—
230C22×C97—22no—
231C22×C97—22no—
232C22×C97—22no—
233C22×C97—41no—
234C22×C97—41no—
235C22×C97—41no—
236C22×C97—41no—
237C22×C97C2×C22⋊C414yescomplemented
238C22×C97C23×C411yescharacteristic
239C22×C97C22×D411yescharacteristic
240C22×C97C22×D411yescharacteristic
241C22×C97C22×D411yescharacteristic
242C22×C97C2×#4111yescharacteristic
243C22×C97H(5,𝔽2)11yescharacteristic
244C22×C97H(5,𝔽2)11yescharacteristic
245C2×D97—22no—
246C2×D97—22no—
247C2×D97—22no—
248C2×D97—22no—
249C2×D97—22no—
250C2×D97—22no—
251C2×D97—22no—
252C2×D97—22no—
253C2×D97—22no—
254C2×D97—22no—
255C2×D97—22no—
256C2×D97—22no—
257C2×D97—22no—
258C2×D97—22no—
259C2×D97—22no—
260C2×D97—416no—
261C2×D97—416no—
262C2×D97—41no—
263C2×D97—41no—
264C2×D97—41no—
265C2×D97—41no—
266C2×D97C2×C22⋊C414yescomplemented
267C2×D97C4×D414yescomplemented
268C4×C97—22no—
269C4×C97—22no—
270C4×C97—41no—
271C4×C97—41no—
272C4×C97—41no—
273Dic97—22no—
274Dic97—41no—
275Dic97—41no—
276Dic97—41no—
277Dic97C2×C22⋊C414yescomplemented
278C2×C97—24no—
279C2×C97—42no—
280C2×C97C22×C22⋊C411yescharacteristic
281C2×C97C2×C4×D411yescharacteristic
282C2×C97#51811yescharacteristic
283C2×C97D4211yescharacteristic
284C2×C97D4211yescharacteristic
285C2×C97#54611yescharacteristic
286C2×C97#54611yescharacteristic
287D97—24no—
288D97—24no—
289D97—42no—
290D4×C22⋊C4—971nomaximal, Sylow
291C97D4×C22⋊C411yescharacteristic, complemented, Sylow
292C23×D4—971no—
293C2×C4×D4—971no—
294C2×C4×D4—971no—
295C22×C22⋊C4—971no—
296C22×C22⋊C4—971no—
297#390—971no—
298#390—971no—
299#386—971no—
300#386—971no—
301#386—971no—
302#386D9711yescharacteristic, complemented
303#380—971no—
304#379—971no—
305#379—971no—
306C4×C22⋊C4—971no—
307C25—971no—
308C25—971no—
309C22×D4—971no—
310C22×D4—971no—
311C22×D4—978no—
312C22×D4—971no—
313C22×D4—1944no—
314C22×D4—1944no—
315C22×D4C2×D9711yescharacteristic
316C23×C4—971no—
317C23×C4—971no—
318C23×C4—971no—
319C23×C4—971no—
320C23×C4C2×D9711yescharacteristic, complemented
321C4×D4—1944no—
322C4×D4—1944no—
323C2×C4⋊C4—971no—
324C2×C4⋊C4—971no—
325C2×C22⋊C4—971no—
326C2×C22⋊C4—971no—
327C2×C22⋊C4—971no—
328C2×C22⋊C4—971no—
329C2×C22⋊C4—971no—
330C2×C22⋊C4—971no—
331C2×C22⋊C4—971no—
332C2×C22⋊C4—971no—
333C2×C22⋊C4—971no—
334C2×C22⋊C4—971no—
335C2×C22⋊C4—971no—
336C2×C22⋊C4—971no—
337C2×C22⋊C4—971no—
338C2×C22⋊C4—1944no—
339C2×C22⋊C4—1944no—
340C2×C22⋊C4C2×D9711yescharacteristic, complemented
341C2×C22⋊C4C2×D9711yescharacteristic, complemented
342C2×C22⋊C4C2×D9711yescharacteristic, complemented
343C2×C42—971no—
344C2×C42—971no—
345#95—971no—
346#95—971no—
347#95C2×D9711yescharacteristic, complemented
348#95C2×D9711yescharacteristic, complemented
349C24—971no—
350C24—971no—
351C24—974no—
352C24—974no—
353C24—971no—
354C24—3881no—
355C24—3881no—
356C24—3881no—
357C24—3881no—
358C24—1942no—
359C24—1942no—
360C24—1942no—
361C24—1942no—
362C24—1942no—
363C24—1942no—
364C24—1942no—
365C24—1942no—
366C24C22×D9711yescharacteristic
367C24C22×D9711yescharacteristic
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—974no—
371C2×D4—974no—
372C2×D4—974no—
373C2×D4—974no—
374C2×D4—974no—
375C2×D4—1942no—
376C2×D4—1942no—
377C2×D4—1942no—
378C2×D4—1942no—
379C2×D4—19432no—
380C2×D4C4×D9714yescomplemented
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—971no—
385C22×C4—971no—
386C22×C4—974no—
387C22×C4—971no—
388C22×C4—971no—
389C22×C4—971no—
390C22×C4—3881no—
391C22×C4—3881no—
392C22×C4—3881no—
393C22×C4—1942no—
394C22×C4—1942no—
395C22×C4—1942no—
396C22×C4—1942no—
397C22×C4—1942no—
398C22×C4—1942no—
399C22×C4—1942no—
400C22×C4—1942no—
401C22×C4C22×D9711yescharacteristic
402C22×C4C22×D9711yescharacteristic
403C22×C4C22×D9711yescharacteristic
404C22×C4C22×D9711yescharacteristic
405C22×C4C22×D9711yescharacteristic
406C4⋊C4—1942no—
407C4⋊C4—1942no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—41no—
411C22⋊C4—974no—
412C22⋊C4—3881no—
413C22⋊C4—3881no—
414C22⋊C4—3881no—
415C22⋊C4—1942no—
416C22⋊C4—1942no—
417C22⋊C4—1942no—
418C22⋊C4—1942no—
419C22⋊C4—1942no—
420C22⋊C4—1942no—
421C22⋊C4—1942no—
422C22⋊C4—1942no—
423C22⋊C4—1942no—
424C42—1942no—
425C42—1942no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—972no—
437C23—972no—
438C23—972no—
439C23—972no—
440C23—972no—
441C23—972no—
442C23—972no—
443C23—3884no—
444C23—3884no—
445C23—3884no—
446C23—3884no—
447C23—3884no—
448C23—3884no—
449C23—3884no—
450C23—3884no—
451C23—3884no—
452C23—3884no—
453C23—1948no—
454C23—1948no—
455C23—1948no—
456C23—1948no—
457C23—1941no—
458C23—1941no—
459C23—1941no—
460C23—1941no—
461C23—1941no—
462C23—1941no—
463C23—1941no—
464C23—1941no—
465C23—1941no—
466C23—1941no—
467C23—1941no—
468C23—1941no—
469C23—1941no—
470C23—1941no—
471C23C2×C4×D9712yes—
472C23C2×C4×D9712yes—
473C23D4×D9712yes—
474C23D4×D9712yes—
475C23C23×D9711yescharacteristic
476D4—28no—
477D4—9716no—
478D4—1948no—
479D4—1948no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—44no—
494C2×C4—972no—
495C2×C4—972no—
496C2×C4—972no—
497C2×C4—972no—
498C2×C4—3884no—
499C2×C4—3884no—
500C2×C4—3884no—
501C2×C4—1948no—
502C2×C4—1941no—
503C2×C4—1941no—
504C2×C4—1941no—
505C2×C4—1941no—
506C2×C4—1941no—
507C2×C4—1941no—
508C2×C4—1941no—
509C2×C4—1941no—
510C2×C4—1941no—
511C2×C4—1941no—
512C2×C4—1941no—
513C2×C4C2×C4×D9712yes—
514C2×C4D4×D9712yescomplemented
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—974no—
529C22—974no—
530C22—38816no—
531C22—38816no—
532C22—3881no—
533C22—3881no—
534C22—3881no—
535C22—3881no—
536C22—1942no—
537C22—1942no—
538C22—1942no—
539C22—1942no—
540C22—1942no—
541C22—1942no—
542C22—1942no—
543C22—1942no—
544C22—1942no—
545C22—1942no—
546C22—1942no—
547C22—1942no—
548C22—1942no—
549C22—1942no—
550C22—1942no—
551C22C22⋊C4×D9714yescomplemented
552C22C22×C4×D9711yescharacteristic
553C22C2×D4×D9711yescharacteristic
554C22C2×D4×D9711yescharacteristic
555C22C2×D4×D9711yescharacteristic
556C22#41×D9711yescharacteristic
557C22#1569085511yescharacteristic
558C22#1569085611yescharacteristic
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—974no—
565C4—3881no—
566C4—3881no—
567C4—3881no—
568C4—1942no—
569C2—24no—
570C2—42no—
571C2—3882no—
572C2—1944no—
573C2—1944no—
574C2C2×C22⋊C4×D9711yescharacteristic
575C2C4×D4×D9711yescharacteristic
576C2#2026120711yescharacteristic
577C2#2026136711yescharacteristic
578C2#2026136911yescharacteristic
579C2#2026137411yescharacteristic
580C2#2026137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC22×D4×D97C2
hypercenter#386D97
hypocenterC97D4×C22⋊C4
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D97C2semidirect
3. C2×#15690770C2semidirect
4. C22×#13095488C2semidirect
5. C2×C22⋊C4×D97C2semidirect
6. #20260635C2semidirect
7. #20260635C2semidirect
8. #20260615C2semidirect
9. #20260613C2semidirect
10. #20260613C2semidirect
11. #20260594C2semidirect
12. #20260590C2semidirect
13. #20260590C2semidirect
14. #20260582C2semidirect
15. #386×C97C2semidirect
23. C2×D4×D97C4semidirect
24. C23×Dic97C22semidirect
26. C22×C4×D97C22semidirect
27. C22×C4×D97C22semidirect
30. C2×#13095490C22semidirect
31. C2×#13095490C22semidirect
32. C2×C22⋊Dic97C22semidirect
33. C2×C22⋊Dic97C22semidirect
34. C2×C22⋊Dic97C22semidirect
40. C2×#13095488C22semidirect
41. C2×#13095488C22semidirect
42. C2×#13095488C22semidirect
43. C2×#13095488C22semidirect
44. C2×#13095488C22semidirect
45. C2×#13095488C22semidirect
47. C2×C4×Dic97C22semidirect
48. C2×C4×Dic97C22semidirect
49. #15690717C22semidirect
50. #15690717C22semidirect
52. C23×C4×C97C22semidirect
53. C2×C22⋊C4×C97C22semidirect
54. C2×C22⋊C4×C97C22semidirect
55. C2×C22⋊C4×C97C22semidirect
56. #95×C97C22semidirect
57. #95×C97C22semidirect
70. C23×D97C2×C4semidirect
71. C23×D97C2×C4semidirect
77. C2×D388C2×C4semidirect
81. C2×C22⋊D97C2×C4semidirect
101. C2×C4×D97C2×C4semidirect
116. #13095488D4semidirect
123. C2×D4×C97C2×C4semidirect
201. C2×C4×C97C2×D4semidirect
203. C22⋊D97C22⋊C4semidirect
219. C2×Dic97C2×D4semidirect
237. C22×C97C2×C22⋊C4semidirect
266. C2×D97C2×C22⋊C4semidirect
267. C2×D97C4×D4semidirect
277. Dic97C2×C22⋊C4semidirect
291. C97D4×C22⋊C4semidirect
302. #386D97semidirect
320. C23×C4C2×D97semidirect
340. C2×C22⋊C4C2×D97semidirect
341. C2×C22⋊C4C2×D97semidirect
342. C2×C22⋊C4C2×D97semidirect
347. #95C2×D97semidirect
348. #95C2×D97semidirect
380. C2×D4C4×D97semidirect
514. C2×C4D4×D97semidirect
551. C22C22⋊C4×D97semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D972
C2×C4×C22⋊D972
C2×#156907702
C22×#130954882
C2×C22⋊C4×D972
#202606352
#202606152
#202606132
#202605942
#202605902
#202605822
#386×C972
D4×C22⋊C497