Finite Group Databasebeta

← #27144713

Subgroups — #27144713

Order 12416.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690778C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C22×#13095488C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260626C211yesmaximal, characteristic, complemented
7#20260625C211yesmaximal, characteristic, complemented
8#20260615C211yesmaximal, characteristic, complemented
9#20260614C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260608C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14C4×#13095488C211yesmaximal, characteristic, complemented
15C2×C4×D4×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22#15690778C418yescomplemented
23C4×C22⋊D97—24no—
24C2×#13095490C2211yescharacteristic, complemented
25C2×#13095490C2211yescharacteristic, complemented
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic, complemented
29C2×C22⋊Dic97C2211yescharacteristic, complemented
30C2×#13095488—24no—
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C22⋊C4×D97—24no—
40C2×C4×Dic97C2211yescharacteristic, complemented
41C2×C4×Dic97C2211yescharacteristic, complemented
42#15690717C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47#15690717C2211yescharacteristic, complemented
48C22×D4×C97C2211yescharacteristic
49C23×C4×C97C2211yescharacteristic
50C23×C4×C97C2211yescharacteristic, complemented
51C4×D4×C97—24no—
52C2×C4⋊C4×C97C2211yescharacteristic, complemented
53C2×C22⋊C4×C97C2211yescharacteristic
54C2×C22⋊C4×C97C2211yescharacteristic, complemented
55C2×C42×C97C2211yescharacteristic, complemented
56C23×D97—22no—
57C23×D97—22no—
58C23×D97—22no—
59C23×D97—22no—
60C23×D97—41no—
61C23×D97—41no—
62C23×D97C2×C414yescomplemented
63C23×D97C2311yescharacteristic
64C23×D97C2311yescharacteristic
65C2×C22⋊D97—22no—
66C2×C22⋊D97—22no—
67C2×C22⋊D97—41no—
68C2×C22⋊D97—41no—
69C2×C22⋊D97C2×C414yescomplemented
70C2×C22⋊D97C2×C414yescomplemented
71C22×Dic97—22no—
72C22×Dic97—22no—
73C22×Dic97—41no—
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C2×C4×D97—22no—
81C2×C4×D97—22no—
82C2×C4×D97—22no—
83C2×C4×D97—22no—
84C2×C4×D97—41no—
85C2×C4×D97—41no—
86#13095490—22no—
87#13095490—41no—
88C22⋊Dic97—22no—
89C22⋊Dic97—22no—
90C22⋊Dic97—41no—
91C22⋊Dic97C2×C414yescomplemented
92#13095488—22no—
93#13095488—22no—
94#13095488—22no—
95#13095488—22no—
96#13095488—22no—
97#13095488C2×C414yescomplemented
98#13095488C2×C414yescomplemented
99#13095488D414yescomplemented
100C4×Dic97—22no—
101C4×Dic97—41no—
102C24×C97C2311yescharacteristic
103C24×C97C2311yescharacteristic
104C2×D4×C97—22no—
105C2×D4×C97—22no—
106C2×D4×C97C2×C414yescomplemented
107C22×C4×C97—22no—
108C22×C4×C97—22no—
109C22×C4×C97—22no—
110C22×C4×C97—22no—
111C22×C4×C97—41no—
112C22×C4×C97—41no—
113C22×C4×C97C2311yescharacteristic
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C4⋊C4×C97—22no—
119C22⋊C4×C97—22no—
120C22⋊C4×C97D414yescomplemented
121C42×C97—22no—
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—21no—
128C23×C97—21no—
129C23×C97—21no—
130C23×C97—21no—
131C23×C97—21no—
132C23×C97C22×C412yes—
133C23×C97C22×C412yes—
134C23×C97C2×D412yes—
135C23×C97#4112yes—
136C23×C97C2411yescharacteristic
137C22×D97—21no—
138C22×D97—21no—
139C22×D97—21no—
140C22×D97—21no—
141C22×D97—28no—
142C22×D97—21no—
143C22×D97—21no—
144C22×D97—21no—
145C22×D97—21no—
146C22×D97—21no—
147C22×D97—21no—
148C22×D97—28no—
149C22×D97—44no—
150C22×D97—44no—
151C22×D97—44no—
152C22×D97—44no—
153C22×D97—44no—
154C22×D97C22×C412yes—
155C22×D97C22×C412yes—
156C22×D97C2×D412yes—
157C22×D97#4112yes—
158D4×C97—28no—
159C2×C4×C97—21no—
160C2×C4×C97—21no—
161C2×C4×C97—21no—
162C2×C4×C97—21no—
163C2×C4×C97—21no—
164C2×C4×C97—21no—
165C2×C4×C97—21no—
166C2×C4×C97—21no—
167C2×C4×C97—21no—
168C2×C4×C97—21no—
169C2×C4×C97—21no—
170C2×C4×C97—44no—
171C2×C4×C97—44no—
172C2×C4×C97C22×C412yes—
173C2×C4×C97C2×D412yes—
174C2×C4×C97C2×D412yes—
175C2×C4×C97C2×D412yes—
176C22⋊D97—28no—
177C22⋊D97—44no—
178C2×Dic97—21no—
179C2×Dic97—21no—
180C2×Dic97—21no—
181C2×Dic97—21no—
182C2×Dic97—21no—
183C2×Dic97—21no—
184C2×Dic97—21no—
185C2×Dic97—21no—
186C2×Dic97—21no—
187C2×Dic97—21no—
188C2×Dic97—21no—
189C2×Dic97—21no—
190C2×Dic97—21no—
191C2×Dic97—21no—
192C2×Dic97—21no—
193C2×Dic97—44no—
194C2×Dic97C22×C412yes—
195C2×Dic97C22×C412yes—
196C2×Dic97C2×D412yes—
197C4×D97—44no—
198C4×D97—44no—
199C22×C97—22no—
200C22×C97—22no—
201C22×C97—22no—
202C22×C97—22no—
203C22×C97—22no—
204C22×C97—22no—
205C22×C97—22no—
206C22×C97—22no—
207C22×C97—22no—
208C22×C97—41no—
209C22×C97—41no—
210C22×C97—41no—
211C22×C97—41no—
212C22×C97C4×D414yescomplemented
213C22×C97C23×C411yescharacteristic
214C22×C97C22×D411yescharacteristic
215C22×C97C22×D411yescharacteristic
216C22×C97C2×#4111yescharacteristic
217C22×C97C2×#4111yescharacteristic
218C22×C97H(5,𝔽2)11yescharacteristic
219C22×C97H(5,𝔽2)11yescharacteristic
220C2×D97—22no—
221C2×D97—22no—
222C2×D97—22no—
223C2×D97—22no—
224C2×D97—22no—
225C2×D97—22no—
226C2×D97—22no—
227C2×D97—22no—
228C2×D97—22no—
229C2×D97—416no—
230C2×D97—41no—
231C2×D97—41no—
232C2×D97—41no—
233C2×D97—41no—
234C2×D97C4×D414yescomplemented
235C4×C97—22no—
236C4×C97—22no—
237C4×C97—22no—
238C4×C97—22no—
239C4×C97—41no—
240Dic97—22no—
241Dic97—22no—
242Dic97—22no—
243Dic97—41no—
244Dic97—41no—
245Dic97—41no—
246C2×C97—24no—
247C2×C97—42no—
248C2×C97C2×C4×D411yescharacteristic
249C2×C97C2×C4×D411yescharacteristic
250C2×C97#51811yescharacteristic
251C2×C97D4211yescharacteristic
252C2×C97#54611yescharacteristic
253C2×C97#54611yescharacteristic
254C2×C97#55111yescharacteristic
255D97—24no—
256D97—42no—
257#2358—971nomaximal, Sylow
258C97#235811yescharacteristic, complemented, Sylow
259C2×#120—971no—
260C2×C4×D4—971no—
261C2×C4×D4D9711yescharacteristic, complemented
262C22×C22⋊C4—971no—
263C22×C22⋊C4—971no—
264#390—971no—
265#388—971no—
266#388—971no—
267#386—971no—
268#386—971no—
269#386—971no—
270#385—971no—
271#385—971no—
272C4×C22⋊C4—971no—
273C4×C22⋊C4—971no—
274C25—971no—
275C22×D4—971no—
276C22×D4—971no—
277C22×D4C2×D9711yescharacteristic
278C23×C4—971no—
279C23×C4—971no—
280C23×C4—971no—
281C23×C4C2×D9711yescharacteristic
282C23×C4C2×D9711yescharacteristic, complemented
283#120—978no—
284C4×D4—24no—
285C4×D4—1944no—
286C2×C4⋊C4—971no—
287C2×C4⋊C4—971no—
288C2×C4⋊C4C2×D9711yescharacteristic, complemented
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—971no—
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4—971no—
295C2×C22⋊C4—971no—
296C2×C22⋊C4—971no—
297C2×C22⋊C4—971no—
298C2×C22⋊C4—971no—
299C2×C22⋊C4—971no—
300C2×C22⋊C4—971no—
301C2×C22⋊C4—1944no—
302C2×C22⋊C4—1944no—
303C2×C22⋊C4C2×D9711yescharacteristic
304C2×C22⋊C4C2×D9711yescharacteristic, complemented
305C2×C42—971no—
306C2×C42—971no—
307C2×C42C2×D9711yescharacteristic, complemented
308#95—971no—
309#95—971no—
310#95—971no—
311#95—971no—
312#95—971no—
313#95—971no—
314C24—971no—
315C24—974no—
316C24—971no—
317C24—3881no—
318C24—3881no—
319C24—1942no—
320C24—1942no—
321C24—1942no—
322C24—1942no—
323C24C22×D9711yescharacteristic
324C24C22×D9711yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—974no—
328C2×D4—974no—
329C2×D4—3881no—
330C2×D4—3881no—
331C2×D4—1942no—
332C2×D4—1942no—
333C2×D4C4×D9714yescomplemented
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—971no—
341C22×C4—971no—
342C22×C4—971no—
343C22×C4—971no—
344C22×C4—971no—
345C22×C4—971no—
346C22×C4—3881no—
347C22×C4—3881no—
348C22×C4—3881no—
349C22×C4—1942no—
350C22×C4—1942no—
351C22×C4—1942no—
352C22×C4—1942no—
353C22×C4—1942no—
354C22×C4—1942no—
355C22×C4C22×D9711yescharacteristic
356C22×C4C22×D9711yescharacteristic
357C22×C4C22×D9711yescharacteristic
358C22×C4C22×D9711yescharacteristic
359C22×C4C22×D9711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—3881no—
362C4⋊C4—1942no—
363C22⋊C4—22no—
364C22⋊C4—974no—
365C22⋊C4—974no—
366C22⋊C4—974no—
367C22⋊C4—974no—
368C22⋊C4—3881no—
369C22⋊C4—1942no—
370C22⋊C4—1942no—
371C22⋊C4—1942no—
372C22⋊C4—1942no—
373C22⋊C4—1942no—
374C22⋊C4—1942no—
375C22⋊C4—1942no—
376C22⋊C4C22⋊D9714yescomplemented
377C42—22no—
378C42—3881no—
379C42—1942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—972no—
391C23—972no—
392C23—972no—
393C23—972no—
394C23—3884no—
395C23—3884no—
396C23—3884no—
397C23—3884no—
398C23—3884no—
399C23—1948no—
400C23—1948no—
401C23—1941no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23—1941no—
409C23—1941no—
410C23—1941no—
411C23C2×C4×D9712yes—
412C23C2×C4×D9712yes—
413C23C2×C22⋊D9712yes—
414C23#1309551112yes—
415C23C23×D9711yescharacteristic
416D4—28no—
417D4—3884no—
418D4—1948no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—972no—
433C2×C4—972no—
434C2×C4—972no—
435C2×C4—3884no—
436C2×C4—3884no—
437C2×C4—3884no—
438C2×C4—1941no—
439C2×C4—1941no—
440C2×C4—1941no—
441C2×C4—1941no—
442C2×C4—1941no—
443C2×C4—1941no—
444C2×C4—1941no—
445C2×C4—1941no—
446C2×C4—1941no—
447C2×C4—1941no—
448C2×C4—1941no—
449C2×C4—1941no—
450C2×C4—1941no—
451C2×C4—1941no—
452C2×C4—1941no—
453C2×C4C2×C4×D9712yes—
454C2×C4D4×D9712yes—
455C2×C4C2×C22⋊D9712yes—
456C2×C4C2×C22⋊D9712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—974no—
471C22—38816no—
472C22—3881no—
473C22—3881no—
474C22—3881no—
475C22—3881no—
476C22—1942no—
477C22—1942no—
478C22—1942no—
479C22—1942no—
480C22—1942no—
481C22—1942no—
482C22—1942no—
483C22—1942no—
484C22—1942no—
485C22C4×C22⋊D9714yescomplemented
486C22C22×C4×D9711yescharacteristic
487C22C2×D4×D9711yescharacteristic
488C22C22×C22⋊D9711yescharacteristic
489C22#41×D9711yescharacteristic
490C22C2×#1309551111yescharacteristic
491C22#1569085511yescharacteristic
492C22#1569085611yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—3881no—
499C4—3881no—
500C4—3881no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C2—24no—
505C2—42no—
506C2—3882no—
507C2—1944no—
508C2C4×D4×D9711yescharacteristic
509C2C2×C4×C22⋊D9711yescharacteristic
510C2#2026120711yescharacteristic
511C2D4×C22⋊D9711yescharacteristic
512C2#2026137311yescharacteristic
513C2#2026138111yescharacteristic
514C2#2026141411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×C4×D4×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690778C2
hypercenterC2×C4×D4D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D97C2semidirect
3. C22×#13095488C2semidirect
4. C2×C22⋊C4×D97C2semidirect
5. #20260635C2semidirect
6. #20260626C2semidirect
7. #20260625C2semidirect
8. #20260615C2semidirect
9. #20260614C2semidirect
10. #20260611C2semidirect
11. #20260608C2semidirect
12. #20260606C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. C4×#13095488C2semidirect
15. C2×C4×D4×C97C2semidirect
22. #15690778C4semidirect
24. C2×#13095490C22semidirect
25. C2×#13095490C22semidirect
28. C2×C22⋊Dic97C22semidirect
29. C2×C22⋊Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. C2×C4×Dic97C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
50. C23×C4×C97C22semidirect
52. C2×C4⋊C4×C97C22semidirect
54. C2×C22⋊C4×C97C22semidirect
55. C2×C42×C97C22semidirect
62. C23×D97C2×C4semidirect
69. C2×C22⋊D97C2×C4semidirect
70. C2×C22⋊D97C2×C4semidirect
91. C22⋊Dic97C2×C4semidirect
97. #13095488C2×C4semidirect
98. #13095488C2×C4semidirect
99. #13095488D4semidirect
106. C2×D4×C97C2×C4semidirect
120. C22⋊C4×C97D4semidirect
212. C22×C97C4×D4semidirect
234. C2×D97C4×D4semidirect
258. C97#2358semidirect
261. C2×C4×D4D97semidirect
282. C23×C4C2×D97semidirect
288. C2×C4⋊C4C2×D97semidirect
304. C2×C22⋊C4C2×D97semidirect
307. C2×C42C2×D97semidirect
333. C2×D4C4×D97semidirect
376. C22⋊C4C22⋊D97semidirect
485. C22C4×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
C2×C4×C22⋊D972
C22×#130954882
C2×C22⋊C4×D972
#202606352
#202606262
#202606252
#202606152
#202606142
#202606112
#202606082
#202606062
C22⋊C4×Dic972
C4×#130954882
C2×C4×D4×C972
#235897