Finite Group Databasebeta

← #27144715

Subgroups — #27144715

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×#15690779C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic97C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260629C211yesmaximal, characteristic, complemented
7#20260623C211yesmaximal, characteristic, complemented
8#20260614C211yesmaximal, characteristic, complemented
9#20260612C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260608C211yesmaximal, characteristic, complemented
12#20260607C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic, complemented
22#15690779C418yescomplemented
23C4×C22⋊D97—24no—
24#15690770—24no—
25C2×#13095490C2211yescharacteristic, complemented
26C2×#13095490C2211yescharacteristic, complemented
27C2×#13095490C2211yescharacteristic, complemented
28C2×C22⋊Dic97—24no—
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×C22⋊Dic97C2211yescharacteristic
33C2×C22⋊Dic97C2211yescharacteristic
34C2×C22⋊Dic97C2211yescharacteristic
35C2×C22⋊Dic97C2211yescharacteristic
36C2×#13095488C2211yescharacteristic, complemented
37C2×#13095488C2211yescharacteristic, complemented
38C2×#13095488C2211yescharacteristic, complemented
39C2×C4×Dic97C2211yescharacteristic, complemented
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
48C25×C97C2211yescharacteristic
49C23×C4×C97C2211yescharacteristic
50C23×C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97—24no—
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C22⋊C4×C97C2211yescharacteristic
54C2×C22⋊C4×C97C2211yescharacteristic
55C2×C22⋊C4×C97C2211yescharacteristic
56C23×D97C2311yescharacteristic
57C2×C22⋊D97—22no—
58C2×C22⋊D97—22no—
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97—41no—
62C2×C22⋊D97—41no—
63C2×C22⋊D97C2×C414yescomplemented
64C2×C22⋊D97C2×C414yescomplemented
65C2×C22⋊D97C2×C414yescomplemented
66C22×Dic97—22no—
67C22×Dic97—22no—
68C22×Dic97—22no—
69C22×Dic97—22no—
70C22×Dic97—41no—
71C22×Dic97—41no—
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C2×C4×D97—22no—
80C2×C4×D97—22no—
81C2×C4×D97—41no—
82#13095490—22no—
83#13095490—22no—
84#13095490—41no—
85C22⋊Dic97—22no—
86C22⋊Dic97—22no—
87C22⋊Dic97—22no—
88C22⋊Dic97C2×C414yescomplemented
89C22⋊Dic97C2×C414yescomplemented
90C22⋊Dic97C2×C414yescomplemented
91C22⋊Dic97D414yescomplemented
92#13095488—22no—
93#13095488—22no—
94#13095488—41no—
95C4×Dic97—22no—
96C4×Dic97—22no—
97C4×Dic97—41no—
98C24×C97—22no—
99C24×C97—22no—
100C24×C97—22no—
101C24×C97—22no—
102C24×C97—41no—
103C24×C97—41no—
104C24×C97C2×C414yescomplemented
105C24×C97C2311yescharacteristic
106C24×C97C2311yescharacteristic
107C24×C97C2311yescharacteristic
108C22×C4×C97—22no—
109C22×C4×C97—22no—
110C22×C4×C97—22no—
111C22×C4×C97—22no—
112C22×C4×C97—41no—
113C22×C4×C97—41no—
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22⋊C4×C97—22no—
119C22⋊C4×C97—22no—
120C22⋊C4×C97—22no—
121C22⋊C4×C97D414yescomplemented
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—28no—
131C23×C97—28no—
132C23×C97—21no—
133C23×C97—21no—
134C23×C97—21no—
135C23×C97—21no—
136C23×C97—21no—
137C23×C97—21no—
138C23×C97—44no—
139C23×C97—44no—
140C23×C97—44no—
141C23×C97—44no—
142C23×C97—44no—
143C23×C97C22×C412yes—
144C23×C97C22×C412yes—
145C23×C97C22×C412yes—
146C23×C97C2×D412yes—
147C23×C97C2×D412yes—
148C23×C97#4112yes—
149C23×C97#4112yes—
150C23×C97C2411yescharacteristic
151C22×D97—21no—
152C22×D97—21no—
153C22×D97—21no—
154C22×D97—21no—
155C22×D97—21no—
156C22×D97—21no—
157C22×D97C22×C412yes—
158C2×C4×C97—21no—
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—44no—
169C2×C4×C97—44no—
170C2×C4×C97C2×D412yes—
171C2×C4×C97C2×D412yes—
172C22⋊D97—28no—
173C22⋊D97—28no—
174C22⋊D97—44no—
175C2×Dic97—21no—
176C2×Dic97—21no—
177C2×Dic97—21no—
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—44no—
192C2×Dic97—44no—
193C2×Dic97C22×C412yes—
194C2×Dic97C22×C412yes—
195C2×Dic97C22×C412yes—
196C2×Dic97C2×D412yes—
197C2×Dic97C2×D412yes—
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—22no—
209C22×C97—22no—
210C22×C97—22no—
211C22×C97—22no—
212C22×C97—22no—
213C22×C97—22no—
214C22×C97—416no—
215C22×C97—41no—
216C22×C97—41no—
217C22×C97—41no—
218C22×C97—41no—
219C22×C97C4×D414yescomplemented
220C22×C97C4×D414yescomplemented
221C22×C97C23×C411yescharacteristic
222C22×C97C22×D411yescharacteristic
223C22×C97C22×D411yescharacteristic
224C22×C97C2×#4111yescharacteristic
225C22×C97C2×#4111yescharacteristic
226C22×C97H(5,𝔽2)11yescharacteristic
227C22×C97H(5,𝔽2)11yescharacteristic
228C2×D97—22no—
229C2×D97—22no—
230C2×D97—22no—
231C2×D97—41no—
232C2×D97—41no—
233C2×D97—41no—
234C2×D97—41no—
235C4×C97—22no—
236C4×C97—22no—
237C4×C97—41no—
238C4×C97—41no—
239Dic97—22no—
240Dic97—22no—
241Dic97—22no—
242Dic97—22no—
243Dic97—22no—
244Dic97—41no—
245Dic97—41no—
246C2×C97—24no—
247C2×C97—24no—
248C2×C97—42no—
249C2×C97C2×C4×D411yescharacteristic
250C2×C97C2×C4×D411yescharacteristic
251C2×C97#51811yescharacteristic
252C2×C97D4211yescharacteristic
253C2×C97#54611yescharacteristic
254C2×C97#54611yescharacteristic
255C2×C97#55111yescharacteristic
256D97—42no—
257#2358—971nomaximal, Sylow
258C97#235811yescharacteristic, complemented, Sylow
259C2×#120—971no—
260C2×C4×D4—971no—
261C2×C4×D4—971no—
262C22×C22⋊C4—971no—
263C22×C22⋊C4D9711yescharacteristic, complemented
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—
274C25C2×D9711yescharacteristic
275C22×D4—971no—
276C22×D4—971no—
277C22×D4—971no—
278C23×C4—971no—
279C23×C4—971no—
280C23×C4—971no—
281C23×C4C2×D9711yescharacteristic
282C23×C4C2×D9711yescharacteristic
283#120—978no—
284C4×D4—1944no—
285C4×D4—1944no—
286C2×C4⋊C4—971no—
287C2×C4⋊C4—971no—
288C2×C4⋊C4—971no—
289C2×C22⋊C4—24no—
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—1944no—
301C2×C22⋊C4C2×D9711yescharacteristic
302C2×C22⋊C4C2×D9711yescharacteristic
303C2×C22⋊C4C2×D9711yescharacteristic
304C2×C22⋊C4C2×D9711yescharacteristic
305C2×C42—971no—
306C2×C42—971no—
307C2×C42—971no—
308#95—971no—
309#95—971no—
310#95—971no—
311#95—971no—
312#95—971no—
313#95—971no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—971no—
321C24C4×D9714yescomplemented
322C24C22×D9711yescharacteristic
323C24C22×D9711yescharacteristic
324C24C22×D9711yescharacteristic
325C2×D4—974no—
326C2×D4—974no—
327C2×D4—974no—
328C2×D4—3881no—
329C2×D4—3881no—
330C2×D4—1942no—
331C2×D4—1942no—
332C2×D4—1942no—
333C2×D4—1942no—
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—971no—
347C22×C4—3881no—
348C22×C4—3881no—
349C22×C4—3881no—
350C22×C4—1942no—
351C22×C4—1942no—
352C22×C4—1942no—
353C22×C4—1942no—
354C22×C4—1942no—
355C22×C4—1942no—
356C22×C4C22×D9711yescharacteristic
357C22×C4C22×D9711yescharacteristic
358C22×C4C22×D9711yescharacteristic
359C22×C4C22×D9711yescharacteristic
360C4⋊C4—3881no—
361C4⋊C4—1942no—
362C4⋊C4—1942no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—974no—
367C22⋊C4—974no—
368C22⋊C4—974no—
369C22⋊C4—974no—
370C22⋊C4—3881no—
371C22⋊C4—1942no—
372C22⋊C4—1942no—
373C22⋊C4—1942no—
374C22⋊C4—1942no—
375C22⋊C4—1942no—
376C22⋊C4C22⋊D9714yescomplemented
377C42—3881no—
378C42—1942no—
379C42—1942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—28no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—972no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23C2×C4×D9712yes—
409C23C2×C4×D9712yes—
410C23C2×C4×D9712yes—
411C23D4×D9712yes—
412C23C2×C22⋊D9712yes—
413C23#1309551012yes—
414C23#1309551112yes—
415C23C23×D9711yescharacteristic
416D4—3884no—
417D4—1948no—
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—44no—
430C2×C4—44no—
431C2×C4—972no—
432C2×C4—972no—
433C2×C4—972no—
434C2×C4—972no—
435C2×C4—972no—
436C2×C4—3884no—
437C2×C4—3884no—
438C2×C4—3884no—
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×C4—1941no—
454C2×C4—1941no—
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—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—3881no—
478C22—3881no—
479C22—3881no—
480C22—3881no—
481C22—1942no—
482C22—1942no—
483C22—1942no—
484C22#1569077014yescomplemented
485C22C4×C22⋊D9714yescomplemented
486C22C22×C4×D9711yescharacteristic
487C22C2×D4×D9711yescharacteristic
488C22C22×C22⋊D9711yescharacteristic
489C22C2×#1309551011yescharacteristic
490C22C2×#1309551111yescharacteristic
491C22#1569085511yescharacteristic
492C22#1569085511yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—3881no—
498C4—3881no—
499C4—1942no—
500C4—1942no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—3882no—
508C2C2×#1569077011yescharacteristic
509C2C2×C4×C22⋊D9711yescharacteristic
510C2#2026120811yescharacteristic
511C2D4×C22⋊D9711yescharacteristic
512C2#2026137511yescharacteristic
513C2#2026137911yescharacteristic
514C2#2026141111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC22×C22⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690779C2
hypercenterC22×C22⋊C4D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D97C2semidirect
3. C2×#15690770C2semidirect
4. C22×C22⋊Dic97C2semidirect
5. #20260635C2semidirect
6. #20260629C2semidirect
7. #20260623C2semidirect
8. #20260614C2semidirect
9. #20260612C2semidirect
10. #20260611C2semidirect
11. #20260608C2semidirect
12. #20260607C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. C4×C22⋊Dic97C2semidirect
15. C22×C22⋊C4×C97C2semidirect
21. C22×C4×D97C22semidirect
22. #15690779C4semidirect
25. C2×#13095490C22semidirect
26. C2×#13095490C22semidirect
27. C2×#13095490C22semidirect
36. C2×#13095488C22semidirect
37. C2×#13095488C22semidirect
38. C2×#13095488C22semidirect
39. C2×C4×Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. C2×C4×Dic97C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
63. C2×C22⋊D97C2×C4semidirect
64. C2×C22⋊D97C2×C4semidirect
65. C2×C22⋊D97C2×C4semidirect
88. C22⋊Dic97C2×C4semidirect
89. C22⋊Dic97C2×C4semidirect
90. C22⋊Dic97C2×C4semidirect
91. C22⋊Dic97D4semidirect
104. C24×C97C2×C4semidirect
121. C22⋊C4×C97D4semidirect
219. C22×C97C4×D4semidirect
220. C22×C97C4×D4semidirect
258. C97#2358semidirect
263. C22×C22⋊C4D97semidirect
321. C24C4×D97semidirect
376. C22⋊C4C22⋊D97semidirect
484. C22#15690770semidirect
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×#156907792
C2×C4×C22⋊D972
C2×#156907702
C22×C22⋊Dic972
#202606352
#202606292
#202606232
#202606142
#202606122
#202606112
#202606082
#202606072
C22⋊C4×Dic972
C4×C22⋊Dic972
C22×C22⋊C4×C972
#235897