Finite Group Databasebeta

← #421×D97 #27150275

Subgroups — #421×D97 #27150275

Order 12416.

772 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C211yesmaximal, characteristic
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#117×D97C211yesmaximal, characteristic, complemented
4#20260753C211yesmaximal, characteristic, complemented
5#20260752C211yesmaximal, characteristic, complemented
6#20260751C211yesmaximal, characteristic, complemented
7#20260750C214yesmaximal, complemented
8#104×D97C214yesmaximal, complemented
9#421×C97C211yesmaximal, characteristic, complemented
10C2×#13095512C411yescharacteristic, complemented
11C2×#13095511—21no—
12C2×#13095511C2211yescharacteristic
13C2×#13095510—21no—
14C2×#13095510C411yescharacteristic, complemented
15#41×D97—21no—
16#41×D97—21no—
17#41×D97—21no—
18#41×D97—21no—
19#41×D97C412yescomplemented
20#41×D97C412yescomplemented
21#41×D97C2212yes—
22#41×D97C2212yes—
23C2×Q8×D97C411yescharacteristic, complemented
24C2×D4×D97—21no—
25C2×D4×D97C411yescharacteristic, complemented
26C22×C4×D97—21no—
27C22×C4×D97C2211yescharacteristic
28C2×#13095495C2211yescharacteristic, complemented
29C2×#13095494—21no—
30M16×D97—21no—
31M16×D97—21no—
32M16×D97C2212yescomplemented
33M16×D97C2212yescomplemented
34C2×C8×D97—21no—
35#15690769C2211yescharacteristic, complemented
36#15690768—21no—
37#15690767C2212yescomplemented
38#15690766—21no—
39C4⋊C4×D97—21no—
40C22⋊C4×D97—21no—
41C42×D97C2212yescomplemented
42#15690738C2214yescomplemented
43#15690737C2214yescomplemented
44#15690736C2214yescomplemented
45C2×#41×C97C2211yescharacteristic
46C2×M16×C97C2211yescharacteristic, complemented
47#117×C97C2211yescharacteristic, complemented
48#104×C97C2214yescomplemented
49C23×D97—21no—
50C23×D97D411yescharacteristic
51#13095512—21no—
52#13095512—21no—
53#13095512C2×C412yes—
54#13095512C2×C412yescomplemented
55#13095511—21no—
56#13095511—21no—
57#13095511—22no—
58#13095511—22no—
59#13095511—22no—
60#13095511—22no—
61#13095511C2×C412yescomplemented
62#13095511C2312yes—
63#13095510—22no—
64#13095510—22no—
65#13095510—22no—
66#13095510—22no—
67#13095510—21no—
68#13095510—21no—
69#13095510C2×C412yes—
70#13095510C2×C412yescomplemented
71C2×Dic194—21no—
72C2×Dic194C2×C411yescharacteristic, complemented
73Q8×D97—21no—
74Q8×D97—21no—
75Q8×D97C2×C412yes—
76Q8×D97C2×C412yescomplemented
77C2×D388—21no—
78C2×D388C2×C411yescharacteristic, complemented
79C2×C22⋊D97—21no—
80C2×C22⋊D97—21no—
81C2×C22⋊D97—21no—
82D4×D97—22no—
83D4×D97—22no—
84D4×D97—22no—
85D4×D97—22no—
86D4×D97—21no—
87D4×D97—21no—
88D4×D97C2×C412yes—
89D4×D97C2×C412yescomplemented
90C22×Dic97—21no—
91C22×Dic97D411yescharacteristic
92C2×C4×D97—21no—
93C2×C4×D97—21no—
94C2×C4×D97—21no—
95C2×C4×D97—22no—
96C2×C4×D97—22no—
97C2×C4×D97—22no—
98C2×C4×D97—22no—
99C2×C4×D97—22no—
100C2×C4×D97C2×C411yescharacteristic
101C2×C4×D97C2×C411yescharacteristic
102C2×C4×D97C2×C411yescharacteristic
103C2×C4×D97C2×C411yescharacteristic
104C2×C4×D97D411yescharacteristic
105C2×C4×D97D411yescharacteristic
106C2×C4×D97D411yescharacteristic
107C2×C4×D97D411yescharacteristic
108C2×C4×D97D411yescharacteristic
109C2×C4×D97D411yescharacteristic
110C2×C4×D97C2311yescharacteristic
111C2×C4×D97C2311yescharacteristic
112#13095495—21no—
113#13095495C2312yescomplemented
114#13095494—22no—
115#13095494—22no—
116C2×#11715895—21no—
117C8×D97—22no—
118C8×D97—22no—
119C4⋊Dic97—21no—
120#13095490—22no—
121C22⋊Dic97—21no—
122#13095488—22no—
123C4×Dic97—22no—
124C4×Dic97C2312yescomplemented
125#41×C97—21no—
126#41×C97—21no—
127#41×C97C2×C412yescomplemented
128#41×C97C2312yes—
129C2×Q8×C97C2×C411yescharacteristic, complemented
130C2×D4×C97—21no—
131C2×D4×C97C2×C411yescharacteristic, complemented
132C22×C4×C97—21no—
133C22×C4×C97C2311yescharacteristic
134M16×C97—21no—
135M16×C97C2312yescomplemented
136C2×C8×C97—21no—
137C4⋊C4×C97—21no—
138C22⋊C4×C97—21no—
139C42×C97C2312yescomplemented
140C23×C97—21no—
141C23×C97C2×D411yescharacteristic
142C22×D97—21no—
143C22×D97—22no—
144C22×D97—22no—
145C22×D97—42no—
146C22×D97—44no—
147C22×D97C22⋊C411yescharacteristic
148C22×D97C22⋊C411yescharacteristic
149C22×D97C22⋊C411yescharacteristic
150C22×D97C22⋊C411yescharacteristic
151C22×D97C2×D411yescharacteristic
152C22×D97C2×D411yescharacteristic
153Q8×C97—21no—
154Q8×C97C22×C412yes—
155D4×C97—22no—
156D4×C97—22no—
157D4×C97—21no—
158D4×C97C22×C412yes—
159C2×C4×C97—21no—
160C2×C4×C97—22no—
161C2×C4×C97—22no—
162C2×C4×C97—22no—
163C2×C4×C97C22×C411yescharacteristic
164C2×C4×C97C22×C411yescharacteristic
165C2×C4×C97C2×D411yescharacteristic
166C2×C4×C97C2×D411yescharacteristic
167C2×C4×C97C2×D411yescharacteristic
168C2×C4×C97C2411yescharacteristic
169C22⋊D97—22no—
170C22⋊D97—22no—
171C22⋊D97—22no—
172C22⋊D97—22no—
173C22⋊D97—22no—
174C22⋊D97—22no—
175C2×Dic97—21no—
176C2×Dic97—22no—
177C2×Dic97—22no—
178C2×Dic97—22no—
179C2×Dic97C22⋊C411yescharacteristic
180C2×Dic97C22⋊C411yescharacteristic
181C2×Dic97C22⋊C411yescharacteristic
182C2×Dic97C22⋊C411yescharacteristic
183C2×Dic97C2×D411yescharacteristic
184C2×Dic97C2×D411yescharacteristic
185D388—22no—
186D388—22no—
187D388—21no—
188D388C22×C412yes—
189C4×D97—22no—
190C4×D97—22no—
191C4×D97—22no—
192C4×D97—22no—
193C4×D97—22no—
194C4×D97—22no—
195C4×D97—44no—
196C4×D97C22⋊C411yescharacteristic
197C4×D97C22⋊C411yescharacteristic
198C4×D97C22⋊C411yescharacteristic
199C4×D97C22⋊C411yescharacteristic
200C4×D97C22⋊C411yescharacteristic
201C4×D97C22⋊C411yescharacteristic
202C4×D97C22⋊C411yescharacteristic
203C4×D97C22⋊C411yescharacteristic
204C4×D97C22×C411yescharacteristic
205C4×D97C22×C411yescharacteristic
206C4×D97C22×C411yescharacteristic
207C4×D97C22×C411yescharacteristic
208C4×D97C2×D411yescharacteristic
209C4×D97C2×D411yescharacteristic
210C4×D97C2×D411yescharacteristic
211C4×D97C2×D411yescharacteristic
212Dic194—21no—
213Dic194—22no—
214Dic194—22no—
215Dic194C22×C412yes—
216C8×C97—22no—
217#11715895—22no—
218C22×C97—22no—
219C22×C97—41no—
220C22×C97—42no—
221C22×C97C2×C22⋊C411yescharacteristic
222C22×C97C2×C22⋊C411yescharacteristic
223C22×C97C22×D411yescharacteristic
224C2×D97—22no—
225C2×D97—22no—
226C2×D97—22no—
227C2×D97—22no—
228C2×D97—41no—
229C2×D97—41no—
230C2×D97—41no—
231C2×D97—44no—
232C2×D97—42no—
233C2×D97—44no—
234C2×D97C2×C22⋊C411yescharacteristic
235C2×D97C2×C22⋊C411yescharacteristic
236C2×D97C2×C22⋊C411yescharacteristic
237C2×D97C2×C22⋊C411yescharacteristic
238C4×C97—22no—
239C4×C97—42no—
240C4×C97C2×C22⋊C411yescharacteristic
241C4×C97C2×C22⋊C411yescharacteristic
242C4×C97C23×C411yescharacteristic
243C4×C97C22×D411yescharacteristic
244Dic97—22no—
245Dic97—42no—
246Dic97C2×C22⋊C411yescharacteristic
247Dic97C2×C22⋊C411yescharacteristic
248Dic97C2×C22⋊C411yescharacteristic
249Dic97C2×C22⋊C411yescharacteristic
250C2×C97—21no—
251C2×C97—21no—
252C2×C97—21no—
253C2×C97—42no—
254C2×C97C22×C22⋊C411yescharacteristic
255D97—21no—
256D97—21no—
257D97—21no—
258D97—42no—
259D97#42112yesdirect product factor
260C2×#421—971nomaximal, Sylow
261C97C2×#42111yescharacteristic, complemented, Sylow
262C22×#41—971no—
263C22×M16—971no—
264C2×#117—971no—
265#421—971no—
266#421—971no—
267#421—971no—
268#421—974no—
269#421D9711yescharacteristic, direct product factor
270C2×#104—974no—
271C2×#41—972no—
272C2×#41—972no—
273C2×#41—972no—
274C2×#41—972no—
275C2×#41—971no—
276C2×#41—971no—
277C2×#41—971no—
278C2×#41—1941no—
279C2×#41—1941no—
280C2×#41—1941no—
281C2×#41—1941no—
282C2×#41—1941no—
283C2×#41—1941no—
284C2×#41C2×D9711yescharacteristic
285C22×Q8—971no—
286C22×D4—971no—
287C22×D4—1941no—
288C23×C4—971no—
289C23×C4—1941no—
290C2×M16—971no—
291C2×M16—972no—
292C2×M16—972no—
293C2×M16—1941no—
294C2×M16—1941no—
295C2×M16—1941no—
296C2×M16C2×D9711yescharacteristic, complemented
297C22×C8—1941no—
298#117—971no—
299#117—972no—
300#117—1941no—
301#117—1941no—
302#117C2×D9711yescharacteristic, complemented
303C2×C4⋊C4—1941no—
304C2×C22⋊C4—1941no—
305C2×C42—972no—
306#104—974no—
307#104—974no—
308#104—974no—
309#104C2×D9714yescomplemented
310C24—971no—
311C24—1941no—
312#41—21no—
313#41—21no—
314#41—972no—
315#41—972no—
316#41—972no—
317#41—972no—
318#41—972no—
319#41—972no—
320#41—1942no—
321#41—1942no—
322#41—1942no—
323#41—1942no—
324#41—1942no—
325#41—1942no—
326#41—1942no—
327#41—1942no—
328#41—1941no—
329#41—1941no—
330#41—1941no—
331#41—1941no—
332#41—1941no—
333#41—1941no—
334#41C4×D9712yescomplemented
335#41C22×D9712yes—
336C2×Q8—972no—
337C2×Q8—972no—
338C2×Q8—971no—
339C2×Q8—1941no—
340C2×Q8—1941no—
341C2×Q8—1941no—
342C2×Q8C4×D9711yescharacteristic, complemented
343C2×D4—21no—
344C2×D4—972no—
345C2×D4—972no—
346C2×D4—971no—
347C2×D4—1941no—
348C2×D4—1941no—
349C2×D4—1941no—
350C2×D4—1941no—
351C2×D4—1942no—
352C2×D4—1942no—
353C2×D4—1942no—
354C2×D4—1942no—
355C2×D4—1941no—
356C2×D4—1941no—
357C2×D4C4×D9711yescharacteristic, complemented
358C22×C4—21no—
359C22×C4—971no—
360C22×C4—971no—
361C22×C4—971no—
362C22×C4—971no—
363C22×C4—971no—
364C22×C4—971no—
365C22×C4—971no—
366C22×C4—971no—
367C22×C4—971no—
368C22×C4—971no—
369C22×C4—971no—
370C22×C4—971no—
371C22×C4—971no—
372C22×C4—1941no—
373C22×C4—1942no—
374C22×C4—1942no—
375C22×C4—1942no—
376C22×C4—1942no—
377C22×C4—1942no—
378C22×C4—1941no—
379C22×C4—1941no—
380C22×C4—1941no—
381C22×C4C22×D9711yescharacteristic
382M16—21no—
383M16—972no—
384M16—1942no—
385M16—1942no—
386M16—1941no—
387M16C22×D9712yescomplemented
388C2×C8—21no—
389C2×C8—1942no—
390C2×C8—1942no—
391C2×C8—1941no—
392C4⋊C4—21no—
393C4⋊C4—1942no—
394C4⋊C4—1941no—
395C22⋊C4—21no—
396C22⋊C4—1942no—
397C22⋊C4—1941no—
398C42—972no—
399C42—1942no—
400C42C22×D9712yescomplemented
401C23—21no—
402C23—971no—
403C23—971no—
404C23—971no—
405C23—971no—
406C23—971no—
407C23—971no—
408C23—3884no—
409C23—3882no—
410C23—1942no—
411C23—1942no—
412C23—1941no—
413C23D4×D9711yescharacteristic
414Q8—21no—
415Q8—972no—
416Q8—1942no—
417Q8—1942no—
418Q8—1941no—
419Q8C2×C4×D9712yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—972no—
424D4—1942no—
425D4—1942no—
426D4—1942no—
427D4—1942no—
428D4—1942no—
429D4—1942no—
430D4—1942no—
431D4—1942no—
432D4—1941no—
433D4C2×C4×D9712yes—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—971no—
439C2×C4—971no—
440C2×C4—971no—
441C2×C4—971no—
442C2×C4—971no—
443C2×C4—971no—
444C2×C4—971no—
445C2×C4—971no—
446C2×C4—971no—
447C2×C4—971no—
448C2×C4—971no—
449C2×C4—971no—
450C2×C4—971no—
451C2×C4—971no—
452C2×C4—971no—
453C2×C4—971no—
454C2×C4—971no—
455C2×C4—971no—
456C2×C4—971no—
457C2×C4—971no—
458C2×C4—971no—
459C2×C4—971no—
460C2×C4—3884no—
461C2×C4—1942no—
462C2×C4—1942no—
463C2×C4—1942no—
464C2×C4—1942no—
465C2×C4—1942no—
466C2×C4—1942no—
467C2×C4—1942no—
468C2×C4—1941no—
469C2×C4—1942no—
470C2×C4—1942no—
471C2×C4C2×C4×D9711yescharacteristic
472C2×C4C2×C4×D9711yescharacteristic
473C2×C4D4×D9711yescharacteristic
474C2×C4D4×D9711yescharacteristic
475C2×C4D4×D9711yescharacteristic
476C2×C4C23×D9711yescharacteristic
477C8—22no—
478C8—1942no—
479C22—22no—
480C22—41no—
481C22—42no—
482C22—971no—
483C22—971no—
484C22—971no—
485C22—971no—
486C22—3882no—
487C22—3881no—
488C22—3881no—
489C22—3881no—
490C22—3884no—
491C22—3884no—
492C22—1942no—
493C22—1942no—
494C22—1942no—
495C22—1942no—
496C22C22⋊C4×D9711yescharacteristic
497C22C22⋊C4×D9711yescharacteristic
498C22C2×D4×D9711yescharacteristic
499C4—22no—
500C4—42no—
501C4—971no—
502C4—971no—
503C4—971no—
504C4—971no—
505C4—3882no—
506C4—1942no—
507C4C22⋊C4×D9711yescharacteristic
508C4C22⋊C4×D9711yescharacteristic
509C4C22×C4×D9711yescharacteristic
510C4C2×D4×D9711yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—972no—
516C2—3882no—
517C2—1941no—
518C2—1941no—
519C2—1941no—
520C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC4C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#421×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#41×D97C2
cosocleC2×C4×C97C24
hypercenter#421D97
hypocenterC97C2×#421
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×M16×D97C2semidirect
3. #117×D97C2semidirect
4. #20260753C2semidirect
5. #20260752C2semidirect
6. #20260751C2semidirect
7. #20260750C2semidirect
8. #104×D97C2semidirect
9. #421×C97C2semidirect
10. C2×#13095512C4semidirect
14. C2×#13095510C4semidirect
19. #41×D97C4semidirect
20. #41×D97C4semidirect
23. C2×Q8×D97C4semidirect
25. C2×D4×D97C4semidirect
28. C2×#13095495C22semidirect
32. M16×D97C22semidirect
33. M16×D97C22semidirect
35. #15690769C22semidirect
37. #15690767C22semidirect
41. C42×D97C22semidirect
42. #15690738C22semidirect
43. #15690737C22semidirect
44. #15690736C22semidirect
46. C2×M16×C97C22semidirect
47. #117×C97C22semidirect
48. #104×C97C22semidirect
54. #13095512C2×C4semidirect
61. #13095511C2×C4semidirect
70. #13095510C2×C4semidirect
72. C2×Dic194C2×C4semidirect
76. Q8×D97C2×C4semidirect
78. C2×D388C2×C4semidirect
89. D4×D97C2×C4semidirect
113. #13095495C23semidirect
124. C4×Dic97C23semidirect
127. #41×C97C2×C4semidirect
129. C2×Q8×C97C2×C4semidirect
131. C2×D4×C97C2×C4semidirect
135. M16×C97C23semidirect
139. C42×C97C23semidirect
259. D97#421direct (N × H)
261. C97C2×#421semidirect
269. #421D97direct (N × H)
296. C2×M16C2×D97semidirect
302. #117C2×D97semidirect
309. #104C2×D97semidirect
334. #41C4×D97semidirect
342. C2×Q8C4×D97semidirect
357. C2×D4C4×D97semidirect
387. M16C22×D97semidirect
400. C42C22×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D972
C2×M16×D972
#117×D972
#202607532
#202607522
#202607512
#202607502
#104×D972
#421×C972
C2×#42197