Finite Group Databasebeta

← #468×D97 #27151320

Subgroups — #468×D97 #27151320

Order 12416.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D97C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3C22⋊Q8×D97C211yesmaximal, characteristic, complemented
4#121×D97C211yesmaximal, characteristic
5#20260984C211yesmaximal, characteristic, complemented
6#20260983C211yesmaximal, characteristic, complemented
7#20260982C211yesmaximal, characteristic, complemented
8#20260981C211yesmaximal, characteristic, complemented
9#20260980C211yesmaximal, characteristic, complemented
10#20260979C211yesmaximal, characteristic, complemented
11#20260978C211yesmaximal, characteristic, complemented
12#107×D97C211yesmaximal, characteristic, complemented
13#103×D97C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#468×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C2×#13095500C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×#13095498C2211yescharacteristic, complemented
23SD16×D97—24no—
24C2×#13095495C2211yescharacteristic, complemented
25C2×#13095494C2211yescharacteristic, complemented
26C2×#13095494C2211yescharacteristic, complemented
27M16×D97—24no—
28C2×C8×D97C2211yescharacteristic
29C2×C8×D97C2211yescharacteristic
30#15690790C2211yescharacteristic, complemented
31#15690789C2211yescharacteristic, complemented
32#15690788—21no—
33#15690787—21no—
34#15690786C2211yescharacteristic, complemented
35#15690785C2211yescharacteristic
36#15690784C2211yescharacteristic
37#15690783—21no—
38#15690782—21no—
39#15690781C2211yescharacteristic
40C4⋊C4×D97—21no—
41C4⋊C4×D97C2211yescharacteristic
42C4⋊C4×D97C2211yescharacteristic
43C22⋊C4×D97—21no—
44C22⋊C4×D97—21no—
45#15690744C2211yescharacteristic, complemented
46#15690743C2211yescharacteristic, complemented
47#15690743C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690734C2211yescharacteristic, complemented
50#15690733C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic, complemented
52#15690731C2211yescharacteristic, complemented
53#15690730C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×M16×C97C2211yescharacteristic, complemented
56C22⋊Q8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic
58#107×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97D411yescharacteristic
63C2×Dic194—21no—
64C2×Dic194C2311yescharacteristic
65Q8×D97—22no—
66Q8×D97—22no—
67C2×D388—21no—
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72D4×D97—22no—
73D4×D97—22no—
74D4×D97—44no—
75C22×Dic97D411yescharacteristic
76C2×C4×D97—22no—
77C2×C4×D97—22no—
78C2×C4×D97—21no—
79C2×C4×D97—21no—
80C2×C4×D97—21no—
81C2×C4×D97D411yescharacteristic
82C2×C4×D97D411yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85#13095500—22no—
86#13095499—22no—
87#13095498—22no—
88#13095495—22no—
89#13095494—22no—
90#13095494—22no—
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C8×D97—22no—
94C8×D97D414yescomplemented
95C4⋊Dic97—21no—
96C4⋊Dic97C2311yescharacteristic
97C4⋊Dic97C2311yescharacteristic
98#13095490—21no—
99#13095490—21no—
100#13095490—21no—
101#13095490—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108C2×Q8×C97C2311yescharacteristic
109C2×D4×C97—21no—
110C2×D4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112SD16×C97—22no—
113M16×C97—22no—
114C2×C8×C97C2311yescharacteristic
115C2×C8×C97C2311yescharacteristic
116C4⋊C4×C97—21no—
117C4⋊C4×C97C2311yescharacteristic
118C4⋊C4×C97C2311yescharacteristic
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C23×C97—21no—
122C23×C97C2×D411yescharacteristic
123C22×D97—22no—
124C22×D97—22no—
125C22×D97—22no—
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97C2×D411yescharacteristic
131C22×D97C2×D411yescharacteristic
132Q8×C97—21no—
133Q8×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—42no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97C2×D411yescharacteristic
143C2×C4×C97C2411yescharacteristic
144C22⋊D97—42no—
145C22⋊D97—42no—
146C22⋊D97—42no—
147C2×Dic97—22no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97C2×D411yescharacteristic
152C2×Dic97C2×D411yescharacteristic
153D388—21no—
154D388—21no—
155D388—42no—
156C4×D97—22no—
157C4×D97—21no—
158C4×D97—21no—
159C4×D97—22no—
160C4×D97—42no—
161C4×D97—42no—
162C4×D97—42no—
163C4×D97C2×D412yes—
164C4×D97#4112yes—
165Dic194—21no—
166Dic194—21no—
167Dic194—42no—
168C8×C97—21no—
169C8×C97C2×D412yescomplemented
170#11715895—21no—
171#11715895C2×D412yescomplemented
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97C22×D411yescharacteristic
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—21no—
182C2×D97—44no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—84no—
187C2×D97#12112yes—
188C2×D97AGL(1,ℤ/8)12yes—
189C2×D97#13712yes—
190C4×C97—21no—
191C4×C97—41no—
192C4×C97—41no—
193C4×C97—41no—
194C4×C97C22×D411yescharacteristic
195C4×C97C2×#4111yescharacteristic
196Dic97—21no—
197Dic97—41no—
198Dic97—41no—
199Dic97—41no—
200Dic97#12112yes—
201C2×C97—41no—
202C2×C97—81no—
203C2×C97C2×#12111yescharacteristic
204C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C97C2×#13711yescharacteristic
206D97—41no—
207D97—81no—
208D97#46814yesdirect product factor
209C2×#468—971nomaximal, Sylow
210C97C2×#46811yescharacteristic, complemented, Sylow
211C22×SD16—971no—
212C22×M16—971no—
213C2×C22⋊Q8—971no—
214C2×#121—971no—
215#468—971no—
216#468—971no—
217#468—971no—
218#468—971no—
219#468—971no—
220#468—971no—
221#468—971no—
222#468D9711yescharacteristic, direct product factor
223C2×#107—971no—
224C2×#103—971no—
225C2×#102—971no—
226C22×Q8—971no—
227C22×D4—971no—
228C22×D4—1941no—
229C23×C4—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—1944no—
234C2×SD16C2×D9711yescharacteristic, complemented
235C2×M16—971no—
236C2×M16—971no—
237C2×M16—971no—
238C2×M16—1944no—
239C2×M16C2×D9711yescharacteristic, complemented
240C22×C8—971no—
241C22×C8—971no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—971no—
245C22⋊Q8—1941no—
246C22⋊Q8—1941no—
247C22⋊Q8C2×D9711yescharacteristic, complemented
248#121—971no—
249#121—971no—
250#121—971no—
251#121—1941no—
252#121—1941no—
253#121C2×D9711yescharacteristic
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—1941no—
257C2×C22⋊C4—1941no—
258C2×C22⋊C4—1941no—
259#107—971no—
260#107—971no—
261#107—971no—
262#107C2×D9711yescharacteristic, complemented
263#103—971no—
264#103—971no—
265#103—971no—
266#103C2×D9711yescharacteristic, complemented
267#102—971no—
268#102—971no—
269#102—971no—
270#102C2×D9711yescharacteristic, complemented
271C24—971no—
272C24—1941no—
273C2×Q8—971no—
274C2×Q8—1941no—
275C2×Q8—1942no—
276C2×Q8—1942no—
277C2×Q8C22×D9711yescharacteristic
278C2×D4—21no—
279C2×D4—971no—
280C2×D4—3884no—
281C2×D4—1941no—
282C2×D4—1942no—
283C2×D4—1942no—
284C2×D4—1941no—
285C2×D4—1941no—
286C2×D4—1941no—
287C2×D4C22×D9711yescharacteristic
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—1942no—
294C22×C4—1942no—
295C22×C4—1941no—
296C22×C4—1941no—
297C22×C4—1941no—
298C22×C4C22×D9711yescharacteristic
299SD16—22no—
300SD16—1942no—
301SD16—1942no—
302SD16—1942no—
303M16—22no—
304M16—1942no—
305M16—1942no—
306M16—1942no—
307C2×C8—971no—
308C2×C8—974no—
309C2×C8—971no—
310C2×C8—1942no—
311C2×C8C22×D9711yescharacteristic
312C2×C8C22×D9711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—971no—
315C4⋊C4—971no—
316C4⋊C4—1941no—
317C4⋊C4—1941no—
318C4⋊C4—1941no—
319C4⋊C4—1941no—
320C4⋊C4—1941no—
321C4⋊C4C22×D9711yescharacteristic
322C4⋊C4C22×D9711yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C23—21no—
332C23—971no—
333C23—971no—
334C23—3882no—
335C23—3882no—
336C23—3882no—
337C23—1942no—
338C23—1942no—
339C23—1942no—
340C23—1941no—
341C23D4×D9711yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—3882no—
345Q8—1941no—
346Q8—1941no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—3882no—
351D4—3882no—
352D4—3882no—
353D4—3882no—
354D4—1941no—
355D4—1941no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—971no—
362C2×C4—972no—
363C2×C4—972no—
364C2×C4—971no—
365C2×C4—3882no—
366C2×C4—3882no—
367C2×C4—3882no—
368C2×C4—1942no—
369C2×C4—1942no—
370C2×C4—1942no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4—1941no—
375C2×C4—1941no—
376C2×C4D4×D9711yescharacteristic
377C2×C4C23×D9711yescharacteristic
378C8—21no—
379C8—972no—
380C8—1941no—
381C8D4×D9712yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—7764no—
389C22—972no—
390C22—972no—
391C22—972no—
392C22—3884no—
393C22—3881no—
394C22—3881no—
395C22—3881no—
396C22—1941no—
397C22—1941no—
398C22—1941no—
399C22C2×D4×D9711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—972no—
405C4—3881no—
406C4—3881no—
407C4—3881no—
408C4—1941no—
409C4C2×D4×D9711yescharacteristic
410C4#41×D9711yescharacteristic
411C2—41no—
412C2—81no—
413C2—7761no—
414C2—974no—
415C2—3881no—
416C2#121×D9711yescharacteristic
417C2AGL(1,ℤ/8)×D9711yescharacteristic
418C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#468×C97C2
socleC22×C97C22×D4
prime radical#121×D97C2
hypercenter#468D97
hypocenterC97C2×#468
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D97C2semidirect
2. C2×M16×D97C2semidirect
3. C22⋊Q8×D97C2semidirect
5. #20260984C2semidirect
6. #20260983C2semidirect
7. #20260982C2semidirect
8. #20260981C2semidirect
9. #20260980C2semidirect
10. #20260979C2semidirect
11. #20260978C2semidirect
12. #107×D97C2semidirect
13. #103×D97C2semidirect
14. #102×D97C2semidirect
15. #468×C97C2semidirect
20. C2×#13095500C22semidirect
21. C2×#13095499C22semidirect
22. C2×#13095498C22semidirect
24. C2×#13095495C22semidirect
25. C2×#13095494C22semidirect
26. C2×#13095494C22semidirect
30. #15690790C22semidirect
31. #15690789C22semidirect
34. #15690786C22semidirect
45. #15690744C22semidirect
46. #15690743C22semidirect
47. #15690743C22semidirect
48. #15690735C22semidirect
49. #15690734C22semidirect
50. #15690733C22semidirect
51. #15690732C22semidirect
52. #15690731C22semidirect
53. #15690730C22semidirect
54. C2×SD16×C97C22semidirect
55. C2×M16×C97C22semidirect
56. C22⋊Q8×C97C22semidirect
58. #107×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
94. C8×D97D4semidirect
169. C8×C97C2×D4semidirect
171. #11715895C2×D4semidirect
208. D97#468direct (N × H)
210. C97C2×#468semidirect
222. #468D97direct (N × H)
234. C2×SD16C2×D97semidirect
239. C2×M16C2×D97semidirect
247. C22⋊Q8C2×D97semidirect
262. #107C2×D97semidirect
266. #103C2×D97semidirect
270. #102C2×D97semidirect
381. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D972
C2×M16×D972
C22⋊Q8×D972
#121×D972
#202609842
#202609832
#202609822
#202609812
#202609802
#202609792
#202609782
#107×D972
#103×D972
#102×D972
#468×C972
C2×#46897