Finite Group Databasebeta

← #548×D97 #27155198

Subgroups — #548×D97 #27155198

Order 12416.

1620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C211yesmaximal, characteristic, complemented
2C22×Q8×D97C211yesmaximal, characteristic, complemented
3#20261397C211yesmaximal, characteristic, complemented
4#20261396C211yesmaximal, characteristic, complemented
5#20261395C213yesmaximal, complemented
6#20261394C213yesmaximal, complemented
7#20261393C213yesmaximal, complemented
8#20261392C211yesmaximal, characteristic, complemented
9#20261391C213yesmaximal, complemented
10#124×D97C213yesmaximal, complemented
11C22⋊Q8×D97C213yesmaximal, complemented
12#121×D97C213yesmaximal, complemented
13C4×Q8×D97C211yesmaximal, characteristic, complemented
14C4×D4×D97C213yesmaximal, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C2×#13095511C2213yescomplemented
18C2×#13095510C2213yescomplemented
19#41×D97—28no—
20C22×Dic194C2213yescomplemented
21C2×Q8×D97—28no—
22C2×Q8×D97C2213yes—
23C2×Q8×D97C2211yescharacteristic
24C2×Q8×D97C2211yescharacteristic, complemented
25C2×Q8×D97C2213yescomplemented
26C2×D4×D97C2213yes—
27C2×D4×D97C2213yes—
28C22×C4×D97C2213yes—
29C22×C4×D97C2213yes—
30#15690799C2213yescomplemented
31#15690798C2213yescomplemented
32#15690797C2213yescomplemented
33#15690796C2216yescomplemented
34#15690796C2216yescomplemented
35#15690790C2213yescomplemented
36#15690789C2213yescomplemented
37#15690788C2216yescomplemented
38#15690787C2216yescomplemented
39#15690786C2213yescomplemented
40#15690785C2213yescomplemented
41#15690784C2213yescomplemented
42#15690783C2216yescomplemented
43#15690782C2216yescomplemented
44#15690781C2213yescomplemented
45Q8×Dic97C2211yescharacteristic, complemented
46C4×Dic194C2213yescomplemented
47#15690775C2213yescomplemented
48C4×D388C2213yescomplemented
49D4×Dic97C2213yescomplemented
50#15690772C2213yescomplemented
51C4×C22⋊D97C2213yescomplemented
52C4×C22⋊D97C2213yescomplemented
53#15690770C2213yescomplemented
54#15690770C2213yescomplemented
55C4⋊C4×D97C2213yes—
56C4⋊C4×D97C2213yes—
57C22⋊C4×D97C2211yescharacteristic
58C22⋊C4×D97C2213yes—
59C22⋊C4×D97C2216yes—
60C42×D97C2213yes—
61C2×#41×C97C2211yescharacteristic, complemented
62C22×Q8×C97C2211yescharacteristic, complemented
63#124×C97C2213yescomplemented
64C22⋊Q8×C97C2213yescomplemented
65#121×C97C2213yescomplemented
66C4×Q8×C97C2211yescharacteristic, complemented
67C4×D4×C97C2213yescomplemented
68C23×D97C2313yes—
69C23×D97C2311yescharacteristic
70#13095512—24no—
71#13095511—212no—
72#13095510—212no—
73C2×Dic194—212no—
74C2×Dic194C2316yes—
75C2×Dic194C2313yes—
76C2×Dic194C2313yes—
77Q8×D97—24no—
78Q8×D97—212no—
79Q8×D97—212no—
80Q8×D97D418yescomplemented
81C2×D388C2313yes—
82C2×D388C2313yes—
83C2×C22⋊D97C2313yes—
84C2×C22⋊D97C2313yes—
85C2×C22⋊D97C2316yes—
86D4×D97—212no—
87D4×D97—212no—
88C22×Dic97C2313yes—
89C22×Dic97C2313yes—
90C2×C4×D97—212no—
91C2×C4×D97—212no—
92C2×C4×D97C2311yescharacteristic
93C2×C4×D97C2313yes—
94C2×C4×D97C2313yes—
95C2×C4×D97C2311yescharacteristic
96C2×C4×D97C2313yes—
97C2×C4×D97C2313yes—
98C2×C4×D97C2313yes—
99C2×C4×D97C2313yes—
100C2×C4×D97C2313yes—
101C4⋊Dic97C2313yes—
102C4⋊Dic97C2313yes—
103#13095490C2313yes—
104#13095490C2313yes—
105#13095490C2316yes—
106C22⋊Dic97C2311yescharacteristic
107C22⋊Dic97C2313yes—
108C22⋊Dic97C2316yes—
109#13095488C2311yescharacteristic
110#13095488C2313yes—
111#13095488C2311yescharacteristic
112#13095488C2313yes—
113#13095488C2316yes—
114#13095488C2316yes—
115C4×Dic97C2313yes—
116C4×Dic97C2313yes—
117C4×Dic97C2313yes—
118#41×C97—24no—
119C2×Q8×C97—24no—
120C2×Q8×C97C2313yes—
121C2×Q8×C97C2311yescharacteristic
122C2×D4×C97C2313yes—
123C2×D4×C97C2313yes—
124C22×C4×C97C2313yes—
125C22×C4×C97C2313yes—
126C4⋊C4×C97C2313yes—
127C4⋊C4×C97C2313yes—
128C22⋊C4×C97C2311yescharacteristic
129C22⋊C4×C97C2313yes—
130C22⋊C4×C97C2316yes—
131C42×C97C2313yes—
132C23×C97C2413yes—
133C23×C97C2411yescharacteristic
134C22×D97—22no—
135C22×D97—22no—
136C22×D97—26no—
137C22×D97—26no—
138C22×D97—26no—
139C22×D97#4114yescomplemented
140C22×D97C2413yes—
141C22×D97C2411yescharacteristic
142C22×D97C2411yescharacteristic
143Q8×C97—26no—
144Q8×C97C2×D414yescomplemented
145D4×C97—26no—
146D4×C97—26no—
147C2×C4×C97—26no—
148C2×C4×C97—26no—
149C2×C4×C97C2411yescharacteristic
150C2×C4×C97C2413yes—
151C2×C4×C97C2413yes—
152C2×C4×C97C2411yescharacteristic
153C2×C4×C97C2413yes—
154C22⋊D97—26no—
155C22⋊D97—26no—
156C22⋊D97—212no—
157C2×Dic97—26no—
158C2×Dic97—26no—
159C2×Dic97C2411yescharacteristic
160C2×Dic97C2413yes—
161C2×Dic97C2413yes—
162C2×Dic97C2411yescharacteristic
163C2×Dic97C2413yes—
164D388—26no—
165D388—26no—
166C4×D97—22no—
167C4×D97—26no—
168C4×D97—26no—
169C4×D97—22no—
170C4×D97—26no—
171C4×D97—26no—
172C4×D97—26no—
173C4×D97—26no—
174C4×D97C2×D4112yes—
175Dic194—26no—
176Dic194—212no—
177Dic194C2×D4112yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—23no—
181C22×C97—23no—
182C22×C97—23no—
183C22×C97C2×#4112yescomplemented
184C22×C97C2511yescharacteristic
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—28no—
188C2×D97—23no—
189C2×D97—23no—
190C2×D97—23no—
191C2×D97—412no—
192C2×D97C22×D412yes—
193C2×D97C2×#4112yes—
194C2×D97C2×#4112yescomplemented
195C2×D972−1+412yes—
196C4×C97—23no—
197C4×C97—21no—
198C4×C97—23no—
199C4×C97—21no—
200C4×C97C22×D416yes—
201Dic97—23no—
202Dic97—21no—
203Dic97—23no—
204Dic97—21no—
205Dic97C22×D416yes—
206C2×C97—22no—
207C2×C97—43no—
208C2×C97C23×D411yescharacteristic
209C2×C97C22×#4111yescharacteristic
210C2×C97C2×2−1+411yescharacteristic
211D97—22no—
212D97—43no—
213D97#54814yesdirect product factor
214C2×#548—971nomaximal, Sylow
215C97C2×#54811yescharacteristic, complemented, Sylow
216C22×#41—971no—
217C23×Q8—971no—
218#548—971no—
219#548—971no—
220#548—973no—
221#548—973no—
222#548—973no—
223#548—971no—
224#548—973no—
225#548D9711yescharacteristic, direct product factor
226C2×#124—973no—
227C2×C22⋊Q8—973no—
228C2×#121—973no—
229C2×C4×Q8—971no—
230C2×C4×D4—973no—
231C2×#41—971no—
232C2×#41—973no—
233C2×#41—973no—
234C2×#41—1948no—
235C2×#41C2×D9711yescharacteristic, complemented
236C22×Q8—971no—
237C22×Q8—973no—
238C22×Q8—973no—
239C22×Q8—971no—
240C22×Q8—973no—
241C22×Q8—1948no—
242C22×Q8C2×D9711yescharacteristic, complemented
243C22×D4—973no—
244C22×D4—973no—
245C23×C4—973no—
246C23×C4—973no—
247#124—973no—
248#124—973no—
249#124—976no—
250#124—976no—
251#124—973no—
252#124C2×D9713yescomplemented
253C22⋊Q8—973no—
254C22⋊Q8—973no—
255C22⋊Q8—976no—
256C22⋊Q8—973no—
257C22⋊Q8—976no—
258C22⋊Q8C2×D9713yescomplemented
259#121—973no—
260#121—976no—
261#121—973no—
262#121—973no—
263#121—976no—
264#121C2×D9713yescomplemented
265C4×Q8—971no—
266C4×Q8—973no—
267C4×Q8—973no—
268C4×Q8C2×D9711yescharacteristic, complemented
269C4×D4—973no—
270C4×D4—973no—
271C4×D4—973no—
272C4×D4—973no—
273C4×D4—973no—
274C4×D4—973no—
275C4×D4—973no—
276C4×D4C2×D9713yescomplemented
277C2×C4⋊C4—973no—
278C2×C4⋊C4—973no—
279C2×C22⋊C4—971no—
280C2×C22⋊C4—973no—
281C2×C22⋊C4—976no—
282C2×C42—973no—
283C24—971no—
284C24—973no—
285#41—24no—
286#41—19412no—
287#41—19412no—
288#41—1944no—
289C2×Q8—24no—
290C2×Q8—978no—
291C2×Q8—973no—
292C2×Q8—976no—
293C2×Q8—973no—
294C2×Q8—19412no—
295C2×Q8—19412no—
296C2×Q8—19412no—
297C2×Q8—1944no—
298C2×Q8C22×D9713yes—
299C2×Q8C22×D9711yescharacteristic
300C2×D4—973no—
301C2×D4—973no—
302C2×D4—976no—
303C2×D4—973no—
304C2×D4—973no—
305C2×D4—19412no—
306C2×D4—19412no—
307C2×D4C22×D9713yes—
308C2×D4C22×D9713yes—
309C22×C4—973no—
310C22×C4—973no—
311C22×C4—973no—
312C22×C4—971no—
313C22×C4—973no—
314C22×C4—973no—
315C22×C4—971no—
316C22×C4—973no—
317C22×C4—973no—
318C22×C4—973no—
319C22×C4—973no—
320C22×C4—19412no—
321C22×C4—19412no—
322C22×C4C22×D9713yes—
323C22×C4C22×D9713yes—
324C4⋊C4—973no—
325C4⋊C4—973no—
326C4⋊C4—976no—
327C4⋊C4—973no—
328C4⋊C4—973no—
329C4⋊C4C22×D9713yes—
330C4⋊C4C22×D9713yes—
331C22⋊C4—971no—
332C22⋊C4—973no—
333C22⋊C4—976no—
334C22⋊C4—971no—
335C22⋊C4—971no—
336C22⋊C4—973no—
337C22⋊C4—973no—
338C22⋊C4—976no—
339C22⋊C4—976no—
340C22⋊C4C22×D9711yescharacteristic
341C22⋊C4C22×D9713yes—
342C22⋊C4C22×D9716yes—
343C42—973no—
344C42—973no—
345C42—973no—
346C42C22×D9713yes—
347C23—971no—
348C23—974no—
349C23—971no—
350C23—973no—
351C23—1946no—
352C23—1946no—
353C23—1946no—
354C23—1942no—
355C23—1942no—
356C23C23×D9713yes—
357C23C23×D9711yescharacteristic
358Q8—26no—
359Q8—9712no—
360Q8—19412no—
361Q8—1946no—
362Q8D4×D9714yescomplemented
363D4—26no—
364D4—26no—
365D4—1946no—
366D4—19412no—
367D4—1946no—
368D4—1946no—
369D4—1946no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—973no—
373C2×C4—9712no—
374C2×C4—971no—
375C2×C4—973no—
376C2×C4—973no—
377C2×C4—971no—
378C2×C4—1946no—
379C2×C4—1946no—
380C2×C4—1942no—
381C2×C4—1946no—
382C2×C4—1946no—
383C2×C4—1942no—
384C2×C4—1946no—
385C2×C4—1946no—
386C2×C4—1946no—
387C2×C4—1946no—
388C2×C4C23×D9711yescharacteristic
389C2×C4C23×D9713yes—
390C2×C4C23×D9713yes—
391C2×C4C23×D9711yescharacteristic
392C2×C4C23×D9713yes—
393C22—21no—
394C22—21no—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—972no—
399C22—972no—
400C22—972no—
401C22—972no—
402C22—38812no—
403C22—1948no—
404C22—1943no—
405C22—1943no—
406C22—1943no—
407C22—1941no—
408C22—1941no—
409C22#41×D9712yescomplemented
410C22C24×D9711yescharacteristic
411C4—21no—
412C4—23no—
413C4—23no—
414C4—21no—
415C4—976no—
416C4—1941no—
417C4—1943no—
418C4—1943no—
419C4—1941no—
420C4C2×D4×D9716yes—
421C2—22no—
422C2—43no—
423C2—974no—
424C2—3883no—
425C2—1942no—
426C2C22×D4×D9711yescharacteristic
427C2C2×#41×D9711yescharacteristic
428C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#548×C97C2
hypercenter#548D97
hypocenterC97C2×#548
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#41×D97C2semidirect
2. C22×Q8×D97C2semidirect
3. #20261397C2semidirect
4. #20261396C2semidirect
5. #20261395C2semidirect
6. #20261394C2semidirect
7. #20261393C2semidirect
8. #20261392C2semidirect
9. #20261391C2semidirect
10. #124×D97C2semidirect
11. C22⋊Q8×D97C2semidirect
12. #121×D97C2semidirect
13. C4×Q8×D97C2semidirect
14. C4×D4×D97C2semidirect
15. #548×C97C2semidirect
16. C2×#13095512C22semidirect
17. C2×#13095511C22semidirect
18. C2×#13095510C22semidirect
20. C22×Dic194C22semidirect
24. C2×Q8×D97C22semidirect
25. C2×Q8×D97C22semidirect
30. #15690799C22semidirect
31. #15690798C22semidirect
32. #15690797C22semidirect
33. #15690796C22semidirect
34. #15690796C22semidirect
35. #15690790C22semidirect
36. #15690789C22semidirect
37. #15690788C22semidirect
38. #15690787C22semidirect
39. #15690786C22semidirect
40. #15690785C22semidirect
41. #15690784C22semidirect
42. #15690783C22semidirect
43. #15690782C22semidirect
44. #15690781C22semidirect
45. Q8×Dic97C22semidirect
46. C4×Dic194C22semidirect
47. #15690775C22semidirect
48. C4×D388C22semidirect
49. D4×Dic97C22semidirect
50. #15690772C22semidirect
51. C4×C22⋊D97C22semidirect
52. C4×C22⋊D97C22semidirect
53. #15690770C22semidirect
54. #15690770C22semidirect
61. C2×#41×C97C22semidirect
62. C22×Q8×C97C22semidirect
63. #124×C97C22semidirect
64. C22⋊Q8×C97C22semidirect
65. #121×C97C22semidirect
66. C4×Q8×C97C22semidirect
67. C4×D4×C97C22semidirect
80. Q8×D97D4semidirect
139. C22×D97#41semidirect
144. Q8×C97C2×D4semidirect
177. Dic194C2×D4semidirect
183. C22×C97C2×#41semidirect
194. C2×D97C2×#41semidirect
213. D97#548direct (N × H)
215. C97C2×#548semidirect
225. #548D97direct (N × H)
235. C2×#41C2×D97semidirect
242. C22×Q8C2×D97semidirect
252. #124C2×D97semidirect
258. C22⋊Q8C2×D97semidirect
264. #121C2×D97semidirect
268. C4×Q8C2×D97semidirect
276. C4×D4C2×D97semidirect
362. Q8D4×D97semidirect
409. C22#41×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D972
C22×Q8×D972
#202613972
#202613962
#202613952
#202613942
#202613932
#202613922
#202613912
#124×D972
C22⋊Q8×D972
#121×D972
C4×Q8×D972
C4×D4×D972
#548×C972
C2×#54897