Finite Group Databasebeta

← #38754521

Subgroups — #38754521

Order 25216.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161730C211yesmaximal, characteristic, complemented
2D4×C22⋊D197C211yesmaximal, characteristic, complemented
3#27161424C211yesmaximal, characteristic, complemented
4#27161392C211yesmaximal, characteristic, complemented
5#27161391C211yesmaximal, characteristic, complemented
6#27161357C211yesmaximal, characteristic, complemented
7#27161315C211yesmaximal, characteristic, complemented
8#27161314C211yesmaximal, characteristic, complemented
9#27161273C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic, complemented
11#27161213C211yesmaximal, characteristic, complemented
12#27161159C211yesmaximal, characteristic, complemented
13C8×C22⋊D197C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15C2×#102×C197C211yesmaximal, characteristic, complemented
16C2×#15691358—21no—
17C22×C22⋊D197—21no—
18C2×D4×D197C2211yescharacteristic
19C2×D1576C2211yescharacteristic, complemented
20C2×#15691343C2211yescharacteristic
21C2×#15691343C2211yescharacteristic, complemented
22C2×C8×D197C2211yescharacteristic, complemented
23#20263325C2211yescharacteristic, complemented
24#20263322C2211yescharacteristic
25#20263311—21no—
26#20263309—21no—
27#20263304—21no—
28#20263301C2211yescharacteristic, complemented
29#20263300C2211yescharacteristic
30#20263300C2211yescharacteristic, complemented
31#20263298—21no—
32#20263297C2211yescharacteristic
33#20263295—21no—
34#20263294—21no—
35D4×Dic197C2211yescharacteristic
36#20263288C2211yescharacteristic, complemented
37C4×C22⋊D197C2211yescharacteristic
38C4⋊C4×D197C2211yescharacteristic, complemented
39#20263260C2211yescharacteristic, complemented
40#20263259C2211yescharacteristic, complemented
41#20263259C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic, complemented
54C2×C4⋊C4×C197C2211yescharacteristic, complemented
55#102×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—44no—
61C2×Dic394—21no—
62C2×D788—21no—
63C2×D788C2311yescharacteristic
64C2×D788C2311yescharacteristic
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—44no—
72C2×C22⋊D197D411yescharacteristic
73D4×D197—22no—
74D4×D197—41no—
75C22×Dic197—21no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79D1576—41no—
80#15691343—22no—
81#15691343—41no—
82C2×#13096802C2311yescharacteristic
83C2×#13096802C2311yescharacteristic
84C8×D197—41no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197D411yescharacteristic
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98C4×Dic197C2311yescharacteristic
99C24×C197—21no—
100C2×D4×C197—22no—
101C2×D4×C197—21no—
102C2×D4×C197—41no—
103C2×D4×C197C2311yescharacteristic
104C2×D4×C197C2311yescharacteristic
105C22×C4×C197—21no—
106C22×C4×C197C2311yescharacteristic
107C2×C8×C197—41no—
108C2×C8×C197C2311yescharacteristic
109C2×C8×C197C2311yescharacteristic
110C4⋊C4×C197—21no—
111C4⋊C4×C197C2311yescharacteristic
112C4⋊C4×C197C2311yescharacteristic
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—81no—
118C22×D197C2×D411yescharacteristic
119Dic394—42no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—42no—
125C22⋊D197—42no—
126C22⋊D197—42no—
127C22⋊D197—42no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C22⋊D197—81no—
131C22⋊D197D814yescomplemented
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—42no—
135C2×Dic197C2×D411yescharacteristic
136C2×Dic197C2×D411yescharacteristic
137C2×Dic197C2×D411yescharacteristic
138C4×D197—21no—
139C4×D197—21no—
140C4×D197—42no—
141C4×D197—42no—
142#13096802—21no—
143#13096802—21no—
144C23×C197—21no—
145C23×C197—21no—
146C23×C197—42no—
147C23×C197—42no—
148C23×C197—81no—
149C23×C197C2×D411yescharacteristic
150D4×C197—21no—
151D4×C197—21no—
152D4×C197—21no—
153D4×C197—42no—
154D4×C197#4112yescomplemented
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197—42no—
160C2×C4×C197C2×D411yescharacteristic
161C2×C4×C197C2411yescharacteristic
162C8×C197—21no—
163C8×C197—21no—
164C22×C197—21no—
165C22×C197—21no—
166C22×C197—22no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197—41no—
170C22×C197—41no—
171C22×C197—41no—
172C22×C197—84no—
173C22×C197C2×D812yes—
174C22×C197C22×D411yescharacteristic
175C2×D197—21no—
176C2×D197—21no—
177C2×D197—41no—
178C2×D197—41no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—84no—
184C2×D197C2×D812yes—
185C4×C197—21no—
186C4×C197—41no—
187C4×C197—41no—
188C4×C197C2×#4111yescharacteristic
189C4×C197H(5,𝔽2)11yescharacteristic
190Dic197—21no—
191Dic197—21no—
192Dic197—41no—
193Dic197—41no—
194Dic197C2×D812yes—
195C2×C197—22no—
196C2×C197—42no—
197C2×C197—81no—
198C2×C197#54611yescharacteristic
199C2×C197C22×D811yescharacteristic
200C2×C197#57711yescharacteristic
201D197—22no—
202D197—81no—
203D197—81no—
204C197#329411yescharacteristic, complemented, Sylow
205#3294—1971nomaximal, Sylow
206#547—1971no—
207D42—1971no—
208#486—1971no—
209#480—1971no—
210#480—1971no—
211#474—1971no—
212#466—1971no—
213#466—1971no—
214#459—1971no—
215#447—1971no—
216#447—1971no—
217C4×D8—1971no—
218C8×D4—1971no—
219C2×#102—1971no—
220C2×#102D19711yescharacteristic, complemented
221C2×#41—3941no—
222C22×D4—3941no—
223C22×D4—1971no—
224C22×D4C2×D19711yescharacteristic
225C2×D8—1971no—
226C2×D8—1971no—
227C2×D8—1971no—
228C22×C8—1971no—
229C22×C8C2×D19711yescharacteristic, complemented
230C4⋊Q8—1971no—
231#127—1971no—
232#123—3941no—
233#123—3941no—
234C22⋊Q8—3941no—
235#121—3941no—
236#121—1971no—
237#121—1971no—
238#121—1971no—
239#121—1971no—
240#120—3941no—
241#120—3941no—
242C4×D4—1971no—
243C4×D4—1971no—
244C4×D4—1971no—
245C2×C4⋊C4—1971no—
246C2×C4⋊C4C2×D19711yescharacteristic, complemented
247#107—1971no—
248#107—1971no—
249#107—1971no—
250C4⋊C8—1971no—
251#102—1971no—
252#102—1971no—
253#102—1971no—
254#102—1971no—
255#102—1971no—
256#102—1971no—
257#102C2×D19711yescharacteristic
258#102C2×D19711yescharacteristic
259#102C2×D19711yescharacteristic, complemented
260#102C2×D19711yescharacteristic, complemented
261C22⋊C8—1971no—
262C22⋊C8—1971no—
263C4×C8—1971no—
264C24—21no—
265C24—3941no—
266#41—7884no—
267C2×Q8—3941no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—7884no—
272C2×D4—7881no—
273C2×D4—3941no—
274C2×D4—3941no—
275C2×D4—3941no—
276C2×D4—3941no—
277C2×D4—3942no—
278C2×D4—3941no—
279C2×D4—3941no—
280C2×D4—3941no—
281C2×D4—1971no—
282C2×D4—1971no—
283C2×D4—1971no—
284C2×D4C22×D19711yescharacteristic
285C2×D4C22×D19711yescharacteristic
286C22×C4—21no—
287C22×C4—3941no—
288C22×C4—3941no—
289C22×C4—3941no—
290C22×C4—1971no—
291C22×C4C22×D19711yescharacteristic
292D8—7881no—
293D8—7881no—
294D8—3942no—
295C2×C8—41no—
296C2×C8—7881no—
297C2×C8—1971no—
298C2×C8—1971no—
299C2×C8C22×D19711yescharacteristic
300C2×C8C22×D19711yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—3941no—
303C4⋊C4—3941no—
304C4⋊C4—1971no—
305C4⋊C4—1971no—
306C4⋊C4—1971no—
307C4⋊C4C22×D19711yescharacteristic
308C4⋊C4C22×D19711yescharacteristic
309C22⋊C4—3941no—
310C22⋊C4—3941no—
311C22⋊C4—3941no—
312C22⋊C4—3941no—
313C22⋊C4—3941no—
314C22⋊C4—3941no—
315C22⋊C4—1971no—
316C22⋊C4—1971no—
317C42—1971no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—7882no—
324C23—7882no—
325C23—15761no—
326C23—3941no—
327C23—3941no—
328C23—1971no—
329C23D4×D19711yescharacteristic
330Q8—7882no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—7882no—
336D4—7882no—
337D4—7882no—
338D4—7882no—
339D4—7882no—
340D4—7882no—
341D4—15761no—
342D4—3941no—
343D4—3941no—
344D4—3941no—
345D4—3941no—
346D4—1974no—
347D4#1569135812yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—7882no—
354C2×C4—7882no—
355C2×C4—7882no—
356C2×C4—3941no—
357C2×C4—3941no—
358C2×C4—3941no—
359C2×C4—3941no—
360C2×C4—1971no—
361C2×C4—1971no—
362C2×C4—1971no—
363C2×C4D4×D19711yescharacteristic
364C2×C4C23×D19711yescharacteristic
365C8—21no—
366C8—21no—
367C8—3941no—
368C8—3941no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—7881no—
379C22—7881no—
380C22—7881no—
381C22—7881no—
382C22—7881no—
383C22—7881no—
384C22—15764no—
385C22—3941no—
386C22—3941no—
387C22—1972no—
388C22D8×D19712yes—
389C22C2×D4×D19711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—7881no—
394C4—7881no—
395C4—3941no—
396C4—3941no—
397C4—1972no—
398C4C2×#1569135811yescharacteristic
399C4#2026337111yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—15761no—
404C2—15761no—
405C2—3942no—
406C2#2716171911yescharacteristic
407C2C2×D8×D19711yescharacteristic
408C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC22×C197C22×D4
hypercenterC2×#102D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #27161730C2semidirect
2. D4×C22⋊D197C2semidirect
3. #27161424C2semidirect
4. #27161392C2semidirect
5. #27161391C2semidirect
6. #27161357C2semidirect
7. #27161315C2semidirect
8. #27161314C2semidirect
9. #27161273C2semidirect
10. #27161214C2semidirect
11. #27161213C2semidirect
12. #27161159C2semidirect
13. C8×C22⋊D197C2semidirect
14. #102×D197C2semidirect
15. C2×#102×C197C2semidirect
19. C2×D1576C22semidirect
21. C2×#15691343C22semidirect
22. C2×C8×D197C22semidirect
23. #20263325C22semidirect
28. #20263301C22semidirect
30. #20263300C22semidirect
36. #20263288C22semidirect
38. C4⋊C4×D197C22semidirect
39. #20263260C22semidirect
40. #20263259C22semidirect
41. #20263259C22semidirect
42. #20263255C22semidirect
43. #20263248C22semidirect
44. #20263248C22semidirect
46. #20263247C22semidirect
47. #20263246C22semidirect
48. #20263246C22semidirect
49. C22⋊#13096802C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
53. C22×C8×C197C22semidirect
54. C2×C4⋊C4×C197C22semidirect
57. #102×C197C22semidirect
58. #102×C197C22semidirect
131. C22⋊D197D8semidirect
154. D4×C197#41semidirect
204. C197#3294semidirect
220. C2×#102D197semidirect
229. C22×C8C2×D197semidirect
246. C2×C4⋊C4C2×D197semidirect
259. #102C2×D197semidirect
260. #102C2×D197semidirect
347. D4#15691358semidirect

Maximal subgroups

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

subgroupindex
#271617302
D4×C22⋊D1972
#271614242
#271613922
#271613912
#271613572
#271613152
#271613142
#271612732
#271612142
#271612132
#271611592
C8×C22⋊D1972
#102×D1972
C2×#102×C1972
#3294197