Finite Group Databasebeta

← #38754672

Subgroups — #38754672

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161732C211yesmaximal, characteristic, complemented
2#27161722C211yesmaximal, characteristic, complemented
3#27161408C211yesmaximal, characteristic, complemented
4#27161396C211yesmaximal, characteristic, complemented
5#27161366C211yesmaximal, characteristic, complemented
6#27161331C211yesmaximal, characteristic, complemented
7#27161311C211yesmaximal, characteristic, complemented
8#27161291C211yesmaximal, characteristic, complemented
9#27161227C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic, complemented
11#27161192C211yesmaximal, characteristic, complemented
12#27161152C211yesmaximal, characteristic, complemented
13#27161079C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#489×C197C211yesmaximal, characteristic, complemented
16C2×#15691359—21no—
17C2×#15691359C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic
19C2×#15691345C2211yescharacteristic, complemented
20C2×D1576C2211yescharacteristic
21C2×#15691343C2211yescharacteristic
22C2×#15691341C2211yescharacteristic, complemented
23C2×C8×D197C2211yescharacteristic
24#20263311—21no—
25#20263311—21no—
26#20263308—21no—
27#20263302—21no—
28#20263302C2211yescharacteristic
29#20263301C2211yescharacteristic
30#20263301C2211yescharacteristic, complemented
31#20263299—21no—
32#20263297C2211yescharacteristic
33#20263296—21no—
34C4×D788C2211yescharacteristic
35C4×D788C2211yescharacteristic
36C4×D788C2211yescharacteristic, complemented
37C4⋊C4×D197C2211yescharacteristic
38C4⋊C4×D197C2211yescharacteristic, complemented
39C22⋊C4×D197—21no—
40#20263258C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43C4⋊#13096802C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263238C2211yescharacteristic, complemented
52C4⋊Q8×C197C2211yescharacteristic, complemented
53#124×C197C2211yescharacteristic
54#103×C197C2211yescharacteristic, complemented
55#103×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691359—22no—
61#15691359—41no—
62#15691359—44no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788D411yescharacteristic
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71D4×D197—22no—
72D4×D197—41no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79C2×C4×D197C2311yescharacteristic
80#15691345—41no—
81D1576—22no—
82#15691343—41no—
83#15691341—41no—
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C8×D197—41no—
87C4⋊Dic197D411yescharacteristic
88C4⋊Dic197C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91#15691337—21no—
92#15691337—21no—
93C22⋊Dic197—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335D411yescharacteristic
100#15691335D411yescharacteristic
101C2×Q8×C197—21no—
102C2×Q8×C197C2311yescharacteristic
103C2×D4×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197—21no—
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C22⋊C4×C197—21no—
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117C22×D197C2×D411yescharacteristic
118C22×D197C2×D411yescharacteristic
119D788—22no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—42no—
124D788—42no—
125D788—42no—
126D788#4112yescomplemented
127C22⋊D197—42no—
128C22⋊D197—81no—
129C2×Dic197—21no—
130C2×Dic197—21no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C4×D197—21no—
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—42no—
138C4×D197—42no—
139C4×D197—42no—
140C4×D197—42no—
141C4×D197—81no—
142#13096802—21no—
143#13096802—21no—
144C23×C197—21no—
145Q8×C197—21no—
146Q8×C197—21no—
147Q8×C197—42no—
148D4×C197—21no—
149D4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197C22×D411yescharacteristic
162C2×D197—21no—
163C2×D197—21no—
164C2×D197—21no—
165C2×D197—22no—
166C2×D197—21no—
167C2×D197—21no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—84no—
174C2×D197AGL(1,ℤ/8)12yes—
175C4×C197—21no—
176C4×C197—21no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197C2×#4111yescharacteristic
181C4×C197H(5,𝔽2)11yescharacteristic
182Dic197—21no—
183Dic197—21no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—81no—
187C2×C197#54611yescharacteristic
188C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C197#57711yescharacteristic
190D197—22no—
191D197—41no—
192D197—42no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489D19711yescharacteristic, complemented
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469—1971no—
203#465—1971no—
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41—1971no—
213C22×D4—1971no—
214C2×SD16—1971no—
215C2×D8—1971no—
216C2×D8—1971no—
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8C2×D19711yescharacteristic, complemented
220#124C2×D19711yescharacteristic
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8—1971no—
226#121—3941no—
227#121—1971no—
228#121—1971no—
229#121—1971no—
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106—1971no—
239#106—1971no—
240C4⋊C8—1971no—
241#103—1971no—
242#103C2×D19711yescharacteristic, complemented
243#103C2×D19711yescharacteristic, complemented
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102C2×D19711yescharacteristic
250#102C2×D19711yescharacteristic
251C22⋊C8—1971no—
252C22⋊C8—1971no—
253#97C2×D19711yescharacteristic, complemented
254C24—3941no—
255#41—7884no—
256#41—7881no—
257#41—3942no—
258C2×Q8—21no—
259C2×Q8C22×D19711yescharacteristic
260C2×D4—7881no—
261C2×D4—3941no—
262C2×D4—3941no—
263C2×D4—3941no—
264C2×D4—3942no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4—1971no—
270C2×D4C22×D19711yescharacteristic
271C22×C4—3941no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4—1971no—
278SD16—7881no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—1971no—
305C22⋊C4—1971no—
306C42C22×D19711yescharacteristic
307C23—21no—
308C23—7882no—
309C23—7882no—
310C23—15761no—
311C23—3941no—
312C23—3941no—
313C23—1971no—
314C23—1971no—
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—21no—
319D4—21no—
320D4—7882no—
321D4—7882no—
322D4—7882no—
323D4—7882no—
324D4—15761no—
325D4—3942no—
326D4—3941no—
327D4—3941no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—7882no—
334C2×C4—7882no—
335C2×C4—7882no—
336C2×C4—7882no—
337C2×C4—15761no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—1971no—
345C2×C4—1971no—
346C2×C4D4×D19711yescharacteristic
347C2×C4D4×D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3941no—
352C8—3941no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22—15764no—
362C22—3941no—
363C22—3941no—
364C22—3942no—
365C22—3941no—
366C22—3941no—
367C22—3941no—
368C22—1972no—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—7881no—
376C4—7881no—
377C4—3941no—
378C4—3941no—
379C4#41×D19711yescharacteristic
380C4#2026337211yescharacteristic
381C2—81no—
382C2—7882no—
383C2—7881no—
384C2—15761no—
385C2—3942no—
386C2#2716172211yescharacteristic
387C2AGL(1,ℤ/8)×D19711yescharacteristic
388C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#489×C197C2
socleC22×C197C22×D4
hypercenter#489D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #27161732C2semidirect
2. #27161722C2semidirect
3. #27161408C2semidirect
4. #27161396C2semidirect
5. #27161366C2semidirect
6. #27161331C2semidirect
7. #27161311C2semidirect
8. #27161291C2semidirect
9. #27161227C2semidirect
10. #27161214C2semidirect
11. #27161192C2semidirect
12. #27161152C2semidirect
13. #27161079C2semidirect
14. #102×D197C2semidirect
15. #489×C197C2semidirect
19. C2×#15691345C22semidirect
22. C2×#15691341C22semidirect
30. #20263301C22semidirect
36. C4×D788C22semidirect
38. C4⋊C4×D197C22semidirect
40. #20263258C22semidirect
41. #20263257C22semidirect
42. #20263257C22semidirect
43. C4⋊#13096802C22semidirect
44. #20263250C22semidirect
46. #20263248C22semidirect
51. #20263238C22semidirect
52. C4⋊Q8×C197C22semidirect
54. #103×C197C22semidirect
55. #103×C197C22semidirect
58. #97×C197C22semidirect
126. D788#41semidirect
194. C197#3306semidirect
198. #489D197semidirect
219. C4⋊Q8C2×D197semidirect
242. #103C2×D197semidirect
243. #103C2×D197semidirect
253. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617322
#271617222
#271614082
#271613962
#271613662
#271613312
#271613112
#271612912
#271612272
#271612142
#271611922
#271611522
#271610792
#102×D1972
#489×C1972
#3306197