Finite Group Databasebeta

← #38754215

Subgroups — #38754215

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
1C2×#20263358C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3Q8×C22⋊D197C211yesmaximal, characteristic, complemented
4#27161720C211yesmaximal, characteristic
5#27161472C211yesmaximal, characteristic, complemented
6#27161328C211yesmaximal, characteristic, complemented
7#27161318C211yesmaximal, characteristic, complemented
8#27161303C211yesmaximal, characteristic, complemented
9#27161283C211yesmaximal, characteristic, complemented
10#27161240C211yesmaximal, characteristic, complemented
11#27161232C211yesmaximal, characteristic, complemented
12#27161219C211yesmaximal, characteristic, complemented
13#27161182C211yesmaximal, characteristic, complemented
14#27161147C211yesmaximal, characteristic, complemented
15#451×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×Q8×D197C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19#20263358—24no—
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691347C2211yescharacteristic
22C2×#15691347C2211yescharacteristic
23C2×#15691345—24no—
24C2×#15691345C2211yescharacteristic
25C2×#15691345C2211yescharacteristic
26C2×#15691345C2211yescharacteristic
27C2×#15691345C2211yescharacteristic
28C2×#15691341C2211yescharacteristic, complemented
29C22×#13096802C2211yescharacteristic
30#20263324—21no—
31#20263324C2211yescharacteristic, complemented
32#20263315C2211yescharacteristic
33#20263311—21no—
34#20263305—21no—
35#20263305C2211yescharacteristic
36#20263302—21no—
37#20263302C2211yescharacteristic, complemented
38#20263301—21no—
39#20263301C2211yescharacteristic
40#20263296—21no—
41Q8×Dic197C2211yescharacteristic, complemented
42#20263288C2211yescharacteristic
43C4×C22⋊D197—21no—
44C4×C22⋊D197C2211yescharacteristic
45C22⋊C4×D197—21no—
46#20263259C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52#20263238C2211yescharacteristic, complemented
53#20263236C2211yescharacteristic, complemented
54C22×Q8×C197C2211yescharacteristic
55C2×Q16×C197C2211yescharacteristic, complemented
56C2×Q16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic
61C23×D197—21no—
62#15691359—22no—
63#15691359—41no—
64C2×Dic394—21no—
65C2×Dic394C2311yescharacteristic
66Q8×D197—22no—
67Q8×D197—41no—
68C2×D788—22no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—21no—
72C2×D788—41no—
73C2×D788C2311yescharacteristic
74C2×D788C2311yescharacteristic
75C2×C22⋊D197D411yescharacteristic
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197—21no—
79C2×C4×D197C2311yescharacteristic
80#15691348—22no—
81#15691347—22no—
82#15691347—41no—
83#15691345—22no—
84#15691345—22no—
85#15691345—22no—
86#15691345D414yescomplemented
87#15691341—22no—
88C2×#13096802—22no—
89C2×#13096802C2311yescharacteristic
90C2×#13096802C2311yescharacteristic
91C4⋊Dic197—21no—
92C4⋊Dic197C2311yescharacteristic
93#15691337—21no—
94#15691337—21no—
95#15691337—21no—
96#15691337—21no—
97#15691337D411yescharacteristic
98C22⋊Dic197—21no—
99C22⋊Dic197D411yescharacteristic
100#15691335—21no—
101#15691335—21no—
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335D411yescharacteristic
106C4×Dic197—21no—
107C4×Dic197C2311yescharacteristic
108C2×Q8×C197—21no—
109C2×Q8×C197—22no—
110C2×Q8×C197—41no—
111C2×Q8×C197C2311yescharacteristic
112C2×Q8×C197C2311yescharacteristic
113C2×Q8×C197C2311yescharacteristic
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—22no—
117Q16×C197—41no—
118C2×C8×C197C2311yescharacteristic
119C2×C8×C197C2311yescharacteristic
120C4⋊C4×C197—21no—
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131Dic394—42no—
132D788—21no—
133D788—21no—
134D788—21no—
135D788—42no—
136D788—42no—
137D788—81no—
138D788C2×D412yes—
139C22⋊D197—22no—
140C2×Dic197—21no—
141C2×Dic197—21no—
142C2×Dic197C2×D411yescharacteristic
143C2×Dic197C2×D411yescharacteristic
144C2×Dic197C2×D411yescharacteristic
145C4×D197—21no—
146C4×D197—21no—
147C4×D197—42no—
148C4×D197—42no—
149C4×D197—81no—
150#13096802—21no—
151#13096802C2×D412yes—
152C23×C197C2×D411yescharacteristic
153Q8×C197—21no—
154Q8×C197—21no—
155Q8×C197—21no—
156Q8×C197—21no—
157Q8×C197—21no—
158Q8×C197—42no—
159Q8×C197C2×D412yes—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—42no—
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2411yescharacteristic
169C8×C197—21no—
170C8×C197—21no—
171C22×C197—21no—
172C22×C197—21no—
173C22×C197#13712yescomplemented
174C22×C197C22×D411yescharacteristic
175C2×D197—21no—
176C2×D197—22no—
177C2×D197—21no—
178C2×D197—21no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—41no—
184C2×D197—84no—
185C4×C197—21no—
186C4×C197—22no—
187C4×C197—41no—
188C4×C197—41no—
189C4×C197—41no—
190C4×C197C22×D411yescharacteristic
191C4×C197H(5,𝔽2)11yescharacteristic
192Dic197—21no—
193Dic197—21no—
194Dic197—21no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—22no—
198C2×C197D4211yescharacteristic
199C2×C197C2×#13711yescharacteristic
200C2×C197#57711yescharacteristic
201D197—41no—
202D197—42no—
203D197—81no—
204C197#326611yescharacteristic, complemented, Sylow
205#3266—1971nomaximal, Sylow
206C2×#137—1971no—
207C22×SD16—1971no—
208D4×Q8—1971no—
209#546—1971no—
210#497—1971no—
211#468—1971no—
212#467—1971no—
213#464—1971no—
214#461—1971no—
215#452—1971no—
216#451D19711yescharacteristic, complemented
217#450—1971no—
218#448—1971no—
219#440—1971no—
220#435—1971no—
221C2×#41—1971no—
222C22×Q8—1971no—
223C22×Q8C2×D19711yescharacteristic
224C22×D4—1971no—
225#137—3944no—
226C2×Q16—1971no—
227C2×Q16C2×D19711yescharacteristic, complemented
228C2×Q16C2×D19711yescharacteristic, complemented
229C2×SD16—3944no—
230C2×SD16—1971no—
231C2×SD16—1971no—
232C2×SD16—1971no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×M16—1971no—
237C22×C8—1971no—
238C4⋊Q8—3941no—
239C4⋊Q8—1971no—
240#124—1971no—
241#123—3941no—
242C22⋊Q8—3941no—
243C22⋊Q8—3941no—
244C22⋊Q8—1971no—
245C22⋊Q8—1971no—
246C22⋊Q8C2×D19711yescharacteristic
247#121—3941no—
248#121—1971no—
249#120—3941no—
250C4×Q8—1971no—
251C4×D4—3941no—
252C4×D4—1971no—
253C4×D4—1971no—
254C2×C22⋊C4—3941no—
255#107—1971no—
256C4⋊C8—1971no—
257#103—1971no—
258#103—1971no—
259#103C2×D19711yescharacteristic, complemented
260#103C2×D19711yescharacteristic, complemented
261#102—1971no—
262#102—1971no—
263C22⋊C8—1971no—
264C22⋊C8C2×D19711yescharacteristic
265#97—1971no—
266C24—3941no—
267#41—7881no—
268#41—3942no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—7881no—
273C2×Q8—3941no—
274C2×Q8—3942no—
275C2×Q8—1971no—
276C2×Q8C22×D19711yescharacteristic
277C2×Q8C22×D19711yescharacteristic
278C2×Q8C22×D19711yescharacteristic
279C2×D4—7881no—
280C2×D4—3941no—
281C2×D4—3942no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—1971no—
285C2×D4—1971no—
286C2×D4—1971no—
287C22×C4—21no—
288C22×C4—3941no—
289C22×C4—3941no—
290C22×C4—3941no—
291C22×C4—1971no—
292C22×C4C22×D19711yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—3942no—
296SD16—7881no—
297SD16—3942no—
298SD16—3942no—
299SD16—3942no—
300SD16—3942no—
301SD16—1974no—
302M16—3942no—
303C2×C8—3942no—
304C2×C8—1971no—
305C2×C8—1971no—
306C2×C8C22×D19711yescharacteristic
307C2×C8C22×D19711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3941no—
310C4⋊C4—3941no—
311C4⋊C4—3941no—
312C4⋊C4—3941no—
313C4⋊C4—3941no—
314C4⋊C4—1971no—
315C4⋊C4—1971no—
316C4⋊C4C22×D19711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3941no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—1971no—
325C22⋊C4—1971no—
326C42—3941no—
327C42—1971no—
328C23—7882no—
329C23—7882no—
330C23—15761no—
331C23—3941no—
332C23—3941no—
333C23—1971no—
334C23D4×D19711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—7882no—
342Q8—3941no—
343Q8—3941no—
344Q8D4×D19712yes—
345D4—7882no—
346D4—7882no—
347D4—15761no—
348D4—3942no—
349D4—3941no—
350D4—3941no—
351D4—3941no—
352D4—1972no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—7882no—
361C2×C4—7882no—
362C2×C4—15761no—
363C2×C4—3941no—
364C2×C4—3941no—
365C2×C4—3941no—
366C2×C4—3941no—
367C2×C4—1971no—
368C2×C4—1971no—
369C2×C4—1971no—
370C2×C4D4×D19711yescharacteristic
371C2×C4C23×D19711yescharacteristic
372C8—21no—
373C8—21no—
374C8—3941no—
375C8—1972no—
376C22—21no—
377C22—21no—
378C22—7881no—
379C22—7881no—
380C22—7881no—
381C22—7881no—
382C22—7881no—
383C22—15764no—
384C22—3941no—
385C22—3941no—
386C22—3941no—
387C22—3942no—
388C22#2026335812yescomplemented
389C22C2×D4×D19711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—7881no—
396C4—7881no—
397C4—3941no—
398C4—3941no—
399C4—3941no—
400C4C2×D4×D19711yescharacteristic
401C4#2026337111yescharacteristic
402C2—22no—
403C2—7882no—
404C2—7881no—
405C2—15761no—
406C2#2716171111yescharacteristic
407C2C2×#2026335811yescharacteristic
408C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#451×C197C2
socleC22×C197C22×D4
prime radical#27161720C2
hypercenter#451D197
hypocenterC197#3266
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#20263358C2semidirect
2. C22×#15691345C2semidirect
3. Q8×C22⋊D197C2semidirect
5. #27161472C2semidirect
6. #27161328C2semidirect
7. #27161318C2semidirect
8. #27161303C2semidirect
9. #27161283C2semidirect
10. #27161240C2semidirect
11. #27161232C2semidirect
12. #27161219C2semidirect
13. #27161182C2semidirect
14. #27161147C2semidirect
15. #451×C197C2semidirect
17. C2×Q8×D197C22semidirect
20. C2×#15691348C22semidirect
28. C2×#15691341C22semidirect
31. #20263324C22semidirect
37. #20263302C22semidirect
41. Q8×Dic197C22semidirect
46. #20263259C22semidirect
47. #20263255C22semidirect
48. #20263250C22semidirect
49. #20263249C22semidirect
52. #20263238C22semidirect
53. #20263236C22semidirect
55. C2×Q16×C197C22semidirect
56. C2×Q16×C197C22semidirect
58. #103×C197C22semidirect
59. #103×C197C22semidirect
86. #15691345D4semidirect
173. C22×C197#137semidirect
204. C197#3266semidirect
216. #451D197semidirect
227. C2×Q16C2×D197semidirect
228. C2×Q16C2×D197semidirect
259. #103C2×D197semidirect
260. #103C2×D197semidirect
388. C22#20263358semidirect

Maximal subgroups

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

subgroupindex
C2×#202633582
C22×#156913452
Q8×C22⋊D1972
#271617202
#271614722
#271613282
#271613182
#271613032
#271612832
#271612402
#271612322
#271612192
#271611822
#271611472
#451×C1972
#3266197