Finite Group Databasebeta

← #38754216

Subgroups — #38754216

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×SD16×D197C211yesmaximal, characteristic, complemented
2Q8×C22⋊D197C211yesmaximal, characteristic, complemented
3#27161717C211yesmaximal, characteristic
4#27161472C211yesmaximal, characteristic, complemented
5#27161325C211yesmaximal, characteristic, complemented
6#27161317C211yesmaximal, characteristic, complemented
7#27161303C211yesmaximal, characteristic, complemented
8#27161282C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161233C211yesmaximal, characteristic, complemented
11#27161228C211yesmaximal, characteristic, complemented
12#27161218C211yesmaximal, characteristic, complemented
13#27161180C211yesmaximal, characteristic, complemented
14#27161148C211yesmaximal, characteristic, complemented
15C2×#137×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691348C2211yescharacteristic, complemented
19C2×#15691347C2211yescharacteristic
20C2×#15691346C2211yescharacteristic
21C2×#15691345C2211yescharacteristic
22C2×#15691345C2211yescharacteristic
23SD16×D197—24no—
24C2×C8×D197C2211yescharacteristic
25#20263324—21no—
26#20263324C2211yescharacteristic, complemented
27#20263315C2211yescharacteristic
28#20263309—21no—
29#20263305—21no—
30#20263305C2211yescharacteristic, complemented
31#20263302—21no—
32#20263302C2211yescharacteristic
33#20263302C2211yescharacteristic
34#20263300C2211yescharacteristic
35#20263298—21no—
36#20263294—21no—
37Q8×Dic197C2211yescharacteristic, complemented
38D4×Dic197C2211yescharacteristic
39C4×C22⋊D197—21no—
40C4×C22⋊D197C2211yescharacteristic
41C2×#15691335—21no—
42#20263260C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263250C2211yescharacteristic, complemented
47#20263250C2211yescharacteristic, complemented
48#20263248C2211yescharacteristic
49#20263247C2211yescharacteristic
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic
52#20263236C2211yescharacteristic, complemented
53C2×#41×C197C2211yescharacteristic
54C22×Q8×C197C2211yescharacteristic, complemented
55#137×C197—24no—
56C2×Q16×C197C2211yescharacteristic, complemented
57C2×Q16×C197C2211yescharacteristic, complemented
58C2×SD16×C197C2211yescharacteristic
59C2×SD16×C197C2211yescharacteristic
60C2×M16×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C2×Dic394—21no—
63C2×Dic394C2311yescharacteristic
64Q8×D197—22no—
65Q8×D197—41no—
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197D411yescharacteristic
70D4×D197—22no—
71D4×D197—41no—
72C22×Dic197—21no—
73C2×C4×D197—21no—
74C2×C4×D197C2311yescharacteristic
75#15691348—41no—
76#15691347—22no—
77#15691346—22no—
78#15691345—22no—
79#15691345—41no—
80C2×#13096802C2311yescharacteristic
81C2×#13096802C2311yescharacteristic
82C8×D197—22no—
83C4⋊Dic197—21no—
84C4⋊Dic197C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337—21no—
88#15691337—21no—
89#15691337—21no—
90#15691337—21no—
91#15691337D411yescharacteristic
92C22⋊Dic197—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197D411yescharacteristic
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335D411yescharacteristic
101C4×Dic197—21no—
102C4×Dic197C2311yescharacteristic
103#41×C197—22no—
104#41×C197—41no—
105C2×Q8×C197—21no—
106C2×Q8×C197—22no—
107C2×Q8×C197—41no—
108C2×Q8×C197C2311yescharacteristic
109C2×Q8×C197C2311yescharacteristic
110C2×Q8×C197C2311yescharacteristic
111C2×D4×C197—21no—
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—22no—
117Q16×C197—22no—
118SD16×C197—22no—
119SD16×C197D414yescomplemented
120M16×C197—22no—
121C2×C8×C197C2311yescharacteristic
122C2×C8×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—42no—
125C22×D197—42no—
126C22×D197—81no—
127C22×D197C2×D411yescharacteristic
128Dic394—21no—
129Dic394—21no—
130Dic394—42no—
131D788—21no—
132D788—21no—
133C22⋊D197—22no—
134C22⋊D197—42no—
135C22⋊D197—81no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—81no—
140C2×Dic197C2×D411yescharacteristic
141C2×Dic197C2×D411yescharacteristic
142C2×Dic197C2×D411yescharacteristic
143C4×D197—21no—
144C4×D197—21no—
145C4×D197—42no—
146#13096802—21no—
147#13096802—21no—
148C23×C197—21no—
149C23×C197C2×D411yescharacteristic
150Q8×C197—21no—
151Q8×C197—21no—
152Q8×C197—21no—
153Q8×C197—21no—
154Q8×C197—21no—
155Q8×C197—42no—
156Q8×C197C2×D412yes—
157D4×C197—21no—
158D4×C197—42no—
159D4×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—42no—
166C2×C4×C197—42no—
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2411yescharacteristic
169C8×C197—21no—
170C8×C197C2×D412yes—
171C22×C197—21no—
172C22×C197—22no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—41no—
176C22×C197—41no—
177C22×C197C22×D411yescharacteristic
178C2×D197—21no—
179C2×D197—21no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—84no—
184C2×D197#13712yescomplemented
185C4×C197—21no—
186C4×C197—22no—
187C4×C197—41no—
188C4×C197—41no—
189C4×C197C22×D411yescharacteristic
190C4×C197H(5,𝔽2)11yescharacteristic
191Dic197—21no—
192Dic197—21no—
193Dic197—21no—
194Dic197—41no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—42no—
198C2×C197—41no—
199C2×C197D4211yescharacteristic
200C2×C197C2×#13711yescharacteristic
201C2×C197#57711yescharacteristic
202D197—22no—
203D197—81no—
204C197#326611yescharacteristic, complemented, Sylow
205#3266—1971nomaximal, Sylow
206C2×#137D19711yescharacteristic, complemented
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#451—1971no—
217#450—1971no—
218#448—1971no—
219#440—1971no—
220#435—1971no—
221C2×#41C2×D19711yescharacteristic
222C22×Q8—1971no—
223C22×Q8C2×D19711yescharacteristic, complemented
224C22×D4—1971no—
225#137—24no—
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×SD16C2×D19711yescharacteristic
235C2×SD16C2×D19711yescharacteristic
236C2×M16C2×D19711yescharacteristic, complemented
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⋊Q8—1971no—
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#103—1971no—
260#103—1971no—
261#102—1971no—
262#102—1971no—
263C22⋊C8—1971no—
264C22⋊C8—1971no—
265#97—1971no—
266C24—3941no—
267#41—22no—
268#41—41no—
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—21no—
280C2×D4—7881no—
281C2×D4—3942no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—1971no—
285C2×D4—1971no—
286C2×D4C22×D19711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3941no—
290C22×C4—3941no—
291C22×C4—1971no—
292C22×C4C22×D19711yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—7881no—
296SD16—22no—
297SD16—7881no—
298SD16—3942no—
299SD16—3942no—
300SD16—3942no—
301SD16C22⋊D19714yescomplemented
302M16—22no—
303C2×C8—3942no—
304C2×C8—1971no—
305C2×C8—1971no—
306C2×C8C22×D19711yescharacteristic
307C2×C8C22×D19711yescharacteristic
308C4⋊C4—3941no—
309C4⋊C4—3941no—
310C4⋊C4—3941no—
311C4⋊C4—3941no—
312C4⋊C4—3941no—
313C4⋊C4—3941no—
314C4⋊C4—1971no—
315C4⋊C4—1971no—
316C4⋊C4—1971no—
317C22⋊C4—3941no—
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—21no—
329C23—7882no—
330C23—7882no—
331C23—15761no—
332C23—3941no—
333C23—1971no—
334C23D4×D19711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—7882no—
342Q8—3941no—
343Q8—3941no—
344Q8C2×C22⋊D19712yes—
345D4—21no—
346D4—42no—
347D4—7882no—
348D4—15761no—
349D4—3942no—
350D4—3941no—
351D4—3941no—
352D4C2×C22⋊D19712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—7882no—
361C2×C4—15761no—
362C2×C4—3941no—
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—3941no—
374C8—3941no—
375C8C2×C22⋊D19712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—7881no—
383C22—7881no—
384C22—7881no—
385C22—15764no—
386C22—3941no—
387C22—3941no—
388C22—1972no—
389C22C2×D4×D19711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—7881no—
395C4—7881no—
396C4—7881no—
397C4—3941no—
398C4—3941no—
399C4—3941no—
400C4C22×C22⋊D19711yescharacteristic
401C4#2026337111yescharacteristic
402C2—41no—
403C2—42no—
404C2—15761no—
405C2—3942no—
406C2D4×C22⋊D19711yescharacteristic
407C2#137×D19711yescharacteristic
408C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197C22×D4
prime radical#27161717C2
hypercenterC2×#137D197
hypocenterC197#3266
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D197C2semidirect
2. Q8×C22⋊D197C2semidirect
4. #27161472C2semidirect
5. #27161325C2semidirect
6. #27161317C2semidirect
7. #27161303C2semidirect
8. #27161282C2semidirect
9. #27161241C2semidirect
10. #27161233C2semidirect
11. #27161228C2semidirect
12. #27161218C2semidirect
13. #27161180C2semidirect
14. #27161148C2semidirect
15. C2×#137×C197C2semidirect
18. C2×#15691348C22semidirect
26. #20263324C22semidirect
30. #20263305C22semidirect
37. Q8×Dic197C22semidirect
42. #20263260C22semidirect
43. #20263255C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263250C22semidirect
47. #20263250C22semidirect
50. C22⋊#13096802C22semidirect
52. #20263236C22semidirect
54. C22×Q8×C197C22semidirect
56. C2×Q16×C197C22semidirect
57. C2×Q16×C197C22semidirect
60. C2×M16×C197C22semidirect
119. SD16×C197D4semidirect
184. C2×D197#137semidirect
204. C197#3266semidirect
206. C2×#137D197semidirect
223. C22×Q8C2×D197semidirect
227. C2×Q16C2×D197semidirect
228. C2×Q16C2×D197semidirect
236. C2×M16C2×D197semidirect
301. SD16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1972
Q8×C22⋊D1972
#271617172
#271614722
#271613252
#271613172
#271613032
#271612822
#271612412
#271612332
#271612282
#271612182
#271611802
#271611482
C2×#137×C1972
#3266197