Finite Group Databasebeta

← #38754237

Subgroups — #38754237

Order 25216.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2#27161728C211yesmaximal, characteristic, complemented
3#27161717C211yesmaximal, characteristic, complemented
4#27161472C211yesmaximal, characteristic, complemented
5#27161334C211yesmaximal, characteristic, complemented
6#27161308C211yesmaximal, characteristic, complemented
7#27161296C211yesmaximal, characteristic, complemented
8#27161282C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161238C211yesmaximal, characteristic, complemented
11#27161228C211yesmaximal, characteristic, complemented
12#27161224C211yesmaximal, characteristic, complemented
13#27161191C211yesmaximal, characteristic, complemented
14#27161150C211yesmaximal, characteristic, complemented
15C2×#135×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18#20263350—24no—
19C2×#15691348C2211yescharacteristic, complemented
20C2×#15691347C2211yescharacteristic
21C2×#15691346C2211yescharacteristic
22C2×#15691345C2211yescharacteristic
23C2×#15691343C2211yescharacteristic
24C2×#15691341C2211yescharacteristic
25#20263324C2211yescharacteristic, complemented
26#20263315C2211yescharacteristic
27#20263309—21no—
28#20263309—21no—
29#20263307—21no—
30#20263305C2211yescharacteristic, complemented
31#20263302C2211yescharacteristic
32#20263302C2211yescharacteristic
33#20263300C2211yescharacteristic
34#20263298—21no—
35#20263298—21no—
36#20263294—21no—
37D4×Dic197C2211yescharacteristic
38D4×Dic197C2211yescharacteristic, complemented
39C4×C22⋊D197C2211yescharacteristic
40C2×#15691337—21no—
41C2×#15691335—21no—
42#20263258C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263250C2211yescharacteristic, complemented
47#20263248C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263247C2211yescharacteristic, complemented
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic
52#20263236C2211yescharacteristic, complemented
53C2×#41×C197C2211yescharacteristic
54C2×#41×C197C2211yescharacteristic, complemented
55#135×C197—24no—
56C2×Q16×C197C2211yescharacteristic, complemented
57C2×SD16×C197C2211yescharacteristic
58C2×SD16×C197C2211yescharacteristic, complemented
59C2×D8×C197C2211yescharacteristic
60C22×C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691357—22no—
63#15691357—41no—
64C2×Dic394C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197D411yescharacteristic
70D4×D197—22no—
71D4×D197—41no—
72C22×Dic197—21no—
73C22×Dic197—21no—
74C22×Dic197—21no—
75C2×C4×D197C2311yescharacteristic
76#15691348—41no—
77#15691347—22no—
78#15691346—22no—
79#15691345—41no—
80#15691343—22no—
81#15691341—22no—
82C2×#13096802C2311yescharacteristic
83C2×#13096802C2311yescharacteristic
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⋊Dic197—21no—
95C22⋊Dic197D411yescharacteristic
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335—21no—
101#15691335D411yescharacteristic
102C4×Dic197C2311yescharacteristic
103#41×C197—22no—
104#41×C197—22no—
105#41×C197—41no—
106#41×C197—41no—
107C2×Q8×C197C2311yescharacteristic
108C2×Q8×C197C2311yescharacteristic
109C2×D4×C197—21no—
110C2×D4×C197—21no—
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—22no—
117SD16×C197—22no—
118SD16×C197—22no—
119D8×C197D414yescomplemented
120C2×C8×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—
130D788—21no—
131D788—21no—
132C22⋊D197—41no—
133C22⋊D197—42no—
134C22⋊D197—42no—
135C22⋊D197—81no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—42no—
140C2×Dic197—42no—
141C2×Dic197—81no—
142C2×Dic197C2×D411yescharacteristic
143C2×Dic197C2×D411yescharacteristic
144C2×Dic197C2×D411yescharacteristic
145C4×D197—21no—
146C4×D197—21no—
147#13096802—21no—
148#13096802—21no—
149C23×C197—21no—
150C23×C197—21no—
151C23×C197C2×D411yescharacteristic
152Q8×C197—21no—
153Q8×C197—21no—
154Q8×C197—21no—
155Q8×C197—21no—
156D4×C197—21no—
157D4×C197—21no—
158D4×C197—42no—
159D4×C197—42no—
160D4×C197C2×D412yes—
161D4×C197C2×D412yes—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—42no—
167C2×C4×C197—42no—
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2411yescharacteristic
170C8×C197—21no—
171C8×C197C2×D412yes—
172C22×C197—21no—
173C22×C197—22no—
174C22×C197—22no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—41no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C22×D411yescharacteristic
182C2×D197—21no—
183C2×D197—21no—
184C2×D197—21no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—84no—
189C4×C197—21no—
190C4×C197—41no—
191C4×C197—41no—
192C4×C197C22×D411yescharacteristic
193C4×C197H(5,𝔽2)11yescharacteristic
194Dic197—21no—
195Dic197—21no—
196Dic197—21no—
197Dic197—41no—
198Dic197—41no—
199Dic197—41no—
200C2×C197—42no—
201C2×C197—41no—
202C2×C197—42no—
203C2×C197D4211yescharacteristic
204C2×C197#57511yescharacteristic
205C2×C197#57711yescharacteristic
206D197—41no—
207D197—81no—
208C197#326711yescharacteristic, complemented, Sylow
209#3267—1971nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1971no—
211C2×#135D19711yescharacteristic, complemented
212#547—1971no—
213#546—1971no—
214#497—1971no—
215#470—1971no—
216#465—1971no—
217#463—1971no—
218#461—1971no—
219#452—1971no—
220#452—1971no—
221#450—1971no—
222#449—1971no—
223#442—1971no—
224#435—1971no—
225C2×#41—1971no—
226C2×#41C2×D19711yescharacteristic
227C2×#41C2×D19711yescharacteristic, complemented
228C22×D4—1971no—
229AGL(1,ℤ/8)—3944no—
230#135—24no—
231C2×Q16—1971no—
232C2×Q16C2×D19711yescharacteristic, complemented
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×SD16C2×D19711yescharacteristic
237C2×SD16C2×D19711yescharacteristic, complemented
238C2×D8—1971no—
239C2×D8C2×D19711yescharacteristic
240C2×M16—1971no—
241C22×C8C2×D19711yescharacteristic, complemented
242C4⋊Q8—1971no—
243#124—1971no—
244#123—3941no—
245#123—3941no—
246#123—3941no—
247C22⋊Q8—1971no—
248C22⋊Q8—1971no—
249C22⋊Q8—1971no—
250#121—3941no—
251#121—3941no—
252#121—1971no—
253#120—3941no—
254C4×D4—1971no—
255C4×D4—1971no—
256C4×D4—1971no—
257C2×C4⋊C4—3941no—
258C2×C22⋊C4—3941no—
259#106—1971no—
260C4⋊C8—1971no—
261#103—1971no—
262#103—1971no—
263#103—1971no—
264#102—1971no—
265#102—1971no—
266#102—1971no—
267C22⋊C8—1971no—
268C22⋊C8—1971no—
269#97—1971no—
270C24—3941no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—7881no—
276#41—3942no—
277C2×Q8—1971no—
278C2×Q8C22×D19711yescharacteristic
279C2×Q8C22×D19711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—7881no—
283C2×D4—3941no—
284C2×D4—3942no—
285C2×D4—3941no—
286C2×D4—3941no—
287C2×D4—1971no—
288C2×D4—1971no—
289C2×D4C22×D19711yescharacteristic
290C2×D4C22×D19711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3941no—
294C22×C4—3941no—
295C22×C4—3941no—
296C22×C4—1971no—
297C22×C4C22×D19711yescharacteristic
298Q16—22no—
299Q16—7881no—
300SD16—22no—
301SD16—22no—
302SD16—7881no—
303SD16—3942no—
304SD16—3942no—
305D8—3942no—
306D8C22⋊D19714yescomplemented
307M16—3942no—
308C2×C8—22no—
309C2×C8—1971no—
310C2×C8—1971no—
311C2×C8C22×D19711yescharacteristic
312C2×C8C22×D19711yescharacteristic
313C4⋊C4—3941no—
314C4⋊C4—3941no—
315C4⋊C4—3941no—
316C4⋊C4—3941no—
317C4⋊C4—3941no—
318C4⋊C4—1971no—
319C4⋊C4—1971no—
320C4⋊C4—1971no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—3941no—
325C22⋊C4—3941no—
326C22⋊C4—3941no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4—1971no—
330C22⋊C4—1971no—
331C42—1971no—
332C23—21no—
333C23—21no—
334C23—7882no—
335C23—7882no—
336C23—15761no—
337C23—3941no—
338C23—1971no—
339C23D4×D19711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3941no—
345Q8—3941no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—7882no—
351D4—7882no—
352D4—7881no—
353D4—15761no—
354D4—3941no—
355D4—3941no—
356D4C2×C22⋊D19712yes—
357D4C2×C22⋊D19712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—7882no—
365C2×C4—7882no—
366C2×C4—15761no—
367C2×C4—3941no—
368C2×C4—3941no—
369C2×C4—3941no—
370C2×C4—3941no—
371C2×C4—3941no—
372C2×C4—1971no—
373C2×C4—1971no—
374C2×C4—1971no—
375C2×C4D4×D19711yescharacteristic
376C2×C4C23×D19711yescharacteristic
377C8—21no—
378C8—3941no—
379C8—3941no—
380C8C2×C22⋊D19712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—7881no—
391C22—7881no—
392C22—7881no—
393C22—15764no—
394C22—3941no—
395C22—3941no—
396C22—3941no—
397C22C2×D4×D19711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—7881no—
402C4—7881no—
403C4—7881no—
404C4—3941no—
405C4—3941no—
406C4—3941no—
407C4C22×C22⋊D19711yescharacteristic
408C4#2026337111yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—7881no—
413C2—15761no—
414C2D4×C22⋊D19711yescharacteristic
415C2#2716187611yescharacteristic
416C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC22×C197C22×D4
hypercenterC2×#135D197
hypocenterC197#3267
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161728C2semidirect
3. #27161717C2semidirect
4. #27161472C2semidirect
5. #27161334C2semidirect
6. #27161308C2semidirect
7. #27161296C2semidirect
8. #27161282C2semidirect
9. #27161241C2semidirect
10. #27161238C2semidirect
11. #27161228C2semidirect
12. #27161224C2semidirect
13. #27161191C2semidirect
14. #27161150C2semidirect
15. C2×#135×C197C2semidirect
19. C2×#15691348C22semidirect
25. #20263324C22semidirect
30. #20263305C22semidirect
38. D4×Dic197C22semidirect
42. #20263258C22semidirect
43. #20263255C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263250C22semidirect
49. #20263247C22semidirect
50. C22⋊#13096802C22semidirect
52. #20263236C22semidirect
54. C2×#41×C197C22semidirect
56. C2×Q16×C197C22semidirect
58. C2×SD16×C197C22semidirect
60. C22×C8×C197C22semidirect
119. D8×C197D4semidirect
208. C197#3267semidirect
211. C2×#135D197semidirect
227. C2×#41C2×D197semidirect
232. C2×Q16C2×D197semidirect
237. C2×SD16C2×D197semidirect
241. C22×C8C2×D197semidirect
306. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617282
#271617172
#271614722
#271613342
#271613082
#271612962
#271612822
#271612412
#271612382
#271612282
#271612242
#271611912
#271611502
C2×#135×C1972
#3267197