Finite Group Databasebeta

← #38752665

Subgroups — #38752665

Order 25216.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161645C211yesmaximal, characteristic
2#27161559C211yesmaximal, characteristic, complemented
3#27161405C211yesmaximal, characteristic, complemented
4#27161403C211yesmaximal, characteristic, complemented
5#27161402C211yesmaximal, characteristic, complemented
6#27161398C211yesmaximal, characteristic, complemented
7#27161396C211yesmaximal, characteristic, complemented
8#27161394C211yesmaximal, characteristic, complemented
9#27161390C211yesmaximal, characteristic, complemented
10#27161120C211yesmaximal, characteristic, complemented
11#27161114C211yesmaximal, characteristic, complemented
12#27161082C211yesmaximal, characteristic, complemented
13#102×D197C211yesmaximal, characteristic, complemented
14#27161044C211yesmaximal, characteristic, complemented
15#482×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19C2×C8×D197C2211yescharacteristic
20#20263322—21no—
21#20263312—21no—
22#20263301C2211yescharacteristic
23#20263300C2211yescharacteristic
24#20263297C2211yescharacteristic
25#20263296—21no—
26#20263294—21no—
27C4×Dic394—21no—
28C4×D788—21no—
29#20263286—21no—
30#20263286—21no—
31#20263284C2211yescharacteristic
32#20263284C2211yescharacteristic
33#20263284C2211yescharacteristic, complemented
34C2×C4⋊Dic197C2211yescharacteristic, complemented
35C4⋊C4×D197C2211yescharacteristic
36C4⋊C4×D197C2211yescharacteristic, complemented
37#20263259C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263258C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52#121×C197C2211yescharacteristic
53#117×C197C2211yescharacteristic, complemented
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—22no—
61#15691358—22no—
62C2×Dic394D411yescharacteristic
63C2×D788—21no—
64C2×D788D411yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197D411yescharacteristic
70C2×C22⋊D197D411yescharacteristic
71D4×D197—22no—
72D4×D197—22no—
73C22×Dic197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197C2311yescharacteristic
77C2×C4×D197C2311yescharacteristic
78#15691341—22no—
79C2×#13096802C2311yescharacteristic
80C2×#13096802C2311yescharacteristic
81C8×D197—22no—
82C4⋊Dic197—21no—
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337—21no—
88C22⋊Dic197—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C2×D4×C197—21no—
98C2×D4×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C22×D197—21no—
109C22×D197—42no—
110C22×D197—81no—
111C22×D197—81no—
112C22×D197C2×D411yescharacteristic
113C22×D197C2×D411yescharacteristic
114Dic394—22no—
115D788—22no—
116D788—21no—
117D788—21no—
118D788—81no—
119C22⋊D197—22no—
120C22⋊D197—22no—
121C22⋊D197—42no—
122C22⋊D197—81no—
123C22⋊D197—81no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—81no—
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—81no—
133C4×D197—81no—
134C4×D197#4112yes—
135C4×D197#4112yes—
136#13096802—21no—
137#13096802—21no—
138C23×C197—21no—
139C23×C197C2×D411yescharacteristic
140D4×C197—21no—
141D4×C197—21no—
142D4×C197—81no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2411yescharacteristic
150C8×C197—21no—
151C8×C197—21no—
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—21no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197C22×D411yescharacteristic
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—21no—
162C2×D197—21no—
163C2×D197—21no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—84no—
168C2×D197#12312yes—
169C4×C197—21no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197—41no—
173C4×C197C2×#4111yescharacteristic
174C4×C197C2×#4111yescharacteristic
175Dic197—21no—
176Dic197—41no—
177Dic197—41no—
178Dic197—41no—
179Dic197#12312yes—
180C2×C197—41no—
181C2×C197—81no—
182C2×C197C2×#12311yescharacteristic
183C2×C197#57611yescharacteristic
184C2×C197#57711yescharacteristic
185D197—22no—
186D197—41no—
187D197—81no—
188C197#309611yescharacteristic, complemented, Sylow
189#3096—1971nomaximal, Sylow
190#535—1971no—
191#520—1971no—
192#482—1971no—
193#482—1971no—
194#482—1971no—
195#482D19711yescharacteristic, complemented
196#481—1971no—
197#481—1971no—
198#480—1971no—
199#480—1971no—
200#428—1971no—
201#427—1971no—
202#418—1971no—
203C2×#102—1971no—
204#408—1971no—
205C2×#41—1971no—
206C22×D4—1971no—
207C2×M16—1971no—
208C22×C8—1971no—
209#127—3941no—
210#124—3941no—
211#121—1971no—
212#121—1971no—
213#121—1971no—
214#121C2×D19711yescharacteristic
215#120—3941no—
216#120—3941no—
217C4×Q8—3941no—
218C4×D4—3941no—
219C4×D4—3941no—
220C4×D4—3941no—
221#117—1971no—
222#117—1971no—
223#117—1971no—
224#117C2×D19711yescharacteristic, complemented
225C2×C4⋊C4—1971no—
226C2×C4⋊C4—1971no—
227C2×C4⋊C4—1971no—
228#107—1971no—
229#107—1971no—
230#107—1971no—
231#107C2×D19711yescharacteristic, complemented
232#106—1971no—
233#106—1971no—
234#106—1971no—
235#106C2×D19711yescharacteristic, complemented
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102—1971no—
242#102C2×D19711yescharacteristic
243#102C2×D19711yescharacteristic
244C22⋊C8—1971no—
245C22⋊C8—1971no—
246C22⋊C8—1971no—
247C22⋊C8C2×D19711yescharacteristic, complemented
248C24—3941no—
249#41—3942no—
250#41—3942no—
251C2×Q8—1971no—
252C2×D4—21no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—3941no—
256C2×D4—3942no—
257C2×D4—3942no—
258C2×D4—3941no—
259C2×D4—1971no—
260C2×D4—1971no—
261C2×D4—1971no—
262C2×D4—1971no—
263C2×D4C22×D19711yescharacteristic
264C22×C4—3941no—
265C22×C4—3941no—
266C22×C4—3941no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4C22×D19711yescharacteristic
270M16—3942no—
271C2×C8—3942no—
272C2×C8—1971no—
273C2×C8—1971no—
274C2×C8C22×D19711yescharacteristic
275C2×C8C22×D19711yescharacteristic
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4—1971no—
282C4⋊C4C22×D19711yescharacteristic
283C4⋊C4C22×D19711yescharacteristic
284C4⋊C4C22×D19711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C42—21no—
294C42—3941no—
295C42—3941no—
296C42—3941no—
297C23—21no—
298C23—7882no—
299C23—15761no—
300C23—15761no—
301C23—3941no—
302C23—1971no—
303C23—1971no—
304C23D4×D19711yescharacteristic
305Q8—3942no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—7882no—
310D4—15761no—
311D4—15761no—
312D4—15761no—
313D4—3942no—
314D4—3942no—
315D4—3942no—
316D4—3941no—
317D4—3941no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—15761no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—3941no—
330C2×C4—3941no—
331C2×C4—1971no—
332C2×C4—1971no—
333C2×C4—1972no—
334C2×C4—1972no—
335C2×C4D4×D19711yescharacteristic
336C2×C4C23×D19711yescharacteristic
337C8—21no—
338C8—21no—
339C8—3941no—
340C8—3941no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—7881no—
348C22—7881no—
349C22—7881no—
350C22—15764no—
351C22—3941no—
352C22—3941no—
353C22—3941no—
354C22—3941no—
355C22—3941no—
356C22—1972no—
357C22C2×D4×D19711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—7881no—
363C4—7881no—
364C4—7881no—
365C4—3941no—
366C4—1972no—
367C4#41×D19711yescharacteristic
368C4#41×D19711yescharacteristic
369C2—41no—
370C2—81no—
371C2—7881no—
372C2—15761no—
373C2—3942no—
374C2#123×D19711yescharacteristic
375C2#2716188311yescharacteristic
376C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#482×C197C2
socleC22×C197C22×D4
prime radical#27161645C2
hypercenter#482D197
hypocenterC197#3096
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #27161559C2semidirect
3. #27161405C2semidirect
4. #27161403C2semidirect
5. #27161402C2semidirect
6. #27161398C2semidirect
7. #27161396C2semidirect
8. #27161394C2semidirect
9. #27161390C2semidirect
10. #27161120C2semidirect
11. #27161114C2semidirect
12. #27161082C2semidirect
13. #102×D197C2semidirect
14. #27161044C2semidirect
15. #482×C197C2semidirect
33. #20263284C22semidirect
34. C2×C4⋊Dic197C22semidirect
36. C4⋊C4×D197C22semidirect
37. #20263259C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263258C22semidirect
42. #20263257C22semidirect
49. C22⋊#13096802C22semidirect
50. #20263238C22semidirect
51. #20263238C22semidirect
53. #117×C197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
58. C22⋊C8×C197C22semidirect
188. C197#3096semidirect
195. #482D197semidirect
224. #117C2×D197semidirect
231. #107C2×D197semidirect
235. #106C2×D197semidirect
247. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616452
#271615592
#271614052
#271614032
#271614022
#271613982
#271613962
#271613942
#271613902
#271611202
#271611142
#271610822
#102×D1972
#271610442
#482×C1972
#3096197