Finite Group Databasebeta

← #16857798

Subgroups — #16857798

Order 3968.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521757C211yesmaximal, characteristic, complemented
2#14521690C211yesmaximal, characteristic
3#14521410C211yesmaximal, characteristic, complemented
4#14521382C211yesmaximal, characteristic, complemented
5#14521364C211yesmaximal, characteristic, complemented
6#14521328C211yesmaximal, characteristic, complemented
7#14521298C211yesmaximal, characteristic, complemented
8#14521273C211yesmaximal, characteristic, complemented
9#14521210C211yesmaximal, characteristic, complemented
10#14521199C211yesmaximal, characteristic, complemented
11#14521152C211yesmaximal, characteristic, complemented
12#14521120C211yesmaximal, characteristic, complemented
13#14521049C211yesmaximal, characteristic, complemented
14C2×#11758409C211yesmaximal, characteristic, complemented
15C31×#467C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic, complemented
19C22×C8×C31C2211yescharacteristic
20C31×C22⋊Q8C2211yescharacteristic
21C31×C22⋊Q8C2211yescharacteristic
22#11758524C2211yescharacteristic
23#11758523C2211yescharacteristic, complemented
24Q8×Dic31C2211yescharacteristic, complemented
25C2×#92481C2211yescharacteristic
26C2×#92481C2211yescharacteristic
27C4×C22⋊D31C2211yescharacteristic
28C2×C4⋊Dic31C2211yescharacteristic
29C2×#92469C2211yescharacteristic, complemented
30#11758474—21no—
31#11758472—21no—
32#11758472C2211yescharacteristic
33#11758471—21no—
34#11758470C2211yescharacteristic
35C4⋊C4×D31C2211yescharacteristic, complemented
36#11758466—21no—
37#11758465C2211yescharacteristic, complemented
38#11758459—21no—
39(C22⋊D31)⋊C4—21no—
40C22⋊C4×D31—21no—
41#11758455—21no—
42C31×#107C2211yescharacteristic, complemented
43C31×#103C2211yescharacteristic, complemented
44C31×#103C2211yescharacteristic, complemented
45#11758423C2211yescharacteristic, complemented
46#11758423C2211yescharacteristic, complemented
47C22⋊(C31⋊C8)C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758409C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758409C2211yescharacteristic
52(C2×D31)⋊C8C2211yescharacteristic, complemented
53#11758403C2211yescharacteristic, complemented
54Dic31⋊C8C2211yescharacteristic, complemented
55#11758397C2211yescharacteristic
56#11758397C2211yescharacteristic
57#11758395C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C2×C22⋊D31D411yescharacteristic
64C22×Dic31—21no—
65#92499—22no—
66#92499—41no—
67C2×D124—22no—
68C2×D124—21no—
69C2×D124—21no—
70C2×D124—21no—
71C2×D124—41no—
72C2×D124C2311yescharacteristic
73C2×D124C2311yescharacteristic
74C2×C4×D31—21no—
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31C2311yescharacteristic
78C2×Dic62—21no—
79Q16×C31—41no—
80C2×C8×C31—41no—
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31C2311yescharacteristic
86C4⋊C4×C31C2311yescharacteristic
87C22⋊C4×C31—21no—
88C22⋊C4×C31—21no—
89C22⋊Dic31D411yescharacteristic
90#92481—22no—
91#92481—41no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4—21no—
97(C2×D31)⋊C4D411yescharacteristic
98C4⋊Dic31—21no—
99C4⋊Dic31—21no—
100C4⋊Dic31C2311yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105Dic31⋊C4D411yescharacteristic
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110#92469—41no—
111C23×C31C2×D411yescharacteristic
112C22×D31—21no—
113C22×D31—21no—
114C22×D31—42no—
115C22×D31—42no—
116C22×D31—81no—
117C22×D31C2×D411yescharacteristic
118Q8×C31—21no—
119Q8×C31—21no—
120Q8×C31—21no—
121Q8×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2411yescharacteristic
130C22⋊D31—22no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—81no—
134C2×Dic31C2×D411yescharacteristic
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137D124—21no—
138D124—21no—
139D124—21no—
140D124—42no—
141D124—42no—
142D124—81no—
143D124#4112yes—
144C4×D31—21no—
145C4×D31—21no—
146C4×D31—42no—
147C4×D31—81no—
148C4×D31—81no—
149Dic62—81no—
150C8×C31—21no—
151C8×C31—21no—
152C31⋊C8—21no—
153C31⋊C8—21no—
154#3314—311nomaximal, Sylow
155C22×C31—21no—
156C22×C31—21no—
157C22×C31#13712yes—
158C22×C31C22×D411yescharacteristic
159C2×D31—21no—
160C2×D31—22no—
161C2×D31—21no—
162C2×D31—21no—
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—84no—
169C4×C31—21no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31C2×#4111yescharacteristic
175C4×C31H(5,𝔽2)11yescharacteristic
176Dic31—21no—
177Dic31—21no—
178Dic31—21no—
179Dic31—41no—
180Dic31—41no—
181#554—311no—
182#546—311no—
183#489—311no—
184#484—311no—
185#480—311no—
186#474—311no—
187#468—311no—
188#467D3111yescharacteristic, complemented
189#464—311no—
190#452—311no—
191#450—311no—
192#440—311no—
193#435—311no—
194#417—311no—
195C2×#102—311no—
196C2×C31—22no—
197C2×C31#54611yescharacteristic
198C2×C31C2×#13711yescharacteristic
199C2×C31#57611yescharacteristic
200D31—41no—
201D31—42no—
202D31—81no—
203C2×#41—311no—
204C22×D4—311no—
205C2×Q16C2×D3111yescharacteristic, complemented
206C2×SD16—311no—
207C2×SD16—311no—
208C2×M16—311no—
209C22×C8C2×D3111yescharacteristic
210C4⋊Q8—311no—
211#125—621no—
212#124—311no—
213#123—621no—
214C22⋊Q8—311no—
215C22⋊Q8—621no—
216C22⋊Q8—621no—
217C22⋊Q8C2×D3111yescharacteristic
218C22⋊Q8C2×D3111yescharacteristic
219#121—311no—
220#121—621no—
221#120—621no—
222C4×Q8—311no—
223C4×D4—311no—
224C4×D4—311no—
225C4×D4—621no—
226C2×C4⋊C4—311no—
227C2×C4⋊C4—311no—
228C2×C22⋊C4—621no—
229#107—311no—
230#107—311no—
231#107C2×D3111yescharacteristic, complemented
232C4⋊C8—311no—
233#103—311no—
234#103—311no—
235#103C2×D3111yescharacteristic, complemented
236#103C2×D3111yescharacteristic, complemented
237#102—311no—
238#102—311no—
239#102—311no—
240#102—311no—
241#102—311no—
242#102—311no—
243C22⋊C8—311no—
244C22⋊C8—311no—
245#97—311no—
246C31#331411yescharacteristic, complemented, Sylow
247C24—621no—
248#41—622no—
249#41—1241no—
250C2×Q8—621no—
251C2×Q8C22×D3111yescharacteristic
252C2×Q8C22×D3111yescharacteristic
253C2×D4—311no—
254C2×D4—311no—
255C2×D4—311no—
256C2×D4—622no—
257C2×D4—621no—
258C2×D4—621no—
259C2×D4—621no—
260C2×D4—1241no—
261C22×C4—311no—
262C22×C4—621no—
263C22×C4—621no—
264C22×C4—621no—
265C22×C4—621no—
266C22×C4C22×D3111yescharacteristic
267Q16—41no—
268SD16—622no—
269SD16—1241no—
270M16—1241no—
271C2×C8—41no—
272C2×C8—311no—
273C2×C8—311no—
274C2×C8C22×D3111yescharacteristic
275C2×C8C22×D3111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—311no—
279C4⋊C4—311no—
280C4⋊C4—311no—
281C4⋊C4—621no—
282C4⋊C4—621no—
283C4⋊C4—621no—
284C4⋊C4—621no—
285C4⋊C4—621no—
286C4⋊C4C22×D3111yescharacteristic
287C4⋊C4C22×D3111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—311no—
291C22⋊C4—311no—
292C22⋊C4—621no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4—621no—
297C42—311no—
298C42—621no—
299C23—311no—
300C23—621no—
301C23—621no—
302C23—1242no—
303C23—1242no—
304C23—2481no—
305C23D4×D3111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—2481no—
311D4—312no—
312D4—622no—
313D4—621no—
314D4—621no—
315D4—621no—
316D4—1242no—
317D4—1242no—
318D4—2481no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—311no—
326C2×C4—311no—
327C2×C4—311no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—621no—
332C2×C4—1242no—
333C2×C4—2481no—
334C2×C4—2481no—
335C2×C4—2481no—
336C2×C4D4×D3111yescharacteristic
337C2×C4C23×D3111yescharacteristic
338C8—21no—
339C8—21no—
340C8—621no—
341C8—621no—
342C22—21no—
343C22—21no—
344C22—621no—
345C22—622no—
346C22—621no—
347C22—621no—
348C22—1241no—
349C22—1241no—
350C22—1241no—
351C22—1241no—
352C22—1241no—
353C22—2484no—
354C22#1175849512yes—
355C22C2×D4×D3111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—621no—
362C4—621no—
363C4—621no—
364C4—1241no—
365C4—1241no—
366C4#1175856111yescharacteristic
367C4#41×D3111yescharacteristic
368C2—22no—
369C2—1241no—
370C2—1242no—
371C2—2481no—
372C2#1452168811yescharacteristic
373C2C2×#1175849511yescharacteristic
374C2#1452185111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#467C2
socleC22×C31C22×D4
prime radical#14521690C2
hypercenter#467D31
hypocenterC31#3314
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #14521757C2semidirect
3. #14521410C2semidirect
4. #14521382C2semidirect
5. #14521364C2semidirect
6. #14521328C2semidirect
7. #14521298C2semidirect
8. #14521273C2semidirect
9. #14521210C2semidirect
10. #14521199C2semidirect
11. #14521152C2semidirect
12. #14521120C2semidirect
13. #14521049C2semidirect
14. C2×#11758409C2semidirect
15. C31×#467C2semidirect
18. C2×Q16×C31C22semidirect
23. #11758523C22semidirect
24. Q8×Dic31C22semidirect
29. C2×#92469C22semidirect
35. C4⋊C4×D31C22semidirect
37. #11758465C22semidirect
42. C31×#107C22semidirect
43. C31×#103C22semidirect
44. C31×#103C22semidirect
45. #11758423C22semidirect
46. #11758423C22semidirect
52. (C2×D31)⋊C8C22semidirect
53. #11758403C22semidirect
54. Dic31⋊C8C22semidirect
57. #11758395C22semidirect
58. #11758395C22semidirect
188. #467D31semidirect
205. C2×Q16C2×D31semidirect
231. #107C2×D31semidirect
235. #103C2×D31semidirect
236. #103C2×D31semidirect
246. C31#3314semidirect

Maximal subgroups

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

subgroupindex
#145217572
#145216902
#145214102
#145213822
#145213642
#145213282
#145212982
#145212732
#145212102
#145211992
#145211522
#145211202
#145210492
C2×#117584092
C31×#4672
#331431