Finite Group Databasebeta

← #16857742

Subgroups — #16857742

Order 3968.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521770C211yesmaximal, characteristic, complemented
2#14521690C211yesmaximal, characteristic
3#14521410C211yesmaximal, characteristic, complemented
4#14521387C211yesmaximal, characteristic, complemented
5#14521375C211yesmaximal, characteristic, complemented
6#14521338C211yesmaximal, characteristic, complemented
7#14521281C211yesmaximal, characteristic, complemented
8#14521273C211yesmaximal, characteristic, complemented
9#14521210C211yesmaximal, characteristic, complemented
10#14521199C211yesmaximal, characteristic, complemented
11#14521152C211yesmaximal, characteristic, complemented
12#14521118C211yesmaximal, characteristic, complemented
13#14521055C211yesmaximal, characteristic, complemented
14#14521044C211yesmaximal, characteristic, complemented
15C31×#470C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic, complemented
19C2×M16×C31C2211yescharacteristic
20C31×C22⋊Q8C2211yescharacteristic
21C31×C22⋊Q8C2211yescharacteristic
22#11758524C2211yescharacteristic
23#11758523C2211yescharacteristic, complemented
24Q8×Dic31C2211yescharacteristic, complemented
25C2×#92481C2211yescharacteristic
26C2×#92481C2211yescharacteristic
27C4×C22⋊D31C2211yescharacteristic
28#11758503C2211yescharacteristic
29C2×C8×D31C2211yescharacteristic, complemented
30#11758475—21no—
31#11758472—21no—
32#11758472C2211yescharacteristic
33#11758471—21no—
34#11758470C2211yescharacteristic
35#11758469C2211yescharacteristic, complemented
36#11758465C2211yescharacteristic, complemented
37#11758464—21no—
38#11758462—21no—
39#11758459—21no—
40C22⋊C4×D31—21no—
41#11758456—21no—
42C31×#106C2211yescharacteristic, 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#11758396C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C2×C22⋊D31D411yescharacteristic
64#92499—22no—
65#92499—41no—
66C2×D124—22no—
67C2×D124—21no—
68C2×D124—21no—
69C2×D124—21no—
70C2×D124—41no—
71C2×D124C2311yescharacteristic
72C2×D124C2311yescharacteristic
73C2×C4×D31—21no—
74C2×C4×D31—21no—
75C2×C4×D31C2311yescharacteristic
76C2×Dic62—21no—
77Q16×C31—41no—
78M16×C31—41no—
79C2×C8×C31C2311yescharacteristic
80C2×C8×C31C2311yescharacteristic
81C4⋊C4×C31—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C22⋊C4×C31—21no—
86C22⋊C4×C31—21no—
87C22⋊Dic31—21no—
88C22⋊Dic31D411yescharacteristic
89#92481—22no—
90#92481—41no—
91(C2×D31)⋊C4—21no—
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⋊Dic31C2311yescharacteristic
100C4⋊Dic31C2311yescharacteristic
101Dic31⋊C4—21no—
102Dic31⋊C4—21no—
103Dic31⋊C4D411yescharacteristic
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C8×D31—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—41no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31C2×D411yescharacteristic
134C2×Dic31C2×D411yescharacteristic
135C2×Dic31C2×D411yescharacteristic
136D124—21no—
137D124—21no—
138D124—21no—
139D124—42no—
140D124—42no—
141D124—81no—
142D124#4112yes—
143C4×D31—21no—
144C4×D31—21no—
145C4×D31—42no—
146C4×D31—81no—
147Dic62—42no—
148C8×C31—21no—
149C8×C31—21no—
150C31⋊C8—21no—
151C31⋊C8—21no—
152#3311—311nomaximal, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31C22×D411yescharacteristic
157C2×D31—21no—
158C2×D31—22no—
159C2×D31—21no—
160C2×D31—21no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—41no—
164C2×D31—41no—
165C2×D31—41no—
166C2×D31—84no—
167C4×C31—21no—
168C4×C31—41no—
169C4×C31—41no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31C2×#4111yescharacteristic
173C4×C31H(5,𝔽2)11yescharacteristic
174Dic31—21no—
175Dic31—21no—
176Dic31—21no—
177Dic31—41no—
178Dic31—41no—
179#556—311no—
180#546—311no—
181#489—311no—
182#485—311no—
183#482—311no—
184#476—311no—
185#470D3111yescharacteristic, complemented
186#465—311no—
187#464—311no—
188#452—311no—
189#450—311no—
190#440—311no—
191#435—311no—
192#418—311no—
193#416—311no—
194C2×C31—41no—
195C2×C31#54611yescharacteristic
196C2×C31#57511yescharacteristic
197C2×C31#57711yescharacteristic
198D31—41no—
199D31—42no—
200D31—81no—
201C2×#41—311no—
202C22×D4—311no—
203C2×Q16C2×D3111yescharacteristic, complemented
204C2×SD16—311no—
205C2×SD16—311no—
206C2×M16C2×D3111yescharacteristic
207C22×C8—311no—
208C4⋊Q8—311no—
209#126—621no—
210#126—621no—
211#124—311no—
212#124—621no—
213#123—621no—
214C22⋊Q8—311no—
215C22⋊Q8C2×D3111yescharacteristic
216C22⋊Q8C2×D3111yescharacteristic
217#121—311no—
218#121—621no—
219#120—621no—
220C4×Q8—311no—
221C4×Q8—621no—
222C4×D4—311no—
223C4×D4—311no—
224#117—311no—
225#117—311no—
226C2×C22⋊C4—621no—
227#107—311no—
228#106—311no—
229#106C2×D3111yescharacteristic, complemented
230C4⋊C8—311no—
231#103—311no—
232#103—311no—
233#103C2×D3111yescharacteristic, complemented
234#103C2×D3111yescharacteristic, complemented
235#102—311no—
236#102—311no—
237#102—311no—
238#102—311no—
239#102—311no—
240#102—311no—
241C22⋊C8—311no—
242C22⋊C8—311no—
243#97—311no—
244C31#331111yescharacteristic, complemented, Sylow
245C24—621no—
246#41—622no—
247#41—1241no—
248C2×Q8—621no—
249C2×Q8C22×D3111yescharacteristic
250C2×Q8C22×D3111yescharacteristic
251C2×D4—311no—
252C2×D4—311no—
253C2×D4—311no—
254C2×D4—622no—
255C2×D4—621no—
256C2×D4—621no—
257C2×D4—621no—
258C2×D4—1241no—
259C22×C4—311no—
260C22×C4—621no—
261C22×C4—621no—
262C22×C4C22×D3111yescharacteristic
263Q16—41no—
264SD16—622no—
265SD16—1241no—
266M16—41no—
267C2×C8—311no—
268C2×C8—311no—
269C2×C8—1241no—
270C2×C8C22×D3111yescharacteristic
271C2×C8C22×D3111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—311no—
275C4⋊C4—311no—
276C4⋊C4—311no—
277C4⋊C4—621no—
278C4⋊C4—621no—
279C4⋊C4—621no—
280C4⋊C4C22×D3111yescharacteristic
281C4⋊C4C22×D3111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—311no—
285C22⋊C4—311no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C22⋊C4—621no—
290C22⋊C4—621no—
291C22⋊C4—621no—
292C22⋊C4—621no—
293C42—311no—
294C42—621no—
295C42—621no—
296C42—621no—
297C23—311no—
298C23—621no—
299C23—621no—
300C23—1242no—
301C23—1242no—
302C23—2481no—
303C23D4×D3111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—1242no—
309D4—312no—
310D4—621no—
311D4—621no—
312D4—621no—
313D4—1242no—
314D4—1242no—
315D4—1241no—
316D4—2481no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—311no—
324C2×C4—311no—
325C2×C4—311no—
326C2×C4—621no—
327C2×C4—621no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—1242no—
331C2×C4—2481no—
332C2×C4D4×D3111yescharacteristic
333C2×C4C23×D3111yescharacteristic
334C8—21no—
335C8—21no—
336C8—621no—
337C8—621no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—621no—
342C22—621no—
343C22—622no—
344C22—621no—
345C22—1241no—
346C22—1241no—
347C22—1241no—
348C22—1241no—
349C22—1241no—
350C22—2484no—
351C22C2×D4×D3111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—621no—
358C4—621no—
359C4—621no—
360C4—1241no—
361C4—1241no—
362C4#1175856111yescharacteristic
363C4#41×D3111yescharacteristic
364C2—41no—
365C2—1241no—
366C2—1242no—
367C2—2481no—
368C2#1452168811yescharacteristic
369C2#1452184811yescharacteristic
370C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#470C2
socleC22×C31C22×D4
prime radical#14521690C2
hypercenter#470D31
hypocenterC31#3311
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #14521770C2semidirect
3. #14521410C2semidirect
4. #14521387C2semidirect
5. #14521375C2semidirect
6. #14521338C2semidirect
7. #14521281C2semidirect
8. #14521273C2semidirect
9. #14521210C2semidirect
10. #14521199C2semidirect
11. #14521152C2semidirect
12. #14521118C2semidirect
13. #14521055C2semidirect
14. #14521044C2semidirect
15. C31×#470C2semidirect
18. C2×Q16×C31C22semidirect
23. #11758523C22semidirect
24. Q8×Dic31C22semidirect
29. C2×C8×D31C22semidirect
35. #11758469C22semidirect
36. #11758465C22semidirect
42. C31×#106C22semidirect
43. C31×#103C22semidirect
44. C31×#103C22semidirect
45. #11758423C22semidirect
46. #11758423C22semidirect
52. (C2×D31)⋊C8C22semidirect
53. #11758403C22semidirect
54. Dic31⋊C8C22semidirect
57. #11758396C22semidirect
58. #11758395C22semidirect
185. #470D31semidirect
203. C2×Q16C2×D31semidirect
229. #106C2×D31semidirect
233. #103C2×D31semidirect
234. #103C2×D31semidirect
244. C31#3311semidirect

Maximal subgroups

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

subgroupindex
#145217702
#145216902
#145214102
#145213872
#145213752
#145213382
#145212812
#145212732
#145212102
#145211992
#145211522
#145211182
#145210552
#145210442
C31×#4702
#331131