Finite Group Databasebeta

← #16855959

Subgroups — #16855959

Order 3968.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521659C211yesmaximal, characteristic, complemented
2#14521523C211yesmaximal, characteristic
3#14521390C211yesmaximal, characteristic, complemented
4#14521352C211yesmaximal, characteristic, complemented
5#14521348C211yesmaximal, characteristic, complemented
6#14521253C211yesmaximal, characteristic, complemented
7#14521253C211yesmaximal, characteristic, complemented
8#14521202C211yesmaximal, characteristic
9#14521199C211yesmaximal, characteristic
10#14521152C211yesmaximal, characteristic, complemented
11#14521148C211yesmaximal, characteristic, complemented
12#14521136C211yesmaximal, characteristic, complemented
13#14521134C211yesmaximal, characteristic, complemented
14#14521107C211yesmaximal, characteristic, complemented
15C31×#485C211yesmaximal, characteristic, complemented
16C22×D124C2211yescharacteristic
17C31×C22⋊Q8C2211yescharacteristic
18C31×#117C2211yescharacteristic
19Q8×Dic31C2211yescharacteristic, complemented
20#11758521C2211yescharacteristic, complemented
21C2×#92481C2211yescharacteristic
22C2×#92481C2211yescharacteristic
23#11758503C2211yescharacteristic
24#11758501C2211yescharacteristic
25C2×#92470C2211yescharacteristic
26C2×#92470C2211yescharacteristic
27#11758470C2211yescharacteristic
28#11758470C2211yescharacteristic
29#11758470C2211yescharacteristic
30#11758470C2211yescharacteristic
31#11758467—21no—
32#11758467C2211yescharacteristic, complemented
33#11758466—21no—
34#11758464C2211yescharacteristic, complemented
35C22⋊C4×D31—21no—
36C22⋊C4×D31—21no—
37#11758456—21no—
38#11758456—21no—
39C4×D124—21no—
40C4×D124—21no—
41C31×#107C2211yescharacteristic, complemented
42C31×#106C2211yescharacteristic, complemented
43C31×#103C2211yescharacteristic, complemented
44C31×#103C2211yescharacteristic, complemented
45C31×C22⋊C8C2211yescharacteristic
46C22⋊(C31⋊C8)C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758403C2211yescharacteristic, complemented
50Dic31⋊C8C2211yescharacteristic, complemented
51Dic31⋊C8C2211yescharacteristic, complemented
52C8×Dic31C2211yescharacteristic, complemented
53#11758398C2211yescharacteristic, complemented
54#11758398C2211yescharacteristic, complemented
55#11758397C2211yescharacteristic
56#11758397C2211yescharacteristic
57#11758396C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C2×D124—21no—
63C2×D124—22no—
64C2×D124—22no—
65C2×D124C2311yescharacteristic
66C2×D124C2311yescharacteristic
67C2×C4×D31—21no—
68C2×C4×D31—21no—
69C2×C4×D31—21no—
70C2×C4×D31—21no—
71C2×Dic62C2311yescharacteristic
72C2×C8×C31C2311yescharacteristic
73C2×C8×C31C2311yescharacteristic
74C4⋊C4×C31—21no—
75C4⋊C4×C31C2311yescharacteristic
76C4⋊C4×C31C2311yescharacteristic
77C4⋊C4×C31C2311yescharacteristic
78C22⋊C4×C31—21no—
79C22⋊C4×C31—21no—
80C42×C31—21no—
81C22⋊Dic31—21no—
82C22⋊Dic31D411yescharacteristic
83C22⋊Dic31D411yescharacteristic
84#92481—22no—
85#92481—22no—
86(C2×D31)⋊C4—21no—
87(C2×D31)⋊C4—21no—
88(C2×D31)⋊C4—21no—
89(C2×D31)⋊C4—21no—
90C4⋊Dic31—21no—
91C4⋊Dic31—21no—
92C4⋊Dic31D411yescharacteristic
93C4⋊Dic31D411yescharacteristic
94C4⋊Dic31C2311yescharacteristic
95Dic31⋊C4—21no—
96Dic31⋊C4—21no—
97Dic31⋊C4—21no—
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100C4×Dic31—21no—
101C4×Dic31—21no—
102C4×Dic31C2311yescharacteristic
103C4×Dic31C2311yescharacteristic
104C2×C31⋊C8C2311yescharacteristic
105C2×C31⋊C8C2311yescharacteristic
106#92470—22no—
107#92470—22no—
108C23×C31C2×D411yescharacteristic
109C22×D31—21no—
110C22×D31—21no—
111C22×D31—24no—
112C22×D31—81no—
113C22×D31—81no—
114Q8×C31—21no—
115Q8×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2411yescharacteristic
124C2×Dic31—21no—
125C2×Dic31—21no—
126C2×Dic31C2×D411yescharacteristic
127C2×Dic31C2×D411yescharacteristic
128C2×Dic31C2×D411yescharacteristic
129C2×Dic31C2×D411yescharacteristic
130D124—24no—
131D124—21no—
132D124—21no—
133D124#4112yes—
134D124#4112yes—
135C4×D31—81no—
136C4×D31—81no—
137C4×D31—81no—
138C4×D31—81no—
139Dic62—21no—
140Dic62—21no—
141C8×C31—21no—
142C8×C31—21no—
143C31⋊C8—21no—
144C31⋊C8—21no—
145#3123—311nomaximal, Sylow
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31C22×D411yescharacteristic
150C2×D31—22no—
151C2×D31—22no—
152C2×D31—41no—
153C2×D31—41no—
154C2×D31—41no—
155C2×D31—41no—
156C2×D31—84no—
157C4×C31—21no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31C2×#4111yescharacteristic
163C4×C31C2×#4111yescharacteristic
164Dic31—21no—
165Dic31—21no—
166Dic31—21no—
167Dic31—21no—
168Dic31—41no—
169Dic31—41no—
170#541—311no—
171#518—311no—
172#485—311no—
173#485D3111yescharacteristic, complemented
174#478—311no—
175#478—311no—
176#461—311no—
177#461—311no—
178#450—311no—
179#450—311no—
180#440—311no—
181#440—311no—
182C4×SD16—311no—
183C4×SD16—311no—
184#433—311no—
185C2×C31—41no—
186C2×C31#52511yescharacteristic
187C2×C31#57711yescharacteristic
188C2×C31#57711yescharacteristic
189D31—42no—
190D31—42no—
191C22×D4—311no—
192C2×SD16—311no—
193C2×SD16—311no—
194C2×SD16—311no—
195C2×SD16—311no—
196C4⋊Q8—311no—
197#126—621no—
198#126—621no—
199#125—311no—
200#125—621no—
201#125—621no—
202C22⋊Q8—311no—
203C22⋊Q8C2×D3111yescharacteristic
204C4×Q8—311no—
205C4×Q8—311no—
206C4×D4—311no—
207C4×D4—311no—
208C4×D4—311no—
209C4×D4—311no—
210C4×D4—621no—
211C4×D4—621no—
212#117—311no—
213#117C2×D3111yescharacteristic
214C2×C22⋊C4—621no—
215C2×C22⋊C4—621no—
216#107—311no—
217#107C2×D3111yescharacteristic, complemented
218#106—311no—
219#106C2×D3111yescharacteristic, complemented
220C4⋊C8—311no—
221C4⋊C8—311no—
222#103—311no—
223#103—311no—
224#103C2×D3111yescharacteristic, complemented
225#103C2×D3111yescharacteristic, complemented
226#102—311no—
227#102—311no—
228#102—311no—
229#102—311no—
230C22⋊C8—311no—
231C22⋊C8C2×D3111yescharacteristic
232#97—311no—
233C4×C8—311no—
234C31#312311yescharacteristic, complemented, Sylow
235C24—621no—
236C2×Q8—311no—
237C2×Q8C22×D3111yescharacteristic
238C2×D4—311no—
239C2×D4—311no—
240C2×D4—622no—
241C2×D4—622no—
242C2×D4—621no—
243C22×C4—621no—
244C22×C4—621no—
245C22×C4—621no—
246C22×C4—621no—
247C22×C4C22×D3111yescharacteristic
248SD16—622no—
249SD16—622no—
250SD16—622no—
251SD16—622no—
252C2×C8—311no—
253C2×C8—311no—
254C2×C8C22×D3111yescharacteristic
255C2×C8C22×D3111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—311no—
258C4⋊C4—311no—
259C4⋊C4—311no—
260C4⋊C4—621no—
261C4⋊C4—621no—
262C4⋊C4—621no—
263C4⋊C4—621no—
264C4⋊C4—621no—
265C4⋊C4—621no—
266C4⋊C4—621no—
267C4⋊C4C22×D3111yescharacteristic
268C4⋊C4C22×D3111yescharacteristic
269C4⋊C4C22×D3111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—311no—
273C22⋊C4—311no—
274C22⋊C4—621no—
275C22⋊C4—621no—
276C22⋊C4—621no—
277C22⋊C4—621no—
278C22⋊C4—621no—
279C42—21no—
280C42—311no—
281C42—311no—
282C42—621no—
283C42—621no—
284C23—624no—
285C23—621no—
286C23—621no—
287C23—2481no—
288C23—2481no—
289C23D4×D3111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—621no—
293Q8—621no—
294D4—312no—
295D4—312no—
296D4—624no—
297D4—621no—
298D4—621no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—311no—
306C2×C4—311no—
307C2×C4—311no—
308C2×C4—311no—
309C2×C4—621no—
310C2×C4—621no—
311C2×C4—2481no—
312C2×C4—2481no—
313C2×C4—2481no—
314C2×C4—2481no—
315C2×C4D4×D3111yescharacteristic
316C2×C4C23×D3111yescharacteristic
317C8—21no—
318C8—21no—
319C8—621no—
320C8—621no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—622no—
325C22—622no—
326C22—1241no—
327C22—1241no—
328C22—1241no—
329C22—1241no—
330C22—2484no—
331C22C2×D4×D3111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—621no—
338C4—621no—
339C4—621no—
340C4—621no—
341C4—1241no—
342C4—1241no—
343C4#41×D3111yescharacteristic
344C4#41×D3111yescharacteristic
345C2—41no—
346C2—1242no—
347C2—1242no—
348C2#1452155711yescharacteristic
349C2#1452185611yescharacteristic
350C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#485C2
socleC22×C31C22×D4
prime radicalC22×D124C22
hypercenter#485D31
hypocenterC31#3123
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #14521659C2semidirect
3. #14521390C2semidirect
4. #14521352C2semidirect
5. #14521348C2semidirect
6. #14521253C2semidirect
7. #14521253C2semidirect
10. #14521152C2semidirect
11. #14521148C2semidirect
12. #14521136C2semidirect
13. #14521134C2semidirect
14. #14521107C2semidirect
15. C31×#485C2semidirect
19. Q8×Dic31C22semidirect
20. #11758521C22semidirect
32. #11758467C22semidirect
34. #11758464C22semidirect
41. C31×#107C22semidirect
42. C31×#106C22semidirect
43. C31×#103C22semidirect
44. C31×#103C22semidirect
49. #11758403C22semidirect
50. Dic31⋊C8C22semidirect
51. Dic31⋊C8C22semidirect
52. C8×Dic31C22semidirect
53. #11758398C22semidirect
54. #11758398C22semidirect
57. #11758396C22semidirect
58. #11758395C22semidirect
173. #485D31semidirect
217. #107C2×D31semidirect
219. #106C2×D31semidirect
224. #103C2×D31semidirect
225. #103C2×D31semidirect
234. C31#3123semidirect

Maximal subgroups

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

subgroupindex
#145216592
#145215232
#145213902
#145213522
#145213482
#145212532
#145212022
#145211992
#145211522
#145211482
#145211362
#145211342
#145211072
C31×#4852
#312331