Finite Group Databasebeta

← #16855537

Subgroups — #16855537

Order 3968.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521676C211yesmaximal, characteristic, complemented
2#14521521C211yesmaximal, characteristic
3#14521351C211yesmaximal, characteristic, complemented
4#14521350C211yesmaximal, characteristic, complemented
5#14521349C211yesmaximal, characteristic, complemented
6#14521348C211yesmaximal, characteristic, complemented
7#14521338C211yesmaximal, characteristic, complemented
8#14521327C211yesmaximal, characteristic, complemented
9#14521326C211yesmaximal, characteristic, complemented
10#14521088C211yesmaximal, characteristic, complemented
11#14521084C211yesmaximal, characteristic, complemented
12#14521074C211yesmaximal, characteristic, complemented
13#14521052C211yesmaximal, characteristic
14#102×D31C211yesmaximal, characteristic
15C31×#476C211yesmaximal, characteristic, complemented
16C2×D4×D31C2211yescharacteristic
17C31×C4⋊Q8C2211yescharacteristic, complemented
18C4×D4×C31C2211yescharacteristic
19#11758523—21no—
20#11758521—21no—
21D4×Dic31C2211yescharacteristic
22C4×C22⋊D31—21no—
23C2×(C2×D31)⋊C4—21no—
24C2×#92469C2211yescharacteristic
25C2×C8×D31C2211yescharacteristic
26#11758473—21no—
27#11758470C2211yescharacteristic
28#11758469C2211yescharacteristic
29#11758469C2211yescharacteristic
30C4⋊C4×D31C2211yescharacteristic
31#11758467C2211yescharacteristic, complemented
32#11758466—21no—
33#11758465C2211yescharacteristic, complemented
34(C22⋊D31)⋊C4—21no—
35C22⋊C4×D31—21no—
36C4×D124C2211yescharacteristic
37#11758449C2211yescharacteristic
38#11758447C2211yescharacteristic, complemented
39C31×#106C2211yescharacteristic, complemented
40C31×#106C2211yescharacteristic, complemented
41C31×C4⋊C8C2211yescharacteristic, complemented
42C31×#102C2211yescharacteristic
43C31×#102C2211yescharacteristic
44#11758420C2211yescharacteristic
45#11758420C2211yescharacteristic
46#11758409C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758406C2211yescharacteristic, complemented
49#11758405C2211yescharacteristic, complemented
50Dic31⋊C8C2211yescharacteristic, complemented
51Dic31⋊C8C2211yescharacteristic, complemented
52#11758397C2211yescharacteristic
53#11758397C2211yescharacteristic
54#11758396C2211yescharacteristic, complemented
55#11758395C2211yescharacteristic, complemented
56#11758395C2211yescharacteristic, complemented
57#11758395C2211yescharacteristic, complemented
58C4⋊(C31⋊C8)C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31—21no—
61C2×D4×C31C2311yescharacteristic
62C22×C4×C31—21no—
63C2×C22⋊D31—21no—
64C22×Dic31—21no—
65D4×D31—22no—
66D4×D31Q814yescomplemented
67C2×D124C2311yescharacteristic
68C2×C4×D31—21no—
69C2×C4×D31—21no—
70C2×C4×D31—21no—
71C2×C4×D31C2311yescharacteristic
72C2×Dic62—21no—
73C2×C8×C31C2311yescharacteristic
74C2×C8×C31C2311yescharacteristic
75C4⋊C4×C31—21no—
76C4⋊C4×C31C2311yescharacteristic
77C4⋊C4×C31C2311yescharacteristic
78C4⋊C4×C31C2311yescharacteristic
79C22⋊C4×C31—21no—
80C42×C31C2311yescharacteristic
81C22⋊Dic31—21no—
82(C2×D31)⋊C4—21no—
83(C2×D31)⋊C4—21no—
84(C2×D31)⋊C4—21no—
85(C2×D31)⋊C4—21no—
86(C2×D31)⋊C4D411yescharacteristic
87(C2×D31)⋊C4D411yescharacteristic
88C4⋊Dic31—21no—
89C4⋊Dic31C2311yescharacteristic
90C4⋊Dic31C2311yescharacteristic
91C4⋊Dic31C2311yescharacteristic
92Dic31⋊C4—21no—
93Dic31⋊C4—21no—
94Dic31⋊C4—21no—
95Dic31⋊C4—21no—
96Dic31⋊C4—21no—
97Dic31⋊C4D411yescharacteristic
98Dic31⋊C4D411yescharacteristic
99C4×Dic31—21no—
100C4×Dic31—21no—
101C4×Dic31C2311yescharacteristic
102C2×C31⋊C8C2311yescharacteristic
103C2×C31⋊C8C2311yescharacteristic
104#92469—22no—
105C8×D31—22no—
106C23×C31—21no—
107C22×D31—21no—
108C22×D31—24no—
109C22×D31—81no—
110C22×D31—81no—
111C22×D31C2×D411yescharacteristic
112Q8×C31—81no—
113D4×C31—21no—
114D4×C31C2×Q812yes—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—81no—
119C2×C4×C31C2×D411yescharacteristic
120C2×C4×C31C2×D411yescharacteristic
121C2×C4×C31C2411yescharacteristic
122C22⋊D31—24no—
123C2×Dic31—21no—
124C2×Dic31—21no—
125C2×Dic31—21no—
126C2×Dic31—81no—
127C2×Dic31C2×D411yescharacteristic
128C2×Dic31C2×D411yescharacteristic
129C2×Dic31C2×D411yescharacteristic
130D124—21no—
131D124C2×Q812yes—
132C4×D31—81no—
133C4×D31—81no—
134C4×D31—81no—
135C4×D31C2×Q812yes—
136C4×D31#4112yes—
137Dic62—81no—
138C8×C31—21no—
139C8×C31—21no—
140C31⋊C8—21no—
141C31⋊C8—21no—
142#3081—311nomaximal, Sylow
143C22×C31—22no—
144C22×C31—41no—
145C22×C31—41no—
146C22×C31C22×D411yescharacteristic
147C2×D31—21no—
148C2×D31—21no—
149C2×D31—22no—
150C2×D31—41no—
151C2×D31—41no—
152C2×D31—84no—
153C2×D31C22⋊Q812yes—
154C4×C31—21no—
155C4×C31—21no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31C22×Q811yescharacteristic
160C4×C31C2×#4111yescharacteristic
161Dic31—21no—
162Dic31—21no—
163Dic31—41no—
164Dic31—41no—
165Dic31—41no—
166Dic31C22⋊Q812yes—
167#544—311no—
168#518—311no—
169#478—311no—
170#478—311no—
171#478—311no—
172#478—311no—
173#476—311no—
174#476D3111yescharacteristic, complemented
175#474—311no—
176#474—311no—
177#428—311no—
178#427—311no—
179#424—311no—
180#418—311no—
181C2×#102—311no—
182C2×C31—42no—
183C2×C31C2×C22⋊Q811yescharacteristic
184C2×C31#57611yescharacteristic
185C2×C31#57711yescharacteristic
186D31—22no—
187D31—42no—
188C22×D4—311no—
189C2×M16—311no—
190C22×C8—311no—
191C4⋊Q8—311no—
192C4⋊Q8—621no—
193C4⋊Q8C2×D3111yescharacteristic, complemented
194#125—311no—
195#125—311no—
196#125—621no—
197C22⋊Q8—621no—
198C22⋊Q8—621no—
199C4×D4—311no—
200C4×D4—311no—
201C4×D4—311no—
202C4×D4—621no—
203C4×D4—621no—
204C4×D4C2×D3111yescharacteristic
205#117—311no—
206#117—311no—
207#117—311no—
208C2×C4⋊C4—311no—
209C2×C22⋊C4—621no—
210C2×C22⋊C4—621no—
211#107—311no—
212#107—311no—
213#107—311no—
214#107—311no—
215#106—311no—
216#106—311no—
217#106C2×D3111yescharacteristic, complemented
218#106C2×D3111yescharacteristic, complemented
219C4⋊C8—311no—
220C4⋊C8—311no—
221C4⋊C8—311no—
222C4⋊C8C2×D3111yescharacteristic, complemented
223#102—311no—
224#102—311no—
225#102—311no—
226#102—311no—
227#102—311no—
228#102—311no—
229#102C2×D3111yescharacteristic
230#102C2×D3111yescharacteristic
231C31#308111yescharacteristic, complemented, Sylow
232C24—621no—
233C2×Q8—21no—
234C2×Q8—621no—
235C2×D4—314no—
236C2×D4—311no—
237C2×D4—622no—
238C2×D4—621no—
239C2×D4C22×D3111yescharacteristic
240C22×C4—21no—
241C22×C4—311no—
242C22×C4—621no—
243C22×C4—621no—
244C22×C4—621no—
245C22×C4—621no—
246M16—622no—
247C2×C8—311no—
248C2×C8—311no—
249C2×C8—622no—
250C2×C8C22×D3111yescharacteristic
251C2×C8C22×D3111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—311no—
254C4⋊C4—311no—
255C4⋊C4—311no—
256C4⋊C4—311no—
257C4⋊C4—311no—
258C4⋊C4—621no—
259C4⋊C4—621no—
260C4⋊C4—621no—
261C4⋊C4—621no—
262C4⋊C4—621no—
263C4⋊C4—621no—
264C4⋊C4C22×D3111yescharacteristic
265C4⋊C4C22×D3111yescharacteristic
266C4⋊C4C22×D3111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—311no—
269C22⋊C4—311no—
270C22⋊C4—621no—
271C22⋊C4—621no—
272C22⋊C4—621no—
273C22⋊C4—621no—
274C22⋊C4—621no—
275C42—311no—
276C42—621no—
277C42—621no—
278C42C22×D3111yescharacteristic
279C23—21no—
280C23—311no—
281C23—624no—
282C23—621no—
283C23—2481no—
284C23—2481no—
285Q8—81no—
286Q8—2481no—
287D4—21no—
288D4—312no—
289D4—624no—
290D4—621no—
291D4Q8×D3112yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—311no—
297C2×C4—312no—
298C2×C4—312no—
299C2×C4—311no—
300C2×C4—311no—
301C2×C4—621no—
302C2×C4—621no—
303C2×C4—621no—
304C2×C4—2481no—
305C2×C4—2481no—
306C2×C4—2481no—
307C2×C4—2481no—
308C2×C4D4×D3111yescharacteristic
309C2×C4D4×D3111yescharacteristic
310C2×C4C23×D3111yescharacteristic
311C8—21no—
312C8—21no—
313C8—621no—
314C8—621no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—312no—
319C22—621no—
320C22—621no—
321C22—622no—
322C22—1241no—
323C22—1241no—
324C22—2484no—
325C22C2×D4×D3111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—312no—
332C4—621no—
333C4—621no—
334C4—1241no—
335C4—1241no—
336C4—1241no—
337C4C2×Q8×D3111yescharacteristic
338C4#41×D3111yescharacteristic
339C2—42no—
340C2—622no—
341C2—1242no—
342C2C22⋊Q8×D3111yescharacteristic
343C2#1452185311yescharacteristic
344C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#476C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
hypercenter#476D31
hypocenterC31#3081
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #14521676C2semidirect
3. #14521351C2semidirect
4. #14521350C2semidirect
5. #14521349C2semidirect
6. #14521348C2semidirect
7. #14521338C2semidirect
8. #14521327C2semidirect
9. #14521326C2semidirect
10. #14521088C2semidirect
11. #14521084C2semidirect
12. #14521074C2semidirect
15. C31×#476C2semidirect
17. C31×C4⋊Q8C22semidirect
31. #11758467C22semidirect
33. #11758465C22semidirect
38. #11758447C22semidirect
39. C31×#106C22semidirect
40. C31×#106C22semidirect
41. C31×C4⋊C8C22semidirect
48. #11758406C22semidirect
49. #11758405C22semidirect
50. Dic31⋊C8C22semidirect
51. Dic31⋊C8C22semidirect
54. #11758396C22semidirect
55. #11758395C22semidirect
56. #11758395C22semidirect
57. #11758395C22semidirect
58. C4⋊(C31⋊C8)C22semidirect
66. D4×D31Q8semidirect
174. #476D31semidirect
193. C4⋊Q8C2×D31semidirect
217. #106C2×D31semidirect
218. #106C2×D31semidirect
222. C4⋊C8C2×D31semidirect
231. C31#3081semidirect

Maximal subgroups

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

subgroupindex
#145216762
#145215212
#145213512
#145213502
#145213492
#145213482
#145213382
#145213272
#145213262
#145210882
#145210842
#145210742
#145210522
#102×D312
C31×#4762
#308131