Finite Group Databasebeta

← #49098661

Subgroups — #49098661

Order 35968.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×D281C211yesmaximal, characteristic, complemented
2C2×#22582856C212yesmaximal, complemented
3C2×#22582839C211yesmaximal, characteristic
4#31823384C211yesmaximal, characteristic, complemented
5#31823382C211yesmaximal, characteristic, complemented
6#31823375C211yesmaximal, characteristic, complemented
7#31823367C212yesmaximal, complemented
8#31823339C212yesmaximal
9#31823337C212yesmaximal, complemented
10#31823322C211yesmaximal, characteristic, complemented
11#393×C281C211yesmaximal, characteristic, complemented
12C22×Dic562C2211yescharacteristic
13C22×Dic562C2212yes—
14C2×Q8×D281—28no—
15C2×Q8×D281—24no—
16C22×C22⋊D281C2211yescharacteristic
17C23×Dic281C2211yescharacteristic
18C22×C4×D281C2212yes—
19C22×C4×D281C2211yescharacteristic
20#22582856—28no—
21#22582839—24no—
22C2×C4⋊Dic281C2212yes—
23C2×#17985117C2212yes—
24C2×#17985117C2211yescharacteristic
25C2×#17985117C2212yes—
26C2×C22⋊Dic281C2212yes—
27C2×#17985115C2211yescharacteristic
28C2×#17985115C2212yes—
29C2×C4×Dic281C2211yescharacteristic
30#22582786C2212yes—
31#22582786C2212yes—
32#22582786C2212yes—
33#22582786C2211yescharacteristic
34C22×Q8×C281C2211yescharacteristic
35C2×C22⋊C4×C281C2211yescharacteristic
36C2×C22⋊C4×C281C2212yes—
37#95×C281C2212yes—
38#95×C281C2211yescharacteristic
39C23×D281C2311yescharacteristic
40C2×Dic562—24no—
41C2×Dic562—24no—
42C2×Dic562—24no—
43C2×Dic562—22no—
44C2×Dic562D418yescomplemented
45Q8×D281—232no—
46C2×C22⋊D281—22no—
47C2×C22⋊D281—22no—
48C2×C22⋊D281Q814yescomplemented
49C22×Dic281—22no—
50C22×Dic281—24no—
51C22×Dic281C2312yes—
52C22×Dic281C2311yescharacteristic
53C22×Dic281C2312yes—
54C22×Dic281C2311yescharacteristic
55C22×Dic281C2311yescharacteristic
56C2×C4×D281—24no—
57C2×C4×D281—24no—
58C2×C4×D281—24no—
59C2×C4×D281—24no—
60C2×C4×D281D414yes—
61C4⋊Dic281—24no—
62#17985117—24no—
63#17985117—22no—
64#17985117—24no—
65C22⋊Dic281—24no—
66#17985115—22no—
67#17985115—24no—
68C4×Dic281—22no—
69C24×C281C2311yescharacteristic
70C2×Q8×C281—24no—
71C2×Q8×C281—22no—
72C22×C4×C281C2311yescharacteristic
73C22×C4×C281C2312yes—
74C22×C4×C281C2312yes—
75C22×C4×C281C2311yescharacteristic
76C22⋊C4×C281—22no—
77C22⋊C4×C281—24no—
78C22×D281—22no—
79C22×D281—22no—
80C22×D281C2×D412yes—
81C22×D281C2×Q812yes—
82C22×D281#4112yes—
83Dic562—28no—
84Dic562—216no—
85C22⋊D281—28no—
86C2×Dic281—21no—
87C2×Dic281—22no—
88C2×Dic281—22no—
89C2×Dic281—21no—
90C2×Dic281—21no—
91C2×Dic281—22no—
92C2×Dic281—22no—
93C2×Dic281—22no—
94C2×Dic281—22no—
95C2×Dic281—44no—
96C2×Dic281C2×D414yes—
97C2×Dic281C2×D412yes—
98C2×Dic281C2×D414yes—
99C2×Dic281C2×Q812yescomplemented
100C4×D281—216no—
101C4×D281—28no—
102C23×C281—22no—
103C23×C281—21no—
104C23×C281—21no—
105C23×C281—22no—
106C23×C281C2×Q812yescomplemented
107C23×C281C2411yescharacteristic
108Q8×C281—28no—
109C2×C4×C281—22no—
110C2×C4×C281—22no—
111C2×C4×C281—22no—
112C2×C4×C281—21no—
113C2×C4×C281—22no—
114C2×C4×C281—22no—
115C2×C4×C281—22no—
116C2×C4×C281—22no—
117C2×C4×C281C2×D414yes—
118C2×C4×C281C2×D412yes—
119C22×C281—22no—
120C22×C281—24no—
121C22×C281—41no—
122C22×C281—41no—
123C22×C281—42no—
124C22×C281C22×D411yescharacteristic
125C22×C281C22×D412yes—
126C22×C281C22×Q811yescharacteristic
127C22×C281C2×#4111yescharacteristic
128C22×C2812−1+412yes—
129C2×D281—22no—
130C2×D281—22no—
131C2×D281—24no—
132C2×D281—24no—
133C2×D281C22⋊Q814yescomplemented
134C4×C281—24no—
135C4×C281—22no—
136C4×C281—41no—
137C4×C281—42no—
138Dic281—24no—
139Dic281—22no—
140Dic281—41no—
141Dic281—42no—
142Dic281#12014yes—
143C2×C281—42no—
144C2×C281C2×#12011yescharacteristic
145C2×C281C2×C22⋊Q811yescharacteristic
146C2×C281#53611yescharacteristic
147C2×C281#54812yes—
148C2×C281D4×Q812yes—
149D281—24no—
150C281#240911yescharacteristic, complemented, Sylow
151#2409—2811nomaximal, Sylow
152C23×Q8—2811no—
153C2×C22⋊Q8—2812no—
154C2×C4×D4—2811no—
155#395—2811no—
156#395—2811no—
157#393—2811no—
158#393D28111yescharacteristic, complemented
159#391—2812no—
160#386—2812no—
161#385—2812no—
162#380—2811no—
163C22×Q8—2812no—
164C22×Q8—2811no—
165C22×Q8—5624no—
166C22×Q8—5628no—
167C22×Q8C2×D28111yescharacteristic
168C22×D4—2811no—
169C23×C4—2811no—
170C23×C4—2811no—
171C23×C4—2812no—
172C22⋊Q8—5628no—
173C4×D4—5624no—
174C2×C4⋊C4—2811no—
175C2×C4⋊C4—2812no—
176C2×C4⋊C4—2812no—
177C2×C4⋊C4—2812no—
178C2×C22⋊C4—2812no—
179C2×C22⋊C4—2811no—
180C2×C22⋊C4—2812no—
181C2×C22⋊C4C2×D28111yescharacteristic
182C2×C22⋊C4C2×D28112yes—
183C2×C42—2811no—
184#95—2811no—
185#95—2812no—
186#95—2812no—
187#95—2812no—
188#95C2×D28111yescharacteristic
189#95C2×D28112yes—
190C24—2811no—
191C24C22×D28111yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—2818no—
195C2×Q8—5624no—
196C2×Q8—5624no—
197C2×Q8—5624no—
198C2×Q8—56232no—
199C2×Q8—5622no—
200C2×D4—2814no—
201C2×D4—5622no—
202C2×D4—5622no—
203C22×C4—2811no—
204C22×C4—2811no—
205C22×C4—2812no—
206C22×C4—2814no—
207C22×C4—2812no—
208C22×C4—2811no—
209C22×C4—5624no—
210C22×C4—5624no—
211C22×C4—5624no—
212C22×C4—5622no—
213C22×C4—5624no—
214C22×C4—5624no—
215C22×C4C22×D28111yescharacteristic
216C22×C4C22×D28112yes—
217C22×C4C22×D28111yescharacteristic
218C22×C4C22×D28112yes—
219C4⋊C4—5624no—
220C4⋊C4—5624no—
221C4⋊C4—5622no—
222C4⋊C4—5624no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—5624no—
226C22⋊C4—5622no—
227C22⋊C4—5624no—
228C42—5622no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—2812no—
234C23—2812no—
235C23—2812no—
236C23—5622no—
237C23—5622no—
238C23Q8×D28112yescomplemented
239C23C23×D28111yescharacteristic
240Q8—28no—
241Q8—56216no—
242Q8—5628no—
243D4—5628no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—2812no—
253C2×C4—2814no—
254C2×C4—2814no—
255C2×C4—2812no—
256C2×C4—5628no—
257C2×C4—56216no—
258C2×C4—5621no—
259C2×C4—5622no—
260C2×C4—5622no—
261C2×C4—5621no—
262C2×C4—5622no—
263C2×C4—5621no—
264C2×C4—5622no—
265C2×C4—5622no—
266C2×C4—5622no—
267C2×C4—11244no—
268C2×C4D4×D28112yes—
269C2×C4D4×D28114yes—
270C22—22no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—2814no—
276C22—5622no—
277C22—5622no—
278C22—5624no—
279C22—5624no—
280C22C2×D4×D28111yescharacteristic
281C22C2×D4×D28112yes—
282C22C2×Q8×D28111yescharacteristic
283C22#41×D28111yescharacteristic
284C22#2258292612yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—2814no—
290C4—5624no—
291C4—5622no—
292C4—11241no—
293C4—11242no—
294C2—42no—
295C2—5624no—
296C2#120×D28111yescharacteristic
297C2C22⋊Q8×D28111yescharacteristic
298C2#3182403311yescharacteristic
299C2#3182412112yes—
300C2#3182412612yes—
subgroupG/H
center, FrattiniC23C23×D281
commutator, socle, cosocleC23×C281C24
Fitting#393×C281C2
prime radicalC22×C22⋊D281C22
hypercenter#393D281
hypocenterC281#2409
LCSG ▷ C23×C281 ▷ C281
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D281C2semidirect
2. C2×#22582856C2semidirect
4. #31823384C2semidirect
5. #31823382C2semidirect
6. #31823375C2semidirect
7. #31823367C2semidirect
9. #31823337C2semidirect
10. #31823322C2semidirect
11. #393×C281C2semidirect
44. C2×Dic562D4semidirect
48. C2×C22⋊D281Q8semidirect
99. C2×Dic281C2×Q8semidirect
106. C23×C281C2×Q8semidirect
133. C2×D281C22⋊Q8semidirect
150. C281#2409semidirect
158. #393D281semidirect
238. C23Q8×D281semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2812
C2×#225828562
C2×#225828392
#318233842
#318233822
#318233752
#318233672
#318233392
#318233372
#318233222
#393×C2812
#2409281