Finite Group Databasebeta

← #49131090

Subgroups — #49131090

Order 36224.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#31827096C211yesmaximal, characteristic, complemented
2#31826939C211yesmaximal, characteristic
3#31826768C212yesmaximal, complemented
4#31826764C212yesmaximal, complemented
5#31826754C212yesmaximal, complemented
6#31826744C212yesmaximal, complemented
7#31826501C211yesmaximal, characteristic, complemented
8#31826493C211yesmaximal, characteristic, complemented
9#31826471C211yesmaximal, characteristic
10C2×#22583530C211yesmaximal, characteristic
11#428×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C22×C283⋊C8C2211yescharacteristic
14#22583609—21no—
15#22583609C2212yescomplemented
16#22583601—21no—
17#22583601C2212yescomplemented
18#22583589—22no—
19C4×D1132—21no—
20C4×D1132—21no—
21#22583572C2212yes—
22#22583572C2212yes—
23#22583569C2211yescharacteristic
24#22583569C2211yescharacteristic
25C22⋊C4×D283—22no—
26#22583543C2211yescharacteristic, complemented
27#22583543C2211yescharacteristic, complemented
28#22583541C2211yescharacteristic, complemented
29#22583541C2211yescharacteristic, complemented
30Dic283⋊C8C2212yescomplemented
31Dic283⋊C8C2212yescomplemented
32#22583532C2212yes—
33#22583532C2212yes—
34#22583530C2212yes—
35#22583530C2212yes—
36C2×M16×C283C2211yescharacteristic
37#117×C283C2211yescharacteristic
38C2×C4⋊C4×C283C2211yescharacteristic
39#107×C283C2212yescomplemented
40#106×C283C2212yescomplemented
41C23×D283—21no—
42C2×Dic566—22no—
43C2×D1132—21no—
44C2×D1132—22no—
45C2×D1132Q814yescomplemented
46C2×D1132C2312yes—
47C2×C4×D283—22no—
48C2×C4×D283—22no—
49C2×C283⋊C8—22no—
50C2×C283⋊C8C2311yescharacteristic
51C2×C283⋊C8C2311yescharacteristic
52C4⋊Dic283—22no—
53C4⋊Dic283D411yescharacteristic
54C4⋊Dic283D411yescharacteristic
55C4⋊Dic283C2311yescharacteristic
56C4⋊Dic283C2311yescharacteristic
57Dic283⋊C4—22no—
58Dic283⋊C4—22no—
59C22⋊Dic283—21no—
60C22⋊Dic283D412yes—
61(C2×D283)⋊C4—22no—
62(C2×D283)⋊C4—22no—
63C4×Dic283—21no—
64C4×Dic283C2312yes—
65C22×C4×C283—21no—
66C22×C4×C283C2311yescharacteristic
67M16×C283—22no—
68C2×C8×C283C2312yes—
69C4⋊C4×C283—21no—
70C4⋊C4×C283C2311yescharacteristic
71C4⋊C4×C283C2311yescharacteristic
72C4⋊C4×C283C2312yes—
73C22⋊C4×C283—21no—
74C42×C283—21no—
75C22×D283—22no—
76C22×D283—24no—
77C22×D283—81no—
78C22×D283—81no—
79Dic566—82no—
80D1132—24no—
81D1132—22no—
82D1132C2×Q814yes—
83C2×Dic283—21no—
84C2×Dic283—21no—
85C2×Dic283C2×D412yes—
86C2×Dic283C2×D412yes—
87C4×D283—82no—
88C4×D283—82no—
89C283⋊C8—21no—
90C283⋊C8—21no—
91C23×C283C2×D411yescharacteristic
92C2×C4×C283—21no—
93C2×C4×C283—21no—
94C2×C4×C283—22no—
95C2×C4×C283—81no—
96C2×C4×C283C2×D411yescharacteristic
97C2×C4×C283C2×Q812yes—
98C2×C4×C283#4112yes—
99C2×C4×C283C2411yescharacteristic
100C8×C283—22no—
101C22×C283—21no—
102C22×C283—21no—
103C22×C283C22⋊Q812yes—
104C22×C283C22×D411yescharacteristic
105C2×D283—24no—
106C2×D283—42no—
107C2×D283—42no—
108C2×D283—84no—
109C4×C283—42no—
110C4×C283—41no—
111C4×C283—41no—
112C4×C283C22⋊Q812yes—
113C4×C283C22×Q811yescharacteristic
114C4×C283C2×#4111yescharacteristic
115Dic283—22no—
116Dic283—22no—
117Dic283—41no—
118Dic283—41no—
119C2×C283—22no—
120C2×C283C2×C22⋊Q811yescharacteristic
121C2×C283#57611yescharacteristic
122C2×C283#57711yescharacteristic
123D283—44no—
124C283#308111yescharacteristic, complemented, Sylow
125#3081—2831nomaximal, Sylow
126#544—2831no—
127#518—2831no—
128#478—2832no—
129#478—2832no—
130#476—2832no—
131#474—2832no—
132#428D28311yescharacteristic, complemented
133#427—2831no—
134#424—2831no—
135#418—2831no—
136C2×#102—2831no—
137C22×D4—2831no—
138C2×M16C2×D28311yescharacteristic
139C22×C8—2831no—
140C4⋊Q8—2832no—
141C4⋊Q8—5661no—
142#125—2832no—
143#125—5661no—
144C22⋊Q8—5662no—
145C4×D4—2832no—
146C4×D4—2832no—
147C4×D4—5661no—
148C4×D4—5661no—
149#117—2831no—
150#117—2831no—
151#117C2×D28311yescharacteristic
152C2×C4⋊C4C2×D28311yescharacteristic
153C2×C22⋊C4—5662no—
154#107—2831no—
155#107—2831no—
156#107C2×D28312yescomplemented
157#106—2831no—
158#106—2831no—
159#106C2×D28312yescomplemented
160C4⋊C8—2832no—
161C4⋊C8—2832no—
162#102—2832no—
163#102—2832no—
164#102—2832no—
165#102—2832no—
166C24—5661no—
167C2×Q8—5662no—
168C2×D4—2834no—
169C2×D4—2832no—
170C2×D4—5661no—
171C2×D4—5662no—
172C22×C4—21no—
173C22×C4—5662no—
174C22×C4—5662no—
175C22×C4C22×D28311yescharacteristic
176M16—22no—
177C2×C8—2831no—
178C2×C8—2831no—
179C2×C8—5662no—
180C2×C8C22×D28312yes—
181C4⋊C4—21no—
182C4⋊C4—2831no—
183C4⋊C4—2831no—
184C4⋊C4—2831no—
185C4⋊C4—2831no—
186C4⋊C4—5662no—
187C4⋊C4—5662no—
188C4⋊C4—5662no—
189C4⋊C4C22×D28312yes—
190C4⋊C4C22×D28311yescharacteristic
191C4⋊C4C22×D28311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—2832no—
194C22⋊C4—5662no—
195C22⋊C4—5662no—
196C22⋊C4—5661no—
197C42—21no—
198C42—2832no—
199C42—5661no—
200C23—5662no—
201C23—5664no—
202C23—22641no—
203C23—22641no—
204C23D4×D28311yescharacteristic
205Q8—22642no—
206D4—2834no—
207D4—5664no—
208D4—5662no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—2832no—
214C2×C4—2832no—
215C2×C4—5661no—
216C2×C4—5661no—
217C2×C4—22642no—
218C2×C4—22642no—
219C2×C4D4×D28311yescharacteristic
220C2×C4Q8×D28312yes—
221C2×C4#1798539212yes—
222C2×C4C23×D28311yescharacteristic
223C8—22no—
224C8—5661no—
225C8—5661no—
226C22—21no—
227C22—21no—
228C22—5664no—
229C22—11322no—
230C22—11322no—
231C22—22644no—
232C22#2258358812yes—
233C22C2×D4×D28311yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—5662no—
238C4—5662no—
239C4—11321no—
240C4—11321no—
241C4#2258358812yes—
242C4C2×Q8×D28311yescharacteristic
243C4C2×#1798539211yescharacteristic
244C2—22no—
245C2—11324no—
246C2C2×#2258358811yescharacteristic
247C2#3182726911yescharacteristic
248C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radicalC22×D1132C22
hypercenter#428D283
hypocenterC283#3081
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #31827096C2semidirect
3. #31826768C2semidirect
4. #31826764C2semidirect
5. #31826754C2semidirect
6. #31826744C2semidirect
7. #31826501C2semidirect
8. #31826493C2semidirect
11. #428×C283C2semidirect
15. #22583609C22semidirect
17. #22583601C22semidirect
26. #22583543C22semidirect
27. #22583543C22semidirect
28. #22583541C22semidirect
29. #22583541C22semidirect
30. Dic283⋊C8C22semidirect
31. Dic283⋊C8C22semidirect
39. #107×C283C22semidirect
40. #106×C283C22semidirect
45. C2×D1132Q8semidirect
124. C283#3081semidirect
132. #428D283semidirect
156. #107C2×D283semidirect
159. #106C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270962
#318269392
#318267682
#318267642
#318267542
#318267442
#318265012
#318264932
#318264712
C2×#225835302
#428×C2832
#3081283