Finite Group Databasebeta

← #49131095

Subgroups — #49131095

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#31827093C211yesmaximal, characteristic, complemented
2#31826939C211yesmaximal, characteristic
3#31826768C212yesmaximal, complemented
4#31826764C212yesmaximal, complemented
5#31826754C212yesmaximal, complemented
6#31826744C212yesmaximal, complemented
7#31826508C211yesmaximal, characteristic, complemented
8#31826491C211yesmaximal, characteristic, complemented
9#31826469C211yesmaximal, characteristic
10C2×#22583532C211yesmaximal, characteristic
11#427×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C2×#17985376C2211yescharacteristic
14#22583610—21no—
15#22583609C2212yescomplemented
16#22583602—21no—
17#22583601C2212yescomplemented
18#22583587—22no—
19C4×D1132—21no—
20C4×D1132—21no—
21#22583572C2212yes—
22#22583572C2212yes—
23#22583569C2211yescharacteristic
24C2×C4⋊Dic283C2211yescharacteristic
25C22⋊C4×D283—22no—
26#22583543C2212yescomplemented
27#22583541C2212yescomplemented
28Dic283⋊C8C2212yescomplemented
29Dic283⋊C8C2212yescomplemented
30#22583532C2212yes—
31#22583532C2212yes—
32#22583530C2212yes—
33#22583530C2212yes—
34C22×C8×C283C2211yescharacteristic
35#117×C283C2211yescharacteristic
36#117×C283C2211yescharacteristic
37#107×C283C2211yescharacteristic, complemented
38#107×C283C2211yescharacteristic, complemented
39#106×C283C2211yescharacteristic, complemented
40#106×C283C2211yescharacteristic, complemented
41C23×D283—21no—
42C2×Dic566—22no—
43C2×D1132—21no—
44C2×D1132—22no—
45C2×D1132Q814yescomplemented
46C2×D1132C2312yes—
47C22×Dic283—21no—
48C2×C4×D283—22no—
49C2×C4×D283—22no—
50#17985376—22no—
51C2×C283⋊C8C2312yes—
52C4⋊Dic283—22no—
53C4⋊Dic283—21no—
54C4⋊Dic283D411yescharacteristic
55C4⋊Dic283D411yescharacteristic
56C4⋊Dic283C2312yes—
57Dic283⋊C4—22no—
58Dic283⋊C4—22no—
59C22⋊Dic283D412yes—
60(C2×D283)⋊C4—22no—
61(C2×D283)⋊C4—22no—
62C4×Dic283C2312yes—
63C22×C4×C283C2311yescharacteristic
64C2×C8×C283—22no—
65C2×C8×C283C2311yescharacteristic
66C2×C8×C283C2311yescharacteristic
67C4⋊C4×C283C2311yescharacteristic
68C4⋊C4×C283C2311yescharacteristic
69C4⋊C4×C283C2311yescharacteristic
70C4⋊C4×C283C2311yescharacteristic
71C22⋊C4×C283—21no—
72C22⋊C4×C283—21no—
73C42×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—22no—
84C2×Dic283—81no—
85C2×Dic283C2×D412yes—
86C2×Dic283C2×D412yes—
87C4×D283—82no—
88C4×D283—82no—
89C283⋊C8—22no—
90C23×C283C2×D411yescharacteristic
91C2×C4×C283—21no—
92C2×C4×C283—21no—
93C2×C4×C283—21no—
94C2×C4×C283—21no—
95C2×C4×C283C2×D411yescharacteristic
96C2×C4×C283C2×Q812yes—
97C2×C4×C283#4112yes—
98C2×C4×C283C2411yescharacteristic
99C8×C283—21no—
100C8×C283—21no—
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—41no—
110C4×C283—41no—
111C4×C283—41no—
112C4×C283—41no—
113C4×C283C22⋊Q812yes—
114C4×C283C22×Q811yescharacteristic
115C4×C283C2×#4111yescharacteristic
116Dic283—22no—
117Dic283—22no—
118Dic283—42no—
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#428—2831no—
133#427D28311yescharacteristic, complemented
134#424—2831no—
135#418—2831no—
136C2×#102—2831no—
137C22×D4—2831no—
138C2×M16—2831no—
139C22×C8C2×D28311yescharacteristic
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#117C2×D28311yescharacteristic
151#117C2×D28311yescharacteristic
152C2×C4⋊C4—2831no—
153C2×C22⋊C4—5662no—
154#107—2832no—
155#107C2×D28311yescharacteristic, complemented
156#107C2×D28311yescharacteristic, complemented
157#106—2832no—
158#106C2×D28311yescharacteristic, complemented
159#106C2×D28311yescharacteristic, complemented
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—5662no—
173C22×C4—5662no—
174C22×C4—5661no—
175C22×C4C22×D28311yescharacteristic
176M16—5662no—
177C2×C8—22no—
178C2×C8—2832no—
179C2×C8C22×D28311yescharacteristic
180C2×C8C22×D28311yescharacteristic
181C4⋊C4—2831no—
182C4⋊C4—2831no—
183C4⋊C4—2832no—
184C4⋊C4—5662no—
185C4⋊C4—5662no—
186C4⋊C4—5662no—
187C4⋊C4—5661no—
188C4⋊C4C22×D28311yescharacteristic
189C4⋊C4C22×D28311yescharacteristic
190C4⋊C4C22×D28311yescharacteristic
191C4⋊C4C22×D28311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—2832no—
195C22⋊C4—5662no—
196C22⋊C4—5662no—
197C42—21no—
198C42—21no—
199C42—2832no—
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—21no—
212C2×C4—21no—
213C2×C4—2832no—
214C2×C4—2832no—
215C2×C4—5662no—
216C2×C4—22642no—
217C2×C4—22642no—
218C2×C4—22641no—
219C2×C4D4×D28311yescharacteristic
220C2×C4Q8×D28312yes—
221C2×C4#1798539212yes—
222C2×C4C23×D28311yescharacteristic
223C8—21no—
224C8—21no—
225C8—5662no—
226C22—21no—
227C22—21no—
228C22—5664no—
229C22—11322no—
230C22—11322no—
231C22—22644no—
232C22#2258358812yes—
233C22C2×D4×D28311yescharacteristic
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—5662no—
239C4—5662no—
240C4—11322no—
241C4#2258358812yes—
242C4C2×Q8×D28311yescharacteristic
243C4C2×#1798539211yescharacteristic
244C2—22no—
245C2—11324no—
246C2C2×#2258358811yescharacteristic
247C2#3182726711yescharacteristic
248C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#427×C283C2
socleC22×C283C22×D4
prime radicalC22×D1132C22
hypercenter#427D283
hypocenterC283#3081
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #31827093C2semidirect
3. #31826768C2semidirect
4. #31826764C2semidirect
5. #31826754C2semidirect
6. #31826744C2semidirect
7. #31826508C2semidirect
8. #31826491C2semidirect
11. #427×C283C2semidirect
15. #22583609C22semidirect
17. #22583601C22semidirect
26. #22583543C22semidirect
27. #22583541C22semidirect
28. Dic283⋊C8C22semidirect
29. Dic283⋊C8C22semidirect
37. #107×C283C22semidirect
38. #107×C283C22semidirect
39. #106×C283C22semidirect
40. #106×C283C22semidirect
45. C2×D1132Q8semidirect
124. C283#3081semidirect
133. #427D283semidirect
155. #107C2×D283semidirect
156. #107C2×D283semidirect
158. #106C2×D283semidirect
159. #106C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270932
#318269392
#318267682
#318267642
#318267542
#318267442
#318265082
#318264912
#318264692
C2×#225835322
#427×C2832
#3081283