Finite Group Databasebeta

← #49122913

Subgroups — #49122913

Order 36224.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#31827037C211yesmaximal, characteristic
2#31826446C211yesmaximal, characteristic
3#31826437C211yesmaximal, characteristic
4#31826256C211yesmaximal, characteristic, complemented
5#31826255C211yesmaximal, characteristic, complemented
6#31826254C211yesmaximal, characteristic, complemented
7#352×C283C211yesmaximal, characteristic, complemented
8C2×#17985391C2211yescharacteristic
9C22×Dic566C411yescharacteristic, complemented
10M16×D283—21no—
11#22583609—21no—
12#22583596—21no—
13#22583594C411yescharacteristic, complemented
14#22583593C411yescharacteristic, complemented
15#22583591C2211yescharacteristic
16#22583588—21no—
17#22583587—21no—
18#22583566—21no—
19#22583566C411yescharacteristic, complemented
20#22583528—21no—
21#22583527C2211yescharacteristic
22#22583525C2211yescharacteristic
23#22583524—21no—
24#123×C283C2211yescharacteristic
25#100×C283C2211yescharacteristic
26#99×C283C2211yescharacteristic
27#17985391—41no—
28#17985391—41no—
29C2×Dic566—21no—
30C2×Dic566—21no—
31C2×Dic566—41no—
32C2×Dic566—41no—
33C2×Dic566D411yescharacteristic
34C2×C22⋊D283C2×C411yescharacteristic
35C2×C22⋊D283D411yescharacteristic
36C22×Dic283C2×C411yescharacteristic
37C22×Dic283C2×C411yescharacteristic
38C22×Dic283D411yescharacteristic
39C2×C4×D283D411yescharacteristic
40#17985376—21no—
41#17985375—41no—
42C8×D283—41no—
43C4⋊Dic283—21no—
44Dic283⋊C4—21no—
45Dic283⋊C4—21no—
46Dic283⋊C4—21no—
47Dic283⋊C4—41no—
48C22⋊Dic283—21no—
49C22⋊Dic283—21no—
50C22⋊Dic283C2×C411yescharacteristic
51(C2×D283)⋊C4—21no—
52(C2×D283)⋊C4—21no—
53C4×Dic283—21no—
54C4×Dic283—41no—
55C2×D4×C283C2311yescharacteristic
56C22×C4×C283C2×C411yescharacteristic
57M16×C283—21no—
58C4⋊C4×C283—21no—
59C22⋊C4×C283—21no—
60C22⋊C4×C283—21no—
61C22⋊C4×C283C2×C411yescharacteristic
62C22×D283C22⋊C411yescharacteristic
63Dic566—41no—
64Dic566—42no—
65Dic566—41no—
66Dic566—41no—
67C22⋊D283—41no—
68C22⋊D283—41no—
69C2×Dic283—21no—
70C2×Dic283—21no—
71C2×Dic283—21no—
72C2×Dic283—21no—
73C2×Dic283—21no—
74C2×Dic283—21no—
75C2×Dic283—41no—
76C2×Dic283—41no—
77C2×Dic283C22⋊C411yescharacteristic
78C2×Dic283C22⋊C411yescharacteristic
79C2×Dic283C22⋊C411yescharacteristic
80C4×D283—21no—
81C4×D283—21no—
82C283⋊C8—41no—
83C23×C283C22×C411yescharacteristic
84C23×C283C2×D411yescharacteristic
85D4×C283—41no—
86C2×C4×C283—21no—
87C2×C4×C283—21no—
88C2×C4×C283—41no—
89C2×C4×C283—41no—
90C2×C4×C283C2×D411yescharacteristic
91C8×C283—41no—
92C22×C283—21no—
93C22×C283—21no—
94C22×C283—41no—
95C22×C283—41no—
96C22×C283C2×C22⋊C411yescharacteristic
97C2×D283—21no—
98C2×D283—41no—
99C4×C283—21no—
100C4×C283—41no—
101C4×C283—41no—
102C4×C283—81no—
103Dic283—21no—
104Dic283—21no—
105Dic283—41no—
106Dic283—41no—
107Dic283—81no—
108Dic283#9911yescharacteristic
109Dic283#9911yescharacteristic
110C2×C283—21no—
111C2×C283—41no—
112C2×C283—41no—
113C2×C283C2×#9911yescharacteristic
114D283—41no—
115C283#212311yescharacteristic, complemented, Sylow
116#2123—2831nomaximal, Sylow
117#536—2831no—
118#413—2831no—
119#410—2831no—
120#352—2831no—
121#352—2831no—
122#352—2831no—
123#352D28311yescharacteristic, complemented
124C2×#41—2831no—
125C22×Q8—2831no—
126C2×M16—5661no—
127C4⋊Q8—5661no—
128#124—5661no—
129#123—2831no—
130#123—2831no—
131#123—2831no—
132#123C2×D28311yescharacteristic
133C22⋊Q8—5661no—
134C22⋊Q8—5661no—
135#117—2831no—
136#117—5661no—
137#101—5661no—
138#100—2831no—
139#100C2×D28311yescharacteristic
140#99—2831no—
141#99—5661no—
142#99C2×D28311yescharacteristic
143#41—11321no—
144#41—11321no—
145C2×Q8—2831no—
146C2×Q8—5661no—
147C2×Q8—5661no—
148C2×Q8—11321no—
149C2×Q8—11321no—
150C2×D4—2831no—
151C2×D4—2831no—
152C2×D4C22×D28311yescharacteristic
153C22×C4—2831no—
154C22×C4—2831no—
155C22×C4—2831no—
156C22×C4—2831no—
157C22×C4C4×D28311yescharacteristic
158M16—21no—
159M16—5661no—
160M16—11321no—
161C2×C8—11321no—
162C4⋊C4—21no—
163C4⋊C4—5661no—
164C4⋊C4—5661no—
165C4⋊C4—5661no—
166C4⋊C4—5661no—
167C4⋊C4—11321no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—2831no—
171C22⋊C4—5661no—
172C22⋊C4—5661no—
173C22⋊C4—5661no—
174C22⋊C4—5661no—
175C22⋊C4C4×D28311yescharacteristic
176C42—5661no—
177C42—11321no—
178C23—2831no—
179C23C2×C4×D28311yescharacteristic
180C23D4×D28311yescharacteristic
181Q8—11321no—
182Q8—11321no—
183Q8—11321no—
184Q8—11322no—
185D4—41no—
186D4—11321no—
187D4—11321no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—2831no—
193C2×C4—2831no—
194C2×C4—2831no—
195C2×C4—5661no—
196C2×C4—5661no—
197C2×C4—5661no—
198C2×C4—5661no—
199C2×C4—5661no—
200C2×C4—5661no—
201C2×C4—5661no—
202C2×C4—5661no—
203C2×C4—11321no—
204C2×C4—11321no—
205C2×C4D4×D28311yescharacteristic
206C8—41no—
207C8—11321no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—5661no—
213C22—11321no—
214C22C22⋊C4×D28311yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—2831no—
220C4—2831no—
221C4—5661no—
222C4—5661no—
223C4—11321no—
224C4—11321no—
225C4—22641no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—11321no—
230C2#99×D28311yescharacteristic
subgroupG/H
centerC2#99×D283
commutatorC23×C283C22×C4
FrattiniC2×D4C22×D283
Fitting#352×C283C2
socleC2×C283C2×#99
prime radicalC2×#17985391C22
cosocleC2×D4×C283C23
hypercenter#352D283
hypocenterC283#2123
LCSG ▷ C23×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #31826256C2semidirect
5. #31826255C2semidirect
6. #31826254C2semidirect
7. #352×C283C2semidirect
9. C22×Dic566C4semidirect
13. #22583594C4semidirect
14. #22583593C4semidirect
19. #22583566C4semidirect
115. C283#2123semidirect
123. #352D283semidirect

Maximal subgroups

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

subgroupindex
#318270372
#318264462
#318264372
#318262562
#318262552
#318262542
#352×C2832
#2123283