Finite Group Databasebeta

← #49134294

Subgroups — #49134294

Order 36224.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#31827194C211yesmaximal, characteristic, complemented
2#31827188C211yesmaximal, characteristic
3#31826859C212yesmaximal
4#31826845C211yesmaximal, characteristic, complemented
5#31826688C212yesmaximal
6#31826676C211yesmaximal, characteristic, complemented
7#31826676C212yesmaximal, complemented
8#31826671C211yesmaximal, characteristic
9#31826573C212yesmaximal, complemented
10C4×#17985381C211yesmaximal, characteristic
11#446×C283C211yesmaximal, characteristic, complemented
12C2×Dic1132C2212yes—
13C2×Dic1132C2212yes—
14C2×Dic1132C2212yes—
15C2×#17985381C2211yescharacteristic
16C2×#17985381C2212yes—
17#22583610C2211yescharacteristic
18#22583610C2211yescharacteristic
19#22583610C2212yes—
20#22583605—22no—
21#22583601—21no—
22#22583601—22no—
23#22583598C2212yes—
24#22583586—21no—
25Q8×Dic283—21no—
26C4×Dic566C2212yes—
27C4×Dic566C2211yescharacteristic
28C4×Dic566C2212yes—
29C4×D1132C2211yescharacteristic
30#22583568—21no—
31#22583542C2211yescharacteristic
32#22583535C2212yes—
33#22583535C2211yescharacteristic
34#22583535C2212yes—
35#22583532C2211yescharacteristic
36C4×Q8×C283C2211yescharacteristic
37C4⋊C8×C283C2212yes—
38C4⋊C8×C283C2211yescharacteristic
39#97×C283C2212yes—
40C4×C8×C283C2211yescharacteristic
41C2×Dic566C2312yes—
42C2×Dic566C2311yescharacteristic
43C2×Dic566C2312yes—
44C2×D1132C2311yescharacteristic
45C2×C4×D283—21no—
46Dic1132—24no—
47Dic1132—42no—
48Dic1132—42no—
49#17985381—22no—
50#17985381—42no—
51C4⋊Dic283—21no—
52C4⋊Dic283—22no—
53C4⋊Dic283—21no—
54C4⋊Dic283—22no—
55C4⋊Dic283—21no—
56C4⋊Dic283—21no—
57C4⋊Dic283C2311yescharacteristic
58C4⋊Dic283C2311yescharacteristic
59Dic283⋊C4—22no—
60Dic283⋊C4—22no—
61Dic283⋊C4—21no—
62(C2×D283)⋊C4—22no—
63(C2×D283)⋊C4—21no—
64(C2×D283)⋊C4—22no—
65C4×Dic283—22no—
66C4×Dic283—22no—
67C4×Dic283—21no—
68C2×Q8×C283D411yescharacteristic
69C2×C8×C283C2312yes—
70C2×C8×C283C2311yescharacteristic
71C2×C8×C283C2311yescharacteristic
72C4⋊C4×C283D411yescharacteristic
73C4⋊C4×C283D412yes—
74C42×C283C2311yescharacteristic
75C42×C283C2312yes—
76C22×D283—21no—
77Dic566—21no—
78Dic566—22no—
79Dic566—22no—
80Dic566—22no—
81Dic566—22no—
82Dic566—21no—
83D1132—21no—
84D1132—21no—
85C2×Dic283—21no—
86C2×Dic283—21no—
87C2×Dic283—22no—
88C2×Dic283—21no—
89C2×Dic283—22no—
90C4×D283—42no—
91Q8×C283—22no—
92C2×C4×C283C2×D412yes—
93C2×C4×C283C2×D411yescharacteristic
94C2×C4×C283C2×D412yes—
95C2×C4×C283C2×D411yescharacteristic
96C2×C4×C283C2411yescharacteristic
97C8×C283—22no—
98C8×C283—21no—
99C8×C283#4112yes—
100C22×C283C22×D411yescharacteristic
101C2×D283—41no—
102C2×D283—41no—
103C2×D283—41no—
104C4×C283—22no—
105C4×C283—22no—
106C4×C283—21no—
107C4×C283#13712yes—
108C4×C283C2×#4111yescharacteristic
109C4×C283H(5,𝔽2)11yescharacteristic
110Dic283—42no—
111Dic283—41no—
112Dic283—42no—
113Dic283—41no—
114Dic283—41no—
115C2×C283#55011yescharacteristic
116C2×C283C2×#13711yescharacteristic
117C2×C283#57811yescharacteristic
118D283—81no—
119C283#339811yescharacteristic, complemented, Sylow
120#3398—2831nomaximal, Sylow
121#557—2831no—
122#556—2831no—
123#497—2832no—
124#494—2831no—
125#464—2832no—
126#462—2832no—
127#462—2831no—
128#461—2831no—
129#446D28311yescharacteristic, complemented
130#441—2832no—
131C4×SD16—2831no—
132C2×Q16—2832no—
133C2×Q16—2832no—
134C2×Q16—2832no—
135C2×SD16—2831no—
136C2×SD16—2832no—
137C4⋊Q8—2831no—
138C4⋊Q8—2831no—
139C4⋊Q8—2832no—
140#126—5662no—
141#125—5661no—
142#125—5662no—
143#124—2832no—
144C22⋊Q8—5661no—
145C4×Q8—2832no—
146C4×Q8—2831no—
147C4×Q8—2832no—
148C4×Q8—5661no—
149C4×Q8C2×D28311yescharacteristic
150C4×D4—2831no—
151#117—5661no—
152#106—2831no—
153C4⋊C8C2×D28311yescharacteristic
154C4⋊C8C2×D28312yes—
155#103—2831no—
156#103—2832no—
157#103—2832no—
158#102—2831no—
159#97C2×D28312yes—
160C4×C8C2×D28311yescharacteristic
161C2×Q8—2832no—
162C2×Q8—2831no—
163C2×Q8—2832no—
164C2×Q8D113211yescharacteristic
165C2×D4—2831no—
166C22×C4—5661no—
167Q16—5664no—
168Q16—11322no—
169Q16—11322no—
170SD16—5662no—
171SD16—11322no—
172C2×C8C22×D28311yescharacteristic
173C2×C8C22×D28312yes—
174C2×C8C22×D28311yescharacteristic
175C4⋊C4—2831no—
176C4⋊C4—2831no—
177C4⋊C4—5662no—
178C4⋊C4—5662no—
179C4⋊C4—5661no—
180C4⋊C4—5661no—
181C4⋊C4—5662no—
182C4⋊C4—5661no—
183C4⋊C4—5662no—
184C4⋊C4—5661no—
185C4⋊C4—5661no—
186C4⋊C4D113211yescharacteristic
187C4⋊C4D113212yes—
188C22⋊C4—5662no—
189C22⋊C4—5661no—
190C22⋊C4—5662no—
191C42—5662no—
192C42—5662no—
193C42—5661no—
194C42C22×D28311yescharacteristic
195C42C22×D28312yes—
196C23—5661no—
197Q8—22no—
198Q8—5661no—
199Q8—5662no—
200Q8—5662no—
201Q8—5662no—
202Q8—5662no—
203Q8—5661no—
204D4—5661no—
205D4—5661no—
206C2×C4—5661no—
207C2×C4—5661no—
208C2×C4—5662no—
209C2×C4—5661no—
210C2×C4—5662no—
211C2×C4—11322no—
212C2×C4C2×D113212yes—
213C2×C4C2×D113212yes—
214C2×C4C2×D113211yescharacteristic
215C2×C4C2×D113211yescharacteristic
216C2×C4C23×D28311yescharacteristic
217C8—22no—
218C8—21no—
219C8#1798539312yes—
220C22—11321no—
221C22—11321no—
222C22—11321no—
223C22C22×D113211yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—11321no—
228C4—11321no—
229C4—11322no—
230C4—11321no—
231C4—11322no—
232C4#2258364312yes—
233C4C2×#1798539311yescharacteristic
234C4#2258365611yescharacteristic
235C2—22641no—
236C2#3182713711yescharacteristic
237C2C2×#2258364311yescharacteristic
238C2#3182728211yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#446×C283C2
socleC22×C283C22×D4
prime radicalC2×D1132C23
hypercenter#446D283
hypocenterC283#3398
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #31827194C2semidirect
4. #31826845C2semidirect
6. #31826676C2semidirect
7. #31826676C2semidirect
9. #31826573C2semidirect
11. #446×C283C2semidirect
119. C283#3398semidirect
129. #446D283semidirect

Maximal subgroups

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

subgroupindex
#318271942
#318271882
#318268592
#318268452
#318266882
#318266762
#318266712
#318265732
C4×#179853812
#446×C2832
#3398283