Finite Group Databasebeta

← #29704

Subgroups — #29704

Order 256.

469 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 121 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#3461C211yesmaximal, characteristic
2#3079C212yesmaximal, complemented
3#2568C211yesmaximal, characteristic, complemented
4#2061C212yesmaximal, complemented
5#2040C212yesmaximal
6#2038C214yesmaximal, complemented
7#1806C212yesmaximal, complemented
8#1801C211yesmaximal, characteristic, complemented
9#544C2211yescharacteristic
10#542—22no—
11#541—22no—
12#539—22no—
13#537C2211yescharacteristic
14#536C2212yes—
15#533—22no—
16C2×C4⋊Q8C2211yescharacteristic
17#529—22no—
18#529—22no—
19C4×#41C2212yes—
20C2×C4×Q8C2211yescharacteristic
21#477—22no—
22#476—22no—
23#475—22no—
24#474—22no—
25#428C2214yes—
26#423C2212yes—
27C2×#104C2214yes—
28#417C2214yes—
29C2×#101C2212yes—
30#400—22no—
31#391—22no—
32#391C2212yes—
33#382C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35#343C2212yes—
36#339C2214yes—
37C2×#41C2312yes—
38C22×Q8—21no—
39C22×Q8C2311yescharacteristic
40C2×M16C2314yes—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C4⋊Q8—22no—
44C4⋊Q8—22no—
45#126—24no—
46#126—24no—
47#125—22no—
48#125—24no—
49#125—22no—
50#124—22no—
51#124—22no—
52#123—24no—
53#123—24no—
54C22⋊Q8—24no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59#121—22no—
60#121—22no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8D412yes—
65C4×Q8D412yes—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4D412yes—
70C4×D4D412yes—
71#117—24no—
72#117—22no—
73#117C2312yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D412yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C42—22no—
80C2×C42D412yes—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83#107—24no—
84#106—24no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87#104—44no—
88#103—24no—
89#102—24no—
90#101—42no—
91#95—21no—
92#95—24no—
93#95—24no—
94#41—22no—
95#41—22no—
96#41—22no—
97#41—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—44no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8C2×D412yes—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4C2×D412yes—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120M16—24no—
121M16—24no—
122C2×C8—24no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—24no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×D412yes—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42C2×D412yes—
151C42C2×D412yes—
152C23—22no—
153C23C22×D411yescharacteristic
154Q8—22no—
155Q8—22no—
156Q8—42no—
157Q8—42no—
158Q8—82no—
159Q8—82no—
160Q8—82no—
161D4—22no—
162D4—22no—
163D4—42no—
164D4—42no—
165D4—82no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—82no—
185C2×C4#12011yescharacteristic
186C2×C4#12012yes—
187C2×C4#12011yescharacteristic
188C2×C4C22×D412yes—
189C2×C4H(5,𝔽2)11yescharacteristic
190C2×C42−1+411yescharacteristic
191C2×C42−1+412yes—
192C8—44no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22C2×#12011yescharacteristic
199C22#53611yescharacteristic
200C22#53711yescharacteristic
201C4—21no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—82no—
209C4—82no—
210C4#45412yes—
211C4#53612yes—
212C4#53712yes—
213C2—21no—
214C2—21no—
215C2—82no—
216C2#261611yescharacteristic
217C2C2×#45411yescharacteristic
218C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3079C2semidirect
3. #2568C2semidirect
4. #2061C2semidirect
6. #2038C2semidirect
7. #1806C2semidirect
8. #1801C2semidirect

Maximal subgroups

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

subgroupindex
#34612
#30792
#25682
#20612
#20402
#20382
#18062
#18012