Finite Group Databasebeta

← #8493

Subgroups — #8493

Order 256.

374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 71 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#3548C211yesmaximal, characteristic
2#1963C211yesmaximal, characteristic, complemented
3#1559C211yesmaximal, characteristic, complemented
4#1557C211yesmaximal, characteristic, complemented
5#1536C211yesmaximal, characteristic, complemented
6#1506C211yesmaximal, characteristic, complemented
7#1505C211yesmaximal, characteristic, complemented
8#562—22no—
9#554—21no—
10#554C412yescomplemented
11#550—21no—
12#550C412yescomplemented
13#547—21no—
14#547—21no—
15#540—22no—
16#537—22no—
17#537—22no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20#532C2211yescharacteristic
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#423C2211yescharacteristic
24C2×#104—22no—
25#376C2211yescharacteristic
26#333—21no—
27#332—21no—
28#332—21no—
29#331—21no—
30#329—22no—
31#328—22no—
32#327—22no—
33C2×#41—22no—
34C2×#41—22no—
35C2×#41—21no—
36C2×#41D412yes—
37C2×M16—21no—
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127—22no—
43#127C2×C411yescharacteristic
44#127D411yescharacteristic
45#125—22no—
46#125—21no—
47#124—21no—
48#124D412yes—
49#123—24no—
50#123—24no—
51C22⋊Q8—22no—
52C22⋊Q8—24no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55C22⋊Q8—21no—
56C22⋊Q8D412yes—
57#121—24no—
58#121—24no—
59#121—22no—
60#121—24no—
61#121—24no—
62#121—21no—
63#121—21no—
64#121—21no—
65#121—22no—
66#121D412yes—
67C4×Q8—21no—
68C4×Q8C2×C412yes—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4C2×C412yes—
75#117—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84#104—44no—
85C22⋊C8—21no—
86C22⋊C8—21no—
87#97—21no—
88#97—21no—
89#95—21no—
90#95—21no—
91#41—42no—
92#41—42no—
93#41—82no—
94#41—84no—
95#41—82no—
96#41—82no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8C22⋊C412yes—
100C2×Q8C22⋊C412yes—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4C22⋊C412yes—
110C2×D4C22⋊C412yes—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119M16—42no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4C22⋊C412yes—
134C4⋊C4C22⋊C412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42C22×C411yescharacteristic
146C42C2×D411yescharacteristic
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C23—22no—
150C23—22no—
151C23#12011yescharacteristic
152Q8—42no—
153Q8—42no—
154Q8—41no—
155Q8—41no—
156Q8—81no—
157D4—42no—
158D4—42no—
159D4—42no—
160D4—42no—
161D4—41no—
162D4—41no—
163D4—81no—
164D4—82no—
165D4—82no—
166D4—84no—
167D4—84no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4C2×C22⋊C411yescharacteristic
186C2×C4C2×C22⋊C411yescharacteristic
187C2×C4#12011yescharacteristic
188C2×C4#12011yescharacteristic
189C2×C4#12011yescharacteristic
190C8—81no—
191C8—81no—
192C8—81no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22#37911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—82no—
213C4#45311yescharacteristic
214C4#45511yescharacteristic
215C2—41no—
216C2—82no—
217C2—82no—
218C2#179211yescharacteristic
219C2#179511yescharacteristic
220C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1963C2semidirect
3. #1559C2semidirect
4. #1557C2semidirect
5. #1536C2semidirect
6. #1506C2semidirect
7. #1505C2semidirect
10. #554C4semidirect
12. #550C4semidirect

Maximal subgroups

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

subgroupindex
#35482
#19632
#15592
#15572
#15362
#15062
#15052