Finite Group Databasebeta

← #8470

Subgroups — #8470

Order 256.

377 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 75 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#3511C211yesmaximal, characteristic
2#1997C211yesmaximal, characteristic, complemented
3#1966C211yesmaximal, characteristic, complemented
4#1560C211yesmaximal, characteristic, complemented
5#1525C211yesmaximal, characteristic, complemented
6#1489C211yesmaximal, characteristic, complemented
7#1489C211yesmaximal, characteristic, complemented
8#556—21no—
9#555—21no—
10#554—21no—
11#553—22no—
12#548—22no—
13#547—21no—
14#542—22no—
15#537C412yes—
16#533—21no—
17#532—21no—
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#520C412yes—
21C4×#41C2211yescharacteristic
22#423C2211yescharacteristic
23#423C2211yescharacteristic
24C2×#101—22no—
25C4×M16C2211yescharacteristic
26C4×M16C2211yescharacteristic
27#343—22no—
28#333—21no—
29#331—21no—
30#329—22no—
31Q8⋊C8—21no—
32Q8⋊C8—21no—
33D4⋊C8—21no—
34D4⋊C8—21no—
35C2×#41—22no—
36C2×#41C2×C411yescharacteristic
37C2×#41C2×C412yes—
38C22×Q8D412yes—
39C2×M16—21no—
40C2×M16—21no—
41C2×M16—21no—
42C2×M16—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8D411yescharacteristic
48#127D411yescharacteristic
49#126—24no—
50#125—22no—
51#124—22no—
52#124—24no—
53#124D412yes—
54#123—24no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59#121—22no—
60#121—22no—
61#121—22no—
62#121—24no—
63#121—21no—
64#121—21no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8D411yescharacteristic
68C4×Q8D411yescharacteristic
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—24no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4D411yescharacteristic
77C4×D4D411yescharacteristic
78#117—22no—
79#117—22no—
80#117—21no—
81#117C2×C411yescharacteristic
82C2×C4⋊C4—22no—
83C2×C4⋊C4C2×C412yes—
84C2×C4⋊C4D412yes—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91#101—44no—
92#97—21no—
93#97—21no—
94C4×C8—21no—
95C4×C8—21no—
96#41—42no—
97#41—44no—
98#41—82no—
99#41—82no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C22⋊C411yescharacteristic
105C2×Q8C22⋊C412yes—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—24no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4C22⋊C411yescharacteristic
112C2×D4C22⋊C412yes—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4C22⋊C414yes—
119C22×C4C22×C411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122M16—22no—
123M16—22no—
124M16—41no—
125M16—41no—
126M16—42no—
127M16—42no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4C22⋊C411yescharacteristic
143C4⋊C4C22⋊C411yescharacteristic
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C42—24no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42C2×D411yescharacteristic
157C42C2×D411yescharacteristic
158C42C2×D411yescharacteristic
159C42C2×D411yescharacteristic
160C23—21no—
161C23—22no—
162C23C2×C22⋊C411yescharacteristic
163Q8—22no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—82no—
168D4—22no—
169D4—42no—
170D4—42no—
171D4—44no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—84no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12011yescharacteristic
200C2×C4#12011yescharacteristic
201C8—42no—
202C8—42no—
203C8—81no—
204C8—81no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—41no—
209C22—42no—
210C22—41no—
211C22—42no—
212C22—41no—
213C22—42no—
214C22#37911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—41no—
219C4—44no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4#41312yes—
224C4#45311yescharacteristic
225C4#45411yescharacteristic
226C4#45511yescharacteristic
227C4#45611yescharacteristic
228C2—41no—
229C2—81no—
230C2—82no—
231C2#178511yescharacteristic
232C2#179211yescharacteristic
233C2#179311yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3511C2
LCSG ▷ C22×C4 ▷ 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. #1997C2semidirect
3. #1966C2semidirect
4. #1560C2semidirect
5. #1525C2semidirect
6. #1489C2semidirect
7. #1489C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#19972
#19662
#15602
#15252
#14892