Finite Group Databasebeta

← #8432

Subgroups — #8432

Order 256.

370 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 87 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#3501C211yesmaximal, characteristic
2#1804C211yesmaximal, characteristic
3#1542C211yesmaximal, characteristic, complemented
4#1518C211yesmaximal, characteristic, complemented
5#1518C211yesmaximal, characteristic, complemented
6#1481C211yesmaximal, characteristic, complemented
7#1478C211yesmaximal, characteristic
8#554—21no—
9#554—22no—
10#554—22no—
11#554C412yescomplemented
12D4×Q8—21no—
13D4×Q8C412yescomplemented
14#544—21no—
15#543—22no—
16#533C2211yescharacteristic
17#533C2211yescharacteristic
18C2×C22⋊Q8—22no—
19#518—21no—
20C2×C4×D4C2211yescharacteristic
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102—21no—
25#382C2211yescharacteristic
26#340—21no—
27#333—21no—
28#332—21no—
29#328—22no—
30#328—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C22×Q8—22no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C22×C8—21no—
37C4⋊Q8—22no—
38C4⋊Q8—22no—
39C4⋊Q8C2×C411yescharacteristic
40C4⋊Q8C2×C411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#125—22no—
44#125—22no—
45#125—21no—
46#125—21no—
47C22⋊Q8—24no—
48C22⋊Q8—24no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—22no—
52C22⋊Q8—24no—
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8D412yes—
58C22⋊Q8D412yes—
59C4×Q8—21no—
60C4×Q8—21no—
61C4×Q8C2×C412yes—
62C4×Q8D412yes—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4C2×C412yes—
70C4×D4D412yes—
71#117—21no—
72#117—21no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D411yescharacteristic
77C2×C22⋊C4—24no—
78C2×C22⋊C4—21no—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84#102—42no—
85C22⋊C8—21no—
86C22⋊C8—21no—
87C4×C8—21no—
88C4×C8—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C2×Q8—24no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—84no—
96C2×Q8C22⋊C412yes—
97C2×Q8C22⋊C412yes—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4C22⋊C412yes—
102C2×D4SD1612yes—
103C2×D4Q1612yes—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—82no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4C22⋊C412yes—
136C4⋊C4C22⋊C412yes—
137C4⋊C4C22⋊C412yes—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C42—22no—
145C42—22no—
146C42C22×C411yescharacteristic
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C42C2×D411yescharacteristic
150C23—22no—
151C23—22no—
152C23—22no—
153C23—81no—
154C23—81no—
155C23#12011yescharacteristic
156Q8—42no—
157Q8—42no—
158Q8—42no—
159Q8—81no—
160Q8—84no—
161D4—22no—
162D4—22no—
163D4—22no—
164D4#10314yes—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—44no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—84no—
176C2×C4—84no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4C2×C22⋊C411yescharacteristic
181C2×C4C2×C22⋊C411yescharacteristic
182C2×C4C2×C22⋊C411yescharacteristic
183C2×C4#12011yescharacteristic
184C2×C4#12011yescharacteristic
185C2×C4#12011yescharacteristic
186C2×C4C2×SD1611yescharacteristic
187C2×C4C2×Q1611yescharacteristic
188C2×C4#13711yescharacteristic
189C2×C4#13711yescharacteristic
190C8—81no—
191C8—81no—
192C8—81no—
193C22—21no—
194C22—21no—
195C22—24no—
196C22—42no—
197C22—42no—
198C22—82no—
199C22—82no—
200C22#37911yescharacteristic
201C22#44811yescharacteristic
202C22#45111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—161no—
213C4C2×#10311yescharacteristic
214C4#41911yescharacteristic
215C4#44811yescharacteristic
216C4#45111yescharacteristic
217C2—21no—
218C2—21no—
219C2—44no—
220C2#178711yescharacteristic
221C2#179411yescharacteristic
222C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
3. #1542C2semidirect
4. #1518C2semidirect
5. #1518C2semidirect
6. #1481C2semidirect
11. #554C4semidirect
13. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#35012
#18042
#15422
#15182
#14812
#14782