Finite Group Databasebeta

← #8424

Subgroups — #8424

Order 256.

367 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 89 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#3503C211yesmaximal, characteristic
2#1987C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic
4#1550C211yesmaximal, characteristic, complemented
5#1549C211yesmaximal, characteristic, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic
8Q82—21no—
9#557C412yescomplemented
10#554—21no—
11#554—22no—
12D4×Q8—22no—
13D4×Q8C412yescomplemented
14#544—22no—
15#544—21no—
16#533—21no—
17#533C2211yescharacteristic
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20#520—21no—
21C4×#41C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—21no—
26C4×C4⋊C4C2211yescharacteristic
27#340—22no—
28#339—21no—
29#333—21no—
30#332—22no—
31#332—21no—
32#328—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
35C2×#41D411yescharacteristic
36C22×Q8—22no—
37C22×Q8—21no—
38C2×M16—21no—
39C22×C8—21no—
40C4⋊Q8—22no—
41C4⋊Q8—22no—
42C4⋊Q8—24no—
43C4⋊Q8—22no—
44C4⋊Q8—22no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8C2×C412yes—
48C4⋊Q8D411yescharacteristic
49C4⋊Q8D412yes—
50C4⋊Q8D411yescharacteristic
51#125—22no—
52#125—24no—
53#125—22no—
54#125—21no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—21no—
59C22⋊Q8—22no—
60C22⋊Q8—24no—
61C22⋊Q8D412yes—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—21no—
66C4×Q8C2×C411yescharacteristic
67C4×Q8C2×C412yes—
68C4×Q8D411yescharacteristic
69C4×D4—24no—
70C4×D4—22no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4C2×C411yescharacteristic
74C4×D4D411yescharacteristic
75#117—22no—
76#117—21no—
77#117—21no—
78#117D411yescharacteristic
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D412yes—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92#104—42no—
93C22⋊C8—21no—
94C4×C8—21no—
95#95—21no—
96#41—21no—
97#41—21no—
98#41SD1612yes—
99#41Q1612yes—
100C2×Q8—24no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—82no—
105C2×Q8—84no—
106C2×Q8C22⋊C411yescharacteristic
107C2×Q8C22⋊C412yes—
108C2×D4—21no—
109C2×D4C22⋊C411yescharacteristic
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119M16—41no—
120M16—41no—
121C2×C8—42no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4C22⋊C414yes—
142C4⋊C4C22⋊C411yescharacteristic
143C4⋊C4C22⋊C411yescharacteristic
144C4⋊C4C22⋊C412yes—
145C22⋊C4—24no—
146C22⋊C4—22no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C42—24no—
151C42—21no—
152C42—22no—
153C42—21no—
154C42—21no—
155C42—41no—
156C42—41no—
157C42C22×C411yescharacteristic
158C42C2×D411yescharacteristic
159C42C2×D411yescharacteristic
160C42C2×D411yescharacteristic
161C23—21no—
162C23#12011yescharacteristic
163Q8—21no—
164Q8—42no—
165Q8—44no—
166Q8—41no—
167Q8—41no—
168Q8—82no—
169Q8—84no—
170Q8#10312yes—
171D4—22no—
172D4—22no—
173D4—21no—
174D4#10312yes—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4C2×C22⋊C411yescharacteristic
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12011yescharacteristic
200C2×C4C2×SD1611yescharacteristic
201C2×C4C2×Q1611yescharacteristic
202C2×C4#13711yescharacteristic
203C2×C4#13711yescharacteristic
204C8—81no—
205C8—81no—
206C8—81no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—41no—
211C22—41no—
212C22#37911yescharacteristic
213C22#44811yescharacteristic
214C22#45111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—22no—
219C4—41no—
220C4—42no—
221C4—41no—
222C4—44no—
223C4—42no—
224C4—82no—
225C4C2×#10311yescharacteristic
226C4#41911yescharacteristic
227C4#44811yescharacteristic
228C4#45111yescharacteristic
229C4#45411yescharacteristic
230C4#45611yescharacteristic
231C2—21no—
232C2—21no—
233C2—42no—
234C2#178711yescharacteristic
235C2#179311yescharacteristic
236C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
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. #1987C2semidirect
4. #1550C2semidirect
5. #1549C2semidirect
6. #1503C2semidirect
9. #557C4semidirect
13. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#35032
#19872
#18062
#15502
#15492
#15032
#14772