Finite Group Databasebeta

← #129665

Subgroups — #129665

Order 512.

676 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C8×2−1+4C211yesmaximal, characteristic
2#13263C211yesmaximal, characteristic, complemented
3#11621C212yesmaximal, complemented
4#11613C212yesmaximal, complemented
5#11493C211yesmaximal, characteristic, complemented
6C4×2−1+4C411yescharacteristic
7#2988—22no—
8#2988—22no—
9#2988C412yes—
10#2982—22no—
11#2982C411yescharacteristic
12C8×#41—22no—
13C8×#41—22no—
14C8×#41C2212yes—
15C2×C8×Q8—22no—
16C2×C8×Q8C2211yescharacteristic
17#2162C2212yes—
18C2×C4⋊C16C2211yescharacteristic
19C2×C4×C16C2211yescharacteristic
20#1374—22no—
21Q8⋊C16—22no—
22Q8⋊C16—22no—
23D4⋊C16—22no—
24D4⋊C16—22no—
25#1322—22no—
26C2×2−1+4C811yescharacteristic
27#520—22no—
28#520—22no—
29#520C812yes—
30#519—22no—
31#519C811yescharacteristic
32C4×#41—22no—
33C4×#41—22no—
34C4×#41C2×C412yes—
35C2×C4×Q8—22no—
36C2×C4×Q8C2×C411yescharacteristic
37C22×C16—21no—
38C22×C16—21no—
39C8×Q8—22no—
40C8×Q8—24no—
41C8×Q8—22no—
42C8×Q8—22no—
43C8×Q8—22no—
44C8×Q8—22no—
45C8×Q8—22no—
46C8×Q8—22no—
47C8×Q8D412yes—
48C8×Q8D412yes—
49C8×D4—22no—
50C8×D4—24no—
51C8×D4—22no—
52C8×D4—24no—
53C8×D4—22no—
54C8×D4—22no—
55C8×D4—22no—
56C8×D4D412yes—
57C8×D4D412yes—
58#431—24no—
59#431—24no—
60#431—22no—
61#431—22no—
62#431—22no—
63#431C2×C412yes—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2×C411yescharacteristic
67C2×C4⋊C8D412yes—
68C2×C4×C8—24no—
69C2×C4×C8—22no—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C4⋊C16—22no—
73C4⋊C16—22no—
74C22⋊C16—22no—
75C4×C16—22no—
76C4×C16—22no—
772−1+4—24no—
782−1+4—22no—
792−1+4C1614yescomplemented
80C2×#41—22no—
81C2×#41—22no—
82C2×#41C2×C812yes—
83C22×Q8—22no—
84C22×Q8C2×C811yescharacteristic
85C22×C8—24no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8C22⋊C412yes—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C4×Q8—22no—
92C4×Q8—24no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8C22⋊C412yes—
100C4×Q8M1612yes—
101C4×D4—22no—
102C4×D4—24no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—24no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4C22⋊C412yes—
109C4×D4M1612yes—
110#117—24no—
111#117—24no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117C2×C812yes—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C2×C811yescharacteristic
119C2×C4⋊C4M1612yes—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42C22⋊C412yes—
123C2×C42C22×C411yescharacteristic
124C2×C16—22no—
125C2×C16—41no—
126C2×C16—41no—
127C2×C16—42no—
128C2×C16—42no—
129C2×C16—41no—
130C4⋊C8—24no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—24no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8C22⋊C412yes—
138C4⋊C8C22⋊C412yes—
139C22⋊C8—24no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C4×C8—24no—
144C4×C8—24no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8C2×D412yes—
151C4×C8C2×D412yes—
152#41—28no—
153#41—28no—
154#41—28no—
155#41—28no—
156#41—22no—
157#41—22no—
158#41C2×C1614yes—
159#41M3214yes—
160C2×Q8—22no—
161C2×Q8—28no—
162C2×Q8—28no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8C22⋊C812yes—
166C2×Q8C2×C1612yes—
167C2×Q8M3212yes—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4C22⋊C812yes—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4C22⋊C812yes—
177C22×C4C22⋊C812yes—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C22×C811yescharacteristic
180C22×C4C2×M1611yescharacteristic
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—82no—
191C2×C8—84no—
192C2×C8#12011yescharacteristic
193C2×C8#12012yes—
194C2×C8#12011yescharacteristic
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4C22⋊C812yes—
203C4⋊C4C22⋊C812yes—
204C22⋊C4—24no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C42—24no—
209C42—24no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42C2×C22⋊C412yes—
216C42C2×M1612yes—
217C16—42no—
218C16—82no—
219C16—81no—
220C23—22no—
221C23C2×C22⋊C811yescharacteristic
222Q8—28no—
223Q8—24no—
224Q8—22no—
225Q8—24no—
226Q8C22⋊C1614yes—
227D4—28no—
228D4—24no—
229D4—24no—
230D4—22no—
231D4C22⋊C1614yes—
232C2×C4—216no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—84no—
249C2×C4C22⋊C1614yes—
250C2×C4#37911yescharacteristic
251C2×C4C2×C22⋊C812yes—
252C2×C4#40711yescharacteristic
253C2×C4#40712yes—
254C2×C4C22×C1611yescharacteristic
255C2×C4C2×M3212yes—
256C2×C4C2×M3211yescharacteristic
257C8—21no—
258C8—22no—
259C8—42no—
260C8—42no—
261C8—42no—
262C8#45411yescharacteristic
263C8#45611yescharacteristic
264C22—21no—
265C22—21no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22#177911yescharacteristic
270C22C2×C22⋊C1611yescharacteristic
271C22#211211yescharacteristic
272C4—21no—
273C4—24no—
274C4—24no—
275C4—22no—
276C4—42no—
277C4—42no—
278C4—42no—
279C4#179311yescharacteristic
280C4#179411yescharacteristic
281C4C2×C22⋊C1612yes—
282C4#211212yes—
283C2—21no—
284C2—21no—
285C2—44no—
286C2#831911yescharacteristic
287C2#1267911yescharacteristic
288C2#1268511yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC2×2−1+4C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13263C2semidirect
3. #11621C2semidirect
4. #11613C2semidirect
5. #11493C2semidirect
79. 2−1+4C16semidirect

Maximal subgroups

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

subgroupindex
C8×2−1+42
#132632
#116212
#116132
#114932