Finite Group Databasebeta

← #150291

Subgroups — #150291

Order 512.

415 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 74 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#23732C211yesmaximal, characteristic
2C2×#1394C211yesmaximal, characteristic
3C2×#1319C211yesmaximal, characteristic
4#7437C212yesmaximal, complemented
5#7436C212yesmaximal, complemented
6C2×#428—21no—
7C2×#419—21no—
8C22×#101—21no—
9C2×#391C2211yescharacteristic
10#2285C411yescharacteristic
11#2097—21no—
12#2097—21no—
13#2096C411yescharacteristic
14#2096C2211yescharacteristic
15#2095C411yescharacteristic
16#2095C2211yescharacteristic
17#1728C411yescharacteristic
18C2×#324—21no—
19#1394—21no—
20#1394C2212yes—
21#1319—21no—
22#1319C2212yes—
23C23×Q8D411yescharacteristic
24C22×M16—21no—
25C22×M16—41no—
26C2×#117—21no—
27C22×C4⋊C4D411yescharacteristic
28C22×C42C2×C411yescharacteristic
29#428—42no—
30#428—42no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#106—21no—
34C2×#106—21no—
35#419—42no—
36#419—42no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#101—24no—
42C2×#101—22no—
43C2×#101—41no—
44C2×#101—42no—
45C2×C22⋊C8—41no—
46#391—41no—
47#391C2×C411yescharacteristic
48#391D411yescharacteristic
49#391D411yescharacteristic
50#391C2311yescharacteristic
51#384C2×C411yescharacteristic
52#384C2×C411yescharacteristic
53#382—21no—
54#380—41no—
55C4×C4⋊C4—41no—
56C2×#95—41no—
57#336—21no—
58#336C2×C411yescharacteristic
59#336C2×C411yescharacteristic
60#324—22no—
61#324—41no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8C22⋊C411yescharacteristic
68C22×Q8SD1612yes—
69C22×Q8Q1612yes—
70C22×Q8C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—44no—
80C2×M16—42no—
81C2×M16—82no—
82C2×M16—81no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—81no—
86C22×C8—81no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C22⋊C411yescharacteristic
99C2×C4⋊C4C2×D411yescharacteristic
100C2×C22⋊C4—41no—
101C2×C42—21no—
102C2×C42—42no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106#107—42no—
107#107—42no—
108#106—42no—
109#106—42no—
110#103—42no—
111#103—42no—
112#103—42no—
113#103—42no—
114#101—24no—
115#101—44no—
116#101—42no—
117C22⋊C8—42no—
118C22⋊C8—81no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—42no—
124C24#9911yescharacteristic
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—41no—
137C2×Q8—81no—
138C2×Q8—82no—
139C2×Q8—82no—
140C2×Q8—81no—
141C2×Q8#10312yes—
142C2×Q8C2×SD1611yescharacteristic
143C2×Q8C2×Q1611yescharacteristic
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—81no—
165C22×C4—82no—
166C22×C4#9911yescharacteristic
167C22×C4#10312yes—
168C22×C4#10412yes—
169C22×C4C2×C22⋊C411yescharacteristic
170M16—44no—
171M16—44no—
172M16—44no—
173M16—82no—
174M16—84no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—84no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—162no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—81no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C42—41no—
197C42—41no—
198C42—42no—
199C42—81no—
200C42—81no—
201C23—41no—
202C23—41no—
203C23#32812yes—
204C23#35212yes—
205C23#35311yescharacteristic
206C23#35311yescharacteristic
207C23C2×#9911yescharacteristic
208Q8—42no—
209Q8—41no—
210Q8—41no—
211Q8—81no—
212Q8—81no—
213Q8—82no—
214Q8—82no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—84no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—161no—
251C2×C4#32812yes—
252C2×C4C2×#10311yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C8—42no—
255C8—42no—
256C8—84no—
257C8—162no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22#135212yes—
267C22#135212yes—
268C22C2×#32811yescharacteristic
269C22C2×#35211yescharacteristic
270C22C2×#35311yescharacteristic
271C4—21no—
272C4—22no—
273C4—21no—
274C4—42no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—161no—
282C4—161no—
283C2—21no—
284C2—22no—
285C2—22no—
286C2—21no—
287C2C2×#135211yescharacteristic
288C2#1357811yescharacteristic
289C2#1359011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect
5. #7436C2semidirect

Maximal subgroups

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

subgroupindex
#237322
C2×#13942
C2×#13192
#74372
#74362