Finite Group Databasebeta

← #150238

Subgroups — #150238

Order 512.

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

normal subgroups only

This group has 519 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
1C2×#2039C211yesmaximal, characteristic
2C2×#1325C211yesmaximal, characteristic, complemented
3#8040C211yesmaximal, characteristic, complemented
4#7417C214yesmaximal, complemented
5C2×#537C411yescharacteristic
6C2×#432—21no—
7C22×#104—21no—
8C2×#417—21no—
9C2×#384C2211yescharacteristic
10#2039—21no—
11#2039—21no—
12#2039C412yes—
13#2039C2212yes—
14C2×#343C411yescharacteristic
15C2×#324—21no—
16#1325—21no—
17#1325C2212yescomplemented
18#1296—21no—
19#1296C2212yescomplemented
20C22×#41—21no—
21C22×M16—21no—
22#537—21no—
23#537—41no—
24#537—42no—
25#537C2×C412yes—
26C2×C22⋊Q8—41no—
27C2×#121—41no—
28C2×#121—41no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33#432—24no—
34#432—42no—
35C2×C4⋊C8—22no—
36C2×#104—22no—
37C2×#104—22no—
38C2×#104—42no—
39C2×#104—42no—
40#417—22no—
41#417—22no—
42#417—42no—
43C2×#103—41no—
44C2×#102—41no—
45C2×C22⋊C8—41no—
46C2×C22⋊C8—41no—
47C2×#97—41no—
48#384—41no—
49#384C2×C411yescharacteristic
50#384D411yescharacteristic
51#384D411yescharacteristic
52#384C2311yescharacteristic
53#343—21no—
54#343C2×C412yes—
55#324—22no—
56#324—41no—
57C2×#41—22no—
58C2×#41—22no—
59C2×#41—41no—
60C2×#41—42no—
61C2×#41—42no—
62C2×#41—41no—
63C2×#41—42no—
64C2×#41—42no—
65C22×Q8—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—42no—
77C2×M16—42no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8—81no—
81C22⋊Q8—42no—
82C22⋊Q8—44no—
83C22⋊Q8—42no—
84#121—42no—
85#121—42no—
86#121—42no—
87#121—42no—
88#121—42no—
89#121—44no—
90#121—42no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—84no—
103C2×C4⋊C4C22⋊C411yescharacteristic
104C2×C4⋊C4C22⋊C411yescharacteristic
105C2×C4⋊C4C22×C411yescharacteristic
106C2×C4⋊C4C2×D411yescharacteristic
107C2×C22⋊C4—42no—
108C2×C42—21no—
109C2×C42—22no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42D812yes—
112C2×C42SD1612yes—
113C2×C42C2×D411yescharacteristic
114C4⋊C8—24no—
115C4⋊C8—42no—
116#104—44no—
117#104—42no—
118#104—42no—
119#103—42no—
120#103—42no—
121#102—42no—
122#102—42no—
123C22⋊C8—44no—
124C22⋊C8—42no—
125C22⋊C8—81no—
126#97—44no—
127C24—41no—
128C24#9911yescharacteristic
129#41—44no—
130#41—44no—
131#41—42no—
132#41—42no—
133#41—42no—
134#41—42no—
135#41—82no—
136#41—82no—
137#41—82no—
138#41—82no—
139C2×Q8—22no—
140C2×Q8—41no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—41no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—82no—
152C2×D4—82no—
153C2×D4—162no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4#9911yescharacteristic
179C22×C4#10212yes—
180C22×C4#10412yes—
181C22×C4C2×C22⋊C411yescharacteristic
182M16—42no—
183M16—42no—
184M16—82no—
185C2×C8—44no—
186C2×C8—42no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8—162no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C42—22no—
207C42—22no—
208C42—21no—
209C42—21no—
210C42#10212yes—
211C42C2×D811yescharacteristic
212C42C2×SD1611yescharacteristic
213C23—41no—
214C23—41no—
215C23—42no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23#32712yes—
220C23AΣL(1,𝔽16)11yescharacteristic
221C23AΣL(1,𝔽16)11yescharacteristic
222C23#35512yes—
223C23C2×#9911yescharacteristic
224Q8—42no—
225Q8—42no—
226Q8—82no—
227D4—42no—
228D4—42no—
229D4—82no—
230D4—84no—
231D4—82no—
232D4—82no—
233D4—162no—
234D4—162no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—161no—
268C2×C4—161no—
269C2×C4#32712yes—
270C2×C4C2×#10211yescharacteristic
271C2×C4C2×#10411yescharacteristic
272C8—84no—
273C8—82no—
274C8—162no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—164no—
287C22#134512yes—
288C22#134512yes—
289C22C2×#32711yescharacteristic
290C22C2×AΣL(1,𝔽16)11yescharacteristic
291C22C2×#35511yescharacteristic
292C4—22no—
293C4—22no—
294C4—21no—
295C4—22no—
296C4—21no—
297C4—42no—
298C4—82no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C2—21no—
304C2—22no—
305C2—22no—
306C2—21no—
307C2—162no—
308C2C2×#134511yescharacteristic
309C2#1356111yescharacteristic
310C2#1357311yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#384C23
prime radicalC2×#537C4
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
3. #8040C2semidirect
4. #7417C2semidirect
17. #1325C22semidirect
19. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#20392
C2×#13252
#80402
#74172