Finite Group Databasebeta

← #100802

Subgroups — #100802

Order 512.

396 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 88 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#28753C211yesmaximal, characteristic
2#23363C211yesmaximal, characteristic, complemented
3#8593C211yesmaximal, characteristic, complemented
4#7719C211yesmaximal, characteristic
5#7719C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic, complemented
7#7041C211yesmaximal, characteristic, complemented
8C2×#489—21no—
9#2858C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2043C412yes—
12#2041C412yes—
13#1984—22no—
14#1983—22no—
15#1959—21no—
16#1916—22no—
17#1916—22no—
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1767C2211yescharacteristic
23C2×#332—21no—
24#1296—21no—
25#1270—22no—
26#1270—22no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×C4⋊Q8D411yescharacteristic
29C2×#127C2×C411yescharacteristic
30C2×#124—21no—
31C2×#124—21no—
32C2×C4×Q8—21no—
33#489—42no—
34#489—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×#103—22no—
40C2×#103—22no—
41C2×#102—22no—
42C2×#102—22no—
43C2×#97—21no—
44C2×C4×C8—22no—
45#391—21no—
46#390—21no—
47#390—21no—
48#390—21no—
49#390—21no—
50#389—21no—
51#389C2×C412yes—
52#384D411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4D412yes—
55C43C2311yescharacteristic
56#336—24no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336C2×C412yes—
66#336D412yes—
67#336D412yes—
68#332—42no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—81no—
82C4⋊Q8D812yes—
83C4⋊Q8SD1612yes—
84#127—81no—
85#124—44no—
86#124—41no—
87#124—41no—
88#124—41no—
89#124—41no—
90C4×Q8—81no—
91C4×Q8—81no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C22⋊C412yes—
99C2×C4⋊C4C22⋊C412yes—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C42—22no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C4⋊C8—42no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—82no—
118#103—44no—
119#103—82no—
120#102—44no—
121#102—82no—
122#97—41no—
123#97—41no—
124C4×C8—82no—
125#95—22no—
126#95—22no—
127C24—41no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4—81no—
145C2×D4—81no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C4×D411yescharacteristic
151C22×C4C4×D411yescharacteristic
152C22×C4#12011yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4#12111yescharacteristic
155C22×C4#12311yescharacteristic
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8—81no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4#10214yes—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—81no—
184C22⋊C4—81no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C42—22no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—42no—
194C42—42no—
195C42C2×D811yescharacteristic
196C42C2×SD1611yescharacteristic
197C42#13511yescharacteristic
198C42#13511yescharacteristic
199C23—81no—
200C23—81no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23#38611yescharacteristic
207Q8—82no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211D4—82no—
212D4—82no—
213D4—162no—
214D4—162no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4C2×#10211yescharacteristic
242C2×C4#41611yescharacteristic
243C2×C4#45412yes—
244C2×C4#46511yescharacteristic
245C2×C4#46511yescharacteristic
246C2×C4#46611yescharacteristic
247C2×C4#46611yescharacteristic
248C8—82no—
249C8—82no—
250C8—82no—
251C8—161no—
252C8—161no—
253C22—161no—
254C22—161no—
255C22—161no—
256C22—161no—
257C22—161no—
258C22—161no—
259C22—161no—
260C22#188111yescharacteristic
261C22#188111yescharacteristic
262C22#188811yescharacteristic
263C22#189311yescharacteristic
264C22#190411yescharacteristic
265C22#190511yescharacteristic
266C22#191411yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—161no—
280C2—321no—
281C2#900811yescharacteristic
282C2#930511yescharacteristic
283C2#930611yescharacteristic
284C2#933311yescharacteristic
285C2#933411yescharacteristic
286C2#939811yescharacteristic
287C2#943011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23363C2semidirect
3. #8593C2semidirect
6. #7041C2semidirect
7. #7041C2semidirect

Maximal subgroups

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

subgroupindex
#287532
#233632
#85932
#77192
#70412