Finite Group Databasebeta

← #100100

Subgroups — #100100

Order 512.

418 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 96 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#23363C211yesmaximal, characteristic, complemented
2#21646C211yesmaximal, characteristic
3#7604C211yesmaximal, characteristic, complemented
4#7603C211yesmaximal, characteristic
5#7603C211yesmaximal, characteristic
6#7019C211yesmaximal, characteristic, complemented
7#7019C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#1990C411yescharacteristic
12#1984—22no—
13#1957C411yescharacteristic
14#1936—22no—
15#1916—22no—
16#1916—22no—
17#1839C2211yescharacteristic
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1839C2211yescharacteristic
21#1834C411yescharacteristic
22#1762—22no—
23C4×#97C2211yescharacteristic
24#1271—22no—
25#1270—22no—
26#1270—22no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×#127C2×C411yescharacteristic
29C2×C4×Q8—21no—
30C2×C4×D4—21no—
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33#486—22no—
34#486C814yescomplemented
35C2×C4⋊C8—22no—
36C2×#102—22no—
37C2×#102—22no—
38C2×#97—21no—
39C2×#97—21no—
40C2×C4×C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43#391—21no—
44#390—21no—
45#390—21no—
46#390—21no—
47#389—21no—
48#389D412yes—
49#384C2×C411yescharacteristic
50#384C2×C411yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4D412yes—
53C43C2311yescharacteristic
54#340—22no—
55#340—22no—
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#336—21no—
66#336—21no—
67#336D412yes—
68#336D412yes—
69C22×Q8—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C23×C4—41no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2×C812yes—
83C4⋊Q8M1612yes—
84#127—21no—
85#127—21no—
86#127—21no—
87#127C2×C812yes—
88C4×Q8—81no—
89C4×Q8—81no—
90C4×D4—44no—
91C4×D4—82no—
92C4×D4—82no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C22⋊C412yes—
101C2×C4⋊C4C22⋊C412yes—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C42—22no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C4⋊C8—82no—
114#102—28no—
115#102—44no—
116#97—42no—
117#97—42no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—82no—
122C4×C8C2×C812yes—
123#95—22no—
124#95—22no—
125C24—41no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—81no—
129C2×Q8—81no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—81no—
139C2×D4—81no—
140C2×D4—81no—
141C2×D4—81no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D411yescharacteristic
150C22×C4C4×D411yescharacteristic
151C22×C4#12011yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4#12311yescharacteristic
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8—81no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4C22⋊C814yes—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C22⋊C4—82no—
187C42—22no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—42no—
194C42—42no—
195C42C22×C811yescharacteristic
196C42C2×M1611yescharacteristic
197C42#13111yescharacteristic
198C42#13111yescharacteristic
199C23—42no—
200C23—81no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23#38611yescharacteristic
207Q8—82no—
208Q8—82no—
209D4—44no—
210D4—82no—
211D4—82no—
212D4—82no—
213D4—82no—
214D4—82no—
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—24no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
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×C4—164no—
242C2×C4C2×C22⋊C811yescharacteristic
243C2×C4#40811yescharacteristic
244C2×C4C8×D411yescharacteristic
245C2×C4C8×D411yescharacteristic
246C2×C4#43511yescharacteristic
247C2×C4#43511yescharacteristic
248C2×C4#45412yes—
249C8—42no—
250C8—42no—
251C8—82no—
252C8—82no—
253C8—161no—
254C8—161no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—161no—
259C22—161no—
260C22—161no—
261C22—161no—
262C22#186911yescharacteristic
263C22#188411yescharacteristic
264C22#188811yescharacteristic
265C22#190311yescharacteristic
266C22#190311yescharacteristic
267C22#190411yescharacteristic
268C22#190511yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—44no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—82no—
281C4—161no—
282C2—162no—
283C2#896011yescharacteristic
284C2#908911yescharacteristic
285C2#909011yescharacteristic
286C2#941711yescharacteristic
287C2#941811yescharacteristic
288C2#942411yescharacteristic
289C2#943011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23363C2semidirect
3. #7604C2semidirect
6. #7019C2semidirect
7. #7019C2semidirect
34. #486C8semidirect

Maximal subgroups

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

subgroupindex
#233632
#216462
#76042
#76032
#70192