Finite Group Databasebeta

← #329650

Subgroups — #329650

Order 512.

717 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 138 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#29808C211yesmaximal, characteristic, complemented
2#29687C212yesmaximal
3#21789C211yesmaximal, characteristic
4#10621C212yesmaximal, complemented
5#10485C214yesmaximal, complemented
6#10274C212yesmaximal, complemented
7#9140C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#474C2214yes—
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2057—22no—
16#2056—22no—
17#2041—22no—
18#2033—24no—
19#2001—22no—
20#1956C2212yes—
21#1955—22no—
22#1954C2212yes—
23#1954C2212yes—
24#1834C2211yescharacteristic
25#1802C2214yes—
26#1797C2212yes—
27#1780—22no—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C22×#41—21no—
33C22×Q16—22no—
34#532—42no—
35#532—42no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8C2312yes—
38C2×#127D411yescharacteristic
39C2×#124—21no—
40C2×C22⋊Q8—22no—
41C2×#121—21no—
42C2×#121D412yes—
43C2×C4×D4—21no—
44C2×C4×D4C2312yes—
45C22×C42C2311yescharacteristic
46#494—42no—
47#474—44no—
48#474—44no—
49#432—42no—
50#431—42no—
51C2×#107—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2314yes—
55C2×#103—24no—
56C2×#102—24no—
57C2×#102—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8D412yes—
63C2×#97—21no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#391—22no—
67#389—22no—
68#389—22no—
69#384C2312yes—
70#380—22no—
71#380—22no—
72#340—22no—
73#340—24no—
74#336—22no—
75#335—42no—
76#332—44no—
77C2×#41—42no—
78C2×#41—42no—
79C2×#41—84no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×D4—22no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4C2×D412yes—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4111yescharacteristic
90C2×Q16—44no—
91C2×Q16—84no—
92C22×C8—24no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—42no—
99#127—42no—
100#124—44no—
101C22⋊Q8—84no—
102#121—44no—
103#121—44no—
104C4×D4—24no—
105C4×D4—24no—
106C4×D4—42no—
107C4×D4—42no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—42no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42C2411yescharacteristic
125#107—48no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—24no—
129C4⋊C8—42no—
130#103—84no—
131#102—48no—
132#102—84no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—84no—
136C22⋊C8—84no—
137C22⋊C8—82no—
138#97—42no—
139C4×C8—22no—
140C4×C8—22no—
141#95—42no—
142#95—42no—
143C24—22no—
144C24#12111yescharacteristic
145#41—88no—
146#41—88no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—82no—
153C2×D4—24no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—82no—
161C2×D4—82no—
162C2×D4—84no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C2×#4111yescharacteristic
185Q16—88no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C4⋊C4—24no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—84no—
207C42—21no—
208C42—22no—
209C42—21no—
210C42—22no—
211C42H(5,𝔽2)12yes—
212C42H(5,𝔽2)12yes—
213C23—21no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—42no—
219C23—44no—
220C23—42no—
221C23—42no—
222C23—44no—
223C23C2×#12111yescharacteristic
224Q8—84no—
225Q8—84no—
226Q8—84no—
227D4—44no—
228D4—44no—
229D4—84no—
230D4—84no—
231D4—88no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#53411yescharacteristic
257C2×C4#53412yes—
258C2×C4#53511yescharacteristic
259C2×C4#53812yes—
260C2×C4#57614yes—
261C2×C4#57611yescharacteristic
262C2×C4#57811yescharacteristic
263C2×C4#57812yes—
264C8—42no—
265C8—42no—
266C8—84no—
267C8—82no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22—82no—
274C22—82no—
275C22—84no—
276C22—84no—
277C22—82no—
278C22#261711yescharacteristic
279C22#304112yes—
280C22#305911yescharacteristic
281C22#306111yescharacteristic
282C22#306912yes—
283C4—22no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—82no—
288C4—162no—
289C4—162no—
290C4#319912yes—
291C4#320212yes—
292C4#320312yes—
293C4#322012yes—
294C2—81no—
295C2—162no—
296C2#1950712yes—
297C2#1951112yes—
298C2#2972111yescharacteristic
299C2#2973911yescharacteristic
300C2#2975211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29808C2semidirect
4. #10621C2semidirect
5. #10485C2semidirect
6. #10274C2semidirect
7. #9140C2semidirect
8. #8599C2semidirect

Maximal subgroups

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

subgroupindex
#298082
#296872
#217892
#106212
#104852
#102742
#91402
#85992