Finite Group Databasebeta

← #136282

Subgroups — #136282

Order 512.

322 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 66 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#23276C211yesmaximal, characteristic
2#11734C211yesmaximal, characteristic
3#9032C211yesmaximal, characteristic
4#8865C211yesmaximal, characteristic, complemented
5#8838C211yesmaximal, characteristic, complemented
6#7284C211yesmaximal, characteristic, complemented
7#7206C211yesmaximal, characteristic, complemented
8#2983C2211yescharacteristic
9#2398C2211yescharacteristic
10#1985C2211yescharacteristic
11#1984C2211yescharacteristic
12#1924—21no—
13#1923—21no—
14#1870—21no—
15C4×#101C2211yescharacteristic
16C4×#101C2211yescharacteristic
17#1753—21no—
18#1753—21no—
19#1746—21no—
20#1573—21no—
21#1565—21no—
22#1490—21no—
23#1461C2211yescharacteristic
24#1353C411yescharacteristic
25#1353C411yescharacteristic
26#1345C411yescharacteristic
27#1345C411yescharacteristic
28#1302—21no—
29#1300—21no—
30#1278—21no—
31C22×M16—21no—
32C2×C4×Q8C2311yescharacteristic
33C2×C4×D4D411yescharacteristic
34#446—41no—
35#446—41no—
36#432—21no—
37#432—21no—
38#431—21no—
39#431—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×#101—21no—
43C2×#101—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C4×M16—21no—
48C4×M16—21no—
49C2×#97—21no—
50C2×#97D411yescharacteristic
51C2×C4×C8—21no—
52C2×C4×C8D411yescharacteristic
53#400C2×C411yescharacteristic
54#400Q811yescharacteristic
55#397—21no—
56#394C2×C411yescharacteristic
57#394Q811yescharacteristic
58#393—21no—
59C4×C22⋊C4—21no—
60C4×C22⋊C4—21no—
61#376D411yescharacteristic
62#343—21no—
63#343C2×C411yescharacteristic
64#343C2×C411yescharacteristic
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#335—41no—
72#335—41no—
73#334—41no—
74#334—41no—
75D4⋊C8—41no—
76D4⋊C8—41no—
77#324—21no—
78#324C2×C411yescharacteristic
79#324C2×C411yescharacteristic
80C22×Q8#4111yescharacteristic
81C22×D4C4⋊C411yescharacteristic
82C23×C4—21no—
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—41no—
88C2×M16—81no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—41no—
96C4×D4—41no—
97C2×C4⋊C4C4⋊C411yescharacteristic
98C2×C4⋊C4C22×C411yescharacteristic
99C2×C4⋊C4C2×Q811yescharacteristic
100C2×C4⋊C4#4111yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#101—22no—
120#101—42no—
121#101—41no—
122C22⋊C8—21no—
123C22⋊C8—21no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—41no—
127C22⋊C8—81no—
128#97—41no—
129#97—41no—
130#97—41no—
131#97—41no—
132#97—41no—
133#97—42no—
134C4×C8—41no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—42no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95C4⋊C411yescharacteristic
146#95C4⋊C411yescharacteristic
147C24—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×D4—41no—
153C2×D4—81no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4C2×C4⋊C411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170M16—22no—
171M16—41no—
172M16—42no—
173M16—41no—
174M16—41no—
175M16—84no—
176M16—81no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—81no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23#38511yescharacteristic
222Q8—81no—
223Q8—81no—
224D4—42no—
225D4—81no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#44311yescharacteristic
256C2×C4#44411yescharacteristic
257C2×C4B(4,𝔽2)11yescharacteristic
258C2×C4#45811yescharacteristic
259C8—42no—
260C8—42no—
261C8—42no—
262C8—81no—
263C8—81no—
264C8—81no—
265C8—82no—
266C8—81no—
267C8—81no—
268C8—81no—
269C8—81no—
270C22—21no—
271C22—21no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22#186111yescharacteristic
280C22#186611yescharacteristic
281C22#186711yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#220411yescharacteristic
294C4#220411yescharacteristic
295C2—21no—
296C2—21no—
297C2—161no—
298C2#887411yescharacteristic
299C2#1296311yescharacteristic
300C2#1297211yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×C4×Q8C23
socleC22#1867
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8865C2semidirect
5. #8838C2semidirect
6. #7284C2semidirect
7. #7206C2semidirect

Maximal subgroups

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

subgroupindex
#232762
#117342
#90322
#88652
#88382
#72842
#72062