Finite Group Databasebeta

← #136031

Subgroups — #136031

Order 512.

343 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 78 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#21920C211yesmaximal, characteristic
2#11447C211yesmaximal, characteristic
3#11212C211yesmaximal, characteristic
4#8862C211yesmaximal, characteristic
5#8861C211yesmaximal, characteristic
6#7252C211yesmaximal, characteristic
7#7252C211yesmaximal, characteristic
8#2905C2211yescharacteristic
9#2285C2211yescharacteristic
10#1917—21no—
11#1917—21no—
12#1916—21no—
13#1916—21no—
14#1817—21no—
15#1798C2211yescharacteristic
16#1798C2211yescharacteristic
17#1750—21no—
18#1740C2211yescharacteristic
19#1733—21no—
20C2×#339—21no—
21#1571—21no—
22#1570—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1325C412yescomplemented
26#1324C412yescomplemented
27#1301—21no—
28#1301—21no—
29C22×M16—21no—
30C2×#117—21no—
31C2×#117D411yescharacteristic
32C2×#117C2311yescharacteristic
33C22×C42D411yescharacteristic
34#431—21no—
35#431—21no—
36#424—42no—
37#423—21no—
38#423—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×#97—21no—
46C2×C4×C8—21no—
47#384C2×C411yescharacteristic
48#384Q811yescharacteristic
49#382C2×C411yescharacteristic
50#382Q811yescharacteristic
51#380—21no—
52#380—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55#342—21no—
56#342—21no—
57#339—42no—
58#339—42no—
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#335—42no—
68#334—42no—
69#324C2×C412yes—
70#324C2×C412yes—
71#323—21no—
72#323—21no—
73C23×C4—21no—
74C23×C4C2×D411yescharacteristic
75C2×M16—22no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—41no—
86#117—22no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91#117—41no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C22×C411yescharacteristic
102C2×C4⋊C4C2×Q811yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4#4111yescharacteristic
106C2×C22⋊C4#4111yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42C4⋊C411yescharacteristic
115C2×C42C4⋊C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134#97—81no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8—81no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24#12311yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4C4×D411yescharacteristic
159C22×C4#12011yescharacteristic
160C22×C4C22⋊Q811yescharacteristic
161C22×C4C22⋊Q811yescharacteristic
162C22×C4#13511yescharacteristic
163C22×C4#13511yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165C22×C4#13711yescharacteristic
166M16—42no—
167M16—42no—
168M16—81no—
169M16—81no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—81no—
186C2×C8—81no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—42no—
211C42—41no—
212C42—41no—
213C42—41no—
214C42—41no—
215C42—81no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23#38511yescharacteristic
221C23#48211yescharacteristic
222C23#48511yescharacteristic
223C2×C4—21no—
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—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
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—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4—161no—
258C2×C4#42711yescharacteristic
259C2×C4#42811yescharacteristic
260C2×C4#44911yescharacteristic
261C2×C4#45211yescharacteristic
262C2×C4#45411yescharacteristic
263C2×C4#45611yescharacteristic
264C2×C4B(4,𝔽2)11yescharacteristic
265C2×C4#45811yescharacteristic
266C8—82no—
267C8—82no—
268C8—81no—
269C8—82no—
270C8—81no—
271C8—81no—
272C8—81no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22#185411yescharacteristic
283C22#185711yescharacteristic
284C22#186711yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—162no—
296C4#159711yescharacteristic
297C4#160311yescharacteristic
298C4#160911yescharacteristic
299C4#161511yescharacteristic
300C2—21no—
301C2—21no—
302C2—41no—
303C2—41no—
304C2#885511yescharacteristic
305C2#1296911yescharacteristic
306C2#1297511yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC24#123
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1325C4semidirect
26. #1324C4semidirect

Maximal subgroups

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

subgroupindex
#219202
#114472
#112122
#88622
#88612
#72522