Finite Group Databasebeta

← #134906

Subgroups — #134906

Order 512.

346 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 74 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#21638C211yesmaximal, characteristic
2#11823C212yesmaximal
3#8611C211yesmaximal, characteristic
4#8609C211yesmaximal, characteristic
5#7322C212yesmaximal
6C2×#446C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8#2398C2211yescharacteristic
9#1985C411yescharacteristic, complemented
10#1955C2211yescharacteristic
11#1923—21no—
12#1923—21no—
13#1832C411yescharacteristic, complemented
14#1810—21no—
15#1808—21no—
16#1799C2212yes—
17#1752—21no—
18#1752—21no—
19#1726C411yescharacteristic, complemented
20#1465C2212yes—
21#1351—22no—
22#1344—22no—
23#1293—21no—
24#1289—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2311yescharacteristic
29#446—41no—
30#446—41no—
31C8×D4—41no—
32C8×D4—41no—
33C2×#108—41no—
34C2×#107—41no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×#99—22no—
39#407—22no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8D411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×#97—21no—
45C2×#97—21no—
46C2×#97C2×C411yescharacteristic
47C2×C4×C8—21no—
48C2×C4×C8C2×C411yescharacteristic
49#400—21no—
50#397D411yescharacteristic
51#397Q811yescharacteristic
52#394—21no—
53#393D411yescharacteristic
54#393Q811yescharacteristic
55#380—22no—
56C4×C22⋊C4—21no—
57C4×C22⋊C4—21no—
58#376C2×C411yescharacteristic
59#342—22no—
60#341—41no—
61#340—41no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#324—22no—
67#323—22no—
68C22×Q8C4⋊C411yescharacteristic
69C22×D4#4111yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×M16—42no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C4×Q8—41no—
86C4×Q8—41no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4C4⋊C411yescharacteristic
92C2×C4⋊C4#4111yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4C2×D411yescharacteristic
99C2×C22⋊C4C2×Q811yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C22×C411yescharacteristic
103#108—82no—
104#107—82no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—42no—
108#99—82no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#97—41no—
116#97—42no—
117#97—41no—
118#97—41no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—42no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—42no—
128#95C4⋊C411yescharacteristic
129#95C4⋊C411yescharacteristic
130C24#12111yescharacteristic
131C24C22⋊Q811yescharacteristic
132C2×Q8—41no—
133C2×Q8—81no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—41no—
137C2×D4—41no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—82no—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4#11711yescharacteristic
154C22×C4#12111yescharacteristic
155C22×C4C22⋊Q811yescharacteristic
156M16—82no—
157M16—82no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—82no—
176C2×C8#9911yescharacteristic
177C2×C8#9911yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23#38011yescharacteristic
208C23#47211yescharacteristic
209C23#47311yescharacteristic
210Q8—82no—
211D4—42no—
212D4—42no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—162no—
243C2×C4—162no—
244C2×C4—162no—
245C2×C4C2×#9911yescharacteristic
246C2×C4#41011yescharacteristic
247C8—21no—
248C8—42no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—81no—
253C8—162no—
254C8#35412yes—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22#179911yescharacteristic
270C22#181411yescharacteristic
271C22#181611yescharacteristic
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—322no—
282C4C2×#35411yescharacteristic
283C4#212611yescharacteristic
284C2—21no—
285C2—21no—
286C2—41no—
287C2—41no—
288C2—81no—
289C2#862811yescharacteristic
290C2#1279811yescharacteristic
291C2#1280611yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#446C4semidirect
9. #1985C4semidirect
13. #1832C4semidirect
19. #1726C4semidirect

Maximal subgroups

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

subgroupindex
#216382
#118232
#86112
#86092
#73222