Finite Group Databasebeta

← #102039

Subgroups — #102039

Order 512.

378 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#487C211yesmaximal, characteristic
2#8807C211yesmaximal, characteristic
3#8133C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic
5#7590C211yesmaximal, characteristic
6#7588C211yesmaximal, characteristic
7#7588C211yesmaximal, characteristic
8C2×#487C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1990C411yescharacteristic
12#1956C411yescharacteristic
13#1933—21no—
14#1933—21no—
15#1834C411yescharacteristic
16#1831C2211yescharacteristic
17#1831C2211yescharacteristic
18C4×#103—21no—
19C4×#103—21no—
20#1744C2211yescharacteristic
21#1744C2211yescharacteristic
22C42×C8C2211yescharacteristic
23C2×#335—21no—
24C2×#334—21no—
25#1273—21no—
26#1273—21no—
27#1272—22no—
28#1271—22no—
29#1271—22no—
30#1271—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35#487—22no—
36#487C814yescomplemented
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#391—21no—
52#391—21no—
53#389—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#384C2×C411yescharacteristic
58#384C2×C411yescharacteristic
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4—21no—
63C43C2311yescharacteristic
64#340—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336D412yes—
75#336D412yes—
76#335—42no—
77#334—42no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8C2×C812yes—
95C4⋊Q8C2×C812yes—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×Q8—82no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126#103—24no—
127#103—24no—
128#103—42no—
129#103—42no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—41no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—41no—
141C4×C8—41no—
142C4×C8C2×C812yes—
143C4×C8M1612yes—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—81no—
157C2×Q8—81no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12411yescharacteristic
168C22×C4#12711yescharacteristic
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8C22⋊C812yes—
192C2×C8C22⋊C812yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—22no—
215C42—22no—
216C42—41no—
217C42—41no—
218C42—82no—
219C42—81no—
220C42—81no—
221C42C22×C811yescharacteristic
222C42C2×M1611yescharacteristic
223C42#13111yescharacteristic
224C42#13111yescharacteristic
225C42AGL(1,ℤ/8)11yescharacteristic
226C42AGL(1,ℤ/8)11yescharacteristic
227C42#13711yescharacteristic
228C42#13711yescharacteristic
229C23#39011yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233Q8—82no—
234Q8—81no—
235Q8—81no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4C2×C22⋊C811yescharacteristic
261C2×C4#40811yescharacteristic
262C2×C4#41711yescharacteristic
263C2×C4#41711yescharacteristic
264C2×C4#42111yescharacteristic
265C2×C4#42111yescharacteristic
266C2×C4C8×D411yescharacteristic
267C2×C4C8×D411yescharacteristic
268C2×C4#43611yescharacteristic
269C2×C4#43611yescharacteristic
270C2×C4#47112yes—
271C2×C4#47312yes—
272C2×C4#48911yescharacteristic
273C2×C4#48911yescharacteristic
274C2×C4#49611yescharacteristic
275C2×C4#49711yescharacteristic
276C8—22no—
277C8—22no—
278C8—41no—
279C8—41no—
280C8—82no—
281C8—82no—
282C8—161no—
283C8—161no—
284C22#195411yescharacteristic
285C22#195911yescharacteristic
286C22#196311yescharacteristic
287C22#197611yescharacteristic
288C22#197611yescharacteristic
289C22#197711yescharacteristic
290C22#197911yescharacteristic
291C4—21no—
292C4—21no—
293C4—22no—
294C4—22no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—82no—
302C4—82no—
303C4—161no—
304C4—161no—
305C4#147711yescharacteristic
306C4#147711yescharacteristic
307C4#149511yescharacteristic
308C4#149511yescharacteristic
309C2#964011yescharacteristic
310C2#966711yescharacteristic
311C2#966911yescharacteristic
312C2#975811yescharacteristic
313C2#976511yescharacteristic
314C2#976611yescharacteristic
315C2#977511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#390
commutatorC42C22×C8
Frattini, cosocleC43C23
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
36. #487C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4872
#88072
#81332
#75912
#75902
#75882