Finite Group Databasebeta

← #229312

Subgroups — #229312

Order 512.

621 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23162C211yesmaximal, characteristic
2#22965C212yesmaximal
3Q8×#106C211yesmaximal, characteristic
4C2×#1676C211yesmaximal, characteristic
5#9836C211yesmaximal, characteristic
6#9617C211yesmaximal, characteristic
7#9567C212yesmaximal
8#8975C212yesmaximal
9#8927C212yesmaximal
10#8285C211yesmaximal, characteristic
11#8203C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C8×Q8C2211yescharacteristic
15#2369C2212yes—
16Q8×C4⋊C4C2212yes—
17#1990C2212yes—
18#1987C2212yes—
19#1956C2212yes—
20#1948—22no—
21#1948C2212yes—
22#1947C2212yes—
23#1943—21no—
24#1942—22no—
25#1921—22no—
26#1918—22no—
27#1863—22no—
28#1844—21no—
29#1844C2211yescharacteristic
30#1776—21no—
31C4×#106—21no—
32C4×#106C2211yescharacteristic
33#1770C2211yescharacteristic
34C8×C4⋊C4C2212yes—
35C4×#103—22no—
36C4×#103C2211yescharacteristic
37#1676C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#333C2212yes—
40C2×Q8⋊C8C2211yescharacteristic
41C2×#322C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8C2311yescharacteristic
49#500—22no—
50#500C2×C414yes—
51#477—24no—
52#477C2×C418yes—
53C8×Q8—22no—
54C8×Q8C2×C414yes—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#106—21no—
60C2×#106—21no—
61C2×#106—22no—
62C2×#106D411yescharacteristic
63C2×#106D411yescharacteristic
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×#103—22no—
69C2×#103—42no—
70C2×#103D412yes—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—42no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—42no—
82#384—22no—
83#384C2312yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88C4×C4⋊C4C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340D412yes—
96#340D412yes—
97#336—22no—
98#336—21no—
99#335C2×C414yes—
100#333C2×C418yes—
101Q8⋊C8—22no—
102#322—22no—
103C22×Q8—22no—
104C22×Q8C2×D411yescharacteristic
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8C22×C414yes—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—44no—
119C4×Q8—82no—
120C4×Q8C22×C412yes—
121C4×Q8#4112yes—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42C2411yescharacteristic
141#107—24no—
142#107—24no—
143#107—28no—
144#107—82no—
145#106—22no—
146#106—42no—
147#106—82no—
148#106—81no—
149#106Q1614yes—
150C4⋊C8—24no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8C22×C412yes—
156C4⋊C8C22×C414yes—
157#103—28no—
158#103—24no—
159#103—84no—
160#97—42no—
161C4×C8—24no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—82no—
167C4×C8C22×C412yes—
168C4×C8#4112yes—
169#95—22no—
170#95—21no—
171#95—42no—
172#95—42no—
173#95—42no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—24no—
177C2×Q8—42no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8C4×D412yes—
181C2×Q8#13714yes—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4H(5,𝔽2)11yescharacteristic
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8C4×D412yes—
209C2×C8C4×D412yes—
210C2×C8C2×Q1612yes—
211C2×C8#13512yes—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—84no—
229C4⋊C4—82no—
230C4⋊C4—162no—
231C4⋊C4—162no—
232C4⋊C4C4×D412yes—
233C4⋊C4C2×Q1614yes—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—42no—
238C42—42no—
239C42—41no—
240C42—84no—
241C42—162no—
242C42C23×C411yescharacteristic
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C422−1+411yescharacteristic
246C23D4211yescharacteristic
247Q8—24no—
248Q8—22no—
249Q8—84no—
250Q8—162no—
251Q8#44114yes—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C2×C4×D411yescharacteristic
273C2×C4C2×C4×D411yescharacteristic
274C2×C4#52011yescharacteristic
275C2×C4#54711yescharacteristic
276C2×C4#54811yescharacteristic
277C2×C4#55311yescharacteristic
278C2×C4C22×Q1611yescharacteristic
279C2×C4C2×#13511yescharacteristic
280C2×C4C2×#13712yes—
281C2×C4#57712yes—
282C2×C4#57711yescharacteristic
283C2×C4#57811yescharacteristic
284C8—22no—
285C8—42no—
286C8—42no—
287C8—44no—
288C8—81no—
289C8—81no—
290C8C4×Q1614yes—
291C22#236111yescharacteristic
292C22#326612yes—
293C22#328411yescharacteristic
294C22D4×Q1611yescharacteristic
295C22#328812yes—
296C4—21no—
297C4—21no—
298C4—24no—
299C4—42no—
300C4—44no—
301C4—84no—
302C4—82no—
303C4—82no—
304C4—162no—
305C4C2×C4×Q1611yescharacteristic
306C4C2×#44111yescharacteristic
307C4#294711yescharacteristic
308C4#295211yescharacteristic
309C4#334412yes—
310C4#336311yescharacteristic
311C4#336411yescharacteristic
312C2#1981312yes—
313C2#1987712yes—
314C2#2299811yescharacteristic
315C2#2302811yescharacteristic
316C2#2304811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1676C4semidirect

Maximal subgroups

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

subgroupindex
#231622
#229652
Q8×#1062
C2×#16762
#98362
#96172
#95672
#89752
#89272
#82852
#82032