Finite Group Databasebeta

← #329468

Subgroups — #329468

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#29691C212yesmaximal
3#21300C211yesmaximal, characteristic
4#10618C212yesmaximal, complemented
5#10482C214yesmaximal, complemented
6#10279C212yesmaximal, complemented
7#9135C212yesmaximal
8#8595C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#477C2214yes—
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2857C2212yes—
15#2057—22no—
16#2043—22no—
17#2036—24no—
18#1990C2212yes—
19#1984—21no—
20#1984—21no—
21#1984C2212yes—
22#1984C2212yes—
23#1942—21no—
24#1942—21no—
25#1845C2211yescharacteristic
26#1808—21no—
27#1808—21no—
28#1803C2214yes—
29#1797C2212yes—
30#1729—21no—
31#1729—21no—
32C2×#334C2212yes—
33C2×#333C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C22×M16—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39#533—42no—
40#533—42no—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8C2312yes—
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8D412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2312yes—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#500—42no—
53#477—44no—
54#477—44no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—24no—
60#423—42no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2314yes—
63C2×C4⋊C8C2312yes—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8—22no—
67C2×C4×C8—42no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#384C2312yes—
76#380—22no—
77#380—22no—
78#340—22no—
79#340—22no—
80#340—24no—
81#336—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#334—42no—
87#333—44no—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—24no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8Q1614yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113#125—44no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—84no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—42no—
120C4×Q8—42no—
121#117—42no—
122#117—42no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—42no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#107—48no—
145#107—42no—
146#107—42no—
147#107—42no—
148#107—42no—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—24no—
152C4⋊C8—42no—
153#103—48no—
154#103—84no—
155C22⋊C8—82no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—41no—
159C4×C8—41no—
160#95—42no—
161#95—42no—
162C24#12111yescharacteristic
163C2×Q8—24no—
164C2×Q8—24no—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—82no—
171C2×Q8—82no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C2×Q1612yes—
190C22×C4#13712yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C2×#4111yescharacteristic
194M16—84no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8C2×Q1612yes—
213C2×C8C2×Q1612yes—
214C2×C8#13712yes—
215C4⋊C4—24no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—84no—
230C22⋊C4—44no—
231C22⋊C4—84no—
232C42—21no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—42no—
237C42—42no—
238C42H(5,𝔽2)12yes—
239C42H(5,𝔽2)12yes—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23#46212yes—
247C23C2×#12111yescharacteristic
248Q8—48no—
249Q8—84no—
250Q8—164no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#46212yes—
279C2×C4#47212yes—
280C2×C4#53411yescharacteristic
281C2×C4#53412yes—
282C2×C4#53511yescharacteristic
283C2×C4#53812yes—
284C2×C4C22×Q1611yescharacteristic
285C2×C4C2×#13711yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57511yescharacteristic
288C2×C4#57712yes—
289C2×C4#57812yes—
290C8—22no—
291C8—22no—
292C8—22no—
293C8—41no—
294C8—84no—
295C8—82no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22#261711yescharacteristic
304C22C2×#46211yescharacteristic
305C22#303711yescharacteristic
306C22#305911yescharacteristic
307C22#306011yescharacteristic
308C22C2×#47211yescharacteristic
309C22#306711yescharacteristic
310C4—22no—
311C4—22no—
312C4—22no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—162no—
317C4—162no—
318C4#318911yescharacteristic
319C4#319511yescharacteristic
320C4#320114yes—
321C4#320911yescharacteristic
322C4#321511yescharacteristic
323C2—42no—
324C2#1920911yescharacteristic
325C2#1921011yescharacteristic
326C2#1934211yescharacteristic
327C2#1934911yescharacteristic
328C2#2971511yescharacteristic
329C2#2973711yescharacteristic
330C2#2974811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10618C2semidirect
5. #10482C2semidirect
6. #10279C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#296912
#213002
#106182
#104822
#102792
#91352
#85952