Finite Group Databasebeta

← #391797

Subgroups — #391797

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2#20683C211yesmaximal, characteristic, complemented
3#13852C211yesmaximal, characteristic, complemented
4#11370C212yesmaximal, complemented
5#11075C214yesmaximal, complemented
6#11011C212yesmaximal, complemented
7#10971C214yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525C2212yes—
10C22×#117C2211yescharacteristic
11C22×#104C2218yes—
12C2×#407C2212yes—
13#2621C412yescomplemented
14#2620C414yescomplemented
15#2619C412yescomplemented
16#2618—22no—
17#2617—22no—
18#2392—24no—
19#2297C2212yes—
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2212yes—
23#1818—24no—
24C2×#339C2218yes—
25#1728C2212yes—
26#1728C2212yes—
27#1441—22no—
28#1439—22no—
29C4×#95C2212yes—
30C22×#41C2312yes—
31C24×C4C2311yescharacteristic
32C22×M16C2314yes—
33#532—44no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×#127C2×C411yescharacteristic
36C2×#124C2×C412yes—
37#525—48no—
38C2×#123—22no—
39C2×C22⋊Q8C2×C414yes—
40C2×#121—22no—
41C2×#121C2×C414yes—
42C2×#120—22no—
43C2×C4×D4—24no—
44C2×C4×D4—22no—
45C2×#117—44no—
46C2×#117C2×C412yes—
47C2×#117C2×C412yes—
48C2×#117D414yes—
49C2×#117C2312yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D412yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42C2314yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#104—48no—
58C2×#104—48no—
59C2×#104—48no—
60#407—44no—
61C2×C22⋊C8—24no—
62#398—24no—
63#397—24no—
64#396—28no—
65#394—24no—
66#394—28no—
67#392—24no—
68#391C2×C414yes—
69#390C2×C414yes—
70#386—24no—
71#384C2×C412yes—
72#382C2×C412yes—
73#380—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—44no—
76C4×C22⋊C4—24no—
77C4×C22⋊C4—44no—
78C2×#95—24no—
79C2×#95—22no—
80C2×#95—22no—
81#339—48no—
82#336—24no—
83#336—24no—
84C25C2×D411yescharacteristic
85C2×#41—28no—
86C2×#41—28no—
87C2×#41—24no—
88C2×#41—24no—
89C22×Q8C22×C412yes—
90C22×D4—22no—
91C22×D4—24no—
92C22×D4—22no—
93C22×D4C22×C412yes—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4C22×C412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2411yescharacteristic
106C2×M16—28no—
107C2×M16—28no—
108C2×M16—24no—
109C2×M16—24no—
110C22×C8—24no—
111C4⋊Q8—42no—
112#127—42no—
113#124—44no—
114#123—48no—
115C22⋊Q8—48no—
116#121—48no—
117#121—44no—
118#121—44no—
119#120—48no—
120C4×D4—48no—
121C4×D4—416no—
122#117—28no—
123#117—44no—
124#117—44no—
125#117—48no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4C22⋊C414yes—
134C2×C4⋊C4C22⋊C414yes—
135C2×C4⋊C4C22×C414yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4C22⋊C414yes—
146C2×C22⋊C4C22⋊C414yes—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—28no—
152C2×C42—28no—
153C2×C42—24no—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159#104—432no—
160C22⋊C8—48no—
161#95—28no—
162#95—24no—
163#95—48no—
164#95—44no—
165#95—44no—
166C24—21no—
167C24—21no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—44no—
172C24C2×C22⋊C412yes—
173C24C22×D411yescharacteristic
174#41—48no—
175#41—48no—
176#41—416no—
177C2×Q8—24no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—44no—
184C2×D4—48no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—84no—
190C2×D4—88no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4H(5,𝔽2)14yes—
228C22×C4H(5,𝔽2)14yes—
229C22×C4H(5,𝔽2)12yes—
230C22×C42−1+412yes—
231M16—48no—
232M16—48no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C4⋊C4—28no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C4⋊C4—416no—
242C4⋊C4—84no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—48no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C22⋊C4—416no—
249C22⋊C4—44no—
250C22⋊C4—88no—
251C42—24no—
252C42—22no—
253C42—22no—
254C42—24no—
255C42—24no—
256C42—44no—
257C42—48no—
258C42—48no—
259C23—24no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—28no—
264C23—44no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—48no—
272C23—82no—
273C23—82no—
274C23—84no—
275C23C22×C22⋊C411yescharacteristic
276C23#53412yes—
277C23#53512yes—
278C23#53511yescharacteristic
279C23#53611yescharacteristic
280Q8—48no—
281D4—48no—
282D4—88no—
283D4—816no—
284D4—84no—
285D4—84no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—48no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4#51812yes—
317C2×C4#51812yes—
318C2×C4#52012yes—
319C2×C4#53412yes—
320C2×C4#53512yes—
321C2×C4#53712yes—
322C2×C4#56018yes—
323C2×C4#56218yes—
324C8—84no—
325C22—22no—
326C22—22no—
327C22—24no—
328C22—22no—
329C22—22no—
330C22—44no—
331C22—44no—
332C22—42no—
333C22—44no—
334C22—84no—
335C22—84no—
336C22—84no—
337C22—84no—
338C22—82no—
339C22—84no—
340C22#230811yescharacteristic
341C22#230911yescharacteristic
342C22#231111yescharacteristic
343C22#281512yes—
344C22#281612yes—
345C22#290114yes—
346C4—44no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—88no—
351C4—84no—
352C4—82no—
353C4#237714yes—
354C4#281314yes—
355C2—22no—
356C2—22no—
357C2—82no—
358C2—162no—
359C2#1465911yescharacteristic
360C2#2177512yes—
361C2#2181712yes—
362C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20683C2semidirect
3. #13852C2semidirect
4. #11370C2semidirect
5. #11075C2semidirect
6. #11011C2semidirect
7. #10971C2semidirect
13. #2621C4semidirect
14. #2620C4semidirect
15. #2619C4semidirect

Maximal subgroups

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

subgroupindex
#346332
#206832
#138522
#113702
#110752
#110112
#109712