Finite Group Databasebeta

← #387101

Subgroups — #387101

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2#13923C212yesmaximal
3#11381C212yesmaximal
4#11203C214yesmaximal
5#10912C214yesmaximal
6#10801C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9#2502—24no—
10#2502—24no—
11C2×#398C414yescomplemented
12C2×#391C412yescomplemented
13C2×#385C2214yes—
14C2×#384C412yescomplemented
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×#342C2214yes—
19C2×#342C2212yes—
20C2×#342C2212yes—
21C2×#342C2214yes—
22C2×#342C2212yes—
23C2×#342C2212yes—
24#1441—24no—
25#1436—24no—
26C22⋊#95C2214yes—
27C22⋊#95C2214yes—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30C24×C4C2312yes—
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—41no—
34C2×C22⋊Q8—41no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C418yes—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4D414yes—
46C22×C22⋊C4C2314yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2312yes—
50C22×C42—22no—
51C22×C42C2×C412yes—
52#398—24no—
53#398—28no—
54#398—24no—
55#391—22no—
56#391—22no—
57#391—24no—
58#385—24no—
59#385—28no—
60#385—24no—
61#384—24no—
62#384—22no—
63#384—22no—
64#380—42no—
65#380—42no—
66C4×C22⋊C4—24no—
67C4×C22⋊C4—42no—
68C4×C22⋊C4—42no—
69C2×#95—28no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—28no—
73C2×#95C2×C412yes—
74C2×#95C2×C418yes—
75#342—28no—
76#342—24no—
77#342—42no—
78#342—44no—
79#342—42no—
80#342Q818yescomplemented
81C25C2411yescharacteristic
82C22×Q8—22no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—44no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—44no—
96C23×C4—84no—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C414yes—
102C23×C4C22×C414yes—
103C23×C4C2×D414yes—
104C23×C4C2×D412yes—
105C23×C4C2×D414yes—
106C23×C4C2×D412yes—
107C22⋊Q8—48no—
108C22⋊Q8—48no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4—88no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C414yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4C2×Q818yes—
152C2×C22⋊C4C2×Q814yes—
153C2×C22⋊C4#4114yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42—84no—
159#95—24no—
160#95—28no—
161#95—28no—
162#95—416no—
163#95—44no—
164#95—44no—
165#95—88no—
166C24—42no—
167C24—42no—
168C24C23×C411yescharacteristic
169C24C22×D412yes—
170C24C22×Q812yes—
171C24C2×#4112yes—
172C24H(5,𝔽2)12yes—
173C24H(5,𝔽2)11yescharacteristic
174C24H(5,𝔽2)11yescharacteristic
175C242−1+414yes—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—84no—
180C2×Q8—84no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—48no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C4×Q818yes—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—88no—
246C4⋊C4—88no—
247C4⋊C4—88no—
248C22⋊C4—28no—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C42—44no—
261C42—44no—
262C42—88no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—24no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—44no—
274C23—44no—
275C23—48no—
276C23C22×C22⋊C411yescharacteristic
277C23C2×C4×Q812yes—
278C23#51811yescharacteristic
279C23#51811yescharacteristic
280C23#51912yes—
281C23#53412yes—
282C23#53412yes—
283C23#54214yes—
284C23#54414yes—
285C23#54814yes—
286C23D4×Q814yes—
287C23#55412yes—
288C23#55511yescharacteristic
289C23Q8211yescharacteristic
290C23#56312yes—
291C23#56312yes—
292Q8—88no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—416no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—88no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—88no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#41014yes—
339C22—24no—
340C22—24no—
341C22—28no—
342C22—28no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—44no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22#230811yescharacteristic
359C22#233112yes—
360C22Q8×C22⋊C412yes—
361C22#236911yescharacteristic
362C22#238111yescharacteristic
363C22#272914yes—
364C22#273112yes—
365C22#273412yes—
366C22C2×#41012yes—
367C22#288312yes—
368C4—24no—
369C4—48no—
370C4—42no—
371C4—44no—
372C4—44no—
373C4—88no—
374C4—82no—
375C4—82no—
376C4—164no—
377C2—24no—
378C2—22no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—22no—
383C2#1445011yescharacteristic
384C2#2174711yescharacteristic
385C2#2175011yescharacteristic
386C2#2251314yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C2×#398C4semidirect
12. C2×#391C4semidirect
14. C2×#384C4semidirect
80. #342Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#139232
#113812
#112032
#109122
#108012