Finite Group Databasebeta

← #6484840

Subgroups — #6484840

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35345C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#33158C214yesmaximal
4#29978C214yesmaximal
5#29295C212yesmaximal, complemented
6#29294C212yesmaximal, complemented
7#29285C214yesmaximal, complemented
8#29221C212yesmaximal, complemented
9#29210C212yesmaximal, complemented
10#29180C214yesmaximal, complemented
11#21985C212yesmaximal
12#21836C212yesmaximal
13#20809C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C2×#431C2212yes—
16#2864C2212yes—
17#2859C2211yescharacteristic
18#2859C2211yescharacteristic
19#2629C2212yes—
20#2629C2212yes—
21#2608—24no—
22#2606—24no—
23#2370—24no—
24#2345—24no—
25#2330C2212yes—
26#2330C2212yes—
27#2330C2212yes—
28#2330C2212yes—
29#2316C2212yes—
30C4×#125C2212yes—
31C4×#125C2212yes—
32#2285C2211yescharacteristic
33#2283C2212yes—
34#2274C2212yes—
35C4×#117C2211yescharacteristic
36#2100C2214yes—
37#2099C2214yes—
38#2088C2214yes—
39#2088C2214yes—
40#2087C2214yes—
41#2086C2214yes—
42#2059C2214yes—
43#2059C2214yes—
44#1991C2214yes—
45#1991C2214yes—
46#1989C2214yes—
47#1989C2214yes—
48#1989C2214yes—
49#1989C2214yes—
50#1948C2214yes—
51#1948C2214yes—
52#1947C2214yes—
53#1947C2214yes—
54#1921C2214yes—
55#1920C2214yes—
56#1919C2214yes—
57#1918C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1846C2214yes—
61#1843C2211yescharacteristic
62#1843C2211yescharacteristic
63#1835C2214yes—
64#1832C2212yes—
65#1824C2214yes—
66#1823C2214yes—
67#1798C2212yes—
68#1798C2212yes—
69C4×C4⋊C8C2212yes—
70C4×C4⋊C8C2212yes—
71C2×#125—22no—
72C2×#125—24no—
73C2×#125Q818yes—
74C2×#125C2312yes—
75C2×#125C2312yes—
76C2×#125C2312yes—
77C2×#125C2312yes—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2312yes—
83C22×C42C2311yescharacteristic
84#431—28no—
85C2×#107C2314yes—
86C2×#107C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8C2314yes—
97#398—24no—
98#398—24no—
99#396—24no—
100#396—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—24no—
106#389—24no—
107#389—24no—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#389C2314yes—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#384C2312yes—
121#384C2312yes—
122#384C2312yes—
123#383—22no—
124#382C2312yes—
125#382C2312yes—
126#382C2312yes—
127#381—22no—
128#380—22no—
129#380—24no—
130#380C2314yes—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D414yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4D414yes—
142#376—22no—
143#376C2311yescharacteristic
144#376C2311yescharacteristic
145C43C2311yescharacteristic
146C43C2311yescharacteristic
147#340C2314yes—
148#340C2314yes—
149#340C2314yes—
150#340C2314yes—
151#340C2314yes—
152#340C2314yes—
153#340C2314yes—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#336C2314yes—
160#336C2314yes—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2411yescharacteristic
164C22×C8C2414yes—
165C22×C8C2414yes—
166#125—216no—
167#125—24no—
168#125—24no—
169#125—24no—
170#125—24no—
171#125—42no—
172#125—42no—
173#125C2×Q818yes—
174#125C2×Q818yes—
175#117—22no—
176#117—22no—
177#117—41no—
178#117—41no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—216no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q818yes—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4C2×D412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42C2×D412yes—
213C2×C42C2×Q814yes—
214C2×C42#4114yes—
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217C2×C42C2411yescharacteristic
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—44no—
222#107—44no—
223#106—44no—
224#106—44no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C22⋊C8—28no—
232C4×C8—28no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95C2×D414yes—
247#95C2×D414yes—
248C24C22×D411yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—44no—
259C22×C4—84no—
260C22×C4C22⋊Q818yes—
261C22×C4C22×D412yes—
262C22×C4C22×D412yes—
263C22×C4C22×D412yes—
264C22×C4C22×D412yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D412yes—
267C22×C4C22×D412yes—
268C22×C4C22×Q812yes—
269C22×C4C2×#4112yes—
270C22×C4C2511yescharacteristic
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C4⋊C4—216no—
278C4⋊C4—216no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4C22×Q818yes—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—84no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—44no—
310C42—44no—
311C42—44no—
312C42—84no—
313C42—84no—
314C42—84no—
315C42C22⋊Q818yes—
316C42C22×Q812yes—
317C42C22×Q812yes—
318C42C2×#4112yes—
319C42C2×#4112yes—
320C422−1+412yes—
321C422−1+412yes—
322C23—24no—
323C23—21no—
324C23—21no—
325C23C2×C22⋊Q812yes—
326C23C23×D411yescharacteristic
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—88no—
356C2×C4C2×C22⋊Q812yes—
357C2×C4C2×C22⋊Q814yes—
358C2×C4C2×C22⋊Q814yes—
359C2×C4#53614yes—
360C2×C4C23×Q811yescharacteristic
361C2×C4C22×#4111yescharacteristic
362C2×C4C2×2−1+411yescharacteristic
363C2×C4C2×2−1+411yescharacteristic
364C2×C4#58511yescharacteristic
365C2×C4#58511yescharacteristic
366C8—44no—
367C8—44no—
368C22—24no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22C22×C22⋊Q811yescharacteristic
373C22C2×#53611yescharacteristic
374C22#344911yescharacteristic
375C22#358414yes—
376C4—24no—
377C4—24no—
378C4—22no—
379C4—24no—
380C4—48no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4#350211yescharacteristic
388C4#350211yescharacteristic
389C4#350711yescharacteristic
390C4#350711yescharacteristic
391C2—42no—
392C2#3517911yescharacteristic
393C2#6048912yes—
394C2#6088212yes—
395C2#6100912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29295C2semidirect
6. #29294C2semidirect
7. #29285C2semidirect
8. #29221C2semidirect
9. #29210C2semidirect
10. #29180C2semidirect

Maximal subgroups

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

subgroupindex
#353452
#342172
#331582
#299782
#292952
#292942
#292852
#292212
#292102
#291802
#219852
#218362
#208092