Finite Group Databasebeta

← #6482323

Subgroups — #6482323

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#33159C214yesmaximal
4#29975C214yesmaximal
5#26525C214yesmaximal, complemented
6#26523C214yesmaximal, complemented
7#26458C214yesmaximal, complemented
8#26456C214yesmaximal, complemented
9#21956C211yesmaximal, characteristic
10#21838C212yesmaximal
11#21364C212yesmaximal
12C22×#125C2211yescharacteristic
13C2×#432C2212yes—
14#2844—22no—
15#2628—24no—
16#2628C2214yes—
17#2571—21no—
18#2571C2214yes—
19#2551—21no—
20#2551C2214yes—
21#2370C2214yes—
22#2370C2214yes—
23#2345—24no—
24#2330—21no—
25#2330—21no—
26#2316C2212yes—
27C4×#125—22no—
28#2285C2212yes—
29#2285C2211yescharacteristic
30#2274C2212yes—
31C4×#117C2211yescharacteristic
32#2101C2214yes—
33#2101C2214yes—
34#2088C2214yes—
35#2088C2214yes—
36#2087C2214yes—
37#2086C2214yes—
38#2058C2214yes—
39#2057C2214yes—
40#1992C2214yes—
41#1992C2214yes—
42#1989C2214yes—
43#1989C2214yes—
44#1988C2214yes—
45#1987C2214yes—
46#1949C2214yes—
47#1949C2214yes—
48#1948C2212yes—
49#1948C2212yes—
50#1947C2212yes—
51#1947C2212yes—
52#1940C2214yes—
53#1940C2214yes—
54#1921C2214yes—
55#1920C2214yes—
56#1919C2214yes—
57#1918C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1824C2214yes—
61#1823C2214yes—
62#1798C2214yes—
63#1776C2214yes—
64#1776C2214yes—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8C2314yes—
67C2×#125—21no—
68C2×#125—21no—
69C2×#125—24no—
70C2×#125Q818yescomplemented
71C2×#125C2314yes—
72C2×#117—22no—
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2312yes—
78C22×C42C2311yescharacteristic
79#432—28no—
80C2×#107C2314yes—
81C2×#107C2314yes—
82C2×#106C2314yes—
83C2×#106C2314yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2314yes—
87C2×C4⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89C2×#97C2314yes—
90#400—24no—
91#400—24no—
92#400—24no—
93#399—24no—
94#395—24no—
95#395—24no—
96#395—24no—
97#393—24no—
98#389—22no—
99#389—22no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389C2314yes—
105#389C2314yes—
106#389C2314yes—
107#389C2314yes—
108#387—24no—
109#387—24no—
110#387—24no—
111#387—24no—
112#384D411yescharacteristic
113#384D411yescharacteristic
114#384C2312yes—
115#384C2312yes—
116#382—24no—
117#382—22no—
118#382D412yes—
119#382C2314yes—
120#380—22no—
121#380—24no—
122#380D414yes—
123#380C2314yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2314yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2312yes—
134C4×C4⋊C4C2314yes—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—24no—
137#376—21no—
138C43—21no—
139#340C2314yes—
140#340C2314yes—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#340C2314yes—
147#340C2314yes—
148#340C2314yes—
149#340C2314yes—
150#340C2314yes—
151#336C2314yes—
152#336C2314yes—
153C22×Q8—24no—
154C23×C4—22no—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8C2414yes—
158C22×C8C2414yes—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161C4⋊Q8—88no—
162#125—28no—
163#125—28no—
164#125—28no—
165#125C2×Q8116yes—
166#117—24no—
167#117—42no—
168#117—42no—
169#117—42no—
170#117—84no—
171C2×C4⋊C4—216no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×Q818yes—
190C2×C4⋊C4C2411yescharacteristic
191C2×C4⋊C4C2411yescharacteristic
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2412yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4C2×D414yes—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q814yes—
210C2×C42#4114yes—
211C2×C42C2414yes—
212C2×C42C2412yes—
213#107—28no—
214#107—44no—
215#106—28no—
216#106—44no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—44no—
220C4⋊C8—44no—
221C22⋊C8—28no—
222#97—28no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95C2×D414yes—
236#95C2×D414yes—
237C24C22×D411yescharacteristic
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C22⋊Q818yes—
255C22×C4C22×D414yes—
256C22×C4C22×D414yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
260C22×C4C22×Q812yes—
261C22×C4C2×#4112yes—
262C22×C4C2511yescharacteristic
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C4⋊C4—216no—
270C4⋊C4—216no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—84no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4C22×Q818yes—
294C4⋊C42−1+414yes—
295C22⋊C4—24no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—84no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—42no—
303C42—48no—
304C42—82no—
305C42—82no—
306C42—84no—
307C42—84no—
308C42C22⋊Q814yes—
309C42C22⋊Q814yes—
310C42C22×Q814yes—
311C42C2×#4114yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×C22⋊Q812yes—
317C23C23×D411yescharacteristic
318Q8—168no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—88no—
351C2×C4C2×C22⋊Q812yes—
352C2×C4C2×C22⋊Q814yes—
353C2×C4C2×C22⋊Q814yes—
354C2×C4C23×Q811yescharacteristic
355C2×C4C22×#4111yescharacteristic
356C2×C4C2×2−1+412yes—
357C2×C4#58512yes—
358C8—44no—
359C8—44no—
360C22—24no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—42no—
365C22C22×C22⋊Q811yescharacteristic
366C22#348511yescharacteristic
367C22#348611yescharacteristic
368C22#358512yes—
369C22#358612yes—
370C4—24no—
371C4—24no—
372C4—22no—
373C4—48no—
374C4—44no—
375C4—44no—
376C4—44no—
377C4—81no—
378C4—84no—
379C4—81no—
380C4—84no—
381C4#346112yes—
382C4#350212yes—
383C2—42no—
384C2#3515411yescharacteristic
385C2#6049312yes—
386C2#6179711yescharacteristic
387C2#6179811yescharacteristic
388C2#6187812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26525C2semidirect
6. #26523C2semidirect
7. #26458C2semidirect
8. #26456C2semidirect
70. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#342172
#331592
#299752
#265252
#265232
#264582
#264562
#219562
#218382
#213642