Finite Group Databasebeta

← #6498951

Subgroups — #6498951

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C212yesmaximal
2#33297C212yesmaximal, complemented
3#33263C212yesmaximal, complemented
4#30307C212yesmaximal, complemented
5#30305C212yesmaximal, complemented
6#28796C218yesmaximal, complemented
7#28780C214yesmaximal
8#28779C214yesmaximal
9#21984C212yesmaximal
10#20816C211yesmaximal, characteristic
11#20760C211yesmaximal, characteristic
12#20473C211yesmaximal, characteristic
13C2×#431C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2868C2212yes—
16#2859C2212yes—
17#2859C2212yes—
18#2859C2212yes—
19#2629—22no—
20#2628—22no—
21#2570—22no—
22#2570C2214yes—
23#2550—22no—
24#2550C2214yes—
25#2441—22no—
26#2441C2214yes—
27#2439—22no—
28#2439C2214yes—
29#2331—22no—
30#2330—22no—
31#2285C2212yes—
32#2285C2212yes—
33C4×#117C2211yescharacteristic
34#2085C2218yes—
35#2084C2218yes—
36#2059C2218yes—
37#2058C2218yes—
38#1993C2218yes—
39#1992C2218yes—
40#1949C2218yes—
41#1944C2218yes—
42#1943C2218yes—
43#1854C2218yes—
44#1845C2212yes—
45#1844C2212yes—
46#1844C2212yes—
47#1835C2214yes—
48#1834C2211yescharacteristic
49#1834C2212yes—
50#1834C2212yes—
51#1834C2212yes—
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1807C2214yes—
56C4×#106C2214yes—
57#1770C2214yes—
58#1752C2212yes—
59#1752C2212yes—
60#1744C2212yes—
61#1735C2214yes—
62C4×#97C2212yes—
63C23×C8C2312yes—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8C2314yes—
67C2×#125—22no—
68C2×#125—22no—
69C2×#125C2314yes—
70C2×#117—22no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2314yes—
74C22×C42C2311yescharacteristic
75#431Q818yescomplemented
76C2×#107C2318yes—
77C2×#106C2318yes—
78C2×#106C2318yes—
79C2×C4⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×#97C2314yes—
82C2×#97C2314yes—
83C2×C4×C8Q818yescomplemented
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2312yes—
87C2×C4×C8C2312yes—
88#400—24no—
89#399—24no—
90#398—24no—
91#398—24no—
92#396—24no—
93#396—24no—
94#395—24no—
95#393—24no—
96#391—24no—
97#391—24no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
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#384—22no—
117#384—22no—
118#384C2314yes—
119#384C2312yes—
120#384C2312yes—
121#384C2312yes—
122#384C2312yes—
123#382—22no—
124#382—22no—
125#382—22no—
126#382C2314yes—
127#382C2314yes—
128#380—22no—
129#380—22no—
130#380—22no—
131#380C2314yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4D411yescharacteristic
139C4×C22⋊C4D411yescharacteristic
140C4×C22⋊C4C2311yescharacteristic
141C4×C22⋊C4C2311yescharacteristic
142#376D412yes—
143C43C2312yes—
144#340C2314yes—
145#340C2318yes—
146#340C2314yes—
147#340C2318yes—
148#340C2314yes—
149#340C2314yes—
150#340C2318yes—
151#336C2314yes—
152#336C2314yes—
153C22×Q8—24no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—44no—
158C22×C8C2×Q818yes—
159C22×C8C2412yes—
160C22×C8C2412yes—
161C22×C8C2414yes—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164C4⋊Q8—84no—
165#125—48no—
166#125—48no—
167#125—84no—
168#117—42no—
169#117—42no—
170#117—48no—
171#117—84no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4C2×D411yescharacteristic
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C22⋊C4C2×D411yescharacteristic
200C2×C22⋊C4C2×D411yescharacteristic
201C2×C42—24no—
202C2×C42—24no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q814yes—
210C2×C42C2×Q814yes—
211C2×C42C2412yes—
212C2×C42C2412yes—
213C2×C42C2411yescharacteristic
214C2×C42C2411yescharacteristic
215#107—48no—
216#106—48no—
217#106—48no—
218C4⋊C8C2×Q818yes—
219C22⋊C8C2×Q818yes—
220#97—44no—
221#97—44no—
222C4×C8C2×Q814yes—
223C4×C8C2×Q814yes—
224C4×C8C2×Q818yes—
225C4×C8C2×Q818yes—
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#95—24no—
236#95C2×D412yes—
237#95C2×D412yes—
238#95C2×D412yes—
239#95C2×D412yes—
240C24C22×D411yescharacteristic
241C2×Q8—48no—
242C2×Q8—84no—
243C2×Q8—84no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4C4⋊Q814yes—
253C22×C4C4⋊Q814yes—
254C22×C4C22×D412yes—
255C22×C4C22×D412yes—
256C22×C4C22×D412yes—
257C22×C4C22×D412yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q812yes—
264C22×C4C22×Q812yes—
265C22×C4C2511yescharacteristic
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—48no—
271C2×C8C22×Q814yes—
272C2×C8C22×Q814yes—
273C2×C8C22×Q818yes—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—84no—
310C42—84no—
311C42C4⋊Q814yes—
312C42C4⋊Q814yes—
313C42C22×Q812yes—
314C42C22×Q812yes—
315C42C22×Q812yes—
316C42C22×Q812yes—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23C2×C4⋊Q812yes—
322C23C23×D411yescharacteristic
323Q8—168no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
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—44no—
356C2×C4C2×C4⋊Q812yes—
357C2×C4C2×C4⋊Q812yes—
358C2×C4C2×C4⋊Q812yes—
359C2×C4C2×C4⋊Q812yes—
360C2×C4C2×C4⋊Q812yes—
361C2×C4#57512yes—
362C2×C4#57512yes—
363C2×C4#57714yes—
364C2×C4C23×Q811yescharacteristic
365C2×C4C23×Q811yescharacteristic
366C2×C4#58514yes—
367C8—24no—
368C8—24no—
369C8—28no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22C22×C4⋊Q811yescharacteristic
376C22#350012yes—
377C22C2×#57511yescharacteristic
378C22C2×#57511yescharacteristic
379C22C2×#57712yes—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—84no—
388C4—84no—
389C4—84no—
390C4—84no—
391C4#316214yes—
392C4#316314yes—
393C4#345812yes—
394C4#350412yes—
395C2—42no—
396C2#3533011yescharacteristic
397C2C2×#316211yescharacteristic
398C2C2×#316311yescharacteristic
399C2#6254312yes—
400C2#6254412yes—
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
2. #33297C2semidirect
3. #33263C2semidirect
4. #30307C2semidirect
5. #30305C2semidirect
6. #28796C2semidirect
75. #431Q8semidirect
83. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#332972
#332632
#303072
#303052
#287962
#287802
#287792
#219842
#208162
#207602
#204732