Finite Group Databasebeta

← #6498952

Subgroups — #6498952

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 399 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#30304C212yesmaximal, complemented
6#28795C218yesmaximal, complemented
7#28778C214yesmaximal
8#28777C214yesmaximal
9#21984C212yesmaximal
10#20817C211yesmaximal, 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#2083C2218yes—
36#2059C2218yes—
37#2057C2218yes—
38#1993C2218yes—
39#1992C2218yes—
40#1949C2218yes—
41#1944C2218yes—
42#1943C2218yes—
43#1854C2218yes—
44#1845C2212yes—
45#1845C2212yes—
46#1844C2212yes—
47#1835C2214yes—
48#1834C2211yescharacteristic
49#1834C2212yes—
50#1834C2212yes—
51#1834C2212yes—
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1808C2214yes—
56C4×#107C2214yes—
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×#117D412yes—
72C2×#117C2314yes—
73C22×C42C2311yescharacteristic
74#431Q818yescomplemented
75C2×#107C2318yes—
76C2×#107C2318yes—
77C2×#106C2318yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×#97C2314yes—
81C2×#97C2314yes—
82C2×C4×C8Q818yescomplemented
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2312yes—
87#400—24no—
88#399—24no—
89#398—24no—
90#398—24no—
91#396—24no—
92#396—24no—
93#395—24no—
94#393—24no—
95#391—24no—
96#391—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—24no—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#387—24no—
112#387—24no—
113#387—24no—
114#387—24no—
115#384—22no—
116#384—22no—
117#384C2314yes—
118#384C2312yes—
119#384C2312yes—
120#384C2312yes—
121#384C2312yes—
122#382—22no—
123#382—22no—
124#382—22no—
125#382C2314yes—
126#382C2314yes—
127#380—22no—
128#380—22no—
129#380—22no—
130#380C2314yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4C2311yescharacteristic
140C4×C22⋊C4C2311yescharacteristic
141#376D411yescharacteristic
142#376D411yescharacteristic
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—44no—
169#117—48no—
170#117—84no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q814yes—
207C2×C42C2×Q814yes—
208C2×C42C2412yes—
209C2×C42C2412yes—
210C2×C42C2411yescharacteristic
211C2×C42C2411yescharacteristic
212#107—48no—
213#107—48no—
214#106—48no—
215C4⋊C8C2×Q818yes—
216C22⋊C8C2×Q818yes—
217#97—44no—
218#97—44no—
219C4×C8C2×Q814yes—
220C4×C8C2×Q814yes—
221C4×C8C2×Q818yes—
222C4×C8C2×Q818yes—
223#95—24no—
224#95—24no—
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#95C2×D412yes—
234#95C2×D412yes—
235#95C2×D411yescharacteristic
236#95C2×D411yescharacteristic
237#95C2×D411yescharacteristic
238#95C2×D411yescharacteristic
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—84no—
242C2×Q8—84no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4C4⋊Q814yes—
252C22×C4C4⋊Q814yes—
253C22×C4C22×D412yes—
254C22×C4C22×D412yes—
255C22×C4C22×D412yes—
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q812yes—
263C22×C4C22×Q812yes—
264C22×C4C2511yescharacteristic
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—48no—
270C2×C8C22×Q814yes—
271C2×C8C22×Q814yes—
272C2×C8C22×Q818yes—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—84no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—84no—
307C42—84no—
308C42C4⋊Q814yes—
309C42C4⋊Q814yes—
310C42C22×Q812yes—
311C42C22×Q812yes—
312C42C22×Q812yes—
313C42C22×Q812yes—
314C23—22no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23C2×C4⋊Q812yes—
319C23C23×D411yescharacteristic
320Q8—168no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
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—44no—
342C2×C4—44no—
343C2×C4—44no—
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×C4C2×C4⋊Q812yes—
354C2×C4C2×C4⋊Q812yes—
355C2×C4C2×C4⋊Q812yes—
356C2×C4C2×C4⋊Q812yes—
357C2×C4C2×C4⋊Q812yes—
358C2×C4#57514yes—
359C2×C4#57612yes—
360C2×C4#57812yes—
361C2×C4C23×Q811yescharacteristic
362C2×C4C23×Q811yescharacteristic
363C2×C4#58514yes—
364C8—24no—
365C8—24no—
366C8—28no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22C22×C4⋊Q811yescharacteristic
373C22#350012yes—
374C22C2×#57512yes—
375C22C2×#57611yescharacteristic
376C22C2×#57811yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4—84no—
388C4#316214yes—
389C4#316414yes—
390C4#345812yes—
391C4#350412yes—
392C2—42no—
393C2#3533011yescharacteristic
394C2C2×#316211yescharacteristic
395C2C2×#316411yescharacteristic
396C2#6254512yes—
397C2#6254612yes—
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. #30304C2semidirect
6. #28795C2semidirect
74. #431Q8semidirect
82. 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
#303042
#287952
#287782
#287772
#219842
#208172
#207602
#204732