Finite Group Databasebeta

← #6499351

Subgroups — #6499351

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33259C214yesmaximal, complemented
4#30312C212yesmaximal, complemented
5#30310C212yesmaximal, complemented
6#29318C214yesmaximal, complemented
7#29315C214yesmaximal, complemented
8#29304C214yesmaximal
9#29300C214yesmaximal
10#21989C212yesmaximal
11#21989C212yesmaximal
12C4×#432C211yesmaximal, characteristic
13C2×#432C2212yes—
14#2864C2212yes—
15#2864C2212yes—
16#2864C2212yes—
17#2863—22no—
18#2862C2211yescharacteristic
19#2859C2212yes—
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2855—22no—
24#2851—22no—
25#2851—22no—
26#2851C2212yes—
27C4×C4⋊Q8C2212yes—
28C4×#125—22no—
29C4×#125—22no—
30C4×#125C2212yes—
31C4×C22⋊Q8—22no—
32#2296C2212yes—
33#2296C2212yes—
34#2295—22no—
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37#2283C2212yes—
38C2×C43C2211yescharacteristic
39#2085C2218yes—
40#2084C2214yes—
41#2083C2214yes—
42#2059C2218yes—
43#2058C2214yes—
44#2057C2214yes—
45#1949C2218yes—
46#1949C2218yes—
47#1944C2218yes—
48#1943C2218yes—
49#1942C2218yes—
50#1941C2218yes—
51#1854C2218yes—
52#1854C2218yes—
53#1847C2214yes—
54#1847C2214yes—
55#1836C2214yes—
56#1836C2214yes—
57#1767C2212yes—
58C4×C4⋊C8C2212yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61#1745C2212yes—
62C4×#97C2212yes—
63C2×C4⋊Q8C2312yes—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125C2312yes—
67C2×C22⋊Q8—22no—
68C2×C4×Q8—22no—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C42D412yes—
72C22×C42D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#432Q8116yescomplemented
77C2×#107C2318yes—
78C2×#107C2318yes—
79C2×#106C2318yes—
80C2×#106C2318yes—
81C2×C4⋊C8C2314yes—
82C2×C22⋊C8C2314yes—
83C2×#97C2314yes—
84C2×#97C2314yes—
85C2×C4×C8C2314yes—
86#391—22no—
87#391—22no—
88#391—22no—
89#391—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389C2314yes—
99#389C2314yes—
100#389C2314yes—
101#389C2314yes—
102#389C2314yes—
103#389C2314yes—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#387—24no—
112#385—24no—
113#385—24no—
114#385—24no—
115#384C2312yes—
116#384C2312yes—
117#384C2312yes—
118#383—22no—
119#382—22no—
120#382—22no—
121#382C2312yes—
122#382C2312yes—
123#382C2312yes—
124#382C2312yes—
125#382C2312yes—
126#381—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—22no—
132#380C2314yes—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4C2314yes—
142C4×C22⋊C4—22no—
143#376—22no—
144C43D412yes—
145C43D412yes—
146C43C2311yescharacteristic
147C43C2311yescharacteristic
148C43C2311yescharacteristic
149C43C2311yescharacteristic
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C22×Q8—22no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8C2418yes—
163C4⋊Q8—44no—
164#125—44no—
165#125—48no—
166#125—48no—
167C22⋊Q8—48no—
168C4×Q8—48no—
169#117—42no—
170#117—44no—
171#117—42no—
172C2×C4⋊C4—22no—
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⋊C4—22no—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—44no—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q818yes—
213C2×C42C2411yescharacteristic
214C2×C42C2411yescharacteristic
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217C2×C42C2411yescharacteristic
218C2×C42C2411yescharacteristic
219#107—48no—
220#107—48no—
221#106—48no—
222#106—48no—
223C4⋊C8C2×Q8116yes—
224C22⋊C8C2×Q8116yes—
225#97—44no—
226#97C2×Q8116yes—
227C4×C8—44no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245C24C22×D411yescharacteristic
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C4⋊Q814yes—
264C22×C4C4⋊Q814yes—
265C22×C4C22×D412yes—
266C22×C4C22×D412yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C22×Q814yes—
275C22×C4C2511yescharacteristic
276C2×C8—28no—
277C2×C8—28no—
278C2×C8C22×Q8116yes—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—48no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42—44no—
317C42—48no—
318C42C4⋊Q814yes—
319C42C4⋊Q814yes—
320C42C22×Q814yes—
321C42C22×Q814yes—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—22no—
326C23C2×C4⋊Q812yes—
327C23C23×D411yescharacteristic
328Q8—88no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—28no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4C2×C4⋊Q812yes—
363C2×C4C2×C4⋊Q812yes—
364C2×C4C2×C4⋊Q812yes—
365C2×C4C2×C4⋊Q812yes—
366C2×C4C2×C4⋊Q812yes—
367C2×C4#57514yes—
368C2×C4#57514yes—
369C2×C4C23×Q812yes—
370C2×C4#58512yes—
371C2×C4#58512yes—
372C8—216no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22C22×C4⋊Q811yescharacteristic
379C22#344911yescharacteristic
380C22#344911yescharacteristic
381C22C2×#57512yes—
382C22C2×#57512yes—
383C4—22no—
384C4—24no—
385C4—24no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—84no—
393C4—84no—
394C4—84no—
395C4—84no—
396C4#350414yes—
397C2—42no—
398C2#3533511yescharacteristic
399C2#6066311yescharacteristic
400C2#6066311yescharacteristic
401C2#6095412yes—
402C2#6096112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33259C2semidirect
4. #30312C2semidirect
5. #30310C2semidirect
6. #29318C2semidirect
7. #29315C2semidirect
76. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#366402
#366352
#332592
#303122
#303102
#293182
#293152
#293042
#293002
#219892
C4×#4322