Finite Group Databasebeta

← #496412

Subgroups — #496412

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35532C211yesmaximal, characteristic
2#20731C212yesmaximal
3#13303C214yesmaximal, complemented
4#13302C214yesmaximal, complemented
5#12863C212yesmaximal
6#12861C212yesmaximal
7C2×#544C411yescharacteristic
8C2×#520C2211yescharacteristic
9#2921—22no—
10#2567C2212yes—
11#2566C412yes—
12#2565C2212yes—
13#2542C412yes—
14#2449—22no—
15#2449—22no—
16#2441—22no—
17#2440—22no—
18#2317C411yescharacteristic
19#2311C2211yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic
22C2×#384C411yescharacteristic
23#1821—22no—
24#1801C2214yes—
25#1800—22no—
26#1785C2214yes—
27C4×#101C2214yes—
28C4×#100—22no—
29#1394C2214yes—
30#1394C2214yes—
31#1394C2218yes—
32C22×#41C2311yescharacteristic
33#544—44no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×#125—21no—
37C2×#125C2×C412yes—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41#520—44no—
42C2×C4×Q8C2312yes—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C42C2×C411yescharacteristic
56#433—48no—
57#423—24no—
58C2×#101C2318yes—
59C2×#100—24no—
60#407—24no—
61C4×M16—24no—
62#399—22no—
63#399—22no—
64#398—22no—
65#398—22no—
66#396—22no—
67#396—22no—
68#395—22no—
69#395—22no—
70#391C2311yescharacteristic
71#391C2312yes—
72#391C2311yescharacteristic
73#390C2×C411yescharacteristic
74#390C2×C412yes—
75#390C2×C411yescharacteristic
76#389—22no—
77#389—21no—
78#389—21no—
79#389—22no—
80#389—22no—
81#387—22no—
82#387—22no—
83#386—22no—
84#386—22no—
85#385—22no—
86#385—22no—
87#385—21no—
88#385—21no—
89#385—22no—
90#384—22no—
91#384C2×C412yes—
92#384C2×C412yes—
93#384C2×C412yes—
94#382C2×C412yes—
95#380—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C22⋊C4—22no—
99#343—24no—
100#343—24no—
101C2×#41—24no—
102C2×#41—42no—
103C2×#41—42no—
104C22×Q8—22no—
105C22×Q8C2411yescharacteristic
106C22×D4C22×C411yescharacteristic
107C22×D4C2×D412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C2×M16—28no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119#125—44no—
120#125—44no—
121C22⋊Q8—48no—
122C22⋊Q8—84no—
123C22⋊Q8—84no—
124C4×Q8—44no—
125C4×D4—44no—
126C4×D4—44no—
127C4×D4—44no—
128#117—24no—
129#117—28no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—42no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C4⋊C8—48no—
178#101—216no—
179#100—48no—
180C22⋊C8—48no—
181#97—44no—
182C4×C8—44no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191C24C2×C22⋊C412yes—
192C24C2×C22⋊C411yescharacteristic
193#41—416no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—44no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q82−1+414yes—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—24no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—41no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D412yes—
239M16—48no—
240M16—48no—
241C2×C8—48no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4#9914yes—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—41no—
277C42—41no—
278C42—42no—
279C42—42no—
280C42—42no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23#41012yes—
291C23C22×C22⋊C411yescharacteristic
292Q8—44no—
293Q8—164no—
294D4—48no—
295D4—44no—
296C2×C4—28no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—48no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—161no—
329C2×C4—161no—
330C2×C4—162no—
331C2×C4—162no—
332C2×C4—162no—
333C2×C4C2×#9914yes—
334C2×C4C2×#9912yes—
335C2×C4#51912yes—
336C2×C4#53614yes—
337C8—88no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—82no—
350C22#231011yescharacteristic
351C22C22×#9911yescharacteristic
352C22C2×#41011yescharacteristic
353C4—24no—
354C4—22no—
355C4—22no—
356C4—24no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—81no—
364C4—82no—
365C4—81no—
366C2—21no—
367C2—21no—
368C2—42no—
369C2—82no—
370C2#2182511yescharacteristic
371C2#3337211yescharacteristic
372C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13303C2semidirect
4. #13302C2semidirect

Maximal subgroups

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

subgroupindex
#355322
#207312
#133032
#133022
#128632
#128612