Finite Group Databasebeta

← #480695

Subgroups — #480695

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35619C211yesmaximal, characteristic
2#17372C212yesmaximal
3#11556C214yesmaximal, complemented
4#11552C214yesmaximal, complemented
5#11513C212yesmaximal, complemented
6#11510C212yesmaximal, complemented
7C2×#536C411yescharacteristic
8C2×#520C2211yescharacteristic
9#2983—22no—
10#2503C412yes—
11#2502C412yes—
12#2499C2212yes—
13#2499C2212yes—
14#2447—22no—
15#2446—22no—
16#2419—22no—
17#2419—22no—
18#2317C411yescharacteristic
19#2310C2211yescharacteristic
20#2297C411yescharacteristic
21C2×#391C411yescharacteristic
22#2285C2211yescharacteristic
23#1466—22no—
24#1466—22no—
25#1464C2214yes—
26#1464C2214yes—
27#1460—22no—
28#1458C2214yes—
29#1325C2214yes—
30#1324C2214yes—
31#1319C2218yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34#536—44no—
35C2×C4⋊Q8—21no—
36C2×#124—21no—
37C2×#123—22no—
38C2×C22⋊Q8C2×C412yes—
39C2×C22⋊Q8C2×C412yes—
40#520—44no—
41C2×C4×Q8D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2311yescharacteristic
48C2×#117C2312yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C42C2×C411yescharacteristic
52#446—48no—
53#432—24no—
54#432—24no—
55#431—24no—
56#400—22no—
57#400—22no—
58#398—22no—
59#398—22no—
60#397—22no—
61#397—22no—
62#397—22no—
63#397—22no—
64#391—21no—
65#391—22no—
66#391—22no—
67#391C2×C412yes—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2312yes—
71#390—21no—
72#389—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#385—21no—
79#385—21no—
80#385—22no—
81#385—22no—
82#384C2×C412yes—
83#384C2×C411yescharacteristic
84#384C2311yescharacteristic
85#382C2×C412yes—
86#382C2×C411yescharacteristic
87#382C2311yescharacteristic
88#381—22no—
89#381—22no—
90#381—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C412yes—
95C2×#95—21no—
96C2×#95—21no—
97#324C2318yes—
98#323—24no—
99#323—24no—
100#323—24no—
101C2×#41—24no—
102C2×#41—42no—
103C2×#41—42no—
104C22×Q8—21no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8C22⋊C411yescharacteristic
109C22×Q8C22×C412yes—
110C22×D4C22⋊C412yes—
111C22×D4C22⋊C411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C412yes—
120C4⋊Q8—44no—
121#124—44no—
122#123—48no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C4×Q8—28no—
126C4×D4—44no—
127C4×D4—44no—
128C4×D4—44no—
129#117—44no—
130#117—42no—
131#117—42no—
132#117—42no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C2411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4C2×D412yes—
157C2×C22⋊C4C2×D412yes—
158C2×C22⋊C4C2×D412yes—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—42no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C4⋊C8—44no—
169C4⋊C8—44no—
170C4⋊C8—44no—
171C22⋊C8—28no—
172#97—44no—
173#97—44no—
174C4×C8—44no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C24C2×C22⋊C412yes—
190C24C2×C22⋊C411yescharacteristic
191#41—416no—
192C2×Q8—24no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—88no—
201C2×Q8#9914yes—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—41no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241C2×C8—48no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—82no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—22no—
270C42—22no—
271C42—42no—
272C42—42no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23#41312yes—
284C23C22×C22⋊C411yescharacteristic
285Q8—44no—
286Q8—84no—
287Q8—84no—
288D4—48no—
289D4—44no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—48no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—161no—
325C2×C4—161no—
326C2×C4—162no—
327C2×C4—162no—
328C2×C4C2×#9912yes—
329C2×C4C2×#9914yes—
330C2×C4#56812yes—
331C8—88no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22#287911yescharacteristic
345C22C22×#9911yescharacteristic
346C22C2×#41311yescharacteristic
347C4—22no—
348C4—22no—
349C4—24no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—82no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—82no—
364C2#1739211yescharacteristic
365C2#3336011yescharacteristic
366C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11556C2semidirect
4. #11552C2semidirect
5. #11513C2semidirect
6. #11510C2semidirect

Maximal subgroups

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

subgroupindex
#356192
#173722
#115562
#115522
#115132
#115102