Finite Group Databasebeta

← #480968

Subgroups — #480968

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39489C211yesmaximal, characteristic
2#17426C212yesmaximal, complemented
3#11574C212yesmaximal, complemented
4#11549C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11514C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8#11448C212yesmaximal, complemented
9C2×#555C411yescharacteristic
10C2×#554C2211yescharacteristic
11#2974—22no—
12#2868C411yescharacteristic
13#2674—24no—
14#2620C412yes—
15#2567C2211yescharacteristic
16#2563C411yescharacteristic
17#2501C2212yes—
18#2499C412yes—
19#2441—22no—
20#2422—22no—
21#2317C2212yes—
22C4×#124C2211yescharacteristic
23#2293C411yescharacteristic
24#1472—24no—
25#1470—22no—
26#1467C2212yes—
27#1463C2214yes—
28#1461C2212yes—
29C2×#324C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1318C2212yes—
36#1318C2212yes—
37#555—42no—
38#555—42no—
39#554—42no—
40#554—42no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125D412yes—
45C2×#124—22no—
46C2×#124C2×C411yescharacteristic
47C2×#123—24no—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8D414yes—
50C2×#121—21no—
51C2×#121C2×C412yes—
52C2×C4×Q8C2311yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×#117—21no—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C4⋊C4C2312yes—
60#436—42no—
61#435—44no—
62C8×D4—42no—
63#432—22no—
64#431—22no—
65#407—24no—
66C2×C22⋊C8—24no—
67#399—24no—
68#397—24no—
69#396—24no—
70#395—22no—
71#394—24no—
72#393—22no—
73#391C2×C412yes—
74#391C2×C411yescharacteristic
75#390—21no—
76#390C2×C412yes—
77#390C2311yescharacteristic
78#389—21no—
79#389—24no—
80#389—22no—
81#388—22no—
82#388—22no—
83#387—24no—
84#385—24no—
85#384—22no—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2312yes—
89#382C2×C412yes—
90#382C2311yescharacteristic
91#380—22no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—22no—
98#324—24no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103C22×Q8—21no—
104C22×Q8C22⋊C412yes—
105C22×Q8C22×C411yescharacteristic
106C22×D4—22no—
107C22×D4C22⋊C411yescharacteristic
108C22×D4C22×C411yescharacteristic
109C23×C4—21no—
110C23×C4C22×C412yes—
111C23×C4C2×D414yes—
112C2×M16—44no—
113C22×C8—44no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—84no—
119#125—42no—
120#125—42no—
121#124—22no—
122#124—41no—
123#124—48no—
124#123—88no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128#121—44no—
129#121—42no—
130#121—42no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—24no—
137C4×D4—22no—
138C4×D4—41no—
139C4×D4—41no—
140#117—24no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—44no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2411yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C414yes—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D412yes—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C22⋊C8—22no—
182C22⋊C8—24no—
183C22⋊C8—22no—
184C22⋊C8—44no—
185C22⋊C8—44no—
186#97—42no—
187C4×C8—42no—
188#95—22no—
189#95—22no—
190#95—21no—
191#95—24no—
192#95—24no—
193#95—22no—
194C24—21no—
195C24C2×C22⋊C412yes—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—44no—
210C2×D4—84no—
211C2×D4#9912yes—
212C2×D4#9912yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—84no—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C414yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D412yes—
235M16—164no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—44no—
239C2×C8—164no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—48no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C42—21no—
270C42—22no—
271C42—22no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—22no—
276C42—41no—
277C42—42no—
278C42—44no—
279C42—42no—
280C42#13112yes—
281C23—22no—
282C23—22no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×#9914yes—
291C23C22×C22⋊C411yescharacteristic
292Q8—81no—
293Q8—81no—
294Q8—162no—
295Q8—162no—
296D4—24no—
297D4—42no—
298D4—81no—
299D4—81no—
300D4—164no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4C2×#9911yescharacteristic
336C2×C4C2×#9911yescharacteristic
337C2×C4C2×#13111yescharacteristic
338C2×C4#56811yescharacteristic
339C8—82no—
340C8—84no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22#288011yescharacteristic
355C22C22×#9911yescharacteristic
356C22#288911yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—44no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4—81no—
370C4#213312yes—
371C2—21no—
372C2—21no—
373C2—44no—
374C2—161no—
375C2#1741611yescharacteristic
376C2#3335911yescharacteristic
377C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#554 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17426C2semidirect
3. #11574C2semidirect
4. #11549C2semidirect
5. #11530C2semidirect
6. #11514C2semidirect
7. #11476C2semidirect
8. #11448C2semidirect

Maximal subgroups

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

subgroupindex
#394892
#174262
#115742
#115492
#115302
#115142
#114762
#114482