Finite Group Databasebeta

← #481019

Subgroups — #481019

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39744C211yesmaximal, characteristic
2#17420C212yesmaximal
3#11568C214yesmaximal, complemented
4#11564C212yesmaximal
5#11536C212yesmaximal, complemented
6#11531C211yesmaximal, characteristic
7#11482C212yesmaximal, complemented
8#11477C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×#556C2211yescharacteristic
11#2924—22no—
12#2868C411yescharacteristic
13#2721—24no—
14#2658C412yes—
15#2547C2211yescharacteristic
16#2546C411yescharacteristic
17#2546C2212yes—
18#2545C412yes—
19#2441—22no—
20#2423—22no—
21#2327C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24#1473—24no—
25#1472—22no—
26#1467C2214yes—
27#1465C2212yes—
28#1461C2214yes—
29#1459C2212yes—
30#1324C2212yes—
31#1324C2211yescharacteristic
32#1324C2211yescharacteristic
33#1319C2212yes—
34#1319C2214yes—
35#1319C2214yes—
36#1319C2212yes—
37#557—42no—
38#557—42no—
39#556—42no—
40#556—42no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#126D414yes—
48C2×#125—22no—
49C2×#125—21no—
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C412yes—
52C2×#124C2×C412yes—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2312yes—
59C2×C4×D4C2311yescharacteristic
60C2×#117D412yes—
61#433—44no—
62#432—24no—
63#431—24no—
64#424—42no—
65#407—22no—
66C2×C22⋊C8—22no—
67#405—42no—
68#399—24no—
69#399—24no—
70#398—22no—
71#398—24no—
72#397—22no—
73#396—24no—
74#391—21no—
75#391C2×C412yes—
76#391C2312yes—
77#391C2311yescharacteristic
78#390C2×C411yescharacteristic
79#389—22no—
80#389—21no—
81#389—24no—
82#388—24no—
83#388—22no—
84#387—22no—
85#387—24no—
86#387—24no—
87#384—22no—
88#384—22no—
89#382C2311yescharacteristic
90#380—22no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94#376C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—24no—
98#324C2312yes—
99#324C2312yes—
100#324C2314yes—
101#323—22no—
102#323—22no—
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C412yes—
106C22×Q8C22×C412yes—
107C22×D4C22×C411yescharacteristic
108C23×C4C2×D412yes—
109C2×M16—42no—
110C22×C8—42no—
111C4⋊Q8—24no—
112C4⋊Q8—41no—
113C4⋊Q8—41no—
114C4⋊Q8—42no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—44no—
118C4⋊Q8—82no—
119C4⋊Q8—82no—
120#126—44no—
121#126—44no—
122#125—48no—
123#125—42no—
124#125—42no—
125#124—22no—
126#124—44no—
127#124—41no—
128#124—82no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—44no—
134C4×Q8—44no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×D4—41no—
140C4×D4—41no—
141#117—22no—
142#117—22no—
143#117—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2411yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4C22⋊C414yes—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D414yes—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185#97—44no—
186C4×C8—44no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—24no—
195C24C2×C22⋊C412yes—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—44no—
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231M16—84no—
232C2×C8—42no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—84no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—41no—
242C4⋊C4—48no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4#9912yes—
256C4⋊C4#9912yes—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—84no—
264C42—21no—
265C42—24no—
266C42—24no—
267C42—21no—
268C42—21no—
269C42—24no—
270C42—42no—
271C42—41no—
272C42—42no—
273C42—42no—
274C42—44no—
275C42—41no—
276C42—41no—
277C42#13112yes—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—42no—
283C23C22×C22⋊C411yescharacteristic
284Q8—42no—
285Q8—42no—
286Q8—82no—
287Q8—82no—
288Q8—81no—
289Q8—82no—
290Q8—81no—
291Q8—82no—
292D4—81no—
293D4—81no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4C2×#9914yes—
327C2×C4C2×#9911yescharacteristic
328C2×C4C2×#9911yescharacteristic
329C2×C4C2×#13111yescharacteristic
330C2×C4#56811yescharacteristic
331C8—82no—
332C8—82no—
333C8—84no—
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#288011yescharacteristic
342C22C22×#9911yescharacteristic
343C22#288811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—44no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—82no—
355C4—81no—
356C4—82no—
357C4—81no—
358C4#212912yes—
359C2—21no—
360C2—21no—
361C2—82no—
362C2#1741911yescharacteristic
363C2C2×#212911yescharacteristic
364C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11568C2semidirect
5. #11536C2semidirect
7. #11482C2semidirect

Maximal subgroups

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

subgroupindex
#397442
#174202
#115682
#115642
#115362
#115312
#114822
#114772