Finite Group Databasebeta

← #331538

Subgroups — #331538

Order 512.

693 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#33289C211yesmaximal, characteristic, complemented
3#33277C211yesmaximal, characteristic
4#33230C211yesmaximal, characteristic
5#30304C211yesmaximal, characteristic, complemented
6#30203C211yesmaximal, characteristic
7#10676C212yesmaximal, complemented
8#10675C212yesmaximal, complemented
9#10672C212yesmaximal
10#10671C212yesmaximal
11#8132C211yesmaximal, characteristic, complemented
12C2×#501—22no—
13C2×#489—22no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2858C2211yescharacteristic
18#2083—22no—
19#2071—22no—
20#2057—22no—
21#2023—22no—
22#1990—22no—
23#1990—22no—
24#1990C2211yescharacteristic
25#1990C2212yes—
26#1990C2211yescharacteristic
27#1959—22no—
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1956C2212yes—
33#1934—22no—
34#1934—22no—
35#1847—22no—
36#1845C2212yes—
37#1845C2212yes—
38#1844C2212yes—
39#1834C2212yes—
40#1834C2212yes—
41#1834C2212yes—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45#1273C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×#127C2311yescharacteristic
55C2×#125—21no—
56C2×#124—21no—
57C2×#124—21no—
58#501—48no—
59#489—48no—
60C2×#107—24no—
61C2×#107—24no—
62C2×#106—24no—
63C2×#103—24no—
64C2×#102—24no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—24no—
68C2×#97D412yes—
69C2×#97D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8D412yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2312yes—
91#384C2311yescharacteristic
92#384C2312yes—
93#384C2311yescharacteristic
94#382—21no—
95C43C2311yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#336D412yes—
103#336D412yes—
104C22×Q8—21no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×C8C2×D414yes—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C22×C8C2×D414yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—44no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4⋊Q8—82no—
121C4⋊Q8—82no—
122#127—22no—
123#127—22no—
124#125—44no—
125#124—44no—
126#124—44no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2411yescharacteristic
160#107—48no—
161#107—84no—
162#106—48no—
163#103—48no—
164#102—44no—
165#102—44no—
166#102—48no—
167#102—84no—
168#97—24no—
169#97D818yes—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—42no—
177C4×C8—42no—
178#95—41no—
179#95—41no—
180C24—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4#12711yescharacteristic
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8C2×D818yes—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42C2×D814yes—
256C42#13714yes—
257C422−1+412yes—
258C422−1+412yes—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23C2×#12711yescharacteristic
267Q8—84no—
268Q8—164no—
269Q8—164no—
270D4—84no—
271D4—82no—
272D4—82no—
273D4—84no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4#49312yes—
298C2×C4#49312yes—
299C2×C4#49712yes—
300C2×C4#49712yes—
301C2×C4#53612yes—
302C2×C4#53611yescharacteristic
303C2×C4#53611yescharacteristic
304C2×C4#53612yes—
305C2×C4C22×D812yes—
306C2×C4C2×#13712yes—
307C2×C4#57512yes—
308C2×C4#57512yes—
309C2×C4#57712yes—
310C2×C4#57712yes—
311C8—28no—
312C8—42no—
313C8—42no—
314C8—44no—
315C22—161no—
316C22—161no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22—161no—
321C22—161no—
322C22#265811yescharacteristic
323C22C2×#49311yescharacteristic
324C22C2×#49711yescharacteristic
325C22#315311yescharacteristic
326C22#315311yescharacteristic
327C22#315411yescharacteristic
328C22#315411yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—161no—
334C4—161no—
335C4—161no—
336C4—162no—
337C4—162no—
338C4#322614yes—
339C4#324012yes—
340C4#324012yes—
341C4#324414yes—
342C2—321no—
343C2#1963111yescharacteristic
344C2#1963111yescharacteristic
345C2#1968511yescharacteristic
346C2#1968611yescharacteristic
347C2#3027611yescharacteristic
348C2#3028611yescharacteristic
349C2#3029011yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33316C2semidirect
2. #33289C2semidirect
5. #30304C2semidirect
7. #10676C2semidirect
8. #10675C2semidirect
11. #8132C2semidirect

Maximal subgroups

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

subgroupindex
#333162
#332892
#332772
#332302
#303042
#302032
#106762
#106752
#106722
#106712
#81322