Finite Group Databasebeta

← #331457

Subgroups — #331457

Order 512.

727 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33294C211yesmaximal, characteristic, complemented
2#33252C211yesmaximal, characteristic
3#30280C211yesmaximal, characteristic, complemented
4#30263C211yesmaximal, characteristic
5#30229C211yesmaximal, characteristic, complemented
6#30219C211yesmaximal, characteristic
7#10317C212yesmaximal, complemented
8#10315C212yesmaximal
9#10313C212yesmaximal, complemented
10#10310C212yesmaximal
11#8134C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#498C2212yes—
14C2×#492C2212yes—
15C2×#492C2211yescharacteristic
16C2×#487C2212yes—
17C2×#487C2212yes—
18C2×#487C2211yescharacteristic
19C2×#486C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2101—24no—
25#2096—24no—
26#2042—24no—
27#2013—24no—
28#1992—22no—
29#1990C2212yes—
30#1990C2212yes—
31#1990C2212yes—
32#1990C2212yes—
33#1973—22no—
34#1973—22no—
35#1957C2212yes—
36#1956C2212yes—
37#1836—22no—
38#1744C2211yescharacteristic
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41#1273C2212yes—
42#1273C2212yes—
43C22×SD16—22no—
44C22×SD16—24no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2312yes—
50C2×#127C2311yescharacteristic
51C2×#125—21no—
52C2×#124—22no—
53#500—42no—
54#498—42no—
55#498—42no—
56#492—42no—
57#492—42no—
58#492—42no—
59#487—42no—
60#487—42no—
61#487—42no—
62#487—42no—
63#487—42no—
64#486—42no—
65C2×#107—22no—
66C2×#106—24no—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#102—22no—
72C2×#102—24no—
73C2×#97D412yes—
74C2×#97D412yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79#391—22no—
80#391—21no—
81#391—21no—
82#390—21no—
83#390—21no—
84#390—22no—
85#389—22no—
86#389—22no—
87#384—21no—
88#384—22no—
89#384—21no—
90#384C2312yes—
91#382—21no—
92#382—22no—
93C43C2311yescharacteristic
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#336D414yes—
99#336D414yes—
100C22×Q8—21no—
101C22×Q8—22no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C2×SD16—44no—
108C2×SD16—44no—
109C2×SD16—48no—
110C2×SD16—44no—
111C2×SD16—88no—
112C22×C8C2×D414yes—
113C22×C8C2×D414yes—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—84no—
127#127—22no—
128#127—22no—
129#125—44no—
130#124—84no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151C2×C42C2411yescharacteristic
152#107—44no—
153#106—48no—
154#103—44no—
155#103—48no—
156#103—48no—
157#103—84no—
158#102—44no—
159#102—84no—
160#97—24no—
161#97—42no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168#95—42no—
169#95—42no—
170#95—42no—
171C24—41no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—44no—
181C2×Q8—48no—
182C2×Q8—44no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4#12712yes—
197C22×C4#12712yes—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201SD16—88no—
202SD16—816no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—28no—
213C2×C8#13714yes—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C42—22no—
235C42—22no—
236C42—24no—
237C42—24no—
238C42H(5,𝔽2)12yes—
239C42H(5,𝔽2)12yes—
240C23—81no—
241C23—81no—
242C23—82no—
243C23—82no—
244C23—81no—
245C23C2×#12711yescharacteristic
246Q8—82no—
247Q8—82no—
248Q8—84no—
249Q8—84no—
250Q8—84no—
251Q8—84no—
252D4—82no—
253D4—82no—
254D4—84no—
255D4—84no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4#53511yescharacteristic
275C2×C4#53511yescharacteristic
276C2×C4#53512yes—
277C2×C4#53512yes—
278C2×C4C2×#13712yes—
279C2×C4#57512yes—
280C2×C4#57714yes—
281C2×C4#57714yes—
282C8—24no—
283C8—42no—
284C8—44no—
285C8—44no—
286C22—162no—
287C22—162no—
288C22—161no—
289C22—161no—
290C22—161no—
291C22#265711yescharacteristic
292C22#315211yescharacteristic
293C22#315311yescharacteristic
294C22#315412yes—
295C22#315412yes—
296C4—42no—
297C4—42no—
298C4—162no—
299C4—161no—
300C4—162no—
301C4—161no—
302C4—161no—
303C4#321512yes—
304C4#321612yes—
305C4#322114yes—
306C4#322214yes—
307C2—321no—
308C2#1971412yes—
309C2#1971812yes—
310C2#3026911yescharacteristic
311C2#3027111yescharacteristic
312C2#3027311yescharacteristic
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. #33294C2semidirect
3. #30280C2semidirect
5. #30229C2semidirect
7. #10317C2semidirect
9. #10313C2semidirect
11. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#332942
#332522
#302802
#302632
#302292
#302192
#103172
#103152
#103132
#103102
#81342