Finite Group Databasebeta

← #320656

Subgroups — #320656

Order 512.

822 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 170 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#33232C211yesmaximal, characteristic, complemented
2#29190C212yesmaximal, complemented
3#28716C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10307C211yesmaximal, characteristic
8#10307C211yesmaximal, characteristic
9#9838C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#491—22no—
14C2×#486C2212yes—
15C2×#486C2212yes—
16C2×#442—22no—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19C4×#127C2212yes—
20#2071—24no—
21#2065—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1972—22no—
25#1960—22no—
26#1957C2212yes—
27#1957C2212yes—
28#1957C2212yes—
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1864—24no—
34#1844C2212yes—
35#1844C2211yescharacteristic
36#1834C2212yes—
37#1802—24no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×D8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2312yes—
49C2×#125—21no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53#493—24no—
54#491—48no—
55#486—24no—
56#486—24no—
57#486—42no—
58#486—42no—
59#442—48no—
60C2×#106—22no—
61C2×#106—24no—
62C2×C4⋊C8D414yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#97D412yes—
73C2×#97D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390—22no—
81#389—22no—
82#389—22no—
83#384—22no—
84#384—22no—
85#384C2312yes—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#382—21no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#322—42no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—22no—
100C23×C4—42no—
101C2×D8—28no—
102C2×D8—44no—
103C2×D8—44no—
104C22×C8C2×D414yes—
105C22×C8C2×D414yes—
106C22×C8C2×D412yes—
107C22×C8C2×D412yes—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—44no—
115C4⋊Q8—44no—
116#127—22no—
117#127—22no—
118#127—22no—
119#127#4114yes—
120#125—44no—
121C4×D4—48no—
122C4×D4—48no—
123C4×D4—84no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
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—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2411yescharacteristic
146#106—44no—
147#106—84no—
148C4⋊C8—44no—
149#103—44no—
150#103—44no—
151#103—44no—
152#103—44no—
153#102—48no—
154#102—48no—
155#102—48no—
156#102—48no—
157#102—84no—
158#97—24no—
159#97—24no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—21no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—21no—
167#95—42no—
168C24—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—84no—
195C22×C4—84no—
196C22×C4—84no—
197C22×C4#12711yescharacteristic
198C22×C4#12711yescharacteristic
199C22×C4#12712yes—
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D412yes—
202D8—416no—
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—24no—
213C2×C8—24no—
214C2×C8AGL(1,ℤ/8)14yes—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C42—22no—
235C42—24no—
236C42—24no—
237C42AGL(1,ℤ/8)14yes—
238C42C2×#4112yes—
239C42H(5,𝔽2)11yescharacteristic
240C422−1+411yescharacteristic
241C23—44no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23C2×#12711yescharacteristic
249Q8—82no—
250Q8—82no—
251Q8—82no—
252Q8—82no—
253D4—48no—
254D4—48no—
255D4—48no—
256D4—84no—
257D4—84no—
258D4—84no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—168no—
279C2×C4#49614yes—
280C2×C4#52511yescharacteristic
281C2×C4#53711yescharacteristic
282C2×C4#54612yes—
283C2×C4#54812yes—
284C2×C4C2×AGL(1,ℤ/8)12yes—
285C2×C4C2×AGL(1,ℤ/8)12yes—
286C2×C4#57514yes—
287C2×C4#57714yes—
288C8—24no—
289C8—42no—
290C8—44no—
291C8—44no—
292C22—84no—
293C22—84no—
294C22—84no—
295C22—162no—
296C22—162no—
297C22—162no—
298C22—162no—
299C22#254011yescharacteristic
300C22C2×#49611yescharacteristic
301C22#315111yescharacteristic
302C22#315312yes—
303C22#315412yes—
304C4—24no—
305C4—41no—
306C4—41no—
307C4—161no—
308C4—161no—
309C4—161no—
310C4—161no—
311C4—162no—
312C4#310912yes—
313C4#325312yes—
314C4#330714yes—
315C4#337614yes—
316C2—164no—
317C2#1967812yes—
318C2#1972111yescharacteristic
319C2#1975011yescharacteristic
320C2#2838111yescharacteristic
321C2#2842912yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #33232C2semidirect
2. #29190C2semidirect
3. #28716C2semidirect
5. #10686C2semidirect
6. #10685C2semidirect
9. #9838C2semidirect
10. #9827C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332322
#291902
#287162
#216902
#106862
#106852
#103072
#98382
#98272
#82632