Finite Group Databasebeta

← #330628

Subgroups — #330628

Order 512.

524 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 138 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#30206C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29829C211yesmaximal, characteristic
4#22013C211yesmaximal, characteristic
5#10598C212yesmaximal, complemented
6#10596C212yesmaximal, complemented
7#10484C212yesmaximal
8#10477C212yesmaximal
9#9904C211yesmaximal, characteristic, complemented
10#9708C211yesmaximal, characteristic, complemented
11#8809C211yesmaximal, characteristic, complemented
12C2×#474C2211yescharacteristic
13C2×#461C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2293C2211yescharacteristic
17#2088—22no—
18#2074—21no—
19#2074—21no—
20#2068—22no—
21#2065—22no—
22#2055—22no—
23#2026—22no—
24#1990C2212yes—
25#1988C2211yescharacteristic
26#1981C2211yescharacteristic
27#1956C2212yes—
28#1954C2212yes—
29#1948C2212yes—
30#1939C2212yes—
31#1924—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1818—21no—
35#1802C2212yes—
36#1798—21no—
37#1767C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×D4C2311yescharacteristic
50C2×#117—21no—
51#474—42no—
52#461—42no—
53C2×#107—21no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#102—21no—
67C2×#102—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×C4×C8C2312yes—
72#391—22no—
73#391—21no—
74#391—21no—
75#390D412yes—
76#389—22no—
77#389—22no—
78#389D412yes—
79#388—22no—
80#384C2312yes—
81#382—21no—
82#382—22no—
83C4×C4⋊C4C2311yescharacteristic
84C4×C22⋊C4—21no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—21no—
93#336D411yescharacteristic
94#336D411yescharacteristic
95#336D412yes—
96#334—42no—
97#334—42no—
98#332—42no—
99#332—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×D4C2×D411yescharacteristic
103C23×C4—21no—
104C2×SD16—42no—
105C2×SD16—82no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116#125—81no—
117#125—81no—
118C4×D4—22no—
119C4×D4—22no—
120#117—82no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C42—21no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42#4112yes—
140C2×C42C2411yescharacteristic
141#107—42no—
142#106—82no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—82no—
150C4⋊C8—82no—
151C4⋊C8—82no—
152#103—42no—
153#103—82no—
154#103—82no—
155#103—82no—
156#102—42no—
157#102—82no—
158C22⋊C8—82no—
159C22⋊C8—82no—
160#97—81no—
161C4×C8—22no—
162C4×C8—22no—
163#95—22no—
164#95—21no—
165#95—42no—
166#95—42no—
167#95—42no—
168C24—21no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4#12112yes—
191C22×C4#12112yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C2×#4111yescharacteristic
195SD16—84no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—84no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—21no—
236C42—42no—
237C42—42no—
238C42H(5,𝔽2)12yes—
239C422−1+412yes—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23C2×#12111yescharacteristic
246Q8—81no—
247Q8—81no—
248Q8—82no—
249Q8—82no—
250Q8—82no—
251D4—41no—
252D4—41no—
253D4—42no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4#53511yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#53712yes—
280C2×C4#54212yes—
281C2×C4#57611yescharacteristic
282C2×C4#57712yes—
283C2×C4#57712yes—
284C2×C4#57712yes—
285C2×C4#57711yescharacteristic
286C8—42no—
287C8—42no—
288C8—82no—
289C8—81no—
290C8—81no—
291C8—82no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#262011yescharacteristic
298C22#304211yescharacteristic
299C22#304511yescharacteristic
300C22#306012yes—
301C22#306912yes—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—162no—
308C4—161no—
309C4—161no—
310C4#322111yescharacteristic
311C4#322211yescharacteristic
312C4#324412yes—
313C4#325712yes—
314C4#326312yes—
315C2—161no—
316C2#1951712yes—
317C2#1954112yes—
318C2#2983811yescharacteristic
319C2#2985011yescharacteristic
320C2#2986211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #30206C2semidirect
5. #10598C2semidirect
6. #10596C2semidirect
9. #9904C2semidirect
10. #9708C2semidirect
11. #8809C2semidirect

Maximal subgroups

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

subgroupindex
#302062
#299982
#298292
#220132
#105982
#105962
#104842
#104772
#99042
#97082
#88092