Finite Group Databasebeta

← #330441

Subgroups — #330441

Order 512.

555 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 154 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#33290C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29828C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic
5#10602C212yesmaximal, complemented
6#10601C212yesmaximal, complemented
7#10481C212yesmaximal
8#10476C212yesmaximal
9#9902C211yesmaximal, characteristic, complemented
10#9701C211yesmaximal, characteristic, complemented
11#8806C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#461C2211yescharacteristic
14C2×#433—21no—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17#2293C2211yescharacteristic
18#2088—22no—
19#2066—22no—
20#2065—22no—
21#2055—22no—
22#1991—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1987C2211yescharacteristic
26#1981C2211yescharacteristic
27#1954C2212yes—
28#1948C2212yes—
29#1939C2212yes—
30#1923—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1835—21no—
34#1835—21no—
35#1802C2212yes—
36#1767C2211yescharacteristic
37#1750—21no—
38C42×C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#335C2212yes—
41C2×#333C2212yes—
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#476—42no—
52#461—42no—
53#433—44no—
54C2×#106—21no—
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—21no—
64C2×#103—22no—
65C2×#102—21no—
66C2×#102—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×#97—21no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D412yes—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2312yes—
74#391—22no—
75#391—22no—
76#390D412yes—
77#389—22no—
78#389—22no—
79#389D412yes—
80#388—22no—
81#384C2312yes—
82#382—21no—
83#382—21no—
84#382—21no—
85C4×C4⋊C4C2311yescharacteristic
86C4×C22⋊C4—21no—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—21no—
97#335—42no—
98#335—42no—
99#333—42no—
100#332—42no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×D4C2×D411yescharacteristic
104C23×C4—21no—
105C2×SD16—42no—
106C2×SD16—82no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117#125—44no—
118C4×D4—22no—
119C4×D4—22no—
120#117—42no—
121#117—42no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C42—21no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42#4112yes—
141C2×C42C2411yescharacteristic
142#106—42no—
143#106—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—44no—
151C4⋊C8—82no—
152C4⋊C8—82no—
153#103—42no—
154#103—82no—
155#102—42no—
156#102—82no—
157C22⋊C8—44no—
158C22⋊C8—44no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—42no—
164C4×C8SD1614yescomplemented
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170C24—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—82no—
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—41no—
186C22×C4—42no—
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—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8C2×SD1614yes—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—44no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—44no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—42no—
234C42—41no—
235C42—41no—
236C42C2×SD1612yes—
237C42#13712yes—
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—162no—
250D4—41no—
251D4—41no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#46114yes—
275C2×C4#53511yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#53712yes—
278C2×C4#54212yes—
279C2×C4C22×SD1611yescharacteristic
280C2×C4C2×#13711yescharacteristic
281C2×C4#57512yes—
282C2×C4#57612yes—
283C2×C4#57812yes—
284C8—24no—
285C8—42no—
286C8—81no—
287C8—82no—
288C8—81no—
289C8—82no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—82no—
294C22—82no—
295C22#262011yescharacteristic
296C22C2×#46111yescharacteristic
297C22#304011yescharacteristic
298C22#306112yes—
299C22#306712yes—
300C4—22no—
301C4—24no—
302C4—42no—
303C4—81no—
304C4—82no—
305C4—161no—
306C4—161no—
307C4—162no—
308C4#322011yescharacteristic
309C4#322311yescharacteristic
310C4#324312yes—
311C4#324612yes—
312C4#325212yes—
313C2—161no—
314C2#1919512yes—
315C2#1936712yes—
316C2#2982811yescharacteristic
317C2#2984911yescharacteristic
318C2#2986011yescharacteristic
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. #33290C2semidirect
5. #10602C2semidirect
6. #10601C2semidirect
9. #9902C2semidirect
10. #9701C2semidirect
11. #8806C2semidirect
164. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332902
#299982
#298282
#212662
#106022
#106012
#104812
#104762
#99022
#97012
#88062