Finite Group Databasebeta

← #330531

Subgroups — #330531

Order 512.

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

normal subgroups only

This group has 575 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#33294C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29798C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10602C212yesmaximal, complemented
6#10598C212yesmaximal, complemented
7#10477C212yesmaximal
8#10476C212yesmaximal
9#9897C211yesmaximal, characteristic, complemented
10#9692C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12C2×#461C2211yescharacteristic
13C2×#461C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2293C2211yescharacteristic
18#2088—22no—
19#2065—22no—
20#2043—22no—
21#2013—22no—
22#1992—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1981C2211yescharacteristic
26#1981C2211yescharacteristic
27#1954C2212yes—
28#1948C2212yes—
29#1939C2212yes—
30#1924—22no—
31#1842C2211yescharacteristic
32#1836—21no—
33#1836—21no—
34#1831C2211yescharacteristic
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37#1750—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×SD16—21no—
46C22×SD16—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C4×D4C2311yescharacteristic
52C2×#117—21no—
53#461—42no—
54#461—42no—
55#433—44no—
56C2×#106—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—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97D411yescharacteristic
71C2×#97D412yes—
72C2×#97D411yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8C2312yes—
75#390D412yes—
76#389—22no—
77#389—22no—
78#389D412yes—
79#388—22no—
80#384—21no—
81#384—21no—
82#384—22no—
83#384C2312yes—
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#336—22no—
95#336—22no—
96#336—21no—
97#335—42no—
98#334—42no—
99#332—42no—
100#332—42no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×D4C2×D411yescharacteristic
104C23×C4—21no—
105C2×SD16—42no—
106C2×SD16—42no—
107C2×SD16—82no—
108C2×SD16—82no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—82no—
120C4⋊Q8—82no—
121#125—44no—
122C4×D4—22no—
123C4×D4—22no—
124#117—42no—
125#117—42no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C42—21no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42#4112yes—
147C2×C42C2411yescharacteristic
148#106—82no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—44no—
156C4⋊C8—82no—
157#103—42no—
158#103—42no—
159#103—82no—
160#103—82no—
161#102—82no—
162C22⋊C8—44no—
163C22⋊C8—44no—
164#97—22no—
165#97—42no—
166#97SD1614yescomplemented
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—42no—
170#95—22no—
171#95—21no—
172#95—42no—
173C24—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C2×#4111yescharacteristic
200SD16—84no—
201SD16—84no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8C2×SD1614yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—42no—
235C22⋊C4—44no—
236C22⋊C4—82no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—42no—
241C42—41no—
242C42—41no—
243C42C2×SD1612yes—
244C42#13712yes—
245C42H(5,𝔽2)12yes—
246C422−1+412yes—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23C2×#12111yescharacteristic
253Q8—81no—
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—82no—
258Q8—82no—
259D4—41no—
260D4—41no—
261D4—42no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#46114yes—
284C2×C4#53511yescharacteristic
285C2×C4#53611yescharacteristic
286C2×C4#53712yes—
287C2×C4#54212yes—
288C2×C4C22×SD1611yescharacteristic
289C2×C4C2×#13711yescharacteristic
290C2×C4#57512yes—
291C2×C4#57612yes—
292C2×C4#57712yes—
293C8—24no—
294C8—42no—
295C8—82no—
296C8—81no—
297C8—81no—
298C8—82no—
299C22—82no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#262011yescharacteristic
305C22C2×#46111yescharacteristic
306C22#304011yescharacteristic
307C22#306212yes—
308C22#306912yes—
309C4—24no—
310C4—22no—
311C4—42no—
312C4—82no—
313C4—81no—
314C4—161no—
315C4—161no—
316C4—162no—
317C4#322011yescharacteristic
318C4#322211yescharacteristic
319C4#324112yes—
320C4#324612yes—
321C4#325212yes—
322C2—161no—
323C2#1923212yes—
324C2#1938312yes—
325C2#2982811yescharacteristic
326C2#2985511yescharacteristic
327C2#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. #33294C2semidirect
5. #10602C2semidirect
6. #10598C2semidirect
9. #9897C2semidirect
10. #9692C2semidirect
11. #8808C2semidirect
166. #97SD16semidirect

Maximal subgroups

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

subgroupindex
#332942
#299982
#297982
#212682
#106022
#105982
#104772
#104762
#98972
#96922
#88082