Finite Group Databasebeta

← #330523

Subgroups — #330523

Order 512.

587 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 150 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#29998C211yesmaximal, characteristic
3#29864C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10600C212yesmaximal, complemented
6#10596C212yesmaximal, complemented
7#10485C212yesmaximal
8#10484C212yesmaximal
9#9897C211yesmaximal, characteristic, complemented
10#9694C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12C2×#491—22no—
13C2×#474C2211yescharacteristic
14C2×#474C2211yescharacteristic
15C2×#433—21no—
16#2868C2211yescharacteristic
17#2859C2211yescharacteristic
18#2293C2211yescharacteristic
19#2088—22no—
20#2083—22no—
21#2071—22no—
22#2065—22no—
23#1988C2211yescharacteristic
24#1988C2211yescharacteristic
25#1960—21no—
26#1960—21no—
27#1956C2212yes—
28#1956C2212yes—
29#1954C2212yes—
30#1948C2212yes—
31#1939C2212yes—
32#1924—22no—
33#1842C2211yescharacteristic
34#1831C2211yescharacteristic
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37#1750—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—21no—
50C2×C4×D4C2311yescharacteristic
51C2×#117—21no—
52#491—48no—
53#474—42no—
54#474—42no—
55#433—44no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#102—21no—
70C2×#102—21no—
71C2×#102—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×#97D412yes—
75C2×#97D411yescharacteristic
76C2×#97D411yescharacteristic
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#390D412yes—
80#389—22no—
81#389—22no—
82#389D412yes—
83#388—22no—
84#384—22no—
85#384—22no—
86#384C2312yes—
87#382—21no—
88C4×C4⋊C4C2311yescharacteristic
89C4×C22⋊C4—21no—
90C43C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#335—42no—
99#334—42no—
100#332—42no—
101#332—42no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×D4C2×D411yescharacteristic
105C23×C4—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—44no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119#125—44no—
120C4×D4—22no—
121C4×D4—22no—
122#117—42no—
123#117—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C42—21no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42#4112yes—
145C2×C42C2411yescharacteristic
146#107—42no—
147#107—42no—
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—44no—
158#103—44no—
159#103—44no—
160#103—44no—
161#102—42no—
162#102—42no—
163#102—82no—
164C22⋊C8—44no—
165C22⋊C8—44no—
166#97—24no—
167#97—22no—
168#97—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—42no—
172#95—22no—
173#95—21no—
174#95—42no—
175C24—21no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
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×C8AGL(1,ℤ/8)14yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—42no—
238C22⋊C4—44no—
239C22⋊C4—82no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—42no—
244C42—41no—
245C42—41no—
246C42#13512yes—
247C42AGL(1,ℤ/8)12yes—
248C42H(5,𝔽2)12yes—
249C422−1+412yes—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23C2×#12111yescharacteristic
256Q8—82no—
257Q8—82no—
258Q8—82no—
259Q8—82no—
260D4—41no—
261D4—41no—
262D4—42no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#46314yes—
287C2×C4#53511yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4#53712yes—
290C2×C4#54212yes—
291C2×C4C2×#13511yescharacteristic
292C2×C4C2×AGL(1,ℤ/8)12yes—
293C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
294C2×C4#57712yes—
295C2×C4#57812yes—
296C8—24no—
297C8—42no—
298C8—81no—
299C8—82no—
300C8—81no—
301C8—82no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—82no—
307C22#262011yescharacteristic
308C22C2×#46311yescharacteristic
309C22#303811yescharacteristic
310C22#306312yes—
311C22#306912yes—
312C4—24no—
313C4—22no—
314C4—42no—
315C4—82no—
316C4—81no—
317C4—161no—
318C4—161no—
319C4—162no—
320C4#322111yescharacteristic
321C4#322311yescharacteristic
322C4#324212yes—
323C4#324712yes—
324C4#325312yes—
325C2—161no—
326C2#1923512yes—
327C2#1937712yes—
328C2#2982511yescharacteristic
329C2#2985611yescharacteristic
330C2#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. #33232C2semidirect
5. #10600C2semidirect
6. #10596C2semidirect
9. #9897C2semidirect
10. #9694C2semidirect
11. #8808C2semidirect

Maximal subgroups

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

subgroupindex
#332322
#299982
#298642
#212682
#106002
#105962
#104852
#104842
#98972
#96942
#88082