Finite Group Databasebeta

← #330520

Subgroups — #330520

Order 512.

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

normal subgroups only

This group has 587 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#33232C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic
3#29864C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10592C212yesmaximal, complemented
6#10588C212yesmaximal, complemented
7#10485C212yesmaximal
8#10484C212yesmaximal
9#9899C211yesmaximal, characteristic, complemented
10#9694C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12C2×#491—21no—
13C2×#491—21no—
14C2×#474C2211yescharacteristic
15C2×#474C2211yescharacteristic
16C2×#433—21no—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2293C2211yescharacteristic
20#2086—22no—
21#2083—22no—
22#2071—22no—
23#2067—22no—
24#1988C2211yescharacteristic
25#1988C2211yescharacteristic
26#1960—22no—
27#1956C2212yes—
28#1956C2212yes—
29#1954C2212yes—
30#1947C2212yes—
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—44no—
53#491—44no—
54#474—42no—
55#474—42no—
56#433—44no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#102—21no—
71C2×#102—21no—
72C2×#102—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97D411yescharacteristic
76C2×#97D412yes—
77C2×#97D411yescharacteristic
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#390D412yes—
81#389—22no—
82#389—22no—
83#389D412yes—
84#388—22no—
85#384—22no—
86#384—22no—
87#384C2312yes—
88#382—21no—
89C4×C4⋊C4C2311yescharacteristic
90C4×C22⋊C4—21no—
91C43C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—22no—
97#336—22no—
98#336—21no—
99#335—42no—
100#334—42no—
101#332—42no—
102#332—42no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×D4C2×D411yescharacteristic
106C23×C4—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—44no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120#125—42no—
121#125—42no—
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#107—42no—
149#107—42no—
150#107—82no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—44no—
158C4⋊C8—82no—
159#103—44no—
160#103—44no—
161#103—44no—
162#103—44no—
163#102—42no—
164#102—42no—
165#102—82no—
166C22⋊C8—44no—
167C22⋊C8—44no—
168#97—22no—
169#97—42no—
170#97D814yes—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—42no—
174#95—22no—
175#95—21no—
176#95—42no—
177C24—21no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4#12112yes—
200C22×C4#12112yes—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8C2×D814yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—44no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—41no—
247C42—41no—
248C42C2×D812yes—
249C42AGL(1,ℤ/8)12yes—
250C42H(5,𝔽2)12yes—
251C422−1+412yes—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23C2×#12111yescharacteristic
258Q8—82no—
259Q8—82no—
260Q8—82no—
261Q8—82no—
262D4—41no—
263D4—41no—
264D4—42no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#45914yes—
289C2×C4#53511yescharacteristic
290C2×C4#53611yescharacteristic
291C2×C4#53712yes—
292C2×C4#54212yes—
293C2×C4C22×D811yescharacteristic
294C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
295C2×C4#57512yes—
296C2×C4#57712yes—
297C2×C4#57812yes—
298C8—24no—
299C8—42no—
300C8—81no—
301C8—82no—
302C8—81no—
303C8—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—82no—
309C22#262011yescharacteristic
310C22C2×#45911yescharacteristic
311C22#304011yescharacteristic
312C22#306312yes—
313C22#306912yes—
314C4—24no—
315C4—22no—
316C4—42no—
317C4—82no—
318C4—81no—
319C4—162no—
320C4—161no—
321C4—161no—
322C4#322111yescharacteristic
323C4#322311yescharacteristic
324C4#324212yes—
325C4#324512yes—
326C4#325112yes—
327C2—161no—
328C2#1923112yes—
329C2#1938112yes—
330C2#2982411yescharacteristic
331C2#2985611yescharacteristic
332C2#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. #10592C2semidirect
6. #10588C2semidirect
9. #9899C2semidirect
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
#299942
#298642
#212682
#105922
#105882
#104852
#104842
#98992
#96942
#88082