Finite Group Databasebeta

← #330631

Subgroups — #330631

Order 512.

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

normal subgroups only

This group has 555 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#30204C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic, complemented
3#29825C211yesmaximal, characteristic
4#22013C211yesmaximal, characteristic, complemented
5#10593C212yesmaximal, complemented
6#10591C212yesmaximal, complemented
7#10481C212yesmaximal, complemented
8#10472C212yesmaximal
9#9902C211yesmaximal, characteristic, complemented
10#9707C211yesmaximal, characteristic, complemented
11#8810C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#459C2211yescharacteristic
14#2859C2211yescharacteristic
15#2858C2211yescharacteristic
16#2293C2211yescharacteristic
17#2086—22no—
18#2073—22no—
19#2067—22no—
20#2065—22no—
21#2054—22no—
22#2025—21no—
23#2025—21no—
24#1990C2212yes—
25#1987C2211yescharacteristic
26#1982C2211yescharacteristic
27#1957C2212yes—
28#1954C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1924—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1818—21no—
35#1802C2212yes—
36#1798—21no—
37#1767C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#335C2212yes—
41C2×#333C2212yes—
42C2×#331C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×D8—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#125—21no—
49C2×#124—22no—
50C2×C4×D4C2311yescharacteristic
51C2×#117—21no—
52#476—42no—
53#459—42no—
54C2×#107—22no—
55C2×#106—21no—
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×#102—21no—
64C2×#102—21no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×C4×C8C2312yes—
73#390—22no—
74#390—21no—
75#390—21no—
76#390D412yes—
77#389—22no—
78#389—22no—
79#389D412yes—
80#388—22no—
81#384C2312yes—
82#382—21no—
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#336D412yes—
95#336D411yescharacteristic
96#335—42no—
97#335—42no—
98#333—42no—
99#331—42no—
100C22×Q8—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—21no—
104C22×D4C2×D411yescharacteristic
105C23×C4—21no—
106C2×D8—42no—
107C2×D8—82no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116#127—22no—
117#127—22no—
118#125—82no—
119#124—84no—
120C4×D4—22no—
121C4×D4—22no—
122#117—82no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
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×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#107—82no—
144#106—42no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—82no—
152C4⋊C8—82no—
153C4⋊C8—82no—
154#102—42no—
155#102—42no—
156#102—82no—
157#102—82no—
158#102—82no—
159#102—82no—
160C22⋊C8—82no—
161C22⋊C8—82no—
162#97—81no—
163C4×C8—22no—
164C4×C8—22no—
165#95—22no—
166#95—21no—
167#95—42no—
168C24—21no—
169C24—41no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—82no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C2×#4111yescharacteristic
199D8—84no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—84no—
230C22⋊C4—42no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C42—22no—
236C42—22no—
237C42—21no—
238C42—21no—
239C42—22no—
240C42—42no—
241C42—42no—
242C42H(5,𝔽2)12yes—
243C422−1+412yes—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—42no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—82no—
253C23—82no—
254C23C2×#12111yescharacteristic
255Q8—162no—
256D4—41no—
257D4—41no—
258D4—42no—
259D4—81no—
260D4—81no—
261D4—82no—
262D4—82no—
263D4—82no—
264D4—162no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#53511yescharacteristic
286C2×C4#53611yescharacteristic
287C2×C4#53712yes—
288C2×C4#54212yes—
289C2×C4#57612yes—
290C2×C4#57612yes—
291C2×C4#57711yescharacteristic
292C2×C4#57812yes—
293C2×C4#57811yescharacteristic
294C8—42no—
295C8—42no—
296C8—81no—
297C8—82no—
298C8—81no—
299C8—82no—
300C22—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—162no—
306C22—162no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22#262011yescharacteristic
311C22#304111yescharacteristic
312C22#304611yescharacteristic
313C22#306112yes—
314C22#306812yes—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—81no—
320C4—162no—
321C4—161no—
322C4#322011yescharacteristic
323C4#322311yescharacteristic
324C4#324312yes—
325C4#325712yes—
326C4#326212yes—
327C2—161no—
328C2—321no—
329C2#1951012yes—
330C2#1955112yes—
331C2#2983911yescharacteristic
332C2#2984911yescharacteristic
333C2#2986111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
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. #30204C2semidirect
2. #29994C2semidirect
4. #22013C2semidirect
5. #10593C2semidirect
6. #10591C2semidirect
7. #10481C2semidirect
9. #9902C2semidirect
10. #9707C2semidirect
11. #8810C2semidirect

Maximal subgroups

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

subgroupindex
#302042
#299942
#298252
#220132
#105932
#105912
#104812
#104722
#99022
#97072
#88102