Finite Group Databasebeta

← #330723

Subgroups — #330723

Order 512.

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

normal subgroups only

This group has 538 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
2#29999C211yesmaximal, characteristic, complemented
3#29797C211yesmaximal, characteristic
4#22019C211yesmaximal, characteristic, complemented
5#10594C212yesmaximal, complemented
6#10587C212yesmaximal
7#10479C212yesmaximal, complemented
8#10474C212yesmaximal
9#9899C211yesmaximal, characteristic, complemented
10#9698C211yesmaximal, characteristic
11#8812C211yesmaximal, characteristic, complemented
12C2×#462C2211yescharacteristic
13C2×#460C2211yescharacteristic
14#2859C2211yescharacteristic
15#2858C2211yescharacteristic
16#2295C2211yescharacteristic
17#2087—22no—
18#2073—21no—
19#2073—21no—
20#2068—22no—
21#2042—22no—
22#2025—22no—
23#2015—22no—
24#1990C2212yes—
25#1984C2212yes—
26#1982C2211yescharacteristic
27#1981C2211yescharacteristic
28#1957C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1916—22no—
32#1842C2211yescharacteristic
33#1840—21no—
34#1832—21no—
35#1831C2211yescharacteristic
36#1803C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#331C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×Q16—21no—
46C22×SD16—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#125—21no—
50C2×#124—22no—
51C2×C4×Q8C2311yescharacteristic
52#462—42no—
53#460—42no—
54C2×#107—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×#103—21no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#102—21no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×C4×C8—21no—
69C2×C4×C8C2312yes—
70#391D412yes—
71#390—21no—
72#390—21no—
73#390—22no—
74#389—22no—
75#389—22no—
76#389D412yes—
77#387—22no—
78#384C2312yes—
79#382—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2311yescharacteristic
82#376—21no—
83C43C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#336—22no—
89#336—22no—
90#336—22no—
91#336—22no—
92#336—21no—
93#336D411yescharacteristic
94#336D411yescharacteristic
95#336D412yes—
96#335—42no—
97#334—42no—
98#332—42no—
99#331—42no—
100C22×Q8—21no—
101C22×Q8C2×D411yescharacteristic
102C22×D4—21no—
103C22×D4—21no—
104C22×D4—21no—
105C2×Q16—42no—
106C2×Q16—82no—
107C2×SD16—42no—
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—
117#127—22no—
118#127—22no—
119#125—81no—
120#125—81no—
121#124—82no—
122#124—82no—
123C4×Q8—22no—
124C4×Q8—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4112yes—
143C2×C42C2411yescharacteristic
144#107—82no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—82no—
152C4⋊C8—82no—
153#103—42no—
154#103—82no—
155#103—82no—
156#102—42no—
157#102—82no—
158#102—82no—
159#102—82no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—81no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—21no—
167#95—42no—
168C24—41no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—82no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4#12112yes—
193C22×C4#12112yes—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197Q16—84no—
198SD16—84no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C22⋊C4—84no—
230C22⋊C4—84no—
231C42—21no—
232C42—22no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—42no—
237C42—42no—
238C42—41no—
239C42—41no—
240C42H(5,𝔽2)12yes—
241C422−1+412yes—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—82no—
246C23—82no—
247C23C2×#12111yescharacteristic
248Q8—41no—
249Q8—41no—
250Q8—42no—
251Q8—81no—
252Q8—81no—
253Q8—82no—
254D4—81no—
255D4—81no—
256D4—82no—
257D4—82no—
258D4—82no—
259D4—162no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#53511yescharacteristic
280C2×C4#53611yescharacteristic
281C2×C4#53712yes—
282C2×C4#54212yes—
283C2×C4#57612yes—
284C2×C4#57611yescharacteristic
285C2×C4#57712yes—
286C2×C4#57711yescharacteristic
287C2×C4#57812yes—
288C8—42no—
289C8—42no—
290C8—82no—
291C8—81no—
292C8—81no—
293C8—82no—
294C22—162no—
295C22—162no—
296C22—161no—
297C22—161no—
298C22—161no—
299C22#262011yescharacteristic
300C22#304411yescharacteristic
301C22#304511yescharacteristic
302C22#306212yes—
303C22#306712yes—
304C4—22no—
305C4—42no—
306C4—42no—
307C4—82no—
308C4—81no—
309C4—81no—
310C4—162no—
311C4—161no—
312C4#322011yescharacteristic
313C4#322211yescharacteristic
314C4#324112yes—
315C4#325812yes—
316C4#326212yes—
317C2—321no—
318C2#1957512yes—
319C2#1957612yes—
320C2#2984011yescharacteristic
321C2#2985511yescharacteristic
322C2#2986011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #29999C2semidirect
4. #22019C2semidirect
5. #10594C2semidirect
7. #10479C2semidirect
9. #9899C2semidirect
11. #8812C2semidirect

Maximal subgroups

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

subgroupindex
#302042
#299992
#297972
#220192
#105942
#105872
#104792
#104742
#98992
#96982
#88122