Finite Group Databasebeta

← #330426

Subgroups — #330426

Order 512.

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

normal subgroups only

This group has 622 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#33226C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic, complemented
3#29824C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic, complemented
5#10588C212yesmaximal, complemented
6#10587C212yesmaximal, complemented
7#10484C212yesmaximal, complemented
8#10473C212yesmaximal
9#9904C211yesmaximal, characteristic, complemented
10#9704C211yesmaximal, characteristic, complemented
11#8805C211yesmaximal, characteristic, complemented
12C2×#488—21no—
13C2×#488—21no—
14C2×#474C2211yescharacteristic
15C2×#459C2211yescharacteristic
16C2×#433—21no—
17#2859C2211yescharacteristic
18#2858C2211yescharacteristic
19#2293C2211yescharacteristic
20#2086—22no—
21#2067—22no—
22#2067—22no—
23#2054—22no—
24#1988C2211yescharacteristic
25#1982C2211yescharacteristic
26#1958—22no—
27#1957C2212yes—
28#1956C2212yes—
29#1954C2212yes—
30#1947C2212yes—
31#1939C2212yes—
32#1923—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1802C2212yes—
36#1767C2211yescharacteristic
37#1750—21no—
38C42×C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#331C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×D8—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#125—21no—
49C2×#124—21no—
50C2×#124—21no—
51C2×C4×D4C2311yescharacteristic
52C2×#117—21no—
53#488—44no—
54#488—44no—
55#474—42no—
56#459—42no—
57#433—44no—
58C2×#107—21no—
59C2×#107—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#102—21no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—21no—
73C2×#102—22no—
74C2×#102—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×C4×C8D411yescharacteristic
79C2×C4×C8D412yes—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2312yes—
82#390—22no—
83#390—22no—
84#390D412yes—
85#389—22no—
86#389—22no—
87#389D412yes—
88#388—22no—
89#384C2312yes—
90#382—21no—
91C4×C4⋊C4C2311yescharacteristic
92C4×C22⋊C4—21no—
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#336—22no—
98#336—21no—
99#334—42no—
100#334—42no—
101#332—42no—
102#331—42no—
103C22×Q8—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4C2×D411yescharacteristic
108C23×C4—21no—
109C2×D8—42no—
110C2×D8—82no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119#127—22no—
120#127—22no—
121#125—42no—
122#125—42no—
123#124—44no—
124#124—42no—
125#124—42no—
126C4×D4—22no—
127C4×D4—22no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42—21no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42#4112yes—
149C2×C42C2411yescharacteristic
150#107—42no—
151#107—82no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—44no—
159C4⋊C8—82no—
160C4⋊C8—82no—
161#103—44no—
162#103—44no—
163#102—42no—
164#102—44no—
165#102—42no—
166#102—44no—
167#102—82no—
168#102—82no—
169C22⋊C8—44no—
170C22⋊C8—44no—
171#97—42no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—42no—
176C4×C8D814yescomplemented
177#95—22no—
178#95—21no—
179#95—42no—
180C24—21no—
181C24—41no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4#12112yes—
209C22×C4#12112yes—
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C2×#4111yescharacteristic
213D8—84no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8C2×D814yes—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—44no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C22⋊C4—44no—
245C22⋊C4—42no—
246C22⋊C4—44no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—42no—
253C42—41no—
254C42—41no—
255C42C2×D812yes—
256C42AGL(1,ℤ/8)12yes—
257C42H(5,𝔽2)12yes—
258C422−1+412yes—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—82no—
268C23—82no—
269C23C2×#12111yescharacteristic
270Q8—82no—
271Q8—82no—
272D4—41no—
273D4—41no—
274D4—42no—
275D4—81no—
276D4—81no—
277D4—82no—
278D4—82no—
279D4—82no—
280D4—162no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#45914yes—
302C2×C4#53511yescharacteristic
303C2×C4#53611yescharacteristic
304C2×C4#53712yes—
305C2×C4#54212yes—
306C2×C4C22×D811yescharacteristic
307C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
308C2×C4#57512yes—
309C2×C4#57712yes—
310C2×C4#57712yes—
311C8—24no—
312C8—42no—
313C8—82no—
314C8—81no—
315C8—81no—
316C8—82no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—161no—
323C22—161no—
324C22—161no—
325C22—162no—
326C22—162no—
327C22#262011yescharacteristic
328C22C2×#45911yescharacteristic
329C22#304011yescharacteristic
330C22#306012yes—
331C22#306712yes—
332C4—22no—
333C4—24no—
334C4—42no—
335C4—81no—
336C4—82no—
337C4—161no—
338C4—162no—
339C4#322111yescharacteristic
340C4#322211yescharacteristic
341C4#324412yes—
342C4#324512yes—
343C4#325112yes—
344C2—161no—
345C2—321no—
346C2#1918912yes—
347C2#1936612yes—
348C2#2982411yescharacteristic
349C2#2985011yescharacteristic
350C2#2986011yescharacteristic
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. #33226C2semidirect
2. #29994C2semidirect
4. #21266C2semidirect
5. #10588C2semidirect
6. #10587C2semidirect
7. #10484C2semidirect
9. #9904C2semidirect
10. #9704C2semidirect
11. #8805C2semidirect
176. C4×C8D8semidirect

Maximal subgroups

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

subgroupindex
#332262
#299942
#298242
#212662
#105882
#105872
#104842
#104732
#99042
#97042
#88052