Finite Group Databasebeta

← #491321

Subgroups — #491321

Order 512.

397 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 130 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#33797C211yesmaximal, characteristic
2#33496C211yesmaximal, characteristic
3#19265C211yesmaximal, characteristic
4#19118C211yesmaximal, characteristic
5#12478C212yesmaximal, complemented
6#12477C212yesmaximal, complemented
7#12459C212yesmaximal, complemented
8#12457C212yesmaximal, complemented
9#12034C211yesmaximal, characteristic
10#11973C211yesmaximal, characteristic
11#11950C211yesmaximal, characteristic
12#3170C2211yescharacteristic
13#3160C2211yescharacteristic
14#3113—21no—
15#3104—21no—
16C2×C8×Q8C2211yescharacteristic
17#2254C2211yescharacteristic
18#2252C2211yescharacteristic
19#2241C2211yescharacteristic
20#2241C2211yescharacteristic
21#2237—22no—
22#2229—22no—
23Q8×C16—22no—
24Q8×C16C2212yes—
25Q8×C16C2212yes—
26#2162—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1680—22no—
30#1676C2212yes—
31#1674C2212yes—
32#1674C2212yes—
33#1672—22no—
34#1670C2212yes—
35#1568C2211yescharacteristic
36#1564C2211yescharacteristic
37#1552—21no—
38#1542—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2212yes—
42#1364C2212yes—
43#1364C2212yes—
44#1362C2212yes—
45#533—21no—
46#533—21no—
47C2×C4×Q8D411yescharacteristic
48#500C2311yescharacteristic
49#500C2311yescharacteristic
50#500C2311yescharacteristic
51#499—21no—
52#499—21no—
53#498—21no—
54#494C2311yescharacteristic
55#487—21no—
56#484—41no—
57#483—41no—
58#477—22no—
59#475—22no—
60#467—21no—
61#467—21no—
62#462—22no—
63#462—22no—
64#451—41no—
65#451—41no—
66C8×Q8—21no—
67C8×Q8—21no—
68C8×Q8D412yes—
69C8×Q8C2312yes—
70#441—42no—
71C4×Q16—21no—
72C4×Q16—21no—
73C4×Q16—42no—
74#432—21no—
75#431—21no—
76#427—21no—
77#427—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×C4×C8—21no—
81C2×C4×C8C2311yescharacteristic
82#367—22no—
83#366—22no—
84C4⋊C16—22no—
85C4⋊C16—22no—
86C4⋊C16D412yes—
87C4⋊C16C2312yes—
88#358—22no—
89#358—22no—
90C22⋊C16—21no—
91C22⋊C16D412yes—
92C4×C16—22no—
93C4×C16—22no—
94C4×C16—21no—
95C4×C16C2311yescharacteristic
96C4×C16C2311yescharacteristic
97#335C2312yes—
98#335C2312yes—
99#334—21no—
100#334—21no—
101#333—22no—
102#332—22no—
103#332—22no—
104#332—22no—
105C22×Q8D811yescharacteristic
106C2×Q16—21no—
107C2×Q16—21no—
108C2×Q16—42no—
109C2×Q16—42no—
110C22×C8—21no—
111C22×C8C2×D411yescharacteristic
112C22×C8#4111yescharacteristic
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117#125—41no—
118#125—41no—
119C22⋊Q8—41no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8D812yes—
130C4×Q8C2×D412yes—
131#117—41no—
132#117—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4D811yescharacteristic
135C2×C42—21no—
136C2×C42C2×D411yescharacteristic
137C2×C16—22no—
138C2×C16C2×D412yes—
139C2×C16C2×D412yes—
140#107—21no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—21no—
145#107—21no—
146#107—42no—
147#106—41no—
148#106—41no—
149#106—42no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8C2×D412yes—
156#103—42no—
157#103—42no—
158#103—41no—
159#103—41no—
160#103—42no—
161#103—42no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—41no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—41no—
168C4×C8C2×D411yescharacteristic
169C4×C8C2×D411yescharacteristic
170C4×C8#4111yescharacteristic
171C4×C8C2411yescharacteristic
172C2×Q8—24no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×D812yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4#12111yescharacteristic
182C22×C4C2×D811yescharacteristic
183C22×C4#13511yescharacteristic
184Q16—42no—
185Q16—81no—
186Q16—84no—
187Q16—84no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—81no—
194C2×C8#12111yescharacteristic
195C2×C8#12111yescharacteristic
196C2×C8C22×D411yescharacteristic
197C2×C8C2×#4111yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4C2×D812yes—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C42—22no—
217C42—22no—
218C42—81no—
219C42—81no—
220C42#12111yescharacteristic
221C42C2×D811yescharacteristic
222C42#13511yescharacteristic
223C42C22×D411yescharacteristic
224C16—22no—
225C16—22no—
226C16—42no—
227C23#46611yescharacteristic
228Q8—24no—
229Q8—22no—
230Q8—82no—
231Q8—82no—
232Q8—81no—
233Q8—81no—
234Q8—161no—
235Q8#50814yes—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4#46611yescharacteristic
249C2×C4#46611yescharacteristic
250C2×C4#46611yescharacteristic
251C2×C4C2×#12111yescharacteristic
252C2×C4C22×D811yescharacteristic
253C2×C4C2×#13511yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—42no—
258C8—82no—
259C8—82no—
260C8#57812yes—
261C22—21no—
262C22—21no—
263C22—21no—
264C22C2×#46611yescharacteristic
265C4—24no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—161no—
272C4—161no—
273C4—161no—
274C4—161no—
275C4#304411yescharacteristic
276C4#306911yescharacteristic
277C4C2×#50812yes—
278C2—41no—
279C2#1940411yescharacteristic
280C2#3360011yescharacteristic
281C2#3361911yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #12478C2semidirect
6. #12477C2semidirect
7. #12459C2semidirect
8. #12457C2semidirect

Maximal subgroups

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

subgroupindex
#337972
#334962
#192652
#191182
#124782
#124772
#124592
#124572
#120342
#119732
#119502