Finite Group Databasebeta

← #138749

Subgroups — #138749

Order 512.

352 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 86 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#33664C211yesmaximal, characteristic
2#21276C211yesmaximal, characteristic
3#11743C211yesmaximal, characteristic
4#11658C211yesmaximal, characteristic
5#9845C211yesmaximal, characteristic
6#7296C211yesmaximal, characteristic
7#7148C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×C4×Q16C2211yescharacteristic
10#2295—21no—
11#2236—21no—
12#2227C412yescomplemented
13#2209C412yescomplemented
14#2188—21no—
15C4×Q32—22no—
16#2162C2211yescharacteristic
17#2162C2211yescharacteristic
18#1994—21no—
19#1969—21no—
20#1942—21no—
21#1933—22no—
22C4×#107C2211yescharacteristic
23C4×#103—21no—
24C42×C8C2211yescharacteristic
25#1567C2211yescharacteristic
26#1493—21no—
27#1348—22no—
28#1332—22no—
29C22×Q16D411yescharacteristic
30#533—21no—
31C2×C4×Q8—21no—
32C2×Q32—22no—
33#500C2×C411yescharacteristic
34#494C2×C411yescharacteristic
35#487—21no—
36#467—21no—
37#467—21no—
38C4×Q16—21no—
39C4×Q16—21no—
40C4×Q16—21no—
41C4×Q16—21no—
42C4×Q16C2×C412yes—
43C4×Q16D412yes—
44C2×#107—21no—
45C2×#107D411yescharacteristic
46#423—21no—
47C2×#103—21no—
48C2×#103—22no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—22no—
54#389—22no—
55#387—42no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—41no—
58#376—41no—
59C43D411yescharacteristic
60#366—22no—
61C4⋊C16—21no—
62C4⋊C16C2×C412yes—
63#358—22no—
64#358—22no—
65C22⋊C16—21no—
66C22⋊C16—21no—
67C4×C16—21no—
68C4×C16—21no—
69#340—22no—
70#340D412yes—
71#339—41no—
72#339—41no—
73#335—21no—
74#335—21no—
75Q8⋊C8—42no—
76C22×Q8—21no—
77C2×Q16—21no—
78C2×Q16—21no—
79C2×Q16—21no—
80C2×Q16—42no—
81C2×Q16C22⋊C412yes—
82C2×M16—41no—
83C22×C8—21no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88#125—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97#117—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42D811yescharacteristic
107C2×C42D811yescharacteristic
108C2×C42C2×D411yescharacteristic
109Q32—44no—
110C2×C16—22no—
111C2×C16—41no—
112#107—21no—
113#107—21no—
114#107—21no—
115#107—42no—
116#107C22⋊C412yes—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—22no—
120#103—22no—
121#103—44no—
122#103—41no—
123#103—41no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—42no—
127C4×C8—42no—
128C4×C8C22×C411yescharacteristic
129C4×C8C2×D411yescharacteristic
130C4×C8#4111yescharacteristic
131C4×C8#4111yescharacteristic
132#95—42no—
133#95—42no—
134#95—42no—
135#95—41no—
136C2×Q8—22no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4#12011yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4AGL(1,ℤ/8)11yescharacteristic
150Q16—22no—
151Q16—22no—
152Q16—42no—
153Q16—42no—
154Q16—81no—
155M16—81no—
156M16—81no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—81no—
163C2×C8C2×C22⋊C411yescharacteristic
164C2×C8C4×D411yescharacteristic
165C2×C8#12111yescharacteristic
166C2×C8#12311yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—84no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—41no—
185C42—41no—
186C42—42no—
187C42—42no—
188C42—42no—
189C42—81no—
190C42—81no—
191C42—81no—
192C42Q3212yescomplemented
193C42C4×D411yescharacteristic
194C42#12111yescharacteristic
195C42#13511yescharacteristic
196C42AGL(1,ℤ/8)11yescharacteristic
197C16—42no—
198C16—81no—
199C23#44711yescharacteristic
200Q8—42no—
201Q8—42no—
202Q8—82no—
203Q8—82no—
204Q8—81no—
205Q8—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4#38611yescharacteristic
225C2×C4C4×D811yescharacteristic
226C2×C4#44211yescharacteristic
227C2×C4#44911yescharacteristic
228C2×C4#45911yescharacteristic
229C2×C4#46311yescharacteristic
230C2×C4C2×Q3211yescharacteristic
231C2×C4#50811yescharacteristic
232C2×C4#51011yescharacteristic
233C2×C4#51011yescharacteristic
234C8—21no—
235C8—21no—
236C8—21no—
237C8—42no—
238C8—161no—
239C8#42112yes—
240C22—21no—
241C22—21no—
242C22#187911yescharacteristic
243C22#218811yescharacteristic
244C22#218911yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—82no—
252C4—162no—
253C4—161no—
254C4—161no—
255C4#189711yescharacteristic
256C4#190611yescharacteristic
257C4C4×Q3211yescharacteristic
258C4#217611yescharacteristic
259C4#220911yescharacteristic
260C4#221111yescharacteristic
261C2—21no—
262C2—21no—
263C2#923711yescharacteristic
264C2#1301511yescharacteristic
265C2#1302711yescharacteristic
subgroupG/H
centerC2×C4#459
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1879
prime radicalC23#447
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2227C4semidirect
13. #2209C4semidirect
192. C42Q32semidirect

Maximal subgroups

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

subgroupindex
#336642
#212762
#117432
#116582
#98452
#72962
#71482