Finite Group Databasebeta

← #137498

Subgroups — #137498

Order 512.

302 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 76 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#33749C211yesmaximal, characteristic
2#18090C211yesmaximal, characteristic
3#11745C211yesmaximal, characteristic
4#11653C211yesmaximal, characteristic
5#9720C211yesmaximal, characteristic
6#7174C211yesmaximal, characteristic
7#7164C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3170C2211yescharacteristic
10#2986—21no—
11#2254C411yescharacteristic, complemented
12#2247C411yescharacteristic, complemented
13#2241C411yescharacteristic, complemented
14#2241C411yescharacteristic, complemented
15#2219—22no—
16#2176—22no—
17#2163C2211yescharacteristic
18C4×M32C2211yescharacteristic
19#1986—21no—
20C4×#106C2211yescharacteristic
21#1593—21no—
22#1584—22no—
23#1579—21no—
24#1564C2211yescharacteristic
25#1503—21no—
26#1485—21no—
27#1451C2211yescharacteristic
28#1348—22no—
29#1334—21no—
30#1334—21no—
31#533—21no—
32#533—21no—
33C2×Q32—22no—
34C2×M32—21no—
35#500C2×C411yescharacteristic
36#500C2×C411yescharacteristic
37#494C2×C411yescharacteristic
38#494C2×C411yescharacteristic
39#487—21no—
40#487—21no—
41#467—21no—
42#467—21no—
43#467—21no—
44#467D412yes—
45#446—22no—
46#446—41no—
47C8×Q8—22no—
48C8×Q8—41no—
49C4×Q16—21no—
50C4×Q16—21no—
51C4×Q16—21no—
52C4×Q16D412yes—
53#433—42no—
54#431—21no—
55#431D411yescharacteristic
56C2×#106—21no—
57C4×M16D411yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C4×C4⋊C4—21no—
60#367—22no—
61C4⋊C16—21no—
62#358—22no—
63#358—22no—
64C22⋊C16—21no—
65#346—21no—
66#346—21no—
67C4×C16C2×C411yescharacteristic
68C4×C16C2×C411yescharacteristic
69#340—21no—
70#339—42no—
71#335—21no—
72#334D411yescharacteristic
73#334D411yescharacteristic
74#328—42no—
75Q8⋊C8—22no—
76Q8⋊C8—22no—
77#322—21no—
78#322—21no—
79C2×Q16—21no—
80C2×Q16—21no—
81C2×Q16—21no—
82C2×Q16C22⋊C412yes—
83C2×M16—21no—
84C22×C8C2×D411yescharacteristic
85C22×C8#4111yescharacteristic
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90#125—41no—
91#125—41no—
92C22⋊Q8—22no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C4×Q8—22no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100#117—41no—
101#117—41no—
102C2×C4⋊C4—41no—
103C2×C42—41no—
104C2×C42C2×D411yescharacteristic
105Q32—44no—
106M32—42no—
107C2×C16—21no—
108C2×C16—21no—
109C2×C16—41no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107C22⋊C412yes—
114#106—22no—
115#106—41no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121#103—22no—
122#103—22no—
123#103—41no—
124#103—41no—
125C22⋊C8—41no—
126C22⋊C8D812yes—
127#97—21no—
128#97—42no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—42no—
132C4×C8—41no—
133C4×C8C22×C411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8#4111yescharacteristic
137#95—41no—
138C2×Q8—22no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C22×C4—81no—
143C22×C4#12111yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146Q16—42no—
147Q16—42no—
148Q16—41no—
149Q16—41no—
150Q16—41no—
151Q16—41no—
152M16—42no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8C2×C22⋊C411yescharacteristic
161C2×C8C4×D411yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8#12311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C22⋊C4—42no—
179C22⋊C4—81no—
180C42—42no—
181C42—82no—
182C42—81no—
183C42C4×D411yescharacteristic
184C42#12011yescharacteristic
185C42#13511yescharacteristic
186C42AGL(1,ℤ/8)11yescharacteristic
187C16—22no—
188C16—41no—
189C16—81no—
190C23#45911yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—82no—
194Q8—81no—
195Q8—81no—
196Q8—161no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—161no—
207C2×C4—161no—
208C2×C4#38611yescharacteristic
209C2×C4C4×D811yescharacteristic
210C2×C4#44211yescharacteristic
211C2×C4#44711yescharacteristic
212C2×C4#44911yescharacteristic
213C2×C4#46311yescharacteristic
214C8—21no—
215C8—21no—
216C8—21no—
217C8—42no—
218C8—81no—
219C8—81no—
220C8—81no—
221C8#44311yescharacteristic
222C8#44411yescharacteristic
223C22—21no—
224C22—21no—
225C22—21no—
226C22#187911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—82no—
232C4—82no—
233C4—162no—
234C4—161no—
235C4—161no—
236C4#187611yescharacteristic
237C4#188411yescharacteristic
238C4#218012yes—
239C2—41no—
240C2#905311yescharacteristic
241C2#1303411yescharacteristic
242C2#1303911yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#459
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2254C4semidirect
12. #2247C4semidirect
13. #2241C4semidirect
14. #2241C4semidirect

Maximal subgroups

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

subgroupindex
#337492
#180902
#117452
#116532
#97202
#71742
#71642