Finite Group Databasebeta

← #144476

Subgroups — #144476

Order 512.

399 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 88 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#30291C211yesmaximal, characteristic
2#12866C212yesmaximal
3#11379C211yesmaximal, characteristic
4#7258C212yesmaximal
5#7183C211yesmaximal, characteristic
6C22×#107C2211yescharacteristic
7#2857—21no—
8C2×C22⋊C16C2212yes—
9#2083—22no—
10#2057—22no—
11#1808—21no—
12#1808C2212yes—
13C2×#340—21no—
14C2×#340C2211yescharacteristic
15C2×#336C2211yescharacteristic
16C4⋊#95—21no—
17#1380—22no—
18#1380—22no—
19C23×C8D411yescharacteristic
20C23×C8C2311yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C22⋊Q8—41no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27C2×#107—22no—
28C2×#107—22no—
29C2×#107—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107D412yes—
33C2×#107D412yes—
34C2×#107Q812yescomplemented
35#391—41no—
36#389—22no—
37#389—41no—
38#384—21no—
39#380—22no—
40#380—41no—
41C2×#95—41no—
42C2×#95—41no—
43C22⋊C16—22no—
44C22⋊C16—22no—
45#340—22no—
46#340—22no—
47#340—22no—
48#340—42no—
49#336—22no—
50C22×Q8—41no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4D811yescharacteristic
54C23×C4D811yescharacteristic
55C23×C4C2×D411yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8SD1612yes—
63C22×C8C2×D412yes—
64C22×C8C2×Q811yescharacteristic
65C22×C8#4112yes—
66C22×C8#4111yescharacteristic
67C4⋊Q8—81no—
68C4⋊Q8—81no—
69C22⋊Q8—82no—
70C22⋊Q8—82no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C16—42no—
95C2×C16—42no—
96C2×C16—42no—
97#107—24no—
98#107—42no—
99#107—42no—
100#107—42no—
101#107—81no—
102#107—81no—
103#95—42no—
104#95—42no—
105#95—82no—
106#95—82no—
107C24C2×D811yescharacteristic
108C2×Q8—82no—
109C2×Q8—82no—
110C2×Q8—82no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—81no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4D1612yes—
140C22×C4SD3212yes—
141C22×C4#12011yescharacteristic
142C22×C4C22⋊Q811yescharacteristic
143C22×C4#12411yescharacteristic
144C22×C4C2×SD1611yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146C22×C4#13711yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—81no—
157C2×C8C22⋊Q812yes—
158C2×C8#12412yes—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—84no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C22⋊C4—82no—
176C22⋊C4—82no—
177C42—41no—
178C42—41no—
179C42—81no—
180C16—82no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23#44711yescharacteristic
186C23#47411yescharacteristic
187C23#48611yescharacteristic
188C23C2×D1611yescharacteristic
189C23C2×SD3211yescharacteristic
190C23#50911yescharacteristic
191C23#51011yescharacteristic
192Q8—164no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—161no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4—161no—
229C2×C4—161no—
230C2×C4—161no—
231C2×C4—161no—
232C2×C4—161no—
233C2×C4#39311yescharacteristic
234C2×C4#44811yescharacteristic
235C2×C4#45511yescharacteristic
236C2×C4#45611yescharacteristic
237C2×C4#45811yescharacteristic
238C2×C4#45811yescharacteristic
239C2×C4#47611yescharacteristic
240C2×C4#48911yescharacteristic
241C8—22no—
242C8—22no—
243C8—42no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22#160711yescharacteristic
257C22#161011yescharacteristic
258C22#202311yescharacteristic
259C22#218411yescharacteristic
260C22#218511yescharacteristic
261C22#219111yescharacteristic
262C22#219111yescharacteristic
263C22#222411yescharacteristic
264C22#222611yescharacteristic
265C22#224011yescharacteristic
266C22#224311yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—162no—
272C4—162no—
273C4—162no—
274C4—162no—
275C4#160811yescharacteristic
276C4#161111yescharacteristic
277C4#202811yescharacteristic
278C4#203011yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—21no—
283C2#1034211yescharacteristic
284C2#1227811yescharacteristic
285C2#1227911yescharacteristic
286C2#1228811yescharacteristic
287C2#1228911yescharacteristic
288C2#1336311yescharacteristic
289C2#1336811yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
#302912
#128662
#113792
#72582
#71832