Finite Group Databasebeta

← #136039

Subgroups — #136039

Order 512.

438 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 110 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#21961C211yesmaximal, characteristic
2#12920C212yesmaximal
3C2×#1322C211yesmaximal, characteristic
4#11232C211yesmaximal, characteristic
5#7258C212yesmaximal
6C22×#107C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2285—21no—
9C2×#367—22no—
10C2×C22⋊C16C2212yes—
11#1947—22no—
12#1947—22no—
13#1808—21no—
14#1808C2212yes—
15C2×#340—21no—
16C2×#340C2211yescharacteristic
17C4⋊#95—21no—
18#1380—22no—
19#1322C414yescomplemented
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117—41no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27C2×#107—22no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107—42no—
31C2×#107D412yes—
32C2×#107D412yes—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C412yes—
36C2×C4⋊C8Q812yes—
37#384—21no—
38#384—21no—
39#382—41no—
40#380—22no—
41#380—41no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—41no—
44C2×#95—41no—
45C2×#95—41no—
46#367—44no—
47C22⋊C16—22no—
48C22⋊C16C2×C414yes—
49#340—22no—
50#340—22no—
51#340—22no—
52#340—42no—
53C23×C4—41no—
54C23×C4—41no—
55C23×C4D811yescharacteristic
56C23×C4D811yescharacteristic
57C23×C4C2×D411yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8C4⋊C412yes—
65C22×C8C22×C411yescharacteristic
66C22×C8C2×D412yes—
67C22×C8C2×Q811yescharacteristic
68C22×C8#4112yes—
69#117—82no—
70#117—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—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—42no—
95C2×C42C4⋊C412yes—
96C2×C42Q1612yes—
97C2×C16—24no—
98C2×C16—42no—
99C2×C16—42no—
100#107—24no—
101#107—42no—
102#107—42no—
103#107—42no—
104#107—81no—
105#107—81no—
106C4⋊C8—24no—
107C4⋊C8—24no—
108#95—42no—
109#95—42no—
110#95—82no—
111#95—82no—
112C24C2×D811yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
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×C4—81no—
140C22×C4—81no—
141C22×C4#10712yes—
142C22×C4#10712yes—
143C22×C4SD3212yes—
144C22×C4SD3212yes—
145C22×C4C2×C4⋊C411yescharacteristic
146C22×C4#12011yescharacteristic
147C22×C4#12311yescharacteristic
148C22×C4C2×Q1611yescharacteristic
149C22×C4AGL(1,ℤ/8)11yescharacteristic
150C22×C4#13711yescharacteristic
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—81no—
161C2×C8C4×D412yes—
162C2×C8C22⋊Q812yes—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C22⋊C4—82no—
180C22⋊C4—82no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—84no—
186C16—44no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23C2×#10711yescharacteristic
192C23#44711yescharacteristic
193C23#48011yescharacteristic
194C23C2×SD3211yescharacteristic
195C23C2×SD3211yescharacteristic
196C23#50911yescharacteristic
197C23#51011yescharacteristic
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—161no—
232C2×C4—161no—
233C2×C4—161no—
234C2×C4—161no—
235C2×C4—161no—
236C2×C4—161no—
237C2×C4—161no—
238C2×C4—161no—
239C2×C4#36714yescomplemented
240C2×C4#38511yescharacteristic
241C2×C4#42811yescharacteristic
242C2×C4#45111yescharacteristic
243C2×C4#45411yescharacteristic
244C2×C4#45611yescharacteristic
245C2×C4B(4,𝔽2)11yescharacteristic
246C2×C4#45811yescharacteristic
247C2×C4#48411yescharacteristic
248C8—22no—
249C8—22no—
250C8—42no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22#159811yescharacteristic
264C22#161011yescharacteristic
265C22#185211yescharacteristic
266C22C2×#36711yescharacteristic
267C22#215911yescharacteristic
268C22#218511yescharacteristic
269C22#218711yescharacteristic
270C22#219011yescharacteristic
271C22#219111yescharacteristic
272C22#223311yescharacteristic
273C22#223511yescharacteristic
274C4—24no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—162no—
279C4—162no—
280C4—162no—
281C4—162no—
282C4#160511yescharacteristic
283C4#161711yescharacteristic
284C4#185711yescharacteristic
285C4#186711yescharacteristic
286C2—21no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2#885611yescharacteristic
291C2#1215911yescharacteristic
292C2#1216111yescharacteristic
293C2#1228911yescharacteristic
294C2#1229111yescharacteristic
295C2#1292011yescharacteristic
296C2#1292311yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1322C4semidirect
239. C2×C4#367semidirect

Maximal subgroups

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

subgroupindex
#219612
#129202
C2×#13222
#112322
#72582