Finite Group Databasebeta

← #138973

Subgroups — #138973

Order 512.

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

normal subgroups only

This group has 370 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
1C2×#2236C211yesmaximal, characteristic
2#22612C211yesmaximal, characteristic
3#12867C211yesmaximal, characteristic
4#11887C211yesmaximal, characteristic
5#9782C211yesmaximal, characteristic
6#7230C211yesmaximal, characteristic
7#7199C211yesmaximal, characteristic
8C2×#467C2211yescharacteristic
9C22×#107C2211yescharacteristic
10#2345—21no—
11#2236—22no—
12#2236—21no—
13#2236—21no—
14C2×#366C412yescomplemented
15C2×#358C412yescomplemented
16C2×C22⋊C16C2211yescharacteristic
17C2×C22⋊C16C2211yescharacteristic
18#1996—21no—
19#1942—21no—
20#1921—22no—
21#1824—21no—
22#1808C2211yescharacteristic
23#1808C2211yescharacteristic
24C4×#107—21no—
25C8×C22⋊C4C2211yescharacteristic
26#1520—21no—
27#1380—22no—
28#1348—22no—
29C22×Q16C2×C411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×C22⋊Q8—21no—
32C22×C4⋊C4—21no—
33C22×C16—21no—
34C22×C16C2×C412yes—
35#467—21no—
36#467—21no—
37#467—21no—
38#467—21no—
39C2×#107—21no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#107—21no—
43C2×#107—21no—
44C2×#107C2×C411yescharacteristic
45C2×#107C2×C412yes—
46C2×#107D412yes—
47C2×#107D411yescharacteristic
48C2×#107D411yescharacteristic
49C2×#103—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×C4×C8—21no—
52#389—22no—
53#387—42no—
54#380—21no—
55#380—21no—
56#380—41no—
57C4×C4⋊C4—22no—
58C4×C22⋊C4—41no—
59C4×C22⋊C4D411yescharacteristic
60#366—22no—
61#366—22no—
62#358—22no—
63#358—22no—
64C22⋊C16—21no—
65C22⋊C16—21no—
66C22⋊C16—21no—
67C22⋊C16—21no—
68#342—41no—
69#342—41no—
70#340—22no—
71#340—22no—
72#340—21no—
73#340D412yes—
74#336—21no—
75#328—42no—
76C22×Q8—21no—
77C23×C4—41no—
78C23×C4C2×D411yescharacteristic
79C2×Q16—21no—
80C2×Q16—21no—
81C2×Q16—42no—
82C22×C8—21no—
83C22×C8—42no—
84C22×C8C22×C411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8#4111yescharacteristic
89C22×C8#4111yescharacteristic
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4D811yescharacteristic
110C2×C22⋊C4D811yescharacteristic
111C2×C42—21no—
112C2×C42—42no—
113C2×C16—22no—
114C2×C16—42no—
115C2×C16—42no—
116C2×C16—41no—
117#107—22no—
118#107—22no—
119#107—21no—
120#107—21no—
121#107—21no—
122#107—21no—
123#107—42no—
124#107—42no—
125#107C22⋊C412yes—
126#107C22⋊C412yes—
127#103—41no—
128#103—41no—
129C22⋊C8—21no—
130C22⋊C8—21no—
131C4×C8—42no—
132#95—21no—
133#95—42no—
134#95—42no—
135#95—41no—
136#95—41no—
137C24C2×D811yescharacteristic
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—81no—
141C2×Q8—81no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—81no—
154C22×C4—162no—
155C22×C4C4×D411yescharacteristic
156C22×C4#12011yescharacteristic
157C22×C4#12111yescharacteristic
158C22×C4#13511yescharacteristic
159C22×C4AGL(1,ℤ/8)11yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161Q16—42no—
162Q16—81no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8C2×C22⋊C411yescharacteristic
173C2×C8C4×D411yescharacteristic
174C2×C8#12111yescharacteristic
175C2×C8#12311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—162no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4Q3212yes—
201C42—41no—
202C42—41no—
203C42—162no—
204C16—42no—
205C16—81no—
206C23—41no—
207C23—41no—
208C23C4×D811yescharacteristic
209C23#44711yescharacteristic
210C23#45911yescharacteristic
211C23C2×Q3211yescharacteristic
212C23#50811yescharacteristic
213C23#51011yescharacteristic
214C23#51011yescharacteristic
215Q8—81no—
216Q8—81no—
217Q8—161no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—162no—
243C2×C4—162no—
244C2×C4—161no—
245C2×C4—161no—
246C2×C4—161no—
247C2×C4—161no—
248C2×C4—161no—
249C2×C4#38611yescharacteristic
250C2×C4#44211yescharacteristic
251C2×C4#44911yescharacteristic
252C2×C4#46311yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—81no—
257C8#41012yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22#187911yescharacteristic
269C22C4×Q3211yescharacteristic
270C22#217611yescharacteristic
271C22#218811yescharacteristic
272C22#218911yescharacteristic
273C22#220911yescharacteristic
274C22#221111yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—81no—
280C4—82no—
281C4—82no—
282C4—162no—
283C4—161no—
284C4—161no—
285C4—321no—
286C4#189911yescharacteristic
287C4#190511yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2#926011yescharacteristic
295C2#1301511yescharacteristic
296C2#1303311yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#366C4semidirect
15. C2×#358C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22362
#226122
#128672
#118872
#97822
#72302
#71992