Finite Group Databasebeta

← #139673

Subgroups — #139673

Order 512.

414 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 82 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#33487C211yesmaximal, characteristic, complemented
2#28140C211yesmaximal, characteristic
3#11667C212yesmaximal, complemented
4#8622C211yesmaximal, characteristic, complemented
5#7222C212yesmaximal, complemented
6C2×C4×D8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2537—21no—
9#2168—24no—
10#2163C2212yes—
11#2112—21no—
12C2×C22⋊C16—21no—
13#2075C411yescharacteristic
14#2075C411yescharacteristic
15#1999C411yescharacteristic
16#1999C411yescharacteristic
17#1893—21no—
18#1893—21no—
19#1834C2211yescharacteristic
20#1802—21no—
21#1568C2212yes—
22#1466—21no—
23#1333—24no—
24#1290—21no—
25C22×D8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121—21no—
29C2×#121—21no—
30C2×#121—41no—
31C2×C4×D4—21no—
32C2×C4×D4D411yescharacteristic
33C2×M32—22no—
34C22×C16—22no—
35C4×D8—22no—
36C4×D8—22no—
37C8×D4—22no—
38C8×D4—22no—
39#432—22no—
40C2×#107C2×C411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48#394—41no—
49#394—41no—
50#384—21no—
51#380—21no—
52#380—21no—
53#380—41no—
54C4⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58#346D414yescomplemented
59#340—21no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64#335—22no—
65#334—22no—
66#323—41no—
67#323—41no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4SD1611yescharacteristic
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C2×D8—22no—
76C2×D8—42no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—81no—
82C22×C8—81no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22×C411yescharacteristic
86C22×C8C2×D411yescharacteristic
87#121—42no—
88#121—42no—
89#121—81no—
90#121—81no—
91#121—84no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—42no—
95C4×D4—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4SD1611yescharacteristic
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4D811yescharacteristic
108C2×C22⋊C4D811yescharacteristic
109C2×C42C2×D411yescharacteristic
110M32—82no—
111C2×C16—24no—
112C2×C16—82no—
113#107—22no—
114C4⋊C8—22no—
115C4⋊C8—42no—
116#102—41no—
117#102—41no—
118#102—81no—
119#102—81no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—42no—
123#97—42no—
124C4×C8C2×D412yes—
125C4×C8#4112yes—
126#95—41no—
127#95—41no—
128C24—41no—
129C24#10211yescharacteristic
130C24#10211yescharacteristic
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4—161no—
145C2×D4—161no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4C2×SD1611yescharacteristic
165D8—24no—
166D8—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—82no—
178C2×C8C4×D412yes—
179C2×C8#12011yescharacteristic
180C2×C8#12311yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—82no—
189C4⋊C4—161no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C42#12112yes—
198C42#13512yes—
199C16—44no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23#36112yes—
217C23C2×#10211yescharacteristic
218D4—44no—
219D4—42no—
220D4—42no—
221D4—81no—
222D4—81no—
223D4—164no—
224D4—164no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—164no—
250C2×C4#38611yescharacteristic
251C2×C4#41611yescharacteristic
252C2×C4#46512yes—
253C2×C4#46612yes—
254C8—22no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—162no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22#189311yescharacteristic
280C22C2×#36111yescharacteristic
281C22#214811yescharacteristic
282C4—22no—
283C4—21no—
284C4—22no—
285C4—41no—
286C4—41no—
287C4—161no—
288C4—161no—
289C4—161no—
290C4#189111yescharacteristic
291C4#191111yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—81no—
296C2—162no—
297C2#948511yescharacteristic
298C2#1306711yescharacteristic
299C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#28140C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33487C2semidirect
3. #11667C2semidirect
4. #8622C2semidirect
5. #7222C2semidirect
58. #346D4semidirect

Maximal subgroups

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

subgroupindex
#334872
#281402
#116672
#86222
#72222