Finite Group Databasebeta

← #138539

Subgroups — #138539

Order 512.

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

normal subgroups only

This group has 401 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#29770C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic, complemented
3#11895C212yesmaximal
4#11440C211yesmaximal, characteristic, complemented
5#7235C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7#2920—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2854—21no—
10#2112C2212yes—
11#2086C411yescharacteristic
12#2086C411yescharacteristic
13#2067—21no—
14#2067—21no—
15#2054C411yescharacteristic
16#2054C411yescharacteristic
17#1950—24no—
18#1893—21no—
19#1810—21no—
20#1810C2212yes—
21#1808C2211yescharacteristic
22#1349—24no—
23#1322—21no—
24#1322—21no—
25C22×D8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#123—41no—
29C2×#121—21no—
30C22×C42D411yescharacteristic
31C2×M32—22no—
32#466—22no—
33#466—22no—
34#432—41no—
35#432—41no—
36#431—41no—
37#431—41no—
38C2×#108—22no—
39C2×#108—22no—
40C2×#107—21no—
41C2×#107C2×C411yescharacteristic
42#423—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×#102—21no—
50C2×#102—21no—
51C2×#102—21no—
52#407—22no—
53C4×M16—22no—
54#390—21no—
55#390—21no—
56#389—21no—
57#389—21no—
58#388—41no—
59#388—41no—
60#386—41no—
61#380—21no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#341—22no—
65#341D414yes—
66#340—21no—
67#340—21no—
68#340C2×C411yescharacteristic
69#340C2×C411yescharacteristic
70C22×D4—21no—
71C22×D4—41no—
72C23×C4SD1611yescharacteristic
73C23×C4SD1611yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×D8—22no—
76C2×D8—81no—
77C2×M16—24no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—81no—
81C22×C8—81no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22×C411yescharacteristic
85C22×C8C2×D412yes—
86C22×C8C2×D411yescharacteristic
87C22×C8#4112yes—
88#123—84no—
89#121—41no—
90#121—41no—
91#121—41no—
92#121—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C22⋊C411yescharacteristic
109C2×C42C22⋊C411yescharacteristic
110C2×C42D811yescharacteristic
111C2×C42D811yescharacteristic
112M32—82no—
113C2×C16—42no—
114#108—82no—
115#108—82no—
116#107—22no—
117#107—81no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124#102—42no—
125#102—81no—
126#102—81no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—42no—
130C4×C8—42no—
131#95—41no—
132#95—41no—
133#95—41no—
134C24—41no—
135C24C2×SD1611yescharacteristic
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4—81no—
145C2×D4—81no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—81no—
158C22×C4#10211yescharacteristic
159C22×C4#10211yescharacteristic
160C22×C4#10211yescharacteristic
161C22×C4#10211yescharacteristic
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4#12011yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4AGL(1,ℤ/8)11yescharacteristic
167C22×C4#13711yescharacteristic
168D8—82no—
169M16—44no—
170M16—44no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—44no—
181C2×C8—81no—
182C2×C8C4×D412yes—
183C2×C8#12112yes—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C16—82no—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—41no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23C2×#10211yescharacteristic
220C23#44811yescharacteristic
221C23#48111yescharacteristic
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—162no—
227D4—162no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#36112yes—
260C2×C4#38611yescharacteristic
261C2×C4#41711yescharacteristic
262C2×C4#44711yescharacteristic
263C2×C4#48011yescharacteristic
264C8—22no—
265C8—22no—
266C8—41no—
267C8—41no—
268C8—84no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22—161no—
286C22#187411yescharacteristic
287C22C2×#36111yescharacteristic
288C22#214811yescharacteristic
289C4—22no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—161no—
297C4—161no—
298C4—161no—
299C4#191211yescharacteristic
300C4#191311yescharacteristic
301C2—21no—
302C2—21no—
303C2—41no—
304C2—41no—
305C2—321no—
306C2#920411yescharacteristic
307C2#1299811yescharacteristic
308C2#1300411yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
2. #21969C2semidirect
4. #11440C2semidirect
5. #7235C2semidirect

Maximal subgroups

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

subgroupindex
#297702
#219692
#118952
#114402
#72352