Finite Group Databasebeta

← #134459

Subgroups — #134459

Order 512.

375 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 98 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#21603C211yesmaximal, characteristic
2#11721C212yesmaximal
3#10867C211yesmaximal, characteristic
4#8601C211yesmaximal, characteristic
5#7253C212yesmaximal
6C2×#446C411yescharacteristic
7C22×#108—21no—
8C2×#405C2211yescharacteristic
9#2396C2211yescharacteristic
10#1985C411yescharacteristic
11#1940C411yescharacteristic
12#1917—21no—
13#1851—21no—
14#1798C2211yescharacteristic
15#1771—21no—
16#1770C411yescharacteristic
17#1752—21no—
18#1742C2212yes—
19C2×#338—21no—
20#1564—21no—
21#1458C2212yes—
22#1354—22no—
23#1353—22no—
24#1275—21no—
25C22×M16—22no—
26C22×M16D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×Q8C2×C411yescharacteristic
29C2×#117C2311yescharacteristic
30#446—41no—
31#446—41no—
32#431—22no—
33C2×#108—24no—
34C2×#108—41no—
35C2×#108—41no—
36C2×#108—42no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42#405—41no—
43#405—41no—
44C2×#97—21no—
45C2×#97—21no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#400D411yescharacteristic
51#399Q811yescharacteristic
52#398—21no—
53#396—21no—
54#395D411yescharacteristic
55#393Q811yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2×C411yescharacteristic
58C4×C22⋊C4—21no—
59#343—22no—
60#343—22no—
61#338—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#335—41no—
68#334—41no—
69#324—22no—
70#323—22no—
71C22×Q8C4⋊C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—24no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—42no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8SD1612yes—
94C22×C8Q1612yes—
95C4×Q8—41no—
96C4×Q8—41no—
97#117—41no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C4⋊C4C2×D411yescharacteristic
105C2×C4⋊C4#4111yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×Q811yescharacteristic
109C2×C22⋊C4#4111yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114#108—44no—
115#108—44no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—42no—
124#97—41no—
125#97—41no—
126#97—41no—
127#97—41no—
128#97—42no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—42no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95C4⋊C411yescharacteristic
139#95C4⋊C411yescharacteristic
140C24C22⋊Q811yescharacteristic
141C2×Q8—41no—
142C2×Q8—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4#11711yescharacteristic
154C22×C4#12111yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4C22⋊Q811yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158C22×C4C2×Q1611yescharacteristic
159C22×C4AGL(1,ℤ/8)11yescharacteristic
160C22×C4#13711yescharacteristic
161M16—42no—
162M16—42no—
163M16—44no—
164M16—44no—
165M16—82no—
166M16—82no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—44no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8#9911yescharacteristic
190C2×C8#9911yescharacteristic
191C2×C8#10312yes—
192C2×C8#10312yes—
193C2×C8#10412yes—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—81no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—41no—
213C23—21no—
214C23—21no—
215C23—41no—
216C23—41no—
217C23#38011yescharacteristic
218C23#47511yescharacteristic
219C23#47711yescharacteristic
220Q8—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4C2×#9911yescharacteristic
252C2×C4#41011yescharacteristic
253C2×C4C2×#10311yescharacteristic
254C2×C4#41711yescharacteristic
255C2×C4C2×#10411yescharacteristic
256C2×C4#42111yescharacteristic
257C2×C4#46011yescharacteristic
258C2×C4#46211yescharacteristic
259C8—22no—
260C8—41no—
261C8—41no—
262C8—84no—
263C8—81no—
264C8—81no—
265C8—162no—
266C8#32812yes—
267C8#32812yes—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22#179911yescharacteristic
278C22#180311yescharacteristic
279C22#180611yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4C2×#32811yescharacteristic
291C4#150911yescharacteristic
292C4#151311yescharacteristic
293C4#151811yescharacteristic
294C2—21no—
295C2—21no—
296C2—41no—
297C2—41no—
298C2#857211yescharacteristic
299C2#1280611yescharacteristic
300C2#1281011yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#216032
#117212
#108672
#86012
#72532