Finite Group Databasebeta

← #137713

Subgroups — #137713

Order 512.

427 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 96 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#33419C211yesmaximal, characteristic, complemented
2#21051C211yesmaximal, characteristic
3#12847C211yesmaximal, characteristic, complemented
4#11421C212yesmaximal
5#7184C212yesmaximal, complemented
6C2×#423C411yescharacteristic
7C2×#408C411yescharacteristic
8C2×#408C2211yescharacteristic
9C2×#405C2211yescharacteristic
10#2297C411yescharacteristic
11#2164—24no—
12#2150—21no—
13#2112C2212yes—
14#2112C2212yes—
15C4×M32—21no—
16#1924—21no—
17#1924—21no—
18#1923—21no—
19#1923—21no—
20#1729C411yescharacteristic
21C2×#336C2211yescharacteristic
22#1377—21no—
23#1377—21no—
24#1315—24no—
25C22×#41C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#117C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×M32—22no—
33C2×M32—22no—
34#423—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37#408—22no—
38#408—22no—
39#408—42no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44#405—22no—
45#405—22no—
46C2×#97D411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391C811yescharacteristic
51#391C811yescharacteristic
52#390C811yescharacteristic
53#390C811yescharacteristic
54#386—21no—
55#386—21no—
56C4×C22⋊C4—21no—
57C4×C22⋊C4—21no—
58C4⋊C16—24no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62C22⋊C16D414yes—
63#346—22no—
64C4×C16—22no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41—41no—
75C22×Q8C2×C811yescharacteristic
76C22×D4—21no—
77C22×D4C2×C811yescharacteristic
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22×C411yescharacteristic
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D412yes—
99C22×C8#4112yes—
100#117—22no—
101#117—22no—
102C2×C4⋊C4C2×C811yescharacteristic
103C2×C4⋊C4C2×C811yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4M1611yescharacteristic
109C2×C22⋊C4M1611yescharacteristic
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42C22⋊C411yescharacteristic
115C2×C42C2×C811yescharacteristic
116C2×C42C2×C811yescharacteristic
117M32—42no—
118M32—42no—
119M32—42no—
120M32—42no—
121C2×C16—24no—
122C2×C16—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—41no—
128C22⋊C8—42no—
129C22⋊C8—41no—
130#97—22no—
131#97—22no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140C24—21no—
141C24C2×M1611yescharacteristic
142#41—82no—
143#41—82no—
144#41—82no—
145#41—82no—
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×D4—22no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—81no—
171C22×C4C22⋊C811yescharacteristic
172C22×C4C22⋊C811yescharacteristic
173C22×C4C22⋊C811yescharacteristic
174C22×C4C22⋊C811yescharacteristic
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C4×D412yes—
177C22×C4C22×C811yescharacteristic
178C22×C4#13112yes—
179M16—44no—
180M16—82no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—82no—
199C2×C8#10012yes—
200C2×C8#12011yescharacteristic
201C2×C8#12112yes—
202C2×C8#12311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—41no—
218C42—41no—
219C16—44no—
220C16—82no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C2×C22⋊C811yescharacteristic
233C23#43512yes—
234Q8—82no—
235D4—82no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—84no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#38611yescharacteristic
266C2×C4#40811yescharacteristic
267C2×C4#41011yescharacteristic
268C2×C4#41011yescharacteristic
269C2×C4C2×#10011yescharacteristic
270C2×C4#41311yescharacteristic
271C2×C4C8×D412yes—
272C8—21no—
273C8—22no—
274C8—21no—
275C8—22no—
276C8—41no—
277C8—81no—
278C8—81no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22#145811yescharacteristic
294C22#146011yescharacteristic
295C22#186911yescharacteristic
296C4—22no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—81no—
304C4#146911yescharacteristic
305C4#147311yescharacteristic
306C4#189911yescharacteristic
307C4#190011yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—161no—
312C2#908011yescharacteristic
313C2#1298211yescharacteristic
314C2#1298811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33419C2semidirect
3. #12847C2semidirect
5. #7184C2semidirect

Maximal subgroups

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

subgroupindex
#334192
#210512
#128472
#114212
#71842