Finite Group Databasebeta

← #129391

Subgroups — #129391

Order 512.

603 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2962C211yesmaximal, characteristic
2#12848C212yesmaximal
3#11444C212yesmaximal
4#11440C211yesmaximal, characteristic
5C2×#1322C211yesmaximal, characteristic
6C2×#533C411yescharacteristic
7#2962—24no—
8#2962—24no—
9C2×#446—21no—
10C2×#446—21no—
11C2×#446—21no—
12C2×C8×Q8—21no—
13C2×#424C411yescharacteristic
14C2×#424C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#405C411yescharacteristic
17C2×#405C2211yescharacteristic
18C2×C4×M16C411yescharacteristic
19#2112C2212yes—
20C2×C22⋊C16C2212yes—
21#1379—24no—
22#1323—24no—
23#1322—21no—
24#1322—21no—
25#1322—21no—
26#1322—21no—
27C22×M16C2×C411yescharacteristic
28C22×M16C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C23×C8C2311yescharacteristic
31#533—22no—
32#533—22no—
33#533—22no—
34#533—22no—
35C2×C4⋊Q8C811yescharacteristic
36C2×C4⋊Q8C811yescharacteristic
37C2×#125C811yescharacteristic
38C2×#125C811yescharacteristic
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C2×M32—22no—
47C22×C16—22no—
48#446—28no—
49#446—28no—
50#446—28no—
51C8×Q8—28no—
52#424—22no—
53#424—22no—
54#424—22no—
55#424D414yes—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66#405—22no—
67#405—22no—
68#405—22no—
69#405D414yes—
70C4×M16—22no—
71C4×M16—22no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80C22⋊C16—22no—
81C22⋊C16—22no—
82C22×Q8—21no—
83C22×Q8—21no—
84C23×C4M1611yescharacteristic
85C23×C4M1611yescharacteristic
86C23×C4C22×C411yescharacteristic
87C2×M16—21no—
88C2×M16—24no—
89C2×M16—21no—
90C2×M16—42no—
91C2×M16C22⋊C412yes—
92C2×M16C22⋊C412yes—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D412yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109#125—22no—
110#125—22no—
111#125—22no—
112#125—22no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C22⋊Q8—24no—
116C22⋊Q8—24no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—24no—
121#117—21no—
122#117—21no—
123#117—21no—
124#117—21no—
125#117C22⋊C412yes—
126#117C22⋊C412yes—
127#117M1612yes—
128#117M1612yes—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4C2×C811yescharacteristic
136C2×C4⋊C4C2×C811yescharacteristic
137C2×C4⋊C4C2×C811yescharacteristic
138C2×C4⋊C4C2×C811yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42C22⋊C411yescharacteristic
148C2×C42C22⋊C411yescharacteristic
149C2×C42C2×C811yescharacteristic
150C2×C42C2×C811yescharacteristic
151M32—44no—
152C2×C16—44no—
153C2×C16—42no—
154C2×C16—42no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—24no—
162C4⋊C8—24no—
163#97—24no—
164#97—24no—
165#97—24no—
166C4×C8—24no—
167C4×C8—24no—
168C4×C8—24no—
169C24C2×M1611yescharacteristic
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—82no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—81no—
188C22×C4C22⋊C811yescharacteristic
189C22×C4C22⋊C811yescharacteristic
190C22×C4C22⋊C811yescharacteristic
191C22×C4C22⋊C811yescharacteristic
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C22×C811yescharacteristic
196C22×C4C2×M1611yescharacteristic
197C22×C4C2×M1611yescharacteristic
198M16—24no—
199M16—22no—
200M16—42no—
201M16—42no—
202M16#10114yes—
203C2×C8—24no—
204C2×C8—22no—
205C2×C8—24no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8#12012yes—
218C2×C8#12012yes—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4C22⋊C812yes—
230C4⋊C4C22⋊C812yes—
231C4⋊C4C22⋊C812yes—
232C4⋊C4C22⋊C812yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—22no—
249C16—82no—
250C16—82no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—41no—
255C23C2×C22⋊C811yescharacteristic
256C23#40711yescharacteristic
257C23#40711yescharacteristic
258Q8—48no—
259Q8—84no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#35012yes—
291C2×C4#37911yescharacteristic
292C2×C4C2×C22⋊C811yescharacteristic
293C2×C4C2×C22⋊C811yescharacteristic
294C2×C4#40711yescharacteristic
295C2×C4#41012yes—
296C2×C4C2×#10112yes—
297C8—22no—
298C8—22no—
299C8—24no—
300C8—41no—
301C8—41no—
302C8—42no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#146712yes—
313C22#177911yescharacteristic
314C22C2×#35011yescharacteristic
315C22#211711yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4#146112yes—
327C4#178211yescharacteristic
328C4#178411yescharacteristic
329C2—21no—
330C2—21no—
331C2—22no—
332C2—41no—
333C2#830311yescharacteristic
334C2#1268311yescharacteristic
335C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#29622
#128482
#114442
#114402
C2×#13222