Finite Group Databasebeta

← #480733

Subgroups — #480733

Order 512.

1034 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2546C211yesmaximal, characteristic
2#17365C212yesmaximal
3#11568C218yesmaximal, complemented
4#11506C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C412yes—
7C22×C4×Q8C2211yescharacteristic
8#2920—22no—
9#2546—22no—
10#2546—22no—
11#2546C414yes—
12#2546C2214yes—
13C2×#398—22no—
14C2×#391C2212yes—
15C2×#389—22no—
16C2×C4×C4⋊C4C411yescharacteristic
17#1467—22no—
18#1467C2214yes—
19#1463—22no—
20#1463C2214yes—
21#1461—22no—
22#1461C2214yes—
23#1319C22116yes—
24C23×Q8C2×C412yes—
25C23×Q8C2×C411yescharacteristic
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8—48no—
33C2×C4⋊Q8C2×C414yes—
34C2×C4⋊Q8C2×C418yes—
35C2×C4×Q8—42no—
36C2×C4×Q8—41no—
37C2×C4×Q8—41no—
38C2×C4×Q8C2×C412yes—
39C2×C4×Q8D412yes—
40C2×C4×Q8D412yes—
41C2×C4×Q8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C412yes—
50C22×C42D412yes—
51#432—24no—
52#432—42no—
53#431—24no—
54#431—42no—
55#423—42no—
56#407—24no—
57C4×M16—42no—
58#398—24no—
59#398—24no—
60#398—24no—
61#398—24no—
62#391—24no—
63#391C2×C414yes—
64#391C2314yes—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2×C414yes—
71C2×#95—21no—
72C2×#95—24no—
73#324—24no—
74#324—24no—
75#324—24no—
76#324C2318yes—
77C22×Q8—24no—
78C22×Q8—22no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—42no—
82C22×Q8—44no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8C22⋊C411yescharacteristic
87C22×Q8C22⋊C411yescharacteristic
88C22×Q8C22×C414yes—
89C22×Q8C22×C412yes—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—44no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100C4⋊Q8—44no—
101C4⋊Q8—44no—
102C4⋊Q8—48no—
103C4⋊Q8—48no—
104C4⋊Q8—416no—
105C4×Q8—42no—
106C4×Q8—42no—
107C4×Q8—42no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—42no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—416no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22×C411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2411yescharacteristic
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—44no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22×C412yes—
149C2×C42C22×C414yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C4⋊C8—44no—
153C4⋊C8—44no—
154C22⋊C8—28no—
155C22⋊C8—44no—
156#97—44no—
157C4×C8—44no—
158#95—24no—
159#95—28no—
160#95—28no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—24no—
163C2×Q8—22no—
164C2×Q8—24no—
165C2×Q8—28no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—48no—
172C2×Q8—48no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—88no—
179C2×Q8—88no—
180C2×Q8#10014yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4#9911yescharacteristic
206C22×C4#9912yes—
207C22×C4#9911yescharacteristic
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D412yes—
216M16—88no—
217C2×C8—48no—
218C4⋊C4—216no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—48no—
227C4⋊C4—416no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—416no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—44no—
244C42—44no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23C2×#9911yescharacteristic
250C23C2×#9911yescharacteristic
251C23C22×C22⋊C411yescharacteristic
252Q8—44no—
253Q8—48no—
254Q8—44no—
255Q8—88no—
256Q8—88no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—416no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#35614yes—
293C2×C4C2×#9912yes—
294C2×C4C2×#9912yes—
295C2×C4C2×#10012yes—
296C2×C4C2×#10014yes—
297C2×C4#56812yes—
298C8—88no—
299C22—24no—
300C22—24no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#212912yes—
308C22C2×#35612yes—
309C22#287911yescharacteristic
310C22C22×#9911yescharacteristic
311C22C22×#10011yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—24no—
316C4—48no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—84no—
322C4C2×#35614yes—
323C2—21no—
324C2—21no—
325C2—22no—
326C2—22no—
327C2#1739611yescharacteristic
328C2C2×#212911yescharacteristic
329C2C22×#35611yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11568C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#173652
#115682
#115062