Finite Group Databasebeta

← #215674

Subgroups — #215674

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2#21435C212yesmaximal
3#9895C212yesmaximal, complemented
4#9894C212yesmaximal, complemented
5#9891C212yesmaximal, complemented
6#9890C212yesmaximal, complemented
7#8722C211yesmaximal, characteristic
8#8722C211yesmaximal, characteristic
9#8718C212yesmaximal
10C2×#544C412yes—
11#2862C2212yes—
12#2859C2212yes—
13#2660C412yes—
14#2660C412yes—
15#2550—22no—
16#2550—22no—
17#2549—22no—
18#2549—22no—
19#2317C412yes—
20#2285C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1924—22no—
24#1923—22no—
25#1917—22no—
26#1916—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1749C2212yes—
30#1749C2212yes—
31#1745C2212yes—
32C4×#97C2212yes—
33#1295C2212yes—
34#1295C2212yes—
35#1295C2212yes—
36#1295C2212yes—
37#1295C2211yescharacteristic
38#1295C2212yes—
39#1295C2211yescharacteristic
40#1295C2212yes—
41#1295C2211yescharacteristic
42#1295C2211yescharacteristic
43#544—48no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C412yes—
47C2×#125—22no—
48C2×#125C2×C412yes—
49C2×#125C2×C412yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×#117C2×C412yes—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×#97C2312yes—
63C2×#97C2312yes—
64C2×#97C2312yes—
65C2×#97C2312yes—
66C2×C4×C8—22no—
67#398—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#395—22no—
72#395—22no—
73#395—22no—
74#395—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#385—22no—
86#385—22no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#382—22no—
91#382C2×C412yes—
92#382C2×C412yes—
93#382C2×C411yescharacteristic
94#382C2×C411yescharacteristic
95#382C2311yescharacteristic
96#382C2312yes—
97#382C2311yescharacteristic
98#380—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104#376C2×C412yes—
105C2×#95—22no—
106C43C2312yes—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C23×C4—22no—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D412yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—48no—
125#125—44no—
126#125—44no—
127#125—48no—
128C22⋊Q8—48no—
129C22⋊Q8—48no—
130#117—24no—
131#117—24no—
132#117—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C42—21no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—22no—
162C2×C42C22×C412yes—
163C2×C42C22×C412yes—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2411yescharacteristic
171C4⋊C8—44no—
172C4⋊C8—44no—
173C22⋊C8—44no—
174C22⋊C8—44no—
175#97—22no—
176#97—22no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—44no—
184C4×C8—44no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C22⋊C412yes—
196#95C22⋊C412yes—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199C24C2×C22⋊C411yescharacteristic
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—82no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C23×C411yescharacteristic
228C22×C4C22×D412yes—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4#10114yes—
261C22⋊C4—22no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C22⋊C4#10014yes—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—24no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—44no—
276C422−1+411yescharacteristic
277C422−1+412yes—
278C422−1+411yescharacteristic
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23C2×#10012yes—
284C23C22×C22⋊C411yescharacteristic
285Q8—164no—
286Q8—164no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—164no—
323C2×C4C2×#10014yes—
324C2×C4C2×#10114yes—
325C2×C4C2×#10112yes—
326C2×C4#51911yescharacteristic
327C2×C4#51911yescharacteristic
328C2×C4#53612yes—
329C2×C4#53612yes—
330C8—44no—
331C8—44no—
332C8—82no—
333C8—82no—
334C22—24no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22#231011yescharacteristic
340C22C22×#10011yescharacteristic
341C22C22×#10111yescharacteristic
342C22#290112yes—
343C22#290112yes—
344C4—24no—
345C4—24no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—82no—
355C4—82no—
356C4—82no—
357C4—82no—
358C2—42no—
359C2#1768512yes—
360C2#1768911yescharacteristic
361C2#1769211yescharacteristic
362C2#2182911yescharacteristic
363C2#2184612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9895C2semidirect
4. #9894C2semidirect
5. #9891C2semidirect
6. #9890C2semidirect

Maximal subgroups

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

subgroupindex
#353302
#214352
#98952
#98942
#98912
#98902
#87222
#87182