Finite Group Databasebeta

← #224463

Subgroups — #224463

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36565C211yesmaximal, characteristic
2#21346C212yesmaximal
3#9895C212yesmaximal
4#9894C212yesmaximal
5#9890C212yesmaximal
6#8584C211yesmaximal, characteristic
7#8584C211yesmaximal, characteristic
8#8580C211yesmaximal, characteristic
9#8225C211yesmaximal, characteristic
10#8225C211yesmaximal, characteristic
11C4×#330C211yesmaximal, characteristic
12#2862C2211yescharacteristic
13#2862C2211yescharacteristic
14#2859C2211yescharacteristic
15#2586—22no—
16#2586—22no—
17#2551C412yes—
18#2550C412yes—
19#2550C412yes—
20#2451—22no—
21#2369—22no—
22#2355C412yes—
23C4×#125C2211yescharacteristic
24C4×#125C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1985—22no—
28#1940—22no—
29#1916—22no—
30#1916—22no—
31#1767C2212yes—
32#1767C2212yes—
33#1767C2212yes—
34#1745C2212yes—
35#1745C2212yes—
36C4×#97C2212yes—
37C2×#330C2211yescharacteristic
38C2×#330C2211yescharacteristic
39C2×#330C2211yescharacteristic
40C2×#330C2211yescharacteristic
41#1295C2212yes—
42#1295C2212yes—
43#1295C2212yes—
44#1295C2212yes—
45#1295C2212yes—
46#1295C2212yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×#125C2×C412yes—
51C2×#125C2311yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×#97—22no—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×#97C2312yes—
61C2×#97C2312yes—
62C2×C4×C8—22no—
63#398—22no—
64#398—22no—
65#395—22no—
66#395—22no—
67#395—22no—
68#395—22no—
69#391—22no—
70#391—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#384C2×C412yes—
90#382C2×C412yes—
91#382C2×C412yes—
92#382C2311yescharacteristic
93#382C2311yescharacteristic
94#382C2311yescharacteristic
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C43C2311yescharacteristic
109C43C2311yescharacteristic
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#330—22no—
119#330—22no—
120#330—22no—
121#330Q814yescomplemented
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22⋊C411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—48no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132#125—22no—
133#125—82no—
134#125C2×Q812yes—
135#125#4112yes—
136C4×Q8—24no—
137C4×Q8—44no—
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—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—44no—
175C4⋊C8—44no—
176C4⋊C8—44no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—44no—
185#97C2×Q814yes—
186C4×C8—44no—
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#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D412yes—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4#10114yes—
259C4⋊C4C4×Q814yes—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—42no—
271C42—42no—
272C42—44no—
273C42—82no—
274C42—82no—
275C42C22×Q811yescharacteristic
276C42C2×#4111yescharacteristic
277C422−1+412yes—
278C23C22×C22⋊C411yescharacteristic
279Q8—44no—
280Q8—164no—
281Q8—164no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
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—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4C2×#10112yes—
316C2×C4C2×#10112yes—
317C2×C4C2×#10112yes—
318C2×C4#42114yes—
319C2×C4C2×C4×Q811yescharacteristic
320C2×C4#51911yescharacteristic
321C2×C4#54812yes—
322C2×C4D4×Q812yes—
323C8—44no—
324C8—44no—
325C8—82no—
326C8—82no—
327C22Q8×C22⋊C411yescharacteristic
328C22C22×#10111yescharacteristic
329C22#288811yescharacteristic
330C22C2×#42112yes—
331C22#290112yes—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—24no—
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—44no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—82no—
347C4—82no—
348C4—82no—
349C2#1766411yescharacteristic
350C2#1767912yes—
351C2#1768911yescharacteristic
352C2Q8×#10111yescharacteristic
353C2#2252912yes—
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
121. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365652
#213462
#98952
#98942
#98902
#85842
#85802
#82252
C4×#3302