Finite Group Databasebeta

← #164362

Subgroups — #164362

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36383C211yesmaximal, characteristic
2#20573C212yesmaximal
3#7693C212yesmaximal
4#7689C212yesmaximal
5#7677C212yesmaximal
6#7639C211yesmaximal, characteristic
7#7630C212yesmaximal
8#7571C211yesmaximal, characteristic
9#7562C212yesmaximal
10C2×#446—22no—
11#2864C2211yescharacteristic
12#2863C2212yes—
13#2581C411yescharacteristic
14#2581C411yescharacteristic
15#2533—22no—
16#2524—22no—
17#2506C412yes—
18#2462—22no—
19#2446C412yes—
20#2371—22no—
21#2356C411yescharacteristic
22#2351C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2295C2212yes—
25C42×Q8C2211yescharacteristic
26#1771—22no—
27#1771—22no—
28C8×C4⋊C4—22no—
29#1767C2212yes—
30C4×C4⋊C8C2212yes—
31C4×C4⋊C8C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34#1744C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2211yescharacteristic
39#1272C2212yes—
40#1272C2211yescharacteristic
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—22no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52#446—216no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×#97—22no—
57C2×#97—22no—
58C2×#97—22no—
59C2×C4×C8—22no—
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62#400—21no—
63#400—21no—
64#400—22no—
65#400—22no—
66#398—22no—
67#398—22no—
68#395—21no—
69#395—21no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2311yescharacteristic
74#391C2311yescharacteristic
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2312yes—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#384C2×C411yescharacteristic
96#383—22no—
97#383—21no—
98#382—21no—
99#382C2×C411yescharacteristic
100#382C2×C411yescharacteristic
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110#376—22no—
111#376—21no—
112C43C2311yescharacteristic
113C43C2312yes—
114#336—22no—
115#336—22no—
116#336—22no—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×Q8C22×C411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—41no—
130#125—48no—
131C4×Q8—28no—
132C4×Q8—24no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
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⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—22no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2411yescharacteristic
171C4⋊C8—28no—
172C4⋊C8—28no—
173C4⋊C8—28no—
174#97—28no—
175#97—28no—
176#97—82no—
177C4×C8—28no—
178C4×C8—82no—
179C4×C8—82no—
180#95—21no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—21no—
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#95C22⋊C411yescharacteristic
199#95C22⋊C411yescharacteristic
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202C2×Q8—21no—
203C2×Q8—24no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8#13112yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C2×C8—24no—
228C2×C8—24no—
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—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4#10418yes—
260C42—22no—
261C42—21no—
262C42—22no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—42no—
275C42—41no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—44no—
280C23C22×C22⋊C411yescharacteristic
281Q8—44no—
282Q8—42no—
283Q8—42no—
284Q8—81no—
285Q8—81no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4#41312yes—
322C2×C4C2×#10414yes—
323C2×C4C2×#10414yes—
324C2×C4C2×#10414yes—
325C2×C4C2×#13111yescharacteristic
326C2×C4#56811yescharacteristic
327C2×C4#57712yes—
328C2×C4#57812yes—
329C8—44no—
330C8—44no—
331C8—44no—
332C8—44no—
333C22#288011yescharacteristic
334C22#288411yescharacteristic
335C22C2×#41311yescharacteristic
336C22#289812yes—
337C22C22×#10412yes—
338C4—21no—
339C4—21no—
340C4—24no—
341C4—24no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—24no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—81no—
351C4—82no—
352C4—82no—
353C4—81no—
354C4—81no—
355C2#1742311yescharacteristic
356C2#1751912yes—
357C2#1754912yes—
358C2#1763811yescharacteristic
359C2#1778011yescharacteristic
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 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#363832
#205732
#76932
#76892
#76772
#76392
#76302
#75712
#75622