Finite Group Databasebeta

← #164882

Subgroups — #164882

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36509C211yesmaximal, characteristic
2#22010C212yesmaximal
3#7721C212yesmaximal, complemented
4#7715C212yesmaximal, complemented
5#7677C212yesmaximal, complemented
6#7648C212yesmaximal, complemented
7#7639C211yesmaximal, characteristic, complemented
8#7600C212yesmaximal, complemented
9#7571C211yesmaximal, characteristic, complemented
10#2864C2211yescharacteristic
11#2855C2212yes—
12#2590—22no—
13#2536C411yescharacteristic
14#2536C411yescharacteristic
15#2515—22no—
16#2496C412yes—
17#2466—22no—
18#2429C412yes—
19#2373—22no—
20#2356C411yescharacteristic
21#2351C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C22⋊Q8C2212yes—
24C42×D4C2211yescharacteristic
25#1955—22no—
26#1940—22no—
27#1923—22no—
28#1923—22no—
29#1817C2212yes—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33C4×C4⋊C8C2212yes—
34#1745C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2211yescharacteristic
39#1272C2211yescharacteristic
40#1272C2212yes—
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#126—22no—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C22×C42C2312yes—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×#97—22no—
63C2×C4×C8—22no—
64#400—22no—
65#398—21no—
66#398—21no—
67#398—22no—
68#397—22no—
69#396—22no—
70#394—21no—
71#394—21no—
72#391C2×C412yes—
73#391C2311yescharacteristic
74#391C2311yescharacteristic
75#389—22no—
76#389—22no—
77#389—22no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2312yes—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#385—22no—
93#384C2×C411yescharacteristic
94#383—21no—
95#382—21no—
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#381—22no—
99#380C2×C412yes—
100#380C2×C412yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111#376—21no—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C22×Q8C22×C411yescharacteristic
121C22×Q8C22×C411yescharacteristic
122C22×D4C22⋊C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—41no—
132#126—48no—
133C22⋊Q8—24no—
134C22⋊Q8—42no—
135C22⋊Q8—82no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—82no—
139C4×D4—42no—
140C4×D4—44no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—82no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—44no—
178C4⋊C8—82no—
179C4⋊C8—82no—
180C22⋊C8—82no—
181C22⋊C8—82no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184#97—44no—
185C4×C8—82no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C411yescharacteristic
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#13112yes—
213C2×D4—41no—
214C2×D4—42no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—82no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—24no—
284C42—42no—
285C42—42no—
286C42—41no—
287C42—42no—
288C42—44no—
289C42—42no—
290C42—42no—
291C42—82no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23C22×C22⋊C411yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304D4—44no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#41312yes—
346C2×C4#42114yes—
347C2×C4C2×#13111yescharacteristic
348C2×C4#56811yescharacteristic
349C2×C4#57712yes—
350C2×C4#57812yes—
351C8—82no—
352C8—82no—
353C8—82no—
354C8—82no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22#288011yescharacteristic
363C22#288311yescharacteristic
364C22C2×#41311yescharacteristic
365C22#289812yes—
366C22C2×#42112yes—
367C4—21no—
368C4—21no—
369C4—24no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—82no—
382C4—82no—
383C2—82no—
384C2#1742411yescharacteristic
385C2#1752412yes—
386C2#1760012yes—
387C2#1766511yescharacteristic
388C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7721C2semidirect
4. #7715C2semidirect
5. #7677C2semidirect
6. #7648C2semidirect
7. #7639C2semidirect
8. #7600C2semidirect
9. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#365092
#220102
#77212
#77152
#76772
#76482
#76392
#76002
#75712