Finite Group Databasebeta

← #164874

Subgroups — #164874

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36412C211yesmaximal, characteristic
2#22012C212yesmaximal, complemented
3#7724C214yesmaximal, complemented
4#7679C212yesmaximal, complemented
5#7645C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7597C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×#555C411yescharacteristic
10C2×#550C411yescharacteristic
11#2858C2211yescharacteristic
12#2856C2212yes—
13#2567C411yescharacteristic
14#2555—24no—
15#2541C411yescharacteristic
16#2500C412yes—
17#2431C412yes—
18#2428—22no—
19#2363—22no—
20C4×#124C2211yescharacteristic
21#2297C2212yes—
22C42×D4C2211yescharacteristic
23#1954—22no—
24#1939—22no—
25#1924—24no—
26#1817C2214yes—
27#1797C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2211yescharacteristic
31#1272C2211yescharacteristic
32#1272C2212yes—
33#1270C2212yes—
34#1270C2212yes—
35#1270C2214yes—
36#1270C2214yes—
37C22×#41C2×C412yes—
38#555—44no—
39#550—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127—21no—
42C2×#127C2×C411yescharacteristic
43C2×#127C2×C412yes—
44C2×#124—22no—
45C2×#124—22no—
46C2×#124C2×C411yescharacteristic
47C2×#121—21no—
48C2×#121—24no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121C2×C412yes—
52C2×#121C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C2×#117—21no—
59C2×#117C2×C412yes—
60C22×C42C2312yes—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×#97—22no—
66C2×C4×C8—22no—
67#395—22no—
68#394—24no—
69#392—24no—
70#392—22no—
71#391C2×C412yes—
72#391C2311yescharacteristic
73#390—24no—
74#390—22no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2311yescharacteristic
78#390C2312yes—
79#389—22no—
80#388—22no—
81#388—24no—
82#386—24no—
83#386—24no—
84#384C2×C412yes—
85#382C2×C411yescharacteristic
86C4×C4⋊C4C2×C411yescharacteristic
87C4×C4⋊C4D412yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4C2×C412yes—
91C4×C22⋊C4D414yes—
92C43C2311yescharacteristic
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—44no—
101C2×#41—82no—
102C22×Q8—21no—
103C22×Q8C22×C411yescharacteristic
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—24no—
110C22×D4C22⋊C411yescharacteristic
111C22×D4C22×C411yescharacteristic
112C22×D4C22×C412yes—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4C22×C412yes—
116C23×C4C2×D414yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C4⋊Q8—42no—
121#127—42no—
122#127—44no—
123#127—82no—
124#124—22no—
125#124—41no—
126#124—48no—
127#124—82no—
128#124—82no—
129#121—48no—
130#121—48no—
131#121—44no—
132#121—44no—
133#121—44no—
134#121—84no—
135#121—84no—
136C4×Q8—42no—
137C4×D4—24no—
138C4×D4—44no—
139C4×D4—42no—
140C4×D4—44no—
141#117—44no—
142#117—44no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C414yes—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—82no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D414yes—
171C2×C42C2411yescharacteristic
172C4⋊C8—44no—
173C4⋊C8—84no—
174C22⋊C8—84no—
175C22⋊C8—84no—
176#97—82no—
177C4×C8—44no—
178#95—22no—
179#95—24no—
180#95—22no—
181#95C22⋊C414yes—
182#95C22⋊C412yes—
183C24—21no—
184C24—22no—
185C24C2×C22⋊C412yes—
186#41—88no—
187#41—88no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—82no—
193C2×Q8#13112yes—
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—44no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—48no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—84no—
213C2×D4—82no—
214C2×D4—84no—
215C2×D4—82no—
216C2×D4—84no—
217C2×D4—84no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—82no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D412yes—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—24no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—84no—
258C4⋊C4#10014yes—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—48no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—84no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—24no—
278C42—44no—
279C42—41no—
280C42—42no—
281C42—44no—
282C42—42no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—44no—
295C23—42no—
296C23—44no—
297C23—42no—
298C23C22×C22⋊C411yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—162no—
302D4—44no—
303D4—81no—
304D4—81no—
305D4—82no—
306D4—82no—
307D4—88no—
308D4—84no—
309D4—162no—
310D4—168no—
311D4—164no—
312D4—164no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—88no—
341C2×C4—84no—
342C2×C4C2×#10014yes—
343C2×C4C2×#10012yes—
344C2×C4#42114yes—
345C2×C4C2×#13111yescharacteristic
346C2×C4#56811yescharacteristic
347C2×C4#57612yes—
348C2×C4#57712yes—
349C8—82no—
350C8—84no—
351C8—82no—
352C22—42no—
353C22—44no—
354C22—42no—
355C22—44no—
356C22—42no—
357C22—82no—
358C22—81no—
359C22—82no—
360C22—84no—
361C22—82no—
362C22—81no—
363C22—82no—
364C22—84no—
365C22—81no—
366C22—82no—
367C22#288011yescharacteristic
368C22#288311yescharacteristic
369C22C22×#10011yescharacteristic
370C22#289712yes—
371C22C2×#42112yes—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—42no—
379C4—41no—
380C4—44no—
381C4—81no—
382C4—82no—
383C4—81no—
384C4#153614yes—
385C2—82no—
386C2—162no—
387C2—161no—
388C2#1742111yescharacteristic
389C2#1752612yes—
390C2#1759912yes—
391C2C2×#153611yescharacteristic
392C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36412C2
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
2. #22012C2semidirect
3. #7724C2semidirect
4. #7679C2semidirect
5. #7645C2semidirect
6. #7635C2semidirect
7. #7597C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364122
#220122
#77242
#76792
#76452
#76352
#75972
#75672