Finite Group Databasebeta

← #164811

Subgroups — #164811

Order 512.

1061 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#22012C212yesmaximal, complemented
3#7730C214yesmaximal, complemented
4#7679C212yesmaximal, complemented
5#7640C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7592C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×#554C411yescharacteristic
10C2×D42C411yescharacteristic
11#2858C2211yescharacteristic
12#2850C2212yes—
13#2563C411yescharacteristic
14#2548C411yescharacteristic
15#2504C412yes—
16#2490—22no—
17#2472—24no—
18#2435C412yes—
19D4×C22⋊C4—22no—
20C4×#124C2211yescharacteristic
21C2×#390C2212yes—
22C42×D4C2211yescharacteristic
23#1954—22no—
24#1939—22no—
25#1924—24no—
26#1817C2214yescomplemented
27#1797C2214yescomplemented
28#1767C2212yescomplemented
29#1744C2212yescomplemented
30#1272C2212yescomplemented
31#1272C2212yescomplemented
32#1270C2211yescharacteristic
33#1270C2211yescharacteristic
34#1270C2212yes—
35#1270C2214yescomplemented
36#1270C2214yescomplemented
37C23×D4—21no—
38C23×D4C2×C412yes—
39#554—44no—
40D42—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#127C2×C411yescharacteristic
43C2×#125—21no—
44C2×#124—22no—
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C412yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C412yes—
49C2×#121—21no—
50C2×#121C2×C412yes—
51C2×#120—22no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C22⋊C4—24no—
59C22×C42C2312yes—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#397—24no—
67#394—24no—
68#394—22no—
69#393—22no—
70#391C2311yescharacteristic
71#390—22no—
72#390—22no—
73#390—24no—
74#390C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2312yes—
78#390C2311yescharacteristic
79#388—24no—
80#388—22no—
81#386—24no—
82#386—24no—
83#382C2×C412yes—
84#382C2×C411yescharacteristic
85#380—22no—
86#379—24no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—22no—
90C4×C22⋊C4C2×C412yes—
91C4×C22⋊C4D414yes—
92C43C2311yescharacteristic
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C25—22no—
99C22×Q8—21no—
100C22×Q8C22×C411yescharacteristic
101C22×D4—24no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—42no—
107C22×D4—44no—
108C22×D4—48no—
109C22×D4—44no—
110C22×D4—82no—
111C22×D4C22⋊C411yescharacteristic
112C22×D4C22×C412yes—
113C22×D4C22×C411yescharacteristic
114C23×C4—22no—
115C23×C4—21no—
116C23×C4C22×C412yes—
117C23×C4C2×D414yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—24no—
121C4⋊Q8—42no—
122#127—42no—
123#125—44no—
124#124—22no—
125#124—41no—
126#124—82no—
127#124—82no—
128#124—82no—
129C22⋊Q8—48no—
130C22⋊Q8—44no—
131#121—44no—
132#121—44no—
133#120—48no—
134C4×Q8—42no—
135C4×D4—24no—
136C4×D4—44no—
137C4×D4—42no—
138C4×D4—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—84no—
160C2×C22⋊C4—84no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C414yes—
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—
185C24—44no—
186C24—48no—
187C24—44no—
188C24—84no—
189C24—82no—
190C24C2×C22⋊C412yes—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—42no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—82no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—24no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—44no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—42no—
208C2×D4—48no—
209C2×D4—44no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—82no—
213C2×D4—82no—
214C2×D4—88no—
215C2×D4—88no—
216C2×D4—816no—
217C2×D4#10014yes—
218C2×D4#13112yes—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—84no—
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—22no—
252C4⋊C4—24no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C22⋊C4—24no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—44no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—24no—
280C42—44no—
281C42—41no—
282C42—42no—
283C42—42no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—44no—
297C23—44no—
298C23—88no—
299C23—88no—
300C23—816no—
301C23—164no—
302C23—168no—
303C23C2×#10014yes—
304C23C22×C22⋊C411yescharacteristic
305Q8—81no—
306Q8—81no—
307Q8—162no—
308D4—28no—
309D4—42no—
310D4—42no—
311D4—82no—
312D4—82no—
313D4—88no—
314D4—88no—
315D4—84no—
316D4—162no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—164no—
347C2×C4C2×#10012yes—
348C2×C4#42114yes—
349C2×C4C2×#13111yescharacteristic
350C2×C4#56811yescharacteristic
351C2×C4#57612yes—
352C2×C4#57712yes—
353C8—82no—
354C8—84no—
355C8—82no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—44no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—82no—
368C22—84no—
369C22—84no—
370C22—82no—
371C22—1616no—
372C22#288011yescharacteristic
373C22#288311yescharacteristic
374C22C22×#10011yescharacteristic
375C22#289712yes—
376C22C2×#42112yes—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—22no—
381C4—42no—
382C4—41no—
383C4—42no—
384C4—44no—
385C4—81no—
386C4—81no—
387C4—82no—
388C4—81no—
389C4#153314yes—
390C2—44no—
391C2—82no—
392C2—162no—
393C2#1742111yescharacteristic
394C2#1752212yes—
395C2#1759912yes—
396C2C2×#153311yescharacteristic
397C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×D42C4
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. #7730C2semidirect
4. #7679C2semidirect
5. #7640C2semidirect
6. #7635C2semidirect
7. #7592C2semidirect
8. #7567C2semidirect
26. #1817C22semidirect
27. #1797C22semidirect
28. #1767C22semidirect
29. #1744C22semidirect
30. #1272C22semidirect
31. #1272C22semidirect
35. #1270C22semidirect
36. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#220122
#77302
#76792
#76402
#76352
#75922
#75672