Finite Group Databasebeta

← #213743

Subgroups — #213743

Order 512.

1043 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2#17368C212yesmaximal
3#9110C214yesmaximal, complemented
4#8968C214yesmaximal, complemented
5#8735C212yesmaximal, complemented
6#8303C212yesmaximal, complemented
7C2×#536C411yescharacteristic
8C2×#520C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2973C2214yes—
11#2367—22no—
12#2365—22no—
13#2362—22no—
14#2360—22no—
15#2322C412yes—
16#2322C2212yes—
17#2310C411yescharacteristic
18C4×#123C412yes—
19C4×#123C2212yes—
20#2297C411yescharacteristic
21#2296C411yescharacteristic
22#2274C2211yescharacteristic
23#1817C2214yes—
24#1467—22no—
25#1465—22no—
26#1464—22no—
27#1463—22no—
28#1459C2214yes—
29C2×#324C2214yes—
30C2×#324C2214yes—
31#1282C2218yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34#536—24no—
35#536—24no—
36C2×C4⋊Q8—21no—
37C2×#124—21no—
38C2×#123C2×C412yes—
39C2×#123C2×C412yes—
40C2×C22⋊Q8—22no—
41#520—44no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—44no—
45C2×C4×Q8D412yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4C2312yes—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60#435—48no—
61#432—24no—
62#407—24no—
63C2×C22⋊C8C2318yes—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—22no—
68#390—21no—
69#389—22no—
70#389—21no—
71#389—21no—
72#389—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#386C2×C412yes—
82#386C2×C412yes—
83#385—22no—
84#385—22no—
85#385C2×C412yes—
86#385C2×C412yes—
87#381C2×C412yes—
88#381C2×C412yes—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C412yes—
98C4×C22⋊C4C2312yes—
99#336—44no—
100#324—24no—
101#324—24no—
102#324—24no—
103#324—44no—
104#323—24no—
105C2×#41—22no—
106C2×#41—22no—
107C2×#41—42no—
108C2×#41—42no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8C22⋊C411yescharacteristic
115C22×Q8M1614yescomplemented
116C22×D4C22⋊C411yescharacteristic
117C22×D4C22×C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C2×M16—44no—
129C22×C8—28no—
130C4⋊Q8—28no—
131#124—28no—
132#123—24no—
133#123—24no—
134#123—24no—
135#123—24no—
136C22⋊Q8—216no—
137C4×Q8—44no—
138C4×Q8—48no—
139C4×Q8—48no—
140C4×D4—28no—
141C4×D4—44no—
142C4×D4—44no—
143#117—22no—
144#117—44no—
145#117—42no—
146#117—42no—
147#117M1614yescomplemented
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22×C412yes—
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C22×C412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C22⋊C8—28no—
191C22⋊C8—28no—
192C22⋊C8—44no—
193#97—44no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204C24C2×C22⋊C412yes—
205C24C23×C411yescharacteristic
206#41—48no—
207#41—48no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—48no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—48no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×M1612yes—
261C22×C4C2×M1612yes—
262C22×C4C2×M1612yes—
263C22×C4C2×M1612yes—
264C22×C4C22×D412yes—
265C22×C4H(5,𝔽2)14yes—
266M16—164no—
267C2×C8—48no—
268C2×C8—88no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C22⋊C4—28no—
284C22⋊C4—28no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—82no—
292C22⋊C4C2×M1614yes—
293C42—24no—
294C42—24no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—44no—
300C42—42no—
301C42—44no—
302C42—82no—
303C42—82no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23C22×C22⋊C411yescharacteristic
312C23#51812yes—
313C23C22×M1611yescharacteristic
314C23C22×M1611yescharacteristic
315C23#56812yes—
316Q8—48no—
317Q8—48no—
318Q8—44no—
319D4—44no—
320D4—44no—
321D4—44no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—162no—
368C2×C4#40712yes—
369C2×C4#40712yes—
370C2×C4#53414yes—
371C8—88no—
372C22—21no—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22#230811yescharacteristic
385C22C2×#40711yescharacteristic
386C22#287911yescharacteristic
387C22#297312yes—
388C22#297512yes—
389C4—24no—
390C4—22no—
391C4—22no—
392C4—44no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—82no—
401C4—81no—
402C2—21no—
403C2—21no—
404C2—22no—
405C2—22no—
406C2—82no—
407C2#2174211yescharacteristic
408C2#2177711yescharacteristic
409C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9110C2semidirect
4. #8968C2semidirect
5. #8735C2semidirect
6. #8303C2semidirect
115. C22×Q8M16semidirect
147. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342662
#173682
#91102
#89682
#87352
#83032