Finite Group Databasebeta

← #391911

Subgroups — #391911

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1840C211yesmaximal, characteristic
3#14350C211yesmaximal, characteristic
4#11385C212yesmaximal
5#11384C212yesmaximal
6#11373C211yesmaximal, characteristic
7#11372C211yesmaximal, characteristic
8#11372C211yesmaximal, characteristic
9#11371C211yesmaximal, characteristic
10#11358C212yesmaximal
11#11357C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14#2660—24no—
15#2660C418yescomplemented
16C2×#398—22no—
17C2×#395—22no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1840—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1443—22no—
37#1443—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8C2×C414yes—
49C2×#125—22no—
50C2×#125—42no—
51C2×#125—42no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#398—216no—
68#395—216no—
69#384—22no—
70#384—22no—
71#384C2×C414yes—
72#384C2×C414yes—
73#382—22no—
74#382—22no—
75#382C2×C414yes—
76#382C2×C414yes—
77#376C2×C414yes—
78#376D414yes—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88C2×#95D412yes—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#336—24no—
98#336—24no—
99#336—22no—
100#336—22no—
101C22×Q8—24no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—82no—
105C22×Q8—82no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—48no—
132C4⋊Q8—48no—
133#125—48no—
134#125—48no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42C22×C412yes—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190#95—28no—
191#95—28no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95C22⋊C418yes—
199#95C22⋊C418yes—
200C24C22×D411yescharacteristic
201C2×Q8—48no—
202C2×Q8—88no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—88no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C414yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4H(5,𝔽2)12yes—
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269C22×C42−1+412yes—
270C22×C42−1+412yes—
271C22×C42−1+412yes—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—88no—
306C4⋊C4—88no—
307C42—24no—
308C42—24no—
309C42—22no—
310C42—22no—
311C42—24no—
312C42—22no—
313C42—22no—
314C42—24no—
315C23C22×C22⋊C411yescharacteristic
316C23#53511yescharacteristic
317C23#53511yescharacteristic
318C23#53612yes—
319C23#53612yes—
320C23#57612yes—
321C23#57712yes—
322C23#57712yes—
323C23#57812yes—
324Q8—816no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#51812yes—
384C2×C4#51912yes—
385C2×C4#51911yescharacteristic
386C2×C4#51911yescharacteristic
387C2×C4#53512yes—
388C2×C4#53611yescharacteristic
389C2×C4#53611yescharacteristic
390C2×C4#53612yes—
391C2×C4#56312yes—
392C2×C4#56312yes—
393C2×C4#56312yes—
394C2×C4#56312yes—
395C2×C4#56312yes—
396C2×C4#56312yes—
397C2×C4#56312yes—
398C2×C4#56312yes—
399C8—82no—
400C8—82no—
401C22#230911yescharacteristic
402C22#231011yescharacteristic
403C22#231011yescharacteristic
404C22#282112yes—
405C22#282112yes—
406C22#289712yes—
407C22#289812yes—
408C22#322011yescharacteristic
409C22#322111yescharacteristic
410C22#322211yescharacteristic
411C22#322311yescharacteristic
412C22#322412yes—
413C22#322512yes—
414C22#324112yes—
415C22#324112yes—
416C22#324212yes—
417C22#324212yes—
418C22#324312yes—
419C22#324312yes—
420C22#324412yes—
421C22#324412yes—
422C4—42no—
423C4—42no—
424C4—44no—
425C4—44no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—162no—
431C4—162no—
432C4#238112yes—
433C4#238112yes—
434C4#282112yes—
435C4#282112yes—
436C2#1466311yescharacteristic
437C2#2181311yescharacteristic
438C2#2181411yescharacteristic
439C2#2184312yes—
440C2#2184312yes—
441C2#3284412yes—
442C2#3284512yes—
443C2#3284612yes—
444C2#3284712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2660C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#18402
#143502
#113852
#113842
#113732
#113722
#113712
#113582
#113572