Finite Group Databasebeta

← #393853

Subgroups — #393853

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1987C211yesmaximal, characteristic
6C2×#1843C211yesmaximal, characteristic
7#13750C211yesmaximal, characteristic
8#10850C214yesmaximal
9#10849C214yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2214yes—
15#2660—24no—
16#2660C418yescomplemented
17C2×#398—22no—
18C2×#395—22no—
19C2×#384C2212yes—
20C2×#382C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2214yes—
24C2×#376C2211yescharacteristic
25#1989—24no—
26#1989—24no—
27#1988—24no—
28#1987—24no—
29#1843—24no—
30C2×#340C2214yes—
31C2×#340C2214yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C4×#95—22no—
37C4×#95—22no—
38C23×Q8—21no—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C414yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—24no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—28no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—216no—
75#398—216no—
76#395—216no—
77#384—24no—
78#384C2×C418yes—
79#382—24no—
80#382C2×C418yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—28no—
84C4×C4⋊C4D414yes—
85C4×C4⋊C4D414yes—
86C4×C4⋊C4Q8116yescomplemented
87#376C2×C414yes—
88#376D414yes—
89C2×#95—24no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—28no—
96#340—28no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101C22×Q8—24no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—82no—
105C22×Q8—82no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D414yes—
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—28no—
122C22×C8—28no—
123C4⋊Q8—28no—
124C4⋊Q8—28no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127#125—28no—
128#125—28no—
129#125—44no—
130#125—44no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—216no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×Q818yes—
168C2×C4⋊C4C2×Q818yes—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—216no—
172C2×C42—84no—
173C2×C42—82no—
174C2×C42—84no—
175C2×C42—82no—
176C2×C42C22×C412yes—
177C2×C42C22×C414yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D414yes—
182C2×C42C2×Q818yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4118yes—
186C4⋊C8—416no—
187C4⋊C8—44no—
188C4⋊C8—44no—
189#95—216no—
190#95—24no—
191#95—24no—
192#95—216no—
193#95C22⋊C418yes—
194#95C22⋊C418yes—
195C24C22×D411yescharacteristic
196C2×Q8—48no—
197C2×Q8—88no—
198C2×Q8—84no—
199C2×Q8—84no—
200C2×Q8—88no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C418yes—
237C22×C4#12112yes—
238C22×C4#12112yes—
239C22×C4#12112yes—
240C22×C4#12112yes—
241C22×C4C22⋊Q818yes—
242C22×C4C22⋊Q818yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×Q814yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4114yes—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—816no—
279C4⋊C4C4⋊Q818yes—
280C4⋊C4C4⋊Q818yes—
281C42—816no—
282C42—84no—
283C42—84no—
284C42C4×D414yes—
285C42C4×D414yes—
286C42C4×Q8116yes—
287C42#12414yes—
288C42#12418yes—
289C42#12714yes—
290C42C4⋊Q818yes—
291C23C22×C22⋊C411yescharacteristic
292C23C2×#12111yescharacteristic
293C23C2×#12111yescharacteristic
294C23C2×C22⋊Q814yes—
295C23#57612yes—
296C23#57712yes—
297C23#57712yes—
298C23#57812yes—
299Q8—816no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4#39014yes—
340C2×C4#39014yes—
341C2×C4#391116yes—
342C2×C4C2×C4×D411yescharacteristic
343C2×C4C2×C4×D411yescharacteristic
344C2×C4C2×C4×Q814yes—
345C2×C4C2×#12412yes—
346C2×C4C2×#12411yescharacteristic
347C2×C4C2×#12711yescharacteristic
348C2×C4C2×C4⋊Q812yes—
349C2×C4C2×C4⋊Q812yes—
350C2×C4C2×C4⋊Q812yes—
351C2×C4#53212yes—
352C2×C4#53212yes—
353C2×C4#53314yes—
354C2×C4#53314yes—
355C8—82no—
356C8—82no—
357C22C2×#39011yescharacteristic
358C22C2×#39112yes—
359C22#285712yes—
360C22#285712yes—
361C22#289712yes—
362C22#289812yes—
363C22#304111yescharacteristic
364C22#304211yescharacteristic
365C22#304311yescharacteristic
366C22#304411yescharacteristic
367C22#304512yes—
368C22#304612yes—
369C22#308114yes—
370C22#308214yes—
371C22#308314yes—
372C22#308414yes—
373C4—24no—
374C4—24no—
375C4—216no—
376C4—48no—
377C4—48no—
378C4—82no—
379C4—82no—
380C4—84no—
381C4C4×C4⋊Q812yes—
382C4C4×C4⋊Q812yes—
383C4#285811yescharacteristic
384C4#286212yes—
385C4#286811yescharacteristic
386C2#1471411yescharacteristic
387C2#2126211yescharacteristic
388C2#2126311yescharacteristic
389C2#2131514yes—
390C2#3320312yes—
391C2#3320412yes—
392C2#3320512yes—
393C2#3320612yes—
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
16. #2660C4semidirect
86. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19892
C2×#19882
C2×#19872
C2×#18432
#137502
#108502
#108492