Finite Group Databasebeta

← #516362

Subgroups — #516362

Order 512.

3978 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C212yesmaximal
2C2×#2316C211yesmaximal, characteristic
3C2×#2296C212yesmaximal
4C22×#389C211yesmaximal, characteristic
5C2×#2274C211yesmaximal, characteristic
6#13965C214yesmaximal
7#13954C218yesmaximal, complemented
8#13910C218yesmaximal, complemented
9#13880C214yesmaximal
10C22×#117C2212yes—
11C22×#117C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16#2317—216no—
17#2316—28no—
18#2296—216no—
19C2×#389—28no—
20C2×#389C2214yes—
21C2×#389C2214yes—
22C2×#389C2218yes—
23C2×#389C2214yes—
24C2×#389C2218yes—
25C2×#389C2214yes—
26C2×#385C2218yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#384C2214yes—
30C2×#384C2218yes—
31C2×#382C2214yes—
32C2×#380C2214yes—
33#2274C4116yescomplemented
34C2×C4×C4⋊C4C2218yes—
35C2×C4×C4⋊C4C2218yes—
36C2×C4×C22⋊C4C2214yes—
37C22×#95C2212yes—
38#1444C22116yes—
39#1444C2218yes—
40#1444C2218yes—
41C4⋊#95C2218yes—
42C4⋊#95C22116yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2312yes—
46C24×C4C2312yes—
47C24×C4C2311yescharacteristic
48C2×#117—28no—
49C2×#117—28no—
50C2×#117—28no—
51C2×#117C2×C4116yes—
52C2×#117D4116yes—
53C2×#117Q8116yes—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—48no—
58C22×C4⋊C4C2×C418yes—
59C22×C4⋊C4D418yes—
60C22×C4⋊C4Q818yes—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2318yes—
65C22×C4⋊C4C2318yes—
66C22×C4⋊C4C2318yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C23116yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C42—24no—
75C22×C42—28no—
76C22×C42C2314yes—
77C22×C42C2314yes—
78C22×C42C2318yes—
79C22×C42C2314yes—
80#389—216no—
81#389—216no—
82#389—232no—
83#389—232no—
84#389—216no—
85#389—216no—
86#385—232no—
87#385—216no—
88#385—216no—
89#384—216no—
90#384—416no—
91#382—216no—
92#380—216no—
93C4×C4⋊C4—232no—
94C4×C4⋊C4C2×C4164yescomplemented
95C4×C22⋊C4C2×C4132yes—
96C2×#95—28no—
97C2×#95C2318yes—
98C2×#95C2318yes—
99C2×#95C2318yes—
100C2×#95C2318yes—
101C2×#95C23116yes—
102C2×#95C2318yes—
103C25C2411yescharacteristic
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—28no—
112C23×C4—44no—
113C23×C4—44no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C22×C418yes—
119C23×C4C22×C414yes—
120C23×C4C2×D418yes—
121C23×C4C2×D414yes—
122C23×C4C2×D418yes—
123C23×C4C2×Q818yes—
124C23×C4C2×Q814yes—
125C23×C4C2×Q818yes—
126C23×C4C2414yes—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2412yes—
130C23×C4C2412yes—
131C23×C4C2418yes—
132C23×C4C2414yes—
133C23×C4C2412yes—
134#117—432no—
135#117C4⋊C41128yescomplemented
136C2×C4⋊C4—28no—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—216no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—216no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—232no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—432no—
158C2×C4⋊C4—416no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—416no—
161C2×C4⋊C4—416no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4C4⋊C4164yescomplemented
164C2×C4⋊C4C22×C4116yes—
165C2×C4⋊C4C22×C4116yes—
166C2×C4⋊C4C2×D4116yes—
167C2×C4⋊C4C2×D4116yes—
168C2×C4⋊C4C2×Q8116yes—
169C2×C4⋊C4C2×Q8116yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4C22×C4116yes—
174C2×C22⋊C4C2×D4116yes—
175C2×C22⋊C4C2×Q8116yes—
176C2×C42—28no—
177C2×C42—28no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—48no—
183C2×C42—48no—
184C2×C42—416no—
185C2×C42C22×C4132yes—
186C2×C42C22×C4116yes—
187C2×C42C2×D4116yes—
188C2×C42C2×Q8116yes—
189#95—216no—
190#95—216no—
191#95—216no—
192#95—216no—
193#95—232no—
194#95C22×C4132yes—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—22no—
199C24C23×C412yes—
200C24C22×D414yes—
201C24C22×D412yes—
202C24C22×Q814yes—
203C24C22×Q812yes—
204C24C2511yescharacteristic
205C22×C4—28no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—48no—
236C22×C4—416no—
237C22×C4—48no—
238C22×C4—416no—
239C22×C4—416no—
240C22×C4—416no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—816no—
249C22×C4C2×C4⋊C4132yes—
250C22×C4C2×C4⋊C4116yes—
251C22×C4C2×C4⋊C4132yes—
252C22×C4#127116yes—
253C22×C4C4⋊Q8116yes—
254C22×C4C4⋊Q8132yes—
255C22×C4C23×C4116yes—
256C22×C4C23×C418yes—
257C22×C4C23×C414yes—
258C22×C4C22×D4116yes—
259C22×C4C22×D418yes—
260C22×C4C22×D414yes—
261C22×C4C22×D414yes—
262C22×C4C22×D414yes—
263C22×C4C22×Q8116yes—
264C22×C4C22×Q818yes—
265C22×C4C22×Q814yes—
266C22×C4C22×Q814yes—
267C22×C4C22×Q814yes—
268C22×C4H(5,𝔽2)14yes—
269C22×C42−1+414yes—
270C4⋊C4—432no—
271C4⋊C4—432no—
272C4⋊C4—432no—
273C4⋊C4—416no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—88no—
277C4⋊C4—88no—
278C4⋊C4—832no—
279C4⋊C4C2×C4⋊C4164yes—
280C4⋊C4C2×C4⋊C4164yescomplemented
281C22⋊C4—416no—
282C22⋊C4C2×C4⋊C4164yes—
283C42—432no—
284C42—416no—
285C42—416no—
286C42C2×C4⋊C4164yescomplemented
287C23—24no—
288C23—24no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—28no—
294C23—28no—
295C23—24no—
296C23—24no—
297C23—28no—
298C23—28no—
299C23C22×C4⋊C418yes—
300C23C22×C4⋊C414yes—
301C23C2×#12714yes—
302C23C2×C4⋊Q814yes—
303C23C2×C4⋊Q818yes—
304C23C24×C411yescharacteristic
305C23C23×D411yescharacteristic
306C23C23×D412yes—
307C23C23×Q811yescharacteristic
308C23C23×Q812yes—
309C23C2×H(5,𝔽2)12yes—
310C23C2×2−1+412yes—
311C23#58512yes—
312C23#58512yes—
313C2×C4—216no—
314C2×C4—216no—
315C2×C4—216no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—216no—
319C2×C4—216no—
320C2×C4—216no—
321C2×C4—216no—
322C2×C4—216no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—432no—
328C2×C4—416no—
329C2×C4—416no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—416no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—816no—
351C2×C4#384164yescomplemented
352C2×C4C22×C4⋊C4132yes—
353C2×C4C22×C4⋊C4116yes—
354C2×C4C22×C4⋊C418yes—
355C2×C4C22×C4⋊C418yes—
356C2×C4C22×C4⋊C418yes—
357C2×C4C2×#12718yes—
358C2×C4C2×C4⋊Q818yes—
359C2×C4C2×C4⋊Q818yes—
360C2×C4C2×C4⋊Q818yes—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—216no—
366C22—24no—
367C22—216no—
368C22—24no—
369C22C2×#38418yes—
370C22C23×C4⋊C411yescharacteristic
371C22C23×C4⋊C412yes—
372C22C4×H(5,𝔽2)12yes—
373C22C4×2−1+412yes—
374C22C22×#12711yescharacteristic
375C22C22×C4⋊Q811yescharacteristic
376C22C22×C4⋊Q812yes—
377C22#348411yescharacteristic
378C22#348511yescharacteristic
379C22#348611yescharacteristic
380C22#348711yescharacteristic
381C22#348814yes—
382C22#348914yes—
383C22#350112yes—
384C22#350212yes—
385C22#350314yes—
386C22#350414yes—
387C4—216no—
388C4—232no—
389C4—44no—
390C4—44no—
391C4—84no—
392C4—84no—
393C4—84no—
394C4C2×#384116yes—
395C2—28no—
396C2C22×#38411yescharacteristic
397C2#3424811yescharacteristic
398C2#3424911yescharacteristic
399C2#3425014yes—
400C2#3556511yescharacteristic
401C2#3556614yes—
402C2#3557611yescharacteristic
403C2#3563411yescharacteristic
404C2#3564011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #13954C2semidirect
8. #13910C2semidirect
33. #2274C4semidirect
94. C4×C4⋊C4C2×C4semidirect
135. #117C4⋊C4semidirect
163. C2×C4⋊C4C4⋊C4semidirect
280. C4⋊C4C2×C4⋊C4semidirect
286. C42C2×C4⋊C4semidirect
351. C2×C4#384semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#23162
C2×#22962
C22×#3892
C2×#22742
#139652
#139542
#139102
#138802