Finite Group Databasebeta

← #6532186

Subgroups — #6532186

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27617C214yesmaximal
4#27614C212yesmaximal
5#27611C212yesmaximal
6#26295C218yesmaximal, complemented
7#26293C218yesmaximal, complemented
8#21841C212yesmaximal
9#21822C211yesmaximal, characteristic
10#21363C212yesmaximal
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2212yes—
14#2567—22no—
15#2567—22no—
16#2567C2214yes—
17#2567C2214yes—
18C2×#395—21no—
19C2×#395—21no—
20#2356—22no—
21#2356—22no—
22#2355C2214yes—
23#2355C2214yes—
24#2313—21no—
25#2313—21no—
26#2310—21no—
27C2×#391—21no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×#384—21no—
33#2283C2212yes—
34#2283C2212yes—
35#2274—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#2101C2218yes—
38#2100C2214yes—
39#2099C2214yes—
40#2085C2218yes—
41#2084C2214yes—
42#2083C2214yes—
43#1993C2218yes—
44#1988C2218yes—
45#1985C2214yes—
46#1985C2214yes—
47#1949C2218yes—
48#1921C2218yes—
49#1920C2218yes—
50#1919C2218yes—
51#1918C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1824C2214yes—
55#1824C2214yes—
56#1817C2212yes—
57#1817C2212yes—
58#1776C2218yes—
59C23×Q8—21no—
60C23×Q8D411yescharacteristic
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8C2314yes—
64C2×C4×Q8—28no—
65C2×C4×Q8D412yes—
66C2×C4×Q8D412yes—
67C2×C4×Q8C2312yes—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—28no—
84C2×#107C2318yes—
85C2×#106C2318yes—
86C2×C4⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88C2×#97C2314yes—
89#395—24no—
90#395—24no—
91#395—24no—
92#395—24no—
93#391—24no—
94#391—24no—
95#391—24no—
96#391—24no—
97#389—24no—
98#389—22no—
99#389—22no—
100#389C2314yes—
101#389C2314yes—
102#389C2318yes—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#384—24no—
108#384—24no—
109#384C2312yes—
110#384C2312yes—
111#383—22no—
112#383—22no—
113#382—21no—
114#382—21no—
115#382C2314yes—
116#382C2314yes—
117#381—22no—
118#381—22no—
119#380—21no—
120#380—21no—
121#380C2312yes—
122#380C2312yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2314yes—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135#376—22no—
136#376—22no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#336C2314yes—
147#336C2314yes—
148#336C2318yes—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×Q8—84no—
155C22×Q8C2×D412yes—
156C22×Q8C2×D412yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8C2418yes—
167C4⋊Q8—28no—
168C4⋊Q8—28no—
169C4⋊Q8—48no—
170C4×Q8—28no—
171C4×Q8—28no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176#117—44no—
177#117—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—44no—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42#4118yes—
221C2×C42C2412yes—
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224C2×C42C2412yes—
225#107—216no—
226#106—216no—
227C4⋊C8—28no—
228C22⋊C8—28no—
229#97—28no—
230#95—24no—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—22no—
242#95—24no—
243#95—24no—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—416no—
252C2×Q8—48no—
253C2×Q8—88no—
254C2×Q8—816no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4#12314yes—
279C22×C4#12314yes—
280C22×C4#13714yes—
281C22×C4C22×D412yes—
282C22×C4C22×D412yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4114yes—
291C22×C4C2511yescharacteristic
292C2×C8—28no—
293C2×C8—28no—
294C2×C8—28no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—88no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C42−1+418yes—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—48no—
326C42—44no—
327C42—44no—
328C42—84no—
329C42—84no—
330C42#12318yes—
331C42C2×#4114yes—
332C42C2×#4114yes—
333C23—22no—
334C23—22no—
335C23C2×#12312yes—
336C23C2×#13712yes—
337C23#57512yes—
338C23C23×D411yescharacteristic
339Q8—48no—
340Q8—48no—
341Q8—816no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—28no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—44no—
364C2×C4—416no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—88no—
380C2×C4C2×#12312yes—
381C2×C4C2×#12314yes—
382C2×C4C2×#12314yes—
383C2×C4C2×#13712yes—
384C2×C4C2×#13712yes—
385C2×C4C22×#4112yes—
386C2×C4C2×2−1+414yes—
387C8—48no—
388C22—22no—
389C22—22no—
390C22—24no—
391C22—24no—
392C22#309514yes—
393C22C22×#12311yescharacteristic
394C22#348612yes—
395C22C22×#13711yescharacteristic
396C22C2×#57511yescharacteristic
397C22#358512yes—
398C4—22no—
399C4—24no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—48no—
408C4—84no—
409C4—84no—
410C4—84no—
411C4#350914yes—
412C2—24no—
413C2#3567411yescharacteristic
414C2C2×#309511yescharacteristic
415C2#6051811yescharacteristic
416C2#6183012yes—
417C2#6183112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #26295C2semidirect
7. #26293C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#341872
#276172
#276142
#276112
#262952
#262932
#218412
#218222
#213632