Finite Group Databasebeta

← #194850

Subgroups — #194850

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic
3#21834C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9024C214yesmaximal
6#9022C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#431C2212yes—
11#2628—24no—
12#2628C418yescomplemented
13C2×#400C2212yes—
14C2×#399C2212yes—
15C2×#395—21no—
16C2×#393—21no—
17#2314—24no—
18#2313—21no—
19#2313C2212yes—
20#2312—21no—
21#2312C2212yes—
22#2298—24no—
23C2×#389—21no—
24#2285C2212yes—
25C2×#382C2211yescharacteristic
26#2280—21no—
27C2×#380—21no—
28C4×#117—21no—
29#1990—24no—
30#1987—24no—
31#1824C2218yes—
32#1458—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C24×C4D411yescharacteristic
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C414yes—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—24no—
47C2×#117C2×C414yes—
48C2×#117D414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4C2311yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60#431—24no—
61#431—24no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×C4×C8—22no—
65#400—24no—
66#400C2×C418yes—
67#399—24no—
68#399C2×C418yes—
69#395—28no—
70#393—28no—
71#389—28no—
72#388—28no—
73#388—28no—
74#387—28no—
75#387—28no—
76#384D414yes—
77#382—22no—
78#382—22no—
79#382C2×C414yes—
80#381—22no—
81#380—22no—
82#380—22no—
83#380—28no—
84#380—22no—
85#380C2314yes—
86#379—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D414yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95#376—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2312yes—
100C43—22no—
101#340—216no—
102#324—24no—
103#323—24no—
104C25C2×D411yescharacteristic
105C22×Q8—24no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—84no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C22⋊C414yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—24no—
124C4⋊Q8—28no—
125C4⋊Q8—28no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128#117—28no—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—44no—
133#117—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—84no—
151C2×C4⋊C4C22⋊C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D418yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4C22⋊C414yes—
166C2×C22⋊C4C22×C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42C22⋊C414yes—
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42#4114yes—
180C4⋊C8—44no—
181C22⋊C8—24no—
182C22⋊C8—24no—
183C4×C8—44no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—28no—
191#95—24no—
192#95—22no—
193#95—24no—
194#95C22×C414yes—
195C24—21no—
196C24—22no—
197C24—21no—
198C24—22no—
199C24—42no—
200C24—42no—
201C24C2×C22⋊C412yes—
202C24C22×D411yescharacteristic
203C2×Q8—48no—
204C2×Q8—88no—
205C2×Q8—88no—
206C2×Q8—88no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4#12012yes—
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12312yes—
245C22×C4#12312yes—
246C22×C4#13514yes—
247C22×C4#13714yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4112yes—
252C2×C8—48no—
253C2×C8—44no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—28no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—84no—
265C4⋊C4—88no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—28no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C22⋊C4—88no—
274C42—24no—
275C42—24no—
276C42—48no—
277C42—84no—
278C42—84no—
279C42C4×D414yes—
280C42C4×D414yes—
281C42#12114yes—
282C42#12114yes—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—48no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—81no—
296C23—81no—
297C23—84no—
298C23C22×C22⋊C411yescharacteristic
299C23C2×#12011yescharacteristic
300C23C2×#12311yescharacteristic
301C23C2×#13512yes—
302C23C2×#13712yes—
303Q8—816no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#38614yes—
339C2×C4#38612yes—
340C2×C4#38612yes—
341C2×C4#41614yes—
342C2×C4#41914yes—
343C2×C4#45214yes—
344C2×C4#45214yes—
345C2×C4#48518yes—
346C2×C4C2×C4×D412yes—
347C2×C4C2×#12112yes—
348C8—84no—
349C22—24no—
350C22—24no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—84no—
361C22—84no—
362C22C2×#38611yescharacteristic
363C22C2×#41611yescharacteristic
364C22C2×#41911yescharacteristic
365C22C2×#45212yes—
366C22#302112yes—
367C22#302812yes—
368C22C2×#48512yes—
369C4—24no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—84no—
376C4—88no—
377C4—84no—
378C4#187314yes—
379C4#187314yes—
380C2—21no—
381C2—21no—
382C2—22no—
383C2—82no—
384C2#1831314yes—
385C2C2×#187311yescharacteristic
386C2#2095611yescharacteristic
387C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21834C2semidirect
6. #9022C2semidirect
7. #8026C2semidirect
12. #2628C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#340482
#218342
C2×#14582
#90242
#90222
#80262