Finite Group Databasebeta

← #196257

Subgroups — #196257

Order 512.

1265 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2#35240C211yesmaximal, characteristic
3#17479C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9103C218yesmaximal, complemented
6#7858C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2975—24no—
10#2975—22no—
11#2975C2214yes—
12C2×#436—24no—
13#2920C2212yes—
14C2×#431C2212yes—
15C2×#407—22no—
16#2848—22no—
17#2633—22no—
18#2547C412yes—
19#2542C412yes—
20#2536—22no—
21#2502C412yes—
22#2482—22no—
23#2435C412yes—
24#2326C2214yes—
25#2283C2211yescharacteristic
26#1497—24no—
27#1493—24no—
28#1479C2218yescomplemented
29#1466—22no—
30#1462—22no—
31#1460C2214yes—
32#1274C2218yescomplemented
33C23×D4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36#533—22no—
37#533—22no—
38#533—22no—
39#533—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×#124C2×C414yes—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8C2312yes—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4D412yes—
55C2×C4×D4D412yes—
56C2×C4×D4C2312yes—
57C2×#117—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4D412yes—
60C22×C42C2311yescharacteristic
61#436—216no—
62#436—216no—
63#432—22no—
64#432—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431D414yes—
69#431C2314yes—
70#431C2314yes—
71#423—22no—
72C2×C4⋊C8—22no—
73#407—28no—
74#407—28no—
75#407—22no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—22no—
78C4×M16—22no—
79C2×C4×C8D414yes—
80#399—24no—
81#398—24no—
82#397—24no—
83#394—24no—
84#391C2×C414yes—
85#390C2×C414yes—
86#388—28no—
87#385—24no—
88#385—28no—
89#384C2×C414yes—
90#383—21no—
91#382—21no—
92#381—21no—
93#380—24no—
94#380—21no—
95#379—24no—
96C4×C4⋊C4C2×C412yes—
97C4×C22⋊C4C2×C412yes—
98#376C2312yes—
99Q8⋊C8—216no—
100#323—28no—
101C25C22⋊C412yes—
102C22×Q8—21no—
103C22×Q8C22×C412yes—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—44no—
107C22×D4C22⋊C412yes—
108C22×D4C22×C412yes—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—84no—
114C23×C4C22⋊C414yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D414yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—216no—
121C2×M16—42no—
122C2×M16—88no—
123C22×C8—24no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128#125—42no—
129#125—42no—
130#124—48no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—88no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4C22⋊C414yes—
171C2×C22⋊C4C2×D414yes—
172C2×C42—22no—
173C2×C42C22×C414yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42C2411yescharacteristic
182C4⋊C8—24no—
183C4⋊C8—24no—
184C4⋊C8—24no—
185C4⋊C8—42no—
186C22⋊C8—216no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—42no—
191#97—42no—
192C4×C8—24no—
193C4×C8—24no—
194C4×C8C2×D418yes—
195#95—22no—
196#95—22no—
197#95—28no—
198#95—22no—
199#95—22no—
200C24—22no—
201C24—22no—
202C24—44no—
203C24—44no—
204C24—82no—
205C24C2×C22⋊C411yescharacteristic
206C24C2×C22⋊C414yes—
207C2×Q8—24no—
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—44no—
217C2×D4—48no—
218C2×D4—48no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—84no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12112yes—
247C22×C4#12112yes—
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253M16—816no—
254M16—84no—
255C2×C8—28no—
256C2×C8—48no—
257C2×C8—44no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—88no—
270C22⋊C4—28no—
271C22⋊C4—28no—
272C22⋊C4—44no—
273C22⋊C4—416no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—88no—
277C42—28no—
278C42—44no—
279C42C4×D412yes—
280C42C4×D412yes—
281C42#12012yes—
282C42#12112yes—
283C42#12112yes—
284C42#12312yes—
285C42C22×D412yes—
286C42C2×#4112yes—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—24no—
292C23—41no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23—82no—
300C23—82no—
301C23—816no—
302C23#38611yescharacteristic
303C23#38611yescharacteristic
304C23C22×C22⋊C411yescharacteristic
305Q8—48no—
306Q8—82no—
307Q8—82no—
308D4—48no—
309D4—48no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4#38612yes—
339C2×C4#38611yescharacteristic
340C2×C4#38612yes—
341C2×C4#38611yescharacteristic
342C2×C4C2×C4×D412yes—
343C2×C4C2×#12011yescharacteristic
344C2×C4C2×#12112yes—
345C2×C4C2×#12311yescharacteristic
346C2×C4#56814yes—
347C8—48no—
348C8—84no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—84no—
362C22—88no—
363C22—88no—
364C22#190912yes—
365C22C2×#38611yescharacteristic
366C22#287912yes—
367C4—22no—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—22no—
373C4—44no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4#191112yes—
378C4#293014yes—
379C2—22no—
380C2—22no—
381C2—41no—
382C2—84no—
383C2#2089611yescharacteristic
384C2C2×#190911yescharacteristic
385C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17479C2semidirect
5. #9103C2semidirect
6. #7858C2semidirect
28. #1479C22semidirect
32. #1274C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#352402
#174792
#173642
#91032
#78582