Finite Group Databasebeta

← #194576

Subgroups — #194576

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34348C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic
4#21811C211yesmaximal, characteristic
5C2×#1537C211yesmaximal, characteristic
6#9000C218yesmaximal, complemented
7#8035C211yesmaximal, characteristic
8#8035C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—24no—
12#3165C418yescomplemented
13C2×#501C2214yes—
14C2×#500—21no—
15C2×#498—21no—
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#428—22no—
19C2×C4×M16C2211yescharacteristic
20#2382C2214yes—
21#2381—21no—
22#2381—21no—
23#2378—22no—
24#2378—22no—
25#2298—22no—
26#2295—22no—
27#2295—22no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30#1993C2214yes—
31#1987—22no—
32#1981—22no—
33#1960C2214yes—
34#1823—22no—
35#1787—22no—
36#1537—24no—
37C2×#330C2214yes—
38#1295C2214yes—
39#1295C2214yes—
40C23×Q8—21no—
41C22×M16—21no—
42#533—22no—
43#533—22no—
44#533C2×C414yes—
45#533D414yes—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×C414yes—
53C2×C4⋊Q8D411yescharacteristic
54C2×C4⋊Q8D411yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×#125C2312yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×#117D412yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#501—28no—
66#501C2×C4116yes—
67#500—28no—
68#498—28no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#428—216no—
74C2×#107—24no—
75C2×#106—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×#103—28no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C4×M16—22no—
82C4×M16C2×C414yes—
83C2×#97C2312yes—
84C2×#97C2312yes—
85C2×#97C2312yes—
86C2×C4×C8—21no—
87#391—24no—
88#391—24no—
89#389—24no—
90#389—24no—
91#388—24no—
92#388—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#383—21no—
100#383—21no—
101#382C2312yes—
102#382C2312yes—
103#381—21no—
104#381—21no—
105#380D412yes—
106#380D412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#376—22no—
114#376—21no—
115#376—21no—
116C43—22no—
117#340—28no—
118#330—28no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—84no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—28no—
131C2×M16—24no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8C22⋊C414yes—
147C4⋊Q8C22⋊C414yes—
148C4⋊Q8C22×C414yes—
149#125—22no—
150#125—22no—
151#125C22×C414yes—
152#125C2×D414yes—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161#117—24no—
162#117C22⋊C418yes—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42C22×C412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4112yes—
195C2×C42#4112yes—
196C2×C42C2411yescharacteristic
197#107—216no—
198#106—216no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201#103—88no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204#97—22no—
205#97—22no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97C22×C414yes—
210C4×C8—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C24#12011yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—48no—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—84no—
233C2×Q8—88no—
234C2×Q8—88no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—84no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C4×D412yes—
247C22×C4C4×D412yes—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12112yes—
252C22×C4#12112yes—
253C22×C4#12312yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257M16—216no—
258C2×C8—28no—
259C2×C8—24no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4C2×C22⋊C414yes—
283C4⋊C4C2×C22⋊C414yes—
284C22⋊C4—28no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—28no—
288C42—44no—
289C42—44no—
290C42—84no—
291C42—84no—
292C42C2×C22⋊C412yes—
293C42C4×D412yes—
294C42C4×D412yes—
295C42#12112yes—
296C42#12112yes—
297C42#12312yes—
298C42C23×C411yescharacteristic
299C42C22×D411yescharacteristic
300C42C2×#4111yescharacteristic
301C42C2×#4111yescharacteristic
302C23—22no—
303C23—22no—
304C23—22no—
305C23#38612yes—
306C23C2×#12011yescharacteristic
307Q8—48no—
308Q8—84no—
309Q8—84no—
310Q8—84no—
311Q8—88no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—88no—
335C2×C4—84no—
336C2×C4#38612yes—
337C2×C4#38612yes—
338C2×C4#38612yes—
339C2×C4#45614yes—
340C2×C4C22×C22⋊C411yescharacteristic
341C2×C4C2×C4×D411yescharacteristic
342C2×C4C2×C4×D411yescharacteristic
343C2×C4C2×#12111yescharacteristic
344C2×C4C2×#12111yescharacteristic
345C2×C4C2×#12311yescharacteristic
346C8—28no—
347C8—82no—
348C8—82no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—44no—
353C22C2×#38611yescharacteristic
354C22C2×#45612yes—
355C22#302112yes—
356C22#302212yes—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—21no—
362C4—22no—
363C4—21no—
364C4—21no—
365C4—44no—
366C4—44no—
367C4—84no—
368C4—84no—
369C4—84no—
370C4#188614yes—
371C2—42no—
372C2#1900914yes—
373C2#2094411yescharacteristic
374C2#2094511yescharacteristic
375C2C2×#188611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9000C2semidirect
12. #3165C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#343482
#218402
#218112
C2×#15372
#90002
#80352