Finite Group Databasebeta

← #108202

Subgroups — #108202

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33996C211yesmaximal, characteristic
2#20717C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7949C212yesmaximal, complemented
5#7869C212yesmaximal, complemented
6C22×#117—21no—
7C22×#117C2211yescharacteristic
8#2920C2212yes—
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2274—24no—
12#2274—21no—
13#2274C412yes—
14#2273—21no—
15#2273C412yes—
16C4×#117—21no—
17C4×#117—24no—
18C4×#117C2212yes—
19#1817—21no—
20#1798C414yescomplemented
21#1779—21no—
22#1750—22no—
23#1464C414yescomplemented
24#1463—21no—
25#1462—21no—
26#1458—22no—
27#1285—24no—
28#1276—24no—
29C24×C4D411yescharacteristic
30C2×#117—216no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117C2×C412yes—
40C2×#117C2×C412yes—
41C2×#117D412yes—
42C2×#117Q812yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—22no—
50C22×C42D411yescharacteristic
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432C2×C414yescomplemented
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#423—22no—
61C2×C4⋊C8C2×C412yescomplemented
62C2×C4⋊C8C2×C412yescomplemented
63#407—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—22no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—28no—
73C4×C4⋊C4—28no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4D414yes—
76C4×C22⋊C4—28no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—28no—
81C4×C22⋊C4C2×C414yes—
82#376—22no—
83#376—22no—
84C43—22no—
85C43D414yes—
86#336—24no—
87#324—24no—
88#323—24no—
89C25C22⋊C411yescharacteristic
90C23×C4—28no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C4⋊C412yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—42no—
105C22×C8—22no—
106C22×C8—22no—
107#117—216no—
108#117—216no—
109#117—216no—
110#117—216no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—24no—
118#117—24no—
119C2×C4⋊C4—216no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C4212yes—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C4⋊C412yes—
129C2×C4⋊C4C4⋊C412yes—
130C2×C22⋊C4—216no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4C4212yes—
138C2×C22⋊C4C22⋊C412yes—
139C2×C22⋊C4C4⋊C412yes—
140C2×C22⋊C4C4⋊C412yes—
141C2×C42—24no—
142C2×C42—216no—
143C2×C42—21no—
144C2×C42—24no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C418yes—
153C2×C42C4⋊C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42#4112yes—
161C2×C42#4112yes—
162C4⋊C8—24no—
163C4⋊C8—24no—
164C4⋊C8—24no—
165C4⋊C8—42no—
166C22⋊C8—24no—
167C22⋊C8—48no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—42no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173#97—24no—
174#97—42no—
175C4×C8—42no—
176C4×C8—42no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181C24—21no—
182C24—24no—
183C24—21no—
184C24—42no—
185C24—81no—
186C24#9512yes—
187C24C2×C22⋊C411yescharacteristic
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—28no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4#9512yes—
218C22×C4#9512yes—
219C22×C4#9512yes—
220C22×C4#10418yes—
221C22×C4C2×C4211yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4C2×C4⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4#11712yes—
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12012yes—
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12312yes—
235M16—84no—
236C2×C8—44no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—44no—
240C2×C8—44no—
241C4⋊C4—216no—
242C4⋊C4—216no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C22⋊C4—216no—
248C22⋊C4—216no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C42—216no—
254C42—216no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—48no—
260C42—84no—
261C42—84no—
262C42—84no—
263C42C4×D412yes—
264C42C4×D412yes—
265C42C4×D412yes—
266C42C4×D412yes—
267C42#12112yes—
268C42#12112yes—
269C42C22⋊Q812yes—
270C42C22⋊Q812yes—
271C23—22no—
272C23—22no—
273C23—24no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—88no—
286C23#33914yes—
287C23C2×#9511yescharacteristic
288C23#37911yescharacteristic
289C23#38111yescharacteristic
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—216no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—48no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—816no—
324C2×C4#33914yes—
325C2×C4C4×C22⋊C412yes—
326C2×C4#38012yes—
327C2×C4#38511yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38512yes—
330C2×C4#38612yes—
331C2×C4#38611yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4C2×#10414yes—
334C8—84no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—24no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—82no—
352C22—84no—
353C22C22⋊#9511yescharacteristic
354C22C2×#33912yes—
355C22#179112yes—
356C4—24no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—416no—
364C4—42no—
365C4—42no—
366C4#189612yes—
367C4#189612yes—
368C2—22no—
369C2—22no—
370C2—41no—
371C2—44no—
372C2#1071911yescharacteristic
373C2#1074311yescharacteristic
374C2#1074711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7949C2semidirect
5. #7869C2semidirect
20. #1798C4semidirect
23. #1464C4semidirect
55. #432C2×C4semidirect
61. C2×C4⋊C8C2×C4semidirect
62. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#339962
#207172
#173572
#79492
#78692