Finite Group Databasebeta

← #353693

Subgroups — #353693

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4C2×#1740C214yesmaximal
5#10854C218yesmaximal
6C22×#117C2212yes—
7C22×#117C2212yes—
8C22×#117C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2316—24no—
11#2316—24no—
12#2316—24no—
13#2316C418yescomplemented
14C2×#389—24no—
15C2×#389C418yescomplemented
16C2×#380C2214yes—
17C2×#380C2214yes—
18#2274—24no—
19#2274C418yescomplemented
20C2×C4×C4⋊C4—24no—
21C2×C4×C22⋊C4C2214yes—
22#1740—216no—
23C2×#342C22116yes—
24C24×C4C2312yes—
25C24×C4C2311yescharacteristic
26C2×#117—24no—
27C2×#117—24no—
28C2×#117—24no—
29C2×#117—24no—
30C2×#117—24no—
31C2×#117—24no—
32C2×#117—28no—
33C2×#117—28no—
34C2×#117—28no—
35C2×#117—28no—
36C2×#117C2×C418yes—
37C2×#117C2×C418yes—
38C2×#117D418yes—
39C2×#117Q818yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4C2×C414yes—
44C22×C4⋊C4C2×C418yes—
45C22×C4⋊C4C2×C414yes—
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4D414yes—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4Q814yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2314yes—
55C22×C42—22no—
56C22×C42—22no—
57C22×C42—24no—
58C22×C42C2×C414yes—
59#389—232no—
60#389—232no—
61#380—28no—
62#380—28no—
63#380—28no—
64#380C2×C4116yes—
65C4×C4⋊C4—232no—
66C4×C22⋊C4—28no—
67C4×C22⋊C4C2×C4116yes—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95C2×C418yes—
71#342—232no—
72C25C2411yescharacteristic
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—84no—
76C23×C4C22×C418yes—
77C23×C4C22×C414yes—
78C23×C4C22×C414yes—
79C23×C4C22×C414yes—
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82C23×C4C22×C412yes—
83C23×C4C2×D414yes—
84C23×C4C2×D414yes—
85C23×C4C2×D414yes—
86C23×C4C2×D412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×D414yes—
90C23×C4C2×Q814yes—
91C23×C4C2×Q812yes—
92C23×C4C2×Q814yes—
93C23×C4#4114yes—
94C23×C4#4114yes—
95#117—216no—
96#117—216no—
97#117—216no—
98#117—232no—
99#117—48no—
100#117—48no—
101#117—416no—
102#117—48no—
103#117—48no—
104#117C4⋊C4132yescomplemented
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—216no—
111C2×C4⋊C4—216no—
112C2×C4⋊C4—216no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—88no—
116C2×C4⋊C4—88no—
117C2×C4⋊C4C42116yes—
118C2×C4⋊C4C22⋊C4116yes—
119C2×C4⋊C4C22⋊C4116yes—
120C2×C4⋊C4C22⋊C4116yes—
121C2×C4⋊C4C4⋊C4116yes—
122C2×C4⋊C4C4⋊C4116yes—
123C2×C4⋊C4C4⋊C4116yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4C22×C418yes—
135C2×C22⋊C4C22×C418yes—
136C2×C22⋊C4C2×D418yes—
137C2×C22⋊C4C2×Q818yes—
138C2×C42—28no—
139C2×C42—28no—
140C2×C42—28no—
141C2×C42—28no—
142C2×C42—28no—
143C2×C42—216no—
144C2×C42—216no—
145C2×C42—44no—
146C2×C42—44no—
147C2×C42—88no—
148#95—216no—
149#95—216no—
150#95—216no—
151C24—42no—
152C24—42no—
153C24C23×C411yescharacteristic
154C24C23×C411yescharacteristic
155C24C23×C411yescharacteristic
156C24C22×D411yescharacteristic
157C24C22×D411yescharacteristic
158C24C22×D411yescharacteristic
159C24C22×D412yes—
160C24C22×Q811yescharacteristic
161C24C22×Q812yes—
162C24C2×#4112yes—
163C24C2×#4112yes—
164C22×C4—28no—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—216no—
170C22×C4—48no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4C2×C4218yes—
186C22×C4C2×C4214yes—
187C22×C4C2×C22⋊C418yes—
188C22×C4C2×C22⋊C418yes—
189C22×C4C2×C22⋊C418yes—
190C22×C4C2×C22⋊C414yes—
191C22×C4C2×C22⋊C414yes—
192C22×C4C2×C22⋊C414yes—
193C22×C4C2×C4⋊C418yes—
194C22×C4C2×C4⋊C418yes—
195C22×C4C2×C4⋊C418yes—
196C22×C4C2×C4⋊C414yes—
197C22×C4C2×C4⋊C414yes—
198C22×C4C2×C4⋊C414yes—
199C22×C4C2×C4⋊C418yes—
200C22×C4#11718yes—
201C22×C4C4×D418yes—
202C22×C4C4×D418yes—
203C22×C4C4×Q818yes—
204C22×C4C4×Q818yes—
205C22×C4#12114yes—
206C22×C4#12114yes—
207C22×C4#12118yes—
208C22×C4C22⋊Q814yes—
209C22×C4C22⋊Q814yes—
210C22×C4C22⋊Q818yes—
211C22×C4#12414yes—
212C22×C4#12414yes—
213C22×C4#12518yes—
214C22×C4#12714yes—
215C22×C4C4⋊Q818yes—
216C22×C4C4⋊Q814yes—
217C4⋊C4—232no—
218C4⋊C4—416no—
219C4⋊C4—416no—
220C4⋊C4—816no—
221C4⋊C4—816no—
222C4⋊C4#95164yes—
223C22⋊C4—28no—
224C22⋊C4—28no—
225C22⋊C4—28no—
226C22⋊C4—216no—
227C22⋊C4—48no—
228C22⋊C4—48no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4C2×C4⋊C4116yes—
232C42—232no—
233C42—416no—
234C42—416no—
235C42—816no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—44no—
250C23—44no—
251C23—41no—
252C23—44no—
253C23—44no—
254C23C22×C4211yescharacteristic
255C23C22×C22⋊C411yescharacteristic
256C23C22×C22⋊C411yescharacteristic
257C23C22×C22⋊C411yescharacteristic
258C23C22×C4⋊C411yescharacteristic
259C23C22×C4⋊C411yescharacteristic
260C23C22×C4⋊C411yescharacteristic
261C23C22×C4⋊C412yes—
262C23C2×#11712yes—
263C23C2×C4×D412yes—
264C23C2×C4×D412yes—
265C23C2×C4×Q812yes—
266C23C2×C4×Q812yes—
267C23C2×#12112yes—
268C23C2×#12112yes—
269C23C2×C22⋊Q812yes—
270C23C2×C22⋊Q812yes—
271C23C2×#12411yescharacteristic
272C23C2×#12411yescharacteristic
273C23C2×#12512yes—
274C23C2×#12711yescharacteristic
275C23C2×C4⋊Q811yescharacteristic
276C23C2×C4⋊Q812yes—
277C2×C4—216no—
278C2×C4—216no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—416no—
287C2×C4—48no—
288C2×C4—416no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—88no—
296C2×C4—88no—
297C2×C4—816no—
298C2×C4C2×#95116yes—
299C2×C4C2×#9518yes—
300C2×C4C4×C4⋊C4116yes—
301C2×C4#38018yes—
302C2×C4#38018yes—
303C2×C4#38218yes—
304C2×C4#38418yes—
305C2×C4#389116yes—
306C2×C4#38918yes—
307C2×C4#38918yes—
308C2×C4#39018yes—
309C2×C4#39018yes—
310C2×C4#39118yes—
311C2×C4#39118yes—
312C22—24no—
313C22—22no—
314C22—24no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22C22×#9511yescharacteristic
341C22C2×C4×C4⋊C412yes—
342C22C2×#38012yes—
343C22C2×#38211yescharacteristic
344C22C2×#38411yescharacteristic
345C22C2×#38912yes—
346C22C2×#38912yes—
347C22C2×#39011yescharacteristic
348C22C2×#39011yescharacteristic
349C22C2×#39111yescharacteristic
350C22C2×#39111yescharacteristic
351C22#288312yes—
352C22#288412yes—
353C4—216no—
354C4—416no—
355C4—88no—
356C4—88no—
357C4C4⋊#95116yes—
358C2—22no—
359C2—21no—
360C2—22no—
361C2—21no—
362C2—21no—
363C2—22no—
364C2—22no—
365C2—21no—
366C2C2×C4⋊#9511yescharacteristic
367C2#2035411yescharacteristic
368C2#2035511yescharacteristic
369C2#2121412yes—
370C2#2133412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2316C4semidirect
15. C2×#389C4semidirect
19. #2274C4semidirect
104. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#22742
C2×#17402
#108542