Finite Group Databasebeta

← #215596

Subgroups — #215596

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#20714C212yesmaximal
3#9879C218yesmaximal, complemented
4#8698C214yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C412yes—
8C23×C42C2211yescharacteristic
9C2×#431—22no—
10C2×#389—22no—
11C2×#389—22no—
12C2×#389—22no—
13C2×#389—22no—
14C2×#389C414yes—
15C2×#389C2214yes—
16C22×#95C2212yes—
17#1817—22no—
18#1797—22no—
19#1797C2214yes—
20#1779C2214yes—
21C4×C22⋊C8—22no—
22C4×C22⋊C8C2214yes—
23#1280C22116yes—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C412yes—
27C24×C4C2×C412yes—
28C24×C4C2311yescharacteristic
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—24no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—48no—
41C22×C4⋊C4C818yescomplemented
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4C2×C418yes—
44C22×C4⋊C4C2×C414yes—
45C22×C42—44no—
46C22×C42C2×C412yes—
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42C2312yes—
52#431—48no—
53C2×C4⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8C2318yes—
57C2×C4×C8—24no—
58#389—24no—
59#389—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66C2×#95—24no—
67C2×#95—24no—
68C2×#95C2×C414yes—
69C2×#95C2314yes—
70#336—24no—
71#336—24no—
72C25C22×C411yescharacteristic
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—44no—
87C23×C4—44no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4—44no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C2×C814yes—
98C23×C4C2×C818yes—
99C23×C4M1614yes—
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C22×C414yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C414yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C22×C8—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—216no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—416no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—416no—
127C2×C4⋊C4C2×C8116yes—
128C2×C42—28no—
129C2×C42—44no—
130C2×C42—44no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42—44no—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137C2×C42C22⋊C412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C4⋊C8—48no—
143C22⋊C8—216no—
144C22⋊C8—48no—
145C22⋊C8—48no—
146C4×C8—48no—
147#95—28no—
148#95—28no—
149C24—22no—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24C2×C22⋊C411yescharacteristic
156C24C2×C22⋊C411yescharacteristic
157C24C22×C812yes—
158C24C2×M1612yes—
159C24C23×C411yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—216no—
163C22×C4—216no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—28no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—416no—
198C22×C4—416no—
199C22×C4—44no—
200C22×C4—416no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—416no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—816no—
215C22×C4C22⋊C814yes—
216C22×C4C22⋊C818yes—
217C22×C4C22⋊C814yes—
218C22×C4#9914yes—
219C22×C4#10014yes—
220C22×C4#10114yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C22×C814yes—
227C22×C4C22×C818yes—
228C22×C4C2×M1614yes—
229C22×C4C22×D412yes—
230C22×C42−1+414yes—
231C2×C8—48no—
232C2×C8—48no—
233C2×C8—48no—
234C4⋊C4—216no—
235C4⋊C4—216no—
236C4⋊C4—48no—
237C4⋊C4—416no—
238C4⋊C4—48no—
239C4⋊C4—416no—
240C42—24no—
241C42—24no—
242C42—24no—
243C42—24no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C23—28no—
249C23—28no—
250C23—28no—
251C23—28no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23C2×C22⋊C812yes—
271C23C2×C22⋊C812yes—
272C23C2×#9912yes—
273C23#41012yes—
274C23C2×#10012yes—
275C23C2×#10112yes—
276C23C22×C22⋊C411yescharacteristic
277C23#51912yes—
278C23C23×C811yescharacteristic
279C23C22×M1611yescharacteristic
280C23#56812yes—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—28no—
288C2×C4—216no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—416no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—416no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4#324116yes—
319C2×C4C2×C22⋊C814yes—
320C2×C4C2×C22⋊C814yes—
321C2×C4C2×#9914yes—
322C2×C4C2×#10014yes—
323C2×C4C2×#10114yes—
324C2×C4#53614yes—
325C8—88no—
326C22—216no—
327C22—216no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22C2×#32414yes—
349C22#146014yes—
350C22#231011yescharacteristic
351C22C22×C22⋊C811yescharacteristic
352C22#287911yescharacteristic
353C22C22×#9911yescharacteristic
354C22C2×#41011yescharacteristic
355C22C22×#10011yescharacteristic
356C22C22×#10111yescharacteristic
357C22#298212yes—
358C22#298312yes—
359C4—24no—
360C4—24no—
361C4—24no—
362C4—42no—
363C4—48no—
364C4—42no—
365C4—84no—
366C4C2×#32418yes—
367C2—22no—
368C2—22no—
369C2—24no—
370C2—24no—
371C2C22×#32411yescharacteristic
372C2C2×#146011yescharacteristic
373C2#1739411yescharacteristic
374C2#1739611yescharacteristic
375C2#2181811yescharacteristic
376C2#2182511yescharacteristic
377C2#2182911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9879C2semidirect
41. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#207142
#98792
#86982