Finite Group Databasebeta

← #415314

Subgroups — #415314

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C212yesmaximal
2#35173C211yesmaximal, characteristic
3C2×#1986C212yesmaximal
4#14160C211yesmaximal, characteristic
5#11342C214yesmaximal
6#11342C214yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#533C2212yes—
9C22×#125C2211yescharacteristic
10#3166—28no—
11#3166—28no—
12C2×#501—22no—
13C2×#501—22no—
14C2×#499—24no—
15C2×#428C2214yes—
16C2×#428C2214yes—
17C2×C4×M16C2212yes—
18#2857—22no—
19#2659—22no—
20#2584—24no—
21#2567—22no—
22#2546—22no—
23#2316C2214yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1986—28no—
27C2×#343C2214yes—
28C2×#339C2218yes—
29#1441—22no—
30#1436—22no—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C22×M16C2314yes—
34#533—24no—
35#533—24no—
36#533—24no—
37#533—24no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D412yes—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—24no—
44C2×#125—22no—
45C2×#125D414yes—
46C2×#125D414yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117C2314yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C42D412yes—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61#501—216no—
62#501—216no—
63#499—232no—
64#428—28no—
65#428—28no—
66#428—28no—
67#428—28no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#106—24no—
71C2×#106—24no—
72C4×M16—24no—
73C4×M16D418yes—
74C2×#97—22no—
75C2×C4×C8—22no—
76#398—28no—
77#395—28no—
78#391—24no—
79#389—28no—
80#389—28no—
81#389—28no—
82#384—22no—
83#380—28no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—24no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95D412yes—
91#343Q8116yescomplemented
92#339—216no—
93C22×Q8—22no—
94C22×Q8—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4C2×D414yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2411yescharacteristic
102C2×M16—24no—
103C2×M16—28no—
104C2×M16C2×D418yes—
105C2×M16C2×Q8116yes—
106C22×C8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—84no—
112#125—28no—
113#125—28no—
114#125—28no—
115#125—28no—
116#125—44no—
117#125—44no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8—44no—
126#117—24no—
127#117—24no—
128#117—24no—
129#117—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×Q818yes—
158C2×C22⋊C4—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—22no—
165C2×C42—82no—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D414yes—
170C2×C42#4112yes—
171#107—216no—
172#107—216no—
173#106—216no—
174#106—216no—
175#97—28no—
176C4×C8—28no—
177#95—216no—
178#95—24no—
179#95—48no—
180#95—42no—
181#95—84no—
182C24C22×D411yescharacteristic
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—84no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#12012yes—
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12112yes—
217C22×C4#12112yes—
218C22×C4#12712yes—
219C22×C4#12712yes—
220C22×C4C4⋊Q818yes—
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D412yes—
223C22×C4C22×D412yes—
224C22×C4C22×Q814yes—
225C22×C4C2×#4111yescharacteristic
226C22×C42−1+414yes—
227M16—28no—
228M16—216no—
229M16C4⋊Q8116yescomplemented
230C2×C8—28no—
231C2×C8—216no—
232C4⋊C4—216no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—88no—
250C4⋊C4—88no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C42—44no—
254C42—44no—
255C42—44no—
256C42—48no—
257C42—82no—
258C42#12014yes—
259C42#12014yes—
260C42#12114yes—
261C42#12114yes—
262C23—22no—
263C23—21no—
264C23—21no—
265C23C2×#12011yescharacteristic
266C23C2×#12111yescharacteristic
267C23C2×#12711yescharacteristic
268C23C2×C4⋊Q812yes—
269C23#53612yes—
270Q8—88no—
271Q8—88no—
272C2×C4—24no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—28no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4#39212yes—
308C2×C4#39214yes—
309C2×C4#39212yes—
310C2×C4C2×#12012yes—
311C2×C4C2×#12112yes—
312C2×C4C2×C4⋊Q814yes—
313C2×C4#53614yes—
314C2×C4D4×Q814yes—
315C2×C4D4×Q818yes—
316C2×C4#55714yes—
317C8—216no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22C2×#39211yescharacteristic
327C22#243412yes—
328C22#254612yes—
329C22#265811yescharacteristic
330C22#286811yescharacteristic
331C22#301812yes—
332C22#301911yescharacteristic
333C22#302011yescharacteristic
334C4—24no—
335C4—22no—
336C4—22no—
337C4—48no—
338C4—84no—
339C4—82no—
340C4—84no—
341C4—82no—
342C4—84no—
343C4#243414yes—
344C4#254614yes—
345C2—21no—
346C2—21no—
347C2—22no—
348C2#1519011yescharacteristic
349C2#2340611yescharacteristic
350C2#2340911yescharacteristic
351C2#2582712yes—
352C2#2582812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
91. #343Q8semidirect
229. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#351732
C2×#19862
#141602
#113422
C4×#3432