Finite Group Databasebeta

← #368947

Subgroups — #368947

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic
4#21314C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#11217C214yesmaximal
7#10899C214yesmaximal
8#10857C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×#125C2211yescharacteristic
12#3166—24no—
13#3166C418yescomplemented
14C2×#501—22no—
15C2×#499—22no—
16C2×#428C2214yes—
17C2×#423C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2360—24no—
20#2355—22no—
21Q8×C4⋊C4—22no—
22#2330—22no—
23#2317C2212yes—
24C4×C4⋊Q8—22no—
25C2×#384C2211yescharacteristic
26#2274C2212yes—
27C4×#117C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1995—24no—
30#1989—22no—
31#1989—22no—
32C2×#343C2214yes—
33#1736C2212yes—
34#1736C2212yes—
35C2×#339C2214yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38#1441—24no—
39C4⋊#95C2214yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42#533—22no—
43#533—22no—
44#533C2×C414yes—
45#533D414yes—
46C2×C4⋊Q8D412yes—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C414yes—
54C2×#125D412yes—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—22no—
58C2×#117C2311yescharacteristic
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—22no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#501—216no—
68#499—216no—
69#428—28no—
70#428C2×C4116yes—
71C2×#107—24no—
72C2×#106—24no—
73#423—22no—
74#423—22no—
75C2×C4⋊C8—22no—
76C4×M16—22no—
77C4×M16C2×C414yes—
78C2×#97—21no—
79C2×C4×C8—21no—
80#391—24no—
81#389—24no—
82#389—28no—
83#389—24no—
84#387—28no—
85#385—24no—
86#384—22no—
87#384—21no—
88#384—21no—
89#384—22no—
90#382—22no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98#376—21no—
99#376—22no—
100#376—22no—
101C2×#95—22no—
102C2×#95—28no—
103C43—21no—
104#343—28no—
105#340—22no—
106#340—22no—
107#340—24no—
108#339—24no—
109#339—28no—
110#339Q818yescomplemented
111C22×Q8—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D414yes—
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—24no—
122C2×M16—22no—
123C2×M16—24no—
124C2×M16C22×C414yes—
125C2×M16C2×Q814yes—
126C2×M16#4114yes—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—24no—
130C4⋊Q8C22⋊C418yes—
131#125—24no—
132#125—24no—
133#125—28no—
134#125—24no—
135#125—44no—
136#125—42no—
137#125—42no—
138#125C22⋊C418yes—
139C22⋊Q8—28no—
140C22⋊Q8—28no—
141C4×Q8—28no—
142C4×Q8—28no—
143C4×Q8—88no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117C22×C414yes—
150#117C2×D414yes—
151#117C2×Q814yes—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C22×C414yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—28no—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×Q814yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#107—216no—
199#106—216no—
200C4⋊C8—44no—
201#97—24no—
202C4×C8—24no—
203#95—28no—
204#95—28no—
205#95—22no—
206#95—22no—
207#95—44no—
208#95—88no—
209C24C22×D411yescharacteristic
210C2×Q8—28no—
211C2×Q8—44no—
212C2×Q8—44no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4C4×D414yes—
233C22×C4#12012yes—
234C22×C4#12012yes—
235C22×C4#12114yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×Q812yes—
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4112yes—
243C22×C4C2×#4111yescharacteristic
244C22×C42−1+412yes—
245C22×C42−1+412yes—
246M16—24no—
247M16—24no—
248M16—24no—
249M16—44no—
250M16—44no—
251M16C4×Q818yescomplemented
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—216no—
261C4⋊C4—28no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C4⋊C4C2×C22⋊C418yes—
279C4⋊C4C4⋊Q814yes—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—44no—
290C42—44no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—88no—
295C42C2×C22⋊C412yes—
296C42C2×C22⋊C412yes—
297C42C4×D414yes—
298C42#12114yes—
299C42#12312yes—
300C42#12312yes—
301C23—22no—
302C23—22no—
303C23C2×C4×D411yescharacteristic
304C23C2×#12011yescharacteristic
305C23C2×#12111yescharacteristic
306C23#54712yes—
307C23D4×Q812yes—
308Q8—416no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—48no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—88no—
338C2×C4#38614yes—
339C2×C4#38614yes—
340C2×C4C22×C22⋊C411yescharacteristic
341C2×C4C2×C4×D411yescharacteristic
342C2×C4C2×C4×Q812yes—
343C2×C4#51912yes—
344C2×C4C2×#12111yescharacteristic
345C2×C4C2×#12311yescharacteristic
346C2×C4C2×C4⋊Q811yescharacteristic
347C2×C4#53211yescharacteristic
348C2×C4#53312yes—
349C2×C4#53611yescharacteristic
350C2×C4#53611yescharacteristic
351C2×C4#54112yes—
352C2×C4#54812yes—
353C2×C4#54812yes—
354C2×C4D4×Q812yes—
355C2×C4D4×Q812yes—
356C2×C4#55612yes—
357C2×C4#55712yes—
358C8—28no—
359C8—82no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—24no—
365C22C2×#38611yescharacteristic
366C22#236512yes—
367C22#242912yes—
368C22#243211yescharacteristic
369C22#243411yescharacteristic
370C22#301912yes—
371C22#302012yes—
372C4—24no—
373C4—24no—
374C4—48no—
375C4—44no—
376C4—84no—
377C4—81no—
378C4—84no—
379C4—81no—
380C4—84no—
381C4C4×C4⋊Q811yescharacteristic
382C4#232711yescharacteristic
383C4Q8×C22⋊C412yes—
384C4#244612yes—
385C4#254511yescharacteristic
386C4#254611yescharacteristic
387C2—22no—
388C2—22no—
389C2#1402411yescharacteristic
390C2#2094312yes—
391C2#2569011yescharacteristic
392C2#2569111yescharacteristic
393C2#2583212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×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
13. #3166C4semidirect
110. #339Q8semidirect
251. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#342142
C2×#19952
#213142
#142832
#112172
#108992
#108572
#107012