Finite Group Databasebeta

← #6372226

Subgroups — #6372226

Order 512.

3454 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#33996C211yesmaximal, characteristic
4#21334C218yesmaximal, complemented
5#21333C218yesmaximal
6#21332C218yesmaximal, complemented
7C2×#1740C214yesmaximal
8#3458—24no—
9#3458—24no—
10#3458—24no—
11#3458—24no—
12C2×#544C418yescomplemented
13C2×#544C2214yes—
14C2×#543C2214yes—
15C2×#533C418yescomplemented
16C2×#533C2214yes—
17C22×#117C2212yes—
18C22×#117C2212yes—
19C22×#117C2212yes—
20C22×#117C2211yescharacteristic
21#2317—24no—
22#2316C2214yes—
23#2315C2214yes—
24#2296—24no—
25#2285C2214yes—
26#2274—24no—
27#2274C2214yes—
28#2273C2214yes—
29C4×#117—24no—
30C4×#117C2214yes—
31#1996C22164yes—
32#1740C22116yes—
33#1740C22116yes—
34C2×#342C22116yes—
35C24×C4C2311yescharacteristic
36#544—232no—
37#544—232no—
38#544—216no—
39#544—216no—
40#543—216no—
41#543—216no—
42#533—232no—
43#533—232no—
44#533—28no—
45#533—28no—
46#533—28no—
47#533—28no—
48C2×C4⋊Q8C2×C4116yescomplemented
49C2×C4⋊Q8C2×C418yescomplemented
50C2×#125C2×C4116yescomplemented
51C2×#125C2×C418yescomplemented
52C2×C22⋊Q8C2×C4132yes—
53C2×C22⋊Q8C23116yes—
54C2×C4×Q8C2×C4116yescomplemented
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—28no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—216no—
65C2×#117C2×C414yes—
66C2×#117C2×C414yes—
67C2×#117C2×C414yes—
68C2×#117C2×C414yes—
69C2×#117D414yes—
70C2×#117D414yes—
71C2×#117D414yes—
72C2×#117D414yes—
73C2×#117Q8116yescomplemented
74C2×#117C2314yes—
75C2×#117C2314yes—
76C2×#117C2314yes—
77C2×#117C2314yes—
78C2×#117C2314yes—
79C2×#117C2314yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2×C414yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42C2×C414yes—
91C22×C42C2312yes—
92#389—216no—
93#385—216no—
94#384—24no—
95#382—24no—
96#380C23116yes—
97C4×C4⋊C4—216no—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—28no—
100C4×C22⋊C4—216no—
101C4×C22⋊C4C23116yes—
102#376—24no—
103C43—24no—
104#342C23164yes—
105C25C2411yescharacteristic
106C22×Q8C22×C4116yes—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4C22×C418yes—
110C23×C4C22×C412yes—
111C23×C4C22×C412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×Q818yes—
117C23×C4#4118yes—
118C23×C4C2414yes—
119C23×C4C2411yescharacteristic
120C23×C4C2411yescharacteristic
121C4⋊Q8—264no—
122C4⋊Q8—216no—
123C4⋊Q8—216no—
124#125—264no—
125#125—216no—
126#125—216no—
127C22⋊Q8—232no—
128C22⋊Q8—232no—
129C22⋊Q8—2128no—
130C4×Q8—232no—
131C4×Q8—232no—
132#117—28no—
133#117—28no—
134#117—28no—
135#117—28no—
136#117—216no—
137#117—216no—
138#117—24no—
139#117—24no—
140#117—24no—
141#117—24no—
142#117—28no—
143#117—28no—
144#117—28no—
145#117—28no—
146#117—216no—
147#117—216no—
148#117—48no—
149#117—48no—
150#117C22⋊C4116yes—
151#117C22⋊C4116yes—
152#117C22⋊C4116yes—
153#117C22⋊C4116yes—
154#117C2×Q8116yes—
155#117C2×Q8116yes—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4C22×C418yes—
166C2×C4⋊C4C22×C418yes—
167C2×C4⋊C4C22×C4116yes—
168C2×C4⋊C4C22×C4132yes—
169C2×C4⋊C4C22×C418yes—
170C2×C4⋊C4C2×D418yes—
171C2×C4⋊C4C2×D418yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—216no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C22×C418yes—
179C2×C22⋊C4C22×C418yes—
180C2×C22⋊C4C2×D418yes—
181C2×C22⋊C4C2×D418yes—
182C2×C22⋊C4C2×Q8116yes—
183C2×C22⋊C4C2418yes—
184C2×C22⋊C4C2418yes—
185C2×C22⋊C4C2418yes—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—216no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—216no—
194C2×C42—48no—
195C2×C42C22×C418yes—
196C2×C42C22×C418yes—
197C2×C42C22×C414yes—
198C2×C42C22×C414yes—
199C2×C42C2×D414yes—
200C2×C42C2×D414yes—
201C2×C42C2×D414yes—
202C2×C42C2×D414yes—
203#95—216no—
204#95—216no—
205C24—82no—
206C24C23×C412yes—
207C24C22×D412yes—
208C24C22×D412yes—
209C24C22×Q814yes—
210C24C2×#4114yes—
211C24C2511yescharacteristic
212C2×Q8—232no—
213C2×Q8—816no—
214C22×C4—28no—
215C22×C4—216no—
216C22×C4—216no—
217C22×C4—28no—
218C22×C4—48no—
219C22×C4—84no—
220C22×C4—88no—
221C22×C4—88no—
222C22×C4—88no—
223C22×C4C2×C22⋊C414yes—
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C414yes—
227C22×C4C4×Q8116yes—
228C22×C4C4×Q8116yes—
229C22×C4C22⋊Q818yes—
230C22×C4C22⋊Q818yes—
231C22×C4C22⋊Q818yes—
232C22×C4C22⋊Q818yes—
233C22×C4#12418yes—
234C22×C4#12418yes—
235C22×C4C4⋊Q818yes—
236C22×C4C4⋊Q818yes—
237C22×C4C23×C4116yes—
238C22×C4C23×C418yes—
239C22×C4C23×C412yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D418yes—
243C22×C4C22×D418yes—
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×Q814yes—
251C22×C4C22×Q814yes—
252C22×C4C2×#4114yes—
253C22×C4C2×#4114yes—
254C4⋊C4—216no—
255C4⋊C4—28no—
256C4⋊C4—28no—
257C4⋊C4—232no—
258C4⋊C4—264no—
259C4⋊C4—216no—
260C4⋊C4—216no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—816no—
264C4⋊C4C2×C22⋊C4132yes—
265C22⋊C4—216no—
266C22⋊C4—216no—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—28no—
270C22⋊C4—216no—
271C22⋊C4—48no—
272C22⋊C4C2×C22⋊C4132yes—
273C22⋊C4C22×Q8116yes—
274C42—28no—
275C42—216no—
276C42—28no—
277C42—216no—
278C42—216no—
279C42—48no—
280C42—48no—
281C42—816no—
282C42C2×C22⋊C418yes—
283C42C2×C22⋊C418yes—
284C42C2×C22⋊C418yes—
285C42C2×C22⋊C418yes—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—42no—
290C23—84no—
291C23—84no—
292C23C22×C22⋊C414yes—
293C23C2×C4×Q818yes—
294C23C2×C22⋊Q814yes—
295C23C2×C22⋊Q814yes—
296C23C2×#12414yes—
297C23C2×C4⋊Q814yes—
298C23C24×C411yescharacteristic
299C23C23×D411yescharacteristic
300C23C23×D411yescharacteristic
301C23C23×Q812yes—
302C23C22×#4112yes—
303Q8—832no—
304C2×C4—216no—
305C2×C4—216no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—416no—
310C2×C4—416no—
311C2×C4—416no—
312C2×C4—48no—
313C2×C4—816no—
314C2×C4—816no—
315C2×C4—816no—
316C2×C4#391116yes—
317C2×C4#391116yes—
318C2×C4#391116yes—
319C2×C4#391116yes—
320C2×C4C22×C22⋊C4116yes—
321C2×C4C22×C22⋊C411yescharacteristic
322C2×C4C22×C22⋊C414yes—
323C2×C4C22×C22⋊C411yescharacteristic
324C2×C4C22×C22⋊C411yescharacteristic
325C2×C4C22×C22⋊C411yescharacteristic
326C2×C4C2×C4×Q818yes—
327C2×C4C2×C4×Q814yes—
328C2×C4C2×C4×Q814yes—
329C2×C4C2×C22⋊Q814yes—
330C2×C4C2×C22⋊Q814yes—
331C2×C4C2×C22⋊Q814yes—
332C2×C4C2×C22⋊Q814yes—
333C2×C4C2×#12412yes—
334C2×C4C2×#12414yes—
335C2×C4C2×#12412yes—
336C2×C4C2×C4⋊Q814yes—
337C2×C4C2×C4⋊Q812yes—
338C2×C4C2×C4⋊Q812yes—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—28no—
343C22—44no—
344C22—44no—
345C22—48no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22C2×#39118yes—
351C22C23×C22⋊C411yescharacteristic
352C22C22×C4×Q812yes—
353C22C22×C22⋊Q812yes—
354C22C22×#12411yescharacteristic
355C22C22×C4⋊Q811yescharacteristic
356C4—216no—
357C4—416no—
358C4—416no—
359C4—88no—
360C4C2×#39112yes—
361C4C2×#39118yes—
362C4C2×#39112yes—
363C4C2×#39112yes—
364C4C2×#39112yes—
365C2—22no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2—28no—
370C2C22×#39111yescharacteristic
371C2#6007012yes—
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
socleC22C23×C22⋊C4
prime radicalC25C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×#117 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21334C2semidirect
6. #21332C2semidirect
12. C2×#544C4semidirect
15. C2×#533C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#125C2×C4semidirect
51. C2×#125C2×C4semidirect
54. C2×C4×Q8C2×C4semidirect
73. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
#340522
#339962
#213342
#213332
#213322
C2×#17402