Finite Group Databasebeta

← #6372179

Subgroups — #6372179

Order 512.

3382 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#2873C211yesmaximal, characteristic
3#34052C211yesmaximal, characteristic
4#21307C2116yesmaximal, complemented
5#21306C218yesmaximal, complemented
6C2×#1733C214yesmaximal
7C23×M16C2212yes—
8#3458—24no—
9#3458—24no—
10#3458C418yescomplemented
11#3458C2218yes—
12C2×#544—22no—
13C2×#544C418yescomplemented
14C2×#543—22no—
15C2×#533—24no—
16C2×#533C2214yes—
17C22×#117C2212yes—
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20#2873—24no—
21#2873C2218yes—
22C2×#405—24no—
23C2×C4×M16C2214yes—
24#2317—24no—
25#2316—22no—
26#2315—22no—
27#2296C2218yes—
28#2285C2214yes—
29#1986C22164yescomplemented
30#1733C22116yes—
31#1733C22116yes—
32C2×#339C22116yes—
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C22×M16—24no—
36C22×M16—28no—
37C22×M16Q8116yescomplemented
38C22×M16C2314yes—
39C22×M16C2318yes—
40C23×C8—22no—
41#544—232no—
42#544—216no—
43#544—216no—
44#544C2×C4164yescomplemented
45#543—216no—
46#543—216no—
47#533—232no—
48#533—232no—
49#533—28no—
50#533—28no—
51#533C2×C4116yescomplemented
52#533C23116yes—
53C2×C4⋊Q8—28no—
54C2×C4⋊Q8C2×C418yescomplemented
55C2×#125—28no—
56C2×#125C2×C418yescomplemented
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—216no—
59C2×C4×Q8—28no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—216no—
67C2×#117C2×C414yes—
68C2×#117C2×C418yes—
69C2×#117C2×C414yes—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C2×#117D414yes—
73C2×#117D414yes—
74C2×#117C2314yes—
75C2×#117C2318yes—
76C2×#117C2314yes—
77C2×#117C2314yes—
78C2×#117C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4D412yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—22no—
86C22×C42C2314yes—
87C22×C42C2312yes—
88#405—232no—
89C4×M16—28no—
90C4×M16C23116yes—
91C2×#97—24no—
92C2×C4×C8—24no—
93#389—216no—
94#385—216no—
95#384—24no—
96#382—24no—
97#380—28no—
98C4×C4⋊C4C23116yes—
99#339C23164yes—
100C25C2×D411yescharacteristic
101C22×Q8—28no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4C22×C412yes—
105C23×C4C22×C412yes—
106C23×C4C2×D414yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×Q818yes—
112C23×C4#4118yes—
113C23×C4C2411yescharacteristic
114C23×C4C2412yes—
115C2×M16—216no—
116C2×M16—232no—
117C2×M16—24no—
118C2×M16—28no—
119C2×M16—28no—
120C2×M16—216no—
121C2×M16C2×Q8116yes—
122C2×M16C2×Q8132yescomplemented
123C2×M16C2418yes—
124C22×C8—216no—
125C22×C8—24no—
126C22×C8—28no—
127C4⋊Q8—264no—
128C4⋊Q8—216no—
129C4⋊Q8C22×C4132yes—
130#125—264no—
131#125—216no—
132#125C22×C4132yes—
133C22⋊Q8—232no—
134C22⋊Q8—232no—
135C22⋊Q8—2128no—
136C4×Q8—232no—
137C4×Q8—232no—
138#117—28no—
139#117—28no—
140#117—28no—
141#117—24no—
142#117—28no—
143#117—24no—
144#117—28no—
145#117—216no—
146#117—216no—
147#117—48no—
148#117—48no—
149#117C22⋊C4116yes—
150#117C22⋊C4116yes—
151#117C22×C418yes—
152#117C22×C4116yes—
153#117C22×C4116yes—
154#117C2×D4116yes—
155#117C2418yes—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4C22⋊C418yes—
165C2×C4⋊C4C22⋊C418yes—
166C2×C4⋊C4C22×C418yes—
167C2×C4⋊C4C22×C4116yes—
168C2×C4⋊C4C2×D418yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4C22⋊C418yes—
177C2×C22⋊C4C22⋊C418yes—
178C2×C22⋊C4C2×D418yes—
179C2×C42—24no—
180C2×C42—28no—
181C2×C42—24no—
182C2×C42—28no—
183C2×C42—216no—
184C2×C42—24no—
185C2×C42C22×C414yes—
186C2×C42C22×C414yes—
187C2×C42C2×D414yes—
188C2×C42C2×D414yes—
189C2×C42C2×D414yes—
190C2×C42C2418yes—
191C2×C42C2414yes—
192#97—216no—
193C4×C8—216no—
194#95—216no—
195C24—82no—
196C24C2×C22⋊C412yes—
197C24C2×C22⋊C412yes—
198C24#12414yes—
199C24C4⋊Q814yes—
200C24C22×D411yescharacteristic
201C24C22×D412yes—
202C2×Q8—232no—
203C2×Q8—816no—
204C22×C4—28no—
205C22×C4—216no—
206C22×C4—28no—
207C22×C4—48no—
208C22×C4—84no—
209C22×C4—88no—
210C22×C4—88no—
211C22×C4C2×C22⋊C418yes—
212C22×C4C2×C22⋊C418yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C4×Q8116yes—
218C22×C4C4×Q8116yes—
219C22×C4C22⋊Q818yes—
220C22×C4C22⋊Q818yes—
221C22×C4C22⋊Q8116yes—
222C22×C4#12418yes—
223C22×C4#12414yes—
224C22×C4C4⋊Q818yes—
225C22×C4C4⋊Q814yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C23×C411yescharacteristic
228C22×C4C23×C414yes—
229C22×C4C22×D418yes—
230C22×C4C22×D412yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D414yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×Q814yes—
236C22×C4C22×Q818yes—
237C22×C4C2×#4118yes—
238C22×C4C2×#4114yes—
239C22×C4C2511yescharacteristic
240M16—216no—
241M16—232no—
242M16—28no—
243M16C22×Q8116yes—
244C2×C8—216no—
245C2×C8—232no—
246C2×C8—28no—
247C4⋊C4—216no—
248C4⋊C4—28no—
249C4⋊C4—232no—
250C4⋊C4—264no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—816no—
254C4⋊C4C2×C22⋊C4132yes—
255C4⋊C4C23×C4116yes—
256C22⋊C4—28no—
257C22⋊C4—216no—
258C22⋊C4—28no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4C2×C22⋊C4132yes—
262C42—28no—
263C42—216no—
264C42—28no—
265C42—216no—
266C42—48no—
267C42C2×C22⋊C418yes—
268C42C2×C22⋊C418yes—
269C42C23×C418yes—
270C42C22×D418yes—
271C23—44no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23—84no—
276C23—84no—
277C23#39118yes—
278C23#39118yes—
279C23C22×C22⋊C411yescharacteristic
280C23C22×C22⋊C414yes—
281C23C22×C22⋊C411yescharacteristic
282C23C2×#12412yes—
283C23C2×#12414yes—
284C23C2×C4⋊Q812yes—
285C23C2×C4⋊Q814yes—
286C23C23×D411yescharacteristic
287Q8—832no—
288C2×C4—216no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—416no—
293C2×C4—416no—
294C2×C4—48no—
295C2×C4—816no—
296C2×C4—816no—
297C2×C4#39118yes—
298C2×C4#391116yes—
299C2×C4#391116yes—
300C2×C4#39118yes—
301C2×C4C22×C22⋊C4116yes—
302C2×C4C22×C22⋊C411yescharacteristic
303C2×C4C22×C22⋊C414yes—
304C2×C4C22×C22⋊C411yescharacteristic
305C2×C4C2×C4×Q814yes—
306C2×C4C2×C4×Q814yes—
307C2×C4C2×C4×Q8116yes—
308C2×C4C2×C22⋊Q814yes—
309C2×C4C2×C22⋊Q818yes—
310C2×C4C2×C22⋊Q814yes—
311C2×C4C2×C22⋊Q818yes—
312C2×C4C2×#12414yes—
313C2×C4C2×#12412yes—
314C2×C4C2×C4⋊Q814yes—
315C2×C4C2×C4⋊Q812yes—
316C2×C4C24×C411yescharacteristic
317C2×C4C23×D411yescharacteristic
318C2×C4C23×Q812yes—
319C2×C4C22×#4112yes—
320C8—216no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—28no—
325C22—44no—
326C22—42no—
327C22—42no—
328C22—44no—
329C22—48no—
330C22—48no—
331C22C2×#39112yes—
332C22C2×#39118yes—
333C22C2×#39112yes—
334C22C23×C22⋊C411yescharacteristic
335C22C22×#12411yescharacteristic
336C22C22×C4⋊Q811yescharacteristic
337C4—216no—
338C4—416no—
339C4—88no—
340C4C2×#39112yes—
341C4C2×#39118yes—
342C4C2×#39112yes—
343C4C22×C4×Q812yes—
344C4C22×C22⋊Q812yes—
345C2—21no—
346C2—21no—
347C2—22no—
348C2—22no—
349C2—28no—
350C2C22×#39111yescharacteristic
351C2#6009312yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21307C2semidirect
5. #21306C2semidirect
10. #3458C4semidirect
13. C2×#544C4semidirect
29. #1986C22semidirect
37. C22×M16Q8semidirect
44. #544C2×C4semidirect
51. #533C2×C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
56. C2×#125C2×C4semidirect
122. C2×M16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#28732
#340522
#213072
#213062
C2×#17332