Finite Group Databasebeta

← #6482119

Subgroups — #6482119

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C212yesmaximal
2#35173C211yesmaximal, characteristic
3C2×#2369C211yesmaximal, characteristic
4#33109C214yesmaximal
5#29757C214yesmaximal
6#25455C218yesmaximal, complemented
7#25278C218yesmaximal, complemented
8#21951C211yesmaximal, characteristic
9#21385C212yesmaximal
10C2×#533C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12#3230—216no—
13C2×#500C2214yes—
14C2×#487C2214yes—
15C2×#484C2218yes—
16C2×#477C2218yes—
17C2×#431C2212yes—
18#2857C2214yes—
19#2659—22no—
20#2584—24no—
21#2567C2214yes—
22#2546C2214yes—
23#2369—24no—
24#2369C2218yes—
25#2316—21no—
26#2316—21no—
27C2×#391—22no—
28C2×#389—21no—
29C2×#389C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2083C2218yes—
35#2068C22116yes—
36#2043C2218yes—
37#1987C2214yes—
38#1981C2218yes—
39#1981C2214yes—
40#1947C2218yes—
41#1939C2218yes—
42#1928C22116yes—
43#1872C2214yes—
44#1872C2214yes—
45#1797C2214yes—
46C4×#103C2218yes—
47C23×Q8—21no—
48#533—28no—
49#533—28no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8D418yes—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2314yes—
55C2×C4⋊Q8C2312yes—
56C2×C4⋊Q8C2314yes—
57C2×#125—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2314yes—
60C2×C4×Q8—22no—
61C2×C4×Q8C2314yes—
62C2×#117—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#500—216no—
75#487—216no—
76#484—232no—
77#477—232no—
78#431—28no—
79C2×#107C23116yes—
80C2×C4⋊C8C2314yes—
81C2×C4⋊C8C2318yes—
82C2×#103C23116yes—
83C2×#103C2318yes—
84C2×#103C2318yes—
85C2×C22⋊C8C2314yes—
86C2×C4×C8C2314yes—
87#398—28no—
88#395—28no—
89#391—216no—
90#391C2318yes—
91#389—24no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—28no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389C2318yes—
100#384C2314yes—
101#380—24no—
102#380—24no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C4×C4⋊C4C2318yes—
108C4×C4⋊C4C2314yes—
109C4×C4⋊C4C2314yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340C2318yes—
115#340C2318yes—
116#336C2318yes—
117C22×Q8—28no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—82no—
121C22×Q8—82no—
122C22×Q8C2414yes—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8C2418yes—
132C4⋊Q8—216no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—28no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—48no—
139C4⋊Q8C2×D4116yes—
140#125—48no—
141C22⋊Q8—28no—
142C22⋊Q8—28no—
143C22⋊Q8—48no—
144C4×Q8—28no—
145C4×Q8—28no—
146C4×Q8—48no—
147#117—48no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—84no—
168C2×C4⋊C4Q1618yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D418yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2412yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2412yes—
176C2×C22⋊C4—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—28no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—84no—
183C2×C42C2×D414yes—
184C2×C42C2×D412yes—
185C2×C42C2×D414yes—
186C2×C42#4114yes—
187C2×C42C2414yes—
188C2×C42C2412yes—
189#107—232no—
190C4⋊C8—28no—
191C4⋊C8—48no—
192#103—232no—
193#103—216no—
194#103—48no—
195C22⋊C8—28no—
196C4×C8—28no—
197#95—28no—
198#95—28no—
199#95—24no—
200#95—28no—
201#95—28no—
202#95C2×D418yes—
203C24C22×D411yescharacteristic
204C2×Q8—216no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—816no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—28no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#12114yes—
239C22×C4#12114yes—
240C22×C4C2×Q1618yes—
241C22×C4C2×Q1614yes—
242C22×C4#13714yes—
243C22×C4C22×D414yes—
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C22×D414yes—
248C22×C4C22×D412yes—
249C22×C4C2×#4112yes—
250C22×C4C2511yescharacteristic
251C2×C8—28no—
252C2×C8—28no—
253C2×C8—28no—
254C4⋊C4—24no—
255C4⋊C4—216no—
256C4⋊C4—28no—
257C4⋊C4—28no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—88no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C4⋊C4C2×Q16116yes—
275C4⋊C4C22×D418yes—
276C4⋊C42−1+414yes—
277C22⋊C4—48no—
278C42—28no—
279C42—44no—
280C42—48no—
281C42—48no—
282C42—88no—
283C42#12118yes—
284C42C22×D414yes—
285C42C2×#4114yes—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23C2×#12112yes—
291C23C22×Q1612yes—
292C23C2×#13712yes—
293C23C23×D411yescharacteristic
294Q8—48no—
295Q8—48no—
296Q8—816no—
297Q8—88no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—416no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4#462116yes—
331C2×C4C2×#12112yes—
332C2×C4C2×#12114yes—
333C2×C4C2×#12114yes—
334C2×C4C22×Q1618yes—
335C2×C4C22×Q1614yes—
336C2×C4C2×#13714yes—
337C2×C4C23×D411yescharacteristic
338C2×C4C22×#4111yescharacteristic
339C2×C4C2×2−1+412yes—
340C2×C4#58512yes—
341C8—48no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22C2×#46214yes—
349C22C22×#12111yescharacteristic
350C22#348511yescharacteristic
351C22#348611yescharacteristic
352C22C23×Q1611yescharacteristic
353C22C22×#13711yescharacteristic
354C22#358511yescharacteristic
355C22#358511yescharacteristic
356C4—28no—
357C4—24no—
358C4—22no—
359C4—48no—
360C4—44no—
361C4—44no—
362C4—84no—
363C4—82no—
364C4—84no—
365C4C2×#46218yes—
366C4#346112yes—
367C4#349712yes—
368C2—24no—
369C2#3515311yescharacteristic
370C2C22×#46211yescharacteristic
371C2#6042311yescharacteristic
372C2#6169811yescharacteristic
373C2#6172111yescharacteristic
374C2#6182111yescharacteristic
375C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #25455C2semidirect
7. #25278C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#351732
C2×#23692
#331092
#297572
#254552
#252782
#219512
#213852