Finite Group Databasebeta

← #232369

Subgroups — #232369

Order 512.

1728 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C212yesmaximal, complemented
2#34641C211yesmaximal, characteristic
3C2×#1517C212yesmaximal, complemented
4#14274C211yesmaximal, characteristic, complemented
5#9970C218yesmaximal, complemented
6#7906C211yesmaximal, characteristic, complemented
7C2×#532—22no—
8C2×#532C2212yes—
9C22×C4⋊Q8C2211yescharacteristic
10#3226—28no—
11#3226—28no—
12C2×#500—22no—
13C2×#486—22no—
14C2×#480C2218yes—
15C2×#474—24no—
16C2×#431C2212yes—
17#2658—22no—
18#2657—22no—
19#2656—22no—
20#2656C2214yes—
21#2397—24no—
22C2×#391—22no—
23C2×#382C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#1517—28no—
26C2×#327C2218yes—
27#1445—22no—
28#1445C2214yes—
29#1280C2214yes—
30C22×#41—22no—
31C23×Q8—21no—
32#532—24no—
33#532—24no—
34#532—24no—
35#532—24no—
36#532—48no—
37#532—48no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8D414yescomplemented
45C2×#127—22no—
46C2×#127D412yes—
47C2×#124—24no—
48C2×#124—22no—
49C2×#121—22no—
50C2×#121—28no—
51C2×#121C2314yes—
52C2×C4×D4—28no—
53C2×C4×D4—22no—
54C2×C4×D4D414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60#500—216no—
61#486—216no—
62#480—216no—
63#480—216no—
64#474—232no—
65#431—24no—
66#431D418yescomplemented
67C2×#107—28no—
68C2×C4⋊C8—22no—
69C2×#102—28no—
70C2×C22⋊C8C2314yes—
71C2×C4×C8—22no—
72#399—28no—
73#399—28no—
74#392—28no—
75#392—28no—
76#391—48no—
77#382—24no—
78#376—22no—
79#376—22no—
80C2×#95—21no—
81C2×#95—28no—
82C2×#95D412yes—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#327—216no—
86C2×#41—44no—
87C2×#41—44no—
88C2×#41—48no—
89C2×#41—48no—
90C22×Q8—22no—
91C22×Q8—24no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—42no—
95C22×Q8—44no—
96C22×Q8—44no—
97C22×D4—24no—
98C22×D4—28no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4C2×D414yes—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4C2×D414yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—24no—
110C4⋊Q8—28no—
111C4⋊Q8—28no—
112C4⋊Q8—24no—
113C4⋊Q8—24no—
114C4⋊Q8—48no—
115C4⋊Q8—44no—
116C4⋊Q8—44no—
117#127—24no—
118#127—24no—
119#127—48no—
120#124—416no—
121#124—44no—
122#124—44no—
123#121—24no—
124#121—24no—
125#121—24no—
126#121—24no—
127#121—416no—
128#121—44no—
129#121—44no—
130#121—416no—
131C4×D4—28no—
132C4×D4—28no—
133C4×D4—432no—
134C4×D4—44no—
135C4×D4—44no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D414yes—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42D818yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D414yes—
159C2×C42#4112yes—
160#107—232no—
161C4⋊C8—28no—
162#102—232no—
163C22⋊C8—24no—
164C22⋊C8C2×D418yes—
165C4×C8—28no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—42no—
170#95—416no—
171C24—24no—
172C24C22×D411yescharacteristic
173#41—816no—
174#41—88no—
175#41—88no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—48no—
180C2×Q8—416no—
181C2×Q8—84no—
182C2×Q8—84no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—48no—
188C2×D4—416no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—48no—
192C2×D4—48no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—48no—
196C2×D4—84no—
197C2×D4—88no—
198C2×D4—88no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—416no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4#12012yes—
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12112yes—
221C22×C4#12112yes—
222C22×C4#12712yes—
223C22×C4#12712yes—
224C22×C4C2×D818yes—
225C22×C4C2×D814yes—
226C22×C4AGL(1,ℤ/8)14yes—
227C22×C4C22×D412yes—
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D412yes—
230C22×C4C2×#4111yescharacteristic
231C2×C8—28no—
232C2×C8—48no—
233C4⋊C4—28no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—416no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—44no—
240C4⋊C4—84no—
241C22⋊C4—432no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—48no—
245C42—28no—
246C42—28no—
247C42#12014yes—
248C42#12014yes—
249C42#12114yes—
250C42#12114yes—
251C23—22no—
252C23—22no—
253C23—44no—
254C23—48no—
255C23—48no—
256C23—48no—
257C23C2×#12011yescharacteristic
258C23C2×#12111yescharacteristic
259C23C2×#12711yescharacteristic
260C23C22×D812yes—
261C23C2×AGL(1,ℤ/8)12yes—
262Q8—88no—
263Q8—88no—
264Q8—88no—
265D4—48no—
266D4—84no—
267D4—88no—
268D4—832no—
269D4—84no—
270D4—84no—
271D4—88no—
272D4—88no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—816no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#39214yes—
298C2×C4#39212yes—
299C2×C4#39212yes—
300C2×C4#44714yes—
301C2×C4#44714yes—
302C2×C4#45918yes—
303C2×C4#49314yes—
304C2×C4#49614yes—
305C2×C4C2×#12012yes—
306C2×C4C2×#12112yes—
307C8—48no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—24no—
312C22—21no—
313C22—21no—
314C22—84no—
315C22—84no—
316C22—816no—
317C22—84no—
318C22C2×#39211yescharacteristic
319C22C2×#44712yes—
320C22#302212yes—
321C22#302812yes—
322C22C2×#45912yes—
323C22C2×#49311yescharacteristic
324C22C2×#49611yescharacteristic
325C4—24no—
326C4—22no—
327C4—22no—
328C4—48no—
329C4—84no—
330C4—84no—
331C4—162no—
332C4#201114yes—
333C4#201114yes—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—164no—
338C2#1830814yes—
339C2#2340811yescharacteristic
340C2C2×#201111yescharacteristic
341C2#2344611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#34641C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3226C2semidirect
3. C2×#1517C2semidirect
4. #14274C2semidirect
5. #9970C2semidirect
6. #7906C2semidirect
44. C2×C4⋊Q8D4semidirect
66. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#346412
C2×#15172
#142742
#99702
#79062