Finite Group Databasebeta

← #232428

Subgroups — #232428

Order 512.

1628 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1628 conjugacy classes of subgroups — too many to draw. Only its 210 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×#3234C212yesmaximal, complemented
2#35166C211yesmaximal, characteristic
3C2×#1514C212yesmaximal, complemented
4#14289C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9968C214yesmaximal, complemented
7#7912C211yesmaximal, characteristic, complemented
8C2×#537—22no—
9C2×#532C2212yes—
10C22×#125C2211yescharacteristic
11#3234—28no—
12#3234—28no—
13C2×#501—22no—
14C2×#490—22no—
15C2×#481C2214yes—
16C2×#480C2214yes—
17C2×#478—24no—
18C2×#432C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2542—22no—
22#2541—22no—
23#2540—24no—
24C2×#383C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1514—28no—
27C2×#327C2218yes—
28#1445C2212yes—
29#1445C2212yes—
30#1441—22no—
31#1436—22no—
32#1280C2214yes—
33C22×#41—22no—
34#537—416no—
35#532—24no—
36#532—24no—
37#532—24no—
38#532—24no—
39C2×C4⋊Q8D412yes—
40C2×#127—22no—
41C2×#127D412yes—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125D414yescomplemented
47C2×#124—22no—
48C2×C22⋊Q8—28no—
49C2×#121—28no—
50C2×#121—22no—
51C2×#121—28no—
52C2×#121C2314yes—
53C2×C4×D4—22no—
54C2×C4×D4D414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62#501—216no—
63#490—216no—
64#481—28no—
65#481—28no—
66#480—28no—
67#480—28no—
68#478—232no—
69#432—24no—
70#432D418yescomplemented
71C2×#107—24no—
72C2×#106—24no—
73C2×C4⋊C8—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78#399—28no—
79#392—28no—
80#390—28no—
81#390—24no—
82#384—22no—
83#383—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C2×#95—21no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#327—216no—
93C2×#41—416no—
94C2×#41—44no—
95C2×#41—44no—
96C22×Q8—22no—
97C22×D4—24no—
98C22×D4—28no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4C2×D414yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D414yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2411yescharacteristic
110C22×C8—24no—
111C4⋊Q8—24no—
112C4⋊Q8—24no—
113#127—24no—
114#127—24no—
115#127—84no—
116#125—28no—
117#125—28no—
118#125—44no—
119#125—44no—
120#124—44no—
121#124—44no—
122C22⋊Q8—432no—
123#121—24no—
124#121—24no—
125#121—24no—
126#121—24no—
127#121—432no—
128#121—416no—
129#121—44no—
130#121—44no—
131#121—416no—
132C4×D4—28no—
133C4×D4—28no—
134C4×D4—44no—
135C4×D4—44no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D414yes—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—82no—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D414yes—
166C2×C42#4112yes—
167#107—216no—
168#106—216no—
169C4⋊C8—28no—
170#102—216no—
171#102—216no—
172C22⋊C8—24no—
173C22⋊C8C2×D418yes—
174#97—28no—
175#95—24no—
176#95—24no—
177#95—42no—
178#95—42no—
179#95—48no—
180#95—84no—
181C24—24no—
182C24C22×D411yescharacteristic
183#41—816no—
184#41—816no—
185C2×Q8—48no—
186C2×Q8—44no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—48no—
190C2×D4—48no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—416no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—416no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—88no—
200C2×D4—88no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—416no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12012yes—
229C22×C4#12112yes—
230C22×C4#12112yes—
231C22×C4#12712yes—
232C22×C4#12712yes—
233C22×C4AGL(1,ℤ/8)18yes—
234C22×C4C22×D412yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C2×#4111yescharacteristic
238C2×C8—28no—
239C2×C8—48no—
240C4⋊C4—216no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—416no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C22⋊C4—432no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—48no—
256C42—44no—
257C42—82no—
258C42#12014yes—
259C42#12014yes—
260C42#12114yes—
261C42#12114yes—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—44no—
266C23—48no—
267C23—44no—
268C23—44no—
269C23—48no—
270C23C2×#12011yescharacteristic
271C23C2×#12111yescharacteristic
272C23C2×#12711yescharacteristic
273C23C2×AGL(1,ℤ/8)14yes—
274Q8—88no—
275D4—48no—
276D4—84no—
277D4—88no—
278D4—88no—
279D4—88no—
280D4—88no—
281D4—1616no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—816no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—82no—
317C2×C4#39212yes—
318C2×C4#39212yes—
319C2×C4#39214yes—
320C2×C4C2×#12012yes—
321C2×C4C2×#12112yes—
322C8—48no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—24no—
328C22—24no—
329C22—84no—
330C22—84no—
331C22—84no—
332C22—88no—
333C22—88no—
334C22C2×#39211yescharacteristic
335C22#300312yes—
336C22#301911yescharacteristic
337C22#302011yescharacteristic
338C22#302812yes—
339C22#303812yes—
340C22#315112yes—
341C4—22no—
342C4—22no—
343C4—24no—
344C4—22no—
345C4—22no—
346C4—48no—
347C4—82no—
348C4—82no—
349C4—84no—
350C4—162no—
351C2—22no—
352C2—22no—
353C2—164no—
354C2#1836312yes—
355C2#1837312yes—
356C2#2340911yescharacteristic
357C2#2343011yescharacteristic
358C2#2344611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35166C2
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×#3234C2semidirect
3. C2×#1514C2semidirect
4. #14289C2semidirect
5. #9970C2semidirect
6. #9968C2semidirect
7. #7912C2semidirect
46. C2×#125D4semidirect
70. #432D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
#351662
C2×#15142
#142892
#99702
#99682
#79122