Finite Group Databasebeta

← #232285

Subgroups — #232285

Order 512.

1778 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 218 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×#3112C212yesmaximal, complemented
2#35166C211yesmaximal, characteristic
3C2×#1505C212yesmaximal, complemented
4#13850C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9964C214yesmaximal, complemented
7#7912C211yesmaximal, characteristic, complemented
8C2×#537—22no—
9C2×#532C2212yes—
10C22×C4×Q8C2211yescharacteristic
11#3112—28no—
12#3112—28no—
13C2×#480C2214yes—
14C2×#477—22no—
15C2×#460—22no—
16C2×#448C2214yes—
17C2×#440—24no—
18C2×#432C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2542—22no—
22#2541—22no—
23#2540—24no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1505—28no—
27C2×#327C2218yes—
28#1446C2212yes—
29#1445C2212yes—
30#1441—22no—
31C4×#95—22no—
32#1280C2214yes—
33C22×#41—22no—
34C23×Q8C2311yescharacteristic
35C22×SD16—24no—
36#537—416no—
37#532—24no—
38#532—24no—
39#532—24no—
40#532—24no—
41C2×C4⋊Q8D412yes—
42C2×#127—22no—
43C2×#127D412yes—
44C2×#124—22no—
45C2×C22⋊Q8—28no—
46C2×#121—28no—
47C2×#121—22no—
48C2×#121—28no—
49C2×#121C2314yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8D414yescomplemented
55C2×C4×D4—22no—
56C2×C4×D4D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64#480—28no—
65#480—28no—
66#477—216no—
67#460—216no—
68#448—28no—
69#448—28no—
70#440—232no—
71#432—24no—
72#432D418yescomplemented
73C2×#107—24no—
74C2×C4⋊C8—22no—
75C2×#103—24no—
76C2×#102—24no—
77C2×C22⋊C8C2314yes—
78C2×#97—22no—
79#399—28no—
80#392—28no—
81#390—28no—
82#390—24no—
83#384—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86#376—24no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#327—216no—
93C2×#41—416no—
94C2×#41—44no—
95C2×#41—44no—
96C22×Q8—22no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—44no—
101C22×Q8C2×D412yes—
102C22×D4—24no—
103C22×D4—28no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4C2×D414yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D414yes—
114C23×C4C2411yescharacteristic
115C2×SD16—216no—
116C2×SD16—416no—
117C22×C8—24no—
118C4⋊Q8—24no—
119C4⋊Q8—24no—
120#127—24no—
121#127—24no—
122#127—84no—
123#124—44no—
124#124—44no—
125C22⋊Q8—432no—
126#121—24no—
127#121—24no—
128#121—24no—
129#121—24no—
130#121—432no—
131#121—416no—
132#121—44no—
133#121—44no—
134#121—416no—
135C4×Q8—28no—
136C4×Q8—28no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—44no—
142C4×D4—44no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D414yes—
155C2×C22⋊C4—216no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—42no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—44no—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D414yes—
172C2×C42#4112yes—
173#107—216no—
174C4⋊C8—28no—
175#103—216no—
176#102—216no—
177C22⋊C8—24no—
178C22⋊C8C2×D418yes—
179#97—28no—
180#95—24no—
181#95—24no—
182#95—48no—
183#95—42no—
184#95—48no—
185C24—24no—
186C24C22×D411yescharacteristic
187#41—816no—
188#41—816no—
189C2×Q8—28no—
190C2×Q8—24no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8—42no—
197C2×Q8—44no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—48no—
201C2×D4—48no—
202C2×D4—416no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—416no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—88no—
211C2×D4—88no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—416no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4#12011yescharacteristic
238C22×C4#12012yes—
239C22×C4#12011yescharacteristic
240C22×C4#12112yes—
241C22×C4#12112yes—
242C22×C4#12712yes—
243C22×C4#12712yes—
244C22×C4AGL(1,ℤ/8)18yes—
245C22×C4C22×D412yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C2×#4111yescharacteristic
249SD16—432no—
250C2×C8—28no—
251C2×C8—48no—
252C4⋊C4—216no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—416no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C22⋊C4—432no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C42—48no—
265C42—44no—
266C42—44no—
267C42—44no—
268C42—82no—
269C42#12014yes—
270C42#12014yes—
271C42#12114yes—
272C42#12114yes—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—44no—
277C23—48no—
278C23—44no—
279C23—44no—
280C23—48no—
281C23C2×#12011yescharacteristic
282C23C2×#12111yescharacteristic
283C23C2×#12711yescharacteristic
284C23C2×AGL(1,ℤ/8)14yes—
285Q8—28no—
286Q8—44no—
287Q8—48no—
288Q8—88no—
289D4—48no—
290D4—84no—
291D4—88no—
292D4—88no—
293D4—88no—
294D4—88no—
295D4—1616no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—28no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—816no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#39212yes—
330C2×C4#39214yes—
331C2×C4#39212yes—
332C2×C4#45512yes—
333C2×C4#45612yes—
334C2×C4C2×#12012yes—
335C2×C4C2×#12112yes—
336C8—48no—
337C22—24no—
338C22—24no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—88no—
344C22—88no—
345C22—84no—
346C22—84no—
347C22—84no—
348C22C2×#39211yescharacteristic
349C22#300312yes—
350C22C2×#45511yescharacteristic
351C22C2×#45611yescharacteristic
352C22#302512yes—
353C22#303812yes—
354C22#315112yes—
355C4—22no—
356C4—24no—
357C4—22no—
358C4—22no—
359C4—48no—
360C4—44no—
361C4—84no—
362C4—82no—
363C4—84no—
364C4#200714yes—
365C2—22no—
366C2—22no—
367C2—164no—
368C2#1817912yes—
369C2#1818712yes—
370C2C2×#200711yescharacteristic
371C2#2343011yescharacteristic
372C2#2344311yescharacteristic
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×#3112C2semidirect
3. C2×#1505C2semidirect
4. #13850C2semidirect
5. #9970C2semidirect
6. #9964C2semidirect
7. #7912C2semidirect
54. C2×C4×Q8D4semidirect
72. #432D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
#351662
C2×#15052
#138502
#99702
#99642
#79122