Finite Group Databasebeta

← #238825

Subgroups — #238825

Order 512.

965 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 965 conjugacy classes of subgroups — too many to draw. Only its 180 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×#3245C211yesmaximal, characteristic
2#21977C211yesmaximal, characteristic, complemented
3#17754C212yesmaximal, complemented
4#17595C212yesmaximal, complemented
5#10513C214yesmaximal, complemented
6#10512C214yesmaximal, complemented
7C2×#1290C211yesmaximal, characteristic, complemented
8C2×#532C2212yes—
9#3245—22no—
10#3245—24no—
11#3245C2214yes—
12C2×#474—22no—
13C2×#466—22no—
14C2×#459—22no—
15C2×#432C2212yes—
16C2×#432C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2283C2211yescharacteristic
19#1940—22no—
20#1850—22no—
21#1843C2212yescomplemented
22#1832C2212yescomplemented
23#1558—22no—
24#1555—22no—
25#1551—24no—
26#1551C2214yescomplemented
27#1551C2214yescomplemented
28#1519—24no—
29#1514—24no—
30#1505C2218yescomplemented
31#1290C2214yescomplemented
32C22×#41—22no—
33C22×D8—22no—
34C23×C8D412yes—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40#532—22no—
41#532D414yes—
42#532C2314yes—
43C2×C4⋊Q8D412yes—
44C2×#127D412yes—
45C2×#124—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C22×C42C2311yescharacteristic
51#474—28no—
52#474—28no—
53#466—28no—
54#466—28no—
55#459—28no—
56#459—28no—
57#432—22no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432C2314yes—
63C2×#107—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8D412yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×#102—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—21no—
75C2×#97—22no—
76C2×#97—21no—
77#383D412yes—
78#382—21no—
79#381—21no—
80#380D412yes—
81C4×C4⋊C4—21no—
82C4×C22⋊C4—21no—
83#376C2312yes—
84#336—24no—
85#333—24no—
86#333—24no—
87#331—24no—
88#331—24no—
89#327—216no—
90C2×#41—24no—
91C2×#41—24no—
92C2×#41—42no—
93C2×#41—42no—
94C2×#41—88no—
95C22×Q8—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—42no—
100C23×C4—42no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×D8—28no—
105C2×D8—84no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—42no—
111C22×C8—82no—
112C22×C8C2×D414yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8C2×D414yes—
117#127—22no—
118#127—22no—
119#127—22no—
120#127C2×D414yes—
121#124—24no—
122#124—24no—
123#124—42no—
124#124—42no—
125#121—24no—
126#121—24no—
127#121—24no—
128#121—24no—
129#121—42no—
130#121—42no—
131#121—42no—
132#121—42no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—24no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C42—22no—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42#4112yes—
153C2×C42#4112yes—
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156#107—28no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—24no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—42no—
163C4⋊C8—84no—
164#102—216no—
165C22⋊C8—24no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—42no—
169#97—24no—
170#97—42no—
171#97—42no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95C2×D414yes—
176C24—42no—
177C24#12111yescharacteristic
178#41—88no—
179#41—88no—
180#41—88no—
181#41—88no—
182C2×Q8—24no—
183C2×Q8—42no—
184C2×Q8—84no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—84no—
196C2×D4—84no—
197C2×D4—88no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—88no—
206C22×C4—81no—
207C22×C4#12011yescharacteristic
208C22×C4#12011yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12112yes—
211C22×C4#12112yes—
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12312yes—
215C22×C4#12312yes—
216C22×C4#12411yescharacteristic
217C22×C4#12411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C2×#4111yescharacteristic
221D8—88no—
222C2×C8—24no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—44no—
230C2×C8—82no—
231C2×C8—84no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—82no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—42no—
243C42—42no—
244C42#12012yes—
245C42#12112yes—
246C42#12112yes—
247C42#12312yes—
248C42#12312yes—
249C42#12412yes—
250C42C22×D412yes—
251C42C2×#4112yes—
252C23—41no—
253C23—41no—
254C23—44no—
255C23—84no—
256C23—84no—
257C23—84no—
258C23#39411yescharacteristic
259C23#39411yescharacteristic
260C23#47214yes—
261C23C2×#12111yescharacteristic
262Q8—84no—
263Q8—84no—
264D4—44no—
265D4—44no—
266D4—84no—
267D4—84no—
268D4—84no—
269D4—84no—
270D4—84no—
271D4—88no—
272D4—88no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—84no—
297C2×C4#39412yes—
298C2×C4#39412yes—
299C2×C4#39411yescharacteristic
300C2×C4#39411yescharacteristic
301C2×C4C2×#12011yescharacteristic
302C2×C4C2×#12112yes—
303C2×C4C2×#12312yes—
304C2×C4C2×#12411yescharacteristic
305C8—44no—
306C8—84no—
307C8—81no—
308C8—81no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—44no—
316C22—84no—
317C22—84no—
318C22—84no—
319C22—168no—
320C22#204512yes—
321C22#204512yes—
322C22C2×#39411yescharacteristic
323C22C2×#47212yes—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—84no—
332C4—84no—
333C4#301911yescharacteristic
334C4#302011yescharacteristic
335C2—22no—
336C2—22no—
337C2—22no—
338C2—164no—
339C2#2349911yescharacteristic
340C2C2×#204511yescharacteristic
341C2#2353511yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#3245C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21977C2semidirect
3. #17754C2semidirect
4. #17595C2semidirect
5. #10513C2semidirect
6. #10512C2semidirect
7. C2×#1290C2semidirect
21. #1843C22semidirect
22. #1832C22semidirect
26. #1551C22semidirect
27. #1551C22semidirect
30. #1505C22semidirect
31. #1290C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32452
#219772
#177542
#175952
#105132
#105122
C2×#12902