Finite Group Databasebeta

← #491328

Subgroups — #491328

Order 512.

398 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#33794C211yesmaximal, characteristic
2#33497C211yesmaximal, characteristic, complemented
3#19264C211yesmaximal, characteristic
4#19118C211yesmaximal, characteristic, complemented
5#12480C212yesmaximal, complemented
6#12479C212yesmaximal, complemented
7#12462C212yesmaximal, complemented
8#12461C212yesmaximal, complemented
9#12034C211yesmaximal, characteristic, complemented
10#11973C211yesmaximal, characteristic, complemented
11#11948C211yesmaximal, characteristic
12#3169C2211yescharacteristic
13#3160C2211yescharacteristic
14#3112—21no—
15#3104—21no—
16C2×C8×Q8C2211yescharacteristic
17#2254C2211yescharacteristic
18#2252C2211yescharacteristic
19#2240C2211yescharacteristic
20#2240C2211yescharacteristic
21#2234—22no—
22#2228—22no—
23#2183—22no—
24#2183C2212yes—
25#2183C2212yes—
26#2163—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1678—22no—
30#1676C2212yes—
31#1676C2212yes—
32#1674C2212yes—
33#1672—22no—
34#1668C2212yes—
35#1568C2211yescharacteristic
36#1564C2211yescharacteristic
37#1551—21no—
38#1542—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2212yes—
42#1365C2212yes—
43#1364C2212yes—
44#1361C2212yes—
45#533—21no—
46#532—21no—
47C2×C4×Q8D411yescharacteristic
48#500C2311yescharacteristic
49#500C2311yescharacteristic
50#500C2311yescharacteristic
51#499—21no—
52#499—21no—
53#498—21no—
54#493C2311yescharacteristic
55#486—21no—
56#484—41no—
57#480—41no—
58#477—22no—
59#477—22no—
60#466—21no—
61#466—21no—
62#462—22no—
63#460—22no—
64#451—41no—
65#448—41no—
66C8×Q8—21no—
67C8×Q8—21no—
68C8×Q8D412yes—
69C8×Q8C2312yes—
70#440—42no—
71C4×Q16—42no—
72C4×D8—21no—
73C4×D8—21no—
74#432—21no—
75#431—21no—
76#427—21no—
77#427—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×C4×C8—21no—
81C2×C4×C8C2311yescharacteristic
82#367—22no—
83#366—22no—
84C4⋊C16—22no—
85C4⋊C16—22no—
86C4⋊C16D412yes—
87C4⋊C16C2312yes—
88#357—22no—
89#357—22no—
90C22⋊C16—21no—
91C22⋊C16D412yes—
92#346—22no—
93#346—22no—
94C4×C16—21no—
95C4×C16C2311yescharacteristic
96C4×C16C2311yescharacteristic
97#335C2312yes—
98#335C2312yes—
99#334—21no—
100#334—21no—
101#333—22no—
102#333—22no—
103#332—22no—
104#331—22no—
105C2×#41—41no—
106C22×Q8D811yescharacteristic
107C2×Q16—42no—
108C2×SD16—42no—
109C2×D8—21no—
110C2×D8—21no—
111C22×C8—21no—
112C22×C8C2×D411yescharacteristic
113C22×C8#4111yescharacteristic
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117#127—21no—
118#125—41no—
119#124—41no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122#121—41no—
123#121—41no—
124C4×Q8—21no—
125C4×Q8—21no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8D812yes—
129C4×Q8C2×D412yes—
130C4×D4—41no—
131C4×D4—41no—
132#117—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4D811yescharacteristic
135C2×C42—21no—
136C2×C42C2×D411yescharacteristic
137C2×C16—22no—
138C2×C16C2×D412yes—
139C2×C16C2×D412yes—
140#107—21no—
141#107—21no—
142#107—21no—
143#107—21no—
144#107—21no—
145#107—21no—
146#107—42no—
147#107—42no—
148#106—41no—
149#106—41no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8C2×D412yes—
156#103—42no—
157#103—42no—
158#103—42no—
159#102—41no—
160#102—41no—
161#102—42no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—41no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—41no—
168C4×C8C2×D411yescharacteristic
169C4×C8C2×D411yescharacteristic
170C4×C8#4111yescharacteristic
171C4×C8C2411yescharacteristic
172#41—162no—
173C2×Q8—21no—
174C2×Q8—42no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×D812yes—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—81no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—81no—
186C22×C4#12111yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4#13511yescharacteristic
189Q16—84no—
190SD16—84no—
191D8—42no—
192D8—81no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—81no—
199C2×C8#12111yescharacteristic
200C2×C8#12111yescharacteristic
201C2×C8C22×D411yescharacteristic
202C2×C8C2×#4111yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4C2×D812yes—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—22no—
219C42—22no—
220C42—81no—
221C42#12111yescharacteristic
222C42C2×D811yescharacteristic
223C42#13511yescharacteristic
224C42C22×D411yescharacteristic
225C16—22no—
226C16—22no—
227C16—42no—
228C23—81no—
229C23#46611yescharacteristic
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—82no—
234Q8—161no—
235Q8—161no—
236D4—82no—
237D4—81no—
238D4—81no—
239D4—161no—
240D4—162no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—162no—
253C2×C4#46611yescharacteristic
254C2×C4#46611yescharacteristic
255C2×C4#46611yescharacteristic
256C2×C4C2×#12111yescharacteristic
257C2×C4C22×D811yescharacteristic
258C2×C4C2×#13511yescharacteristic
259C8—21no—
260C8—21no—
261C8—21no—
262C8—42no—
263C8—82no—
264C8—82no—
265C8#57812yes—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—161no—
270C22—161no—
271C22—161no—
272C22C2×#46611yescharacteristic
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—161no—
280C4—161no—
281C4—161no—
282C4#304411yescharacteristic
283C4#306911yescharacteristic
284C4#341412yes—
285C2—41no—
286C2—321no—
287C2#1940411yescharacteristic
288C2#3360411yescharacteristic
289C2#3361811yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33497C2semidirect
4. #19118C2semidirect
5. #12480C2semidirect
6. #12479C2semidirect
7. #12462C2semidirect
8. #12461C2semidirect
9. #12034C2semidirect
10. #11973C2semidirect

Maximal subgroups

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

subgroupindex
#337942
#334972
#192642
#191182
#124802
#124792
#124622
#124612
#120342
#119732
#119482