Finite Group Databasebeta

← #83401

Subgroups — #83401

Order 384.

372 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#536C211yesmaximal, characteristic
2#5519C211yesmaximal, characteristic, complemented
3#5233C211yesmaximal, characteristic
4#5232C211yesmaximal, characteristic
5#5225C211yesmaximal, characteristic, complemented
6C2×#835C211yesmaximal, characteristic, complemented
7#5082C212yesmaximal, complemented
8#5079C212yesmaximal, complemented
9#5056C212yesmaximal, complemented
10#5055C212yesmaximal, complemented
11#4997C211yesmaximal, characteristic, complemented
12#3137—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C3×C4⋊Q8C2212yes—
16C3×#124C2212yes—
17C3×#123—22no—
18C3×C22⋊Q8—21no—
19C3×C22⋊Q8—21no—
20C3×#117C2211yescharacteristic
21#926C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×#208C2211yescharacteristic
24C22×C3⋊C8C2211yescharacteristic
25#890—21no—
26#888—21no—
27#878—22no—
28#870C2212yes—
29#869C2212yes—
30#835C2212yes—
31#835C2212yes—
32#835C2212yes—
33#835C2212yes—
34#835C2212yes—
35#835C2211yescharacteristic
36#835C2211yescharacteristic
37#832C2212yes—
38#832C2211yescharacteristic
39#832C2211yescharacteristic
40#803C2212yes—
41C4×C3⋊C8C2212yes—
42#544—31no—
43#536D311yescharacteristic
44#491—32no—
45#489—32no—
46#488—32no—
47#487—32no—
48#419—31no—
49#417—31no—
50#416—31no—
51C2×#103—31no—
52#405—31no—
53C3×#41—22no—
54C3×#41—22no—
55C2×C3×Q8—21no—
56C2×C3×Q8—22no—
57C2×C3×Q8—22no—
58C2×C3×Q8C2312yes—
59C2×C3×Q8C2311yescharacteristic
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C22×Dic3—21no—
66C2×Dic6—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4D411yescharacteristic
70C3×C4⋊C4D411yescharacteristic
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4D412yes—
74C3×C42C2312yes—
75#217—21no—
76C4⋊Dic3—22no—
77C4⋊Dic3—21no—
78C4⋊Dic3C2311yescharacteristic
79C4⋊Dic3C2312yes—
80C4⋊Dic3C2311yescharacteristic
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—21no—
84#208—22no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2312yes—
88C2×C3⋊C8C2311yescharacteristic
89C2×#41C2×D311yescharacteristic
90C22×Q8C2×D311yescharacteristic
91C2×M16—31no—
92C22×C8—31no—
93C4⋊Q8—32no—
94C4⋊Q8—61no—
95C4⋊Q8C2×D312yes—
96#125—32no—
97#125—61no—
98#124C2×D312yes—
99#123—22no—
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102C22⋊Q8—62no—
103#117—31no—
104#117C2×D311yescharacteristic
105C2×C4⋊C4—31no—
106#103—32no—
107#103—32no—
108#103—31no—
109#103—31no—
110#103—32no—
111#103—32no—
112#103—32no—
113#102—31no—
114#102—31no—
115#102—32no—
116#97—32no—
117C4×C8—32no—
118C23×C3—21no—
119C23×C3C2×D411yescharacteristic
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—21no—
123C3×Q8—21no—
124C3×Q8—42no—
125C3×D4—21no—
126C3×D4—21no—
127C3×D4—42no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4#4112yes—
136C2×C3×C4#4112yes—
137C2×C3×C4C2411yescharacteristic
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—81no—
142Dic6—82no—
143C3⋊C8—21no—
144C3⋊C8—21no—
145C3⋊C8—22no—
146#41—22no—
147#41—22no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—62no—
152C2×Q8C22×D312yes—
153C2×Q8C22×D311yescharacteristic
154C2×D4—21no—
155C2×D4C22×D311yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—61no—
159C22×C4C22×D311yescharacteristic
160M16—62no—
161C2×C8—31no—
162C2×C8—32no—
163C2×C8—31no—
164C2×C8—62no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—32no—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4#6711yescharacteristic
175C4⋊C4#6711yescharacteristic
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—61no—
179C22⋊C4#6712yes—
180C42—61no—
181C42C22×D312yes—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3#12412yes—
188C22×C3C22×D411yescharacteristic
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—41no—
192C3×C4—42no—
193C3×C4#12412yes—
194C3×C4C2×#4111yescharacteristic
195C3×C4C2×#4111yescharacteristic
196Dic3—41no—
197Dic3—42no—
198Dic3—41no—
199C23—21no—
200C23C2×#6711yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—21no—
204Q8—21no—
205Q8—42no—
206Q8—242no—
207D4—21no—
208D4—21no—
209D4—42no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—241no—
218C2×C4#23712yes—
219C2×C4#23712yes—
220C2×C4C2×#6712yes—
221C2×C4C2×#6712yes—
222C2×C4C2×#6711yescharacteristic
223C2×C4C23×D311yescharacteristic
224C8—61no—
225C8—61no—
226C8—62no—
227C2×C3—22no—
228C2×C3—81no—
229C2×C3C2×#12411yescharacteristic
230C2×C3#57711yescharacteristic
231C2×C3#57811yescharacteristic
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22#93612yes—
238C22C22×#6711yescharacteristic
239C4—22no—
240C4—22no—
241C4—41no—
242C4—42no—
243C4—122no—
244C4—121no—
245C4—121no—
246C4#93612yes—
247C4C2×#23711yescharacteristic
248C4C2×#23711yescharacteristic
249C3#313711yescharacteristic, complemented, Sylow
250C2—22no—
251C2—81no—
252C2C2×#93611yescharacteristic
253C2#583711yescharacteristic
254C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #5519C2semidirect
5. #5225C2semidirect
6. C2×#835C2semidirect
7. #5082C2semidirect
8. #5079C2semidirect
9. #5056C2semidirect
10. #5055C2semidirect
11. #4997C2semidirect
249. C3#3137semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#55192
#52332
#52322
#52252
C2×#8352
#50822
#50792
#50562
#50552
#49972
#31373