Finite Group Databasebeta

← #71863

Subgroups — #71863

Order 384.

278 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×#837C211yesmaximal, characteristic
3#5232C211yesmaximal, characteristic
4#5229C211yesmaximal, characteristic, complemented
5C2×#836C211yesmaximal, characteristic, complemented
6#4995C211yesmaximal, characteristic, complemented
7#4532C211yesmaximal, characteristic, complemented
8#2038—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C3×C4⋊Q8—21no—
12C3×#124—21no—
13C3×#123—22no—
14C3×C22⋊Q8—21no—
15C3×C22⋊Q8—21no—
16C3×#117C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4×Dic3C2211yescharacteristic
19C2×#208C2211yescharacteristic
20C2×#208C2211yescharacteristic
21#837—22no—
22#836—22no—
23#835—21no—
24#832—21no—
25C3⋊#95—22no—
26#808—21no—
27#807—21no—
28#536D311yescharacteristic
29#428—31no—
30C2×#104—31no—
31#417—31no—
32C2×#101—31no—
33#391—31no—
34#339—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8D411yescharacteristic
46C2×C3×D4—21no—
47C2×C3×D4D411yescharacteristic
48C22×C3×C4—21no—
49C22×C3×C4D412yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3—21no—
52C22×Dic3—21no—
53C3×C4⋊C4—22no—
54C3×C4⋊C4—22no—
55C3×C4⋊C4D411yescharacteristic
56C3×C4⋊C4D411yescharacteristic
57C3×C22⋊C4—22no—
58C3×C22⋊C4—22no—
59C3×C22⋊C4—21no—
60C3×C42—21no—
61C4⋊Dic3—21no—
62C4⋊Dic3—21no—
63C4×Dic3—21no—
64C4×Dic3—21no—
65#208—21no—
66#208—21no—
67#208—21no—
68#208D412yes—
69C2×C3⋊C8—21no—
70C2×C3⋊C8—21no—
71C2×#41C2×D311yescharacteristic
72C22×Q8C2×D311yescharacteristic
73C2×M16—31no—
74C2×M16—31no—
75C4⋊Q8—21no—
76#124—21no—
77#123—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80#117C2×D311yescharacteristic
81C2×C4⋊C4—31no—
82C2×C42—31no—
83#107—61no—
84#106—61no—
85#104—62no—
86#103—61no—
87#102—61no—
88#101—62no—
89#95—62no—
90C23×C3—21no—
91C23×C3C2×D411yescharacteristic
92C3×Q8—21no—
93C3×Q8—21no—
94C3×Q8—41no—
95C3×Q8—41no—
96C3×Q8—41no—
97C3×Q8—41no—
98C3×D4—21no—
99C3×D4—21no—
100C3×D4—41no—
101C3×D4—41no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4#4111yescharacteristic
114C2×C3×C4#4111yescharacteristic
115C2×C3×C4#4111yescharacteristic
116C2×Dic3—22no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119C2×Dic3—42no—
120C2×Dic3—41no—
121C3⋊C8—22no—
122C3⋊C8—41no—
123#41—21no—
124#41—21no—
125#41—21no—
126#41—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8#6711yescharacteristic
134C2×D4—21no—
135C2×D4#6711yescharacteristic
136C22×C4—21no—
137C22×C4—61no—
138C22×C4—61no—
139C22×C4#6712yes—
140C22×C4C22×D311yescharacteristic
141M16—32no—
142M16—61no—
143M16—61no—
144M16—61no—
145C2×C8—61no—
146C2×C8—61no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4#6711yescharacteristic
152C4⋊C4#6711yescharacteristic
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—21no—
156C42—21no—
157C42—61no—
158C42—61no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3#12011yescharacteristic
165C22×C3#12111yescharacteristic
166C22×C3#12311yescharacteristic
167C3×C4—42no—
168C3×C4—41no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4#12111yescharacteristic
172C3×C4#12111yescharacteristic
173C3×C4#12311yescharacteristic
174C3×C4#12411yescharacteristic
175Dic3—42no—
176Dic3—81no—
177C23—21no—
178C23C2×#6711yescharacteristic
179Q8—21no—
180Q8—21no—
181Q8—41no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—62no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—122no—
201C2×C4—121no—
202C2×C4D3×D411yescharacteristic
203C2×C4#23711yescharacteristic
204C2×C4#23711yescharacteristic
205C2×C4#23711yescharacteristic
206C2×C4C2×#6711yescharacteristic
207C2×C4C2×#6711yescharacteristic
208C8—62no—
209C8—121no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3—81no—
213C2×C3#39411yescharacteristic
214C2×C3#45411yescharacteristic
215C2×C3#45611yescharacteristic
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22#93511yescharacteristic
222C22#93911yescharacteristic
223C22#95311yescharacteristic
224C4—42no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—122no—
229C4—241no—
230C4#93511yescharacteristic
231C4#93611yescharacteristic
232C4#93811yescharacteristic
233C4#93911yescharacteristic
234C3#203811yescharacteristic, complemented, Sylow
235C2—21no—
236C2—21no—
237C2—81no—
238C2#522211yescharacteristic
239C2#524511yescharacteristic
240C2#524611yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#536C2
socleC22×C3#120
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#2038
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
4. #5229C2semidirect
5. C2×#836C2semidirect
6. #4995C2semidirect
7. #4532C2semidirect
234. C3#2038semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
C2×#8372
#52322
#52292
C2×#8362
#49952
#45322
#20383