Finite Group Databasebeta

← #70100

Subgroups — #70100

Order 384.

278 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 69 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#5816C211yesmaximal, characteristic
2C2×C3×#101C211yesmaximal, characteristic, complemented
3#5229C211yesmaximal, characteristic, complemented
4C2×#825C211yesmaximal, characteristic
5#5133C211yesmaximal, characteristic
6#5121C211yesmaximal, characteristic, complemented
7#4555C211yesmaximal, characteristic, complemented
8#2038—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13#947—21no—
14#946—21no—
15#946—21no—
16#944—21no—
17#929—22no—
18#926C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4×Dic3C2211yescharacteristic
21C3×#101—22no—
22C3⋊#95—22no—
23#825—22no—
24#821—21no—
25#818—21no—
26#817—21no—
27#816—21no—
28#536—31no—
29#428—31no—
30C2×#104—31no—
31#417—31no—
32C2×#101D311yescharacteristic, complemented
33#391—31no—
34#339—31no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C22×C3×C4D412yes—
42C22×C3×C4C2311yescharacteristic
43C2×#67—21no—
44C22×Dic3—21no—
45C22×Dic3—21no—
46#235—21no—
47#235—21no—
48#235—21no—
49#235—21no—
50C2×D12D411yescharacteristic
51C2×C4×D3—21no—
52C2×Dic6D411yescharacteristic
53C3×M16—21no—
54C3×M16—21no—
55C3×M16—21no—
56C3×M16D412yes—
57C2×C3×C8—21no—
58C2×C3×C8—21no—
59#217—21no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—21no—
63C4⋊Dic3—21no—
64C4⋊Dic3D411yescharacteristic
65C4⋊Dic3D411yescharacteristic
66Dic3⋊C4—22no—
67Dic3⋊C4—22no—
68C4×Dic3—21no—
69C4×Dic3—21no—
70C4×Dic3—21no—
71C2×#41—31no—
72C22×Q8C2×D311yescharacteristic
73C2×M16C2×D311yescharacteristic
74C2×M16C2×D311yescharacteristic
75C4⋊Q8—61no—
76#124—61no—
77#123—62no—
78C22⋊Q8—61no—
79C22⋊Q8—61no—
80#117—31no—
81C2×C4⋊C4—31no—
82C2×C42—31no—
83#107—61no—
84#106—61no—
85#104—62no—
86#103—61no—
87#102—61no—
88#101—22no—
89#95—62no—
90C23×C3C2×D411yescharacteristic
91C22×D3—21no—
92C3×Q8—41no—
93C3×Q8—41no—
94C3×Q8—41no—
95C3×Q8—41no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4C2×D411yescharacteristic
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4#4111yescharacteristic
103C2×C3×C4#4111yescharacteristic
104C2×C3×C4#4111yescharacteristic
105#67—41no—
106#67—41no—
107C2×Dic3—22no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3—42no—
114C2×Dic3—41no—
115D12—21no—
116D12—21no—
117C4×D3—41no—
118C4×D3—41no—
119Dic6—21no—
120Dic6—21no—
121C3×C8—22no—
122C3×C8—41no—
123#41—61no—
124#41—61no—
125#41—61no—
126#41—61no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—31no—
134C2×D4—31no—
135C2×D4—61no—
136C22×C4—61no—
137C22×C4—61no—
138C22×C4—61no—
139C22×C4D1212yes—
140C22×C4C22×D311yescharacteristic
141M16—21no—
142M16—21no—
143M16—21no—
144M16#6712yes—
145C2×C8—21no—
146C2×C8—21no—
147C4⋊C4—31no—
148C4⋊C4—31no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—61no—
152C4⋊C4—61no—
153C22⋊C4—62no—
154C22⋊C4—62no—
155C22⋊C4—61no—
156C42—61no—
157C42—61no—
158C42—61no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3#12011yescharacteristic
162C22×C3#12111yescharacteristic
163C22×C3#12311yescharacteristic
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—41no—
167C3×C4—42no—
168C3×C4#12111yescharacteristic
169C3×C4#12111yescharacteristic
170C3×C4#12311yescharacteristic
171C3×C4#12411yescharacteristic
172Dic3—42no—
173Dic3—41no—
174Dic3—41no—
175Dic3—41no—
176Dic3—81no—
177C23—61no—
178C23C2×D1211yescharacteristic
179Q8—41no—
180Q8—41no—
181Q8—41no—
182Q8—41no—
183Q8—61no—
184Q8—61no—
185D4—61no—
186D4—61no—
187D4—121no—
188D4—121no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—122no—
201C2×C4—121no—
202C2×C4#23511yescharacteristic
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C2×C4#23711yescharacteristic
206C2×C4#23711yescharacteristic
207C2×C4C2×#6711yescharacteristic
208C8—22no—
209C8—41no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3#39411yescharacteristic
213C2×C3#45411yescharacteristic
214C2×C3#45611yescharacteristic
215D3—81no—
216C22—21no—
217C22—21no—
218C22—121no—
219C22—121no—
220C22—121no—
221C22#88211yescharacteristic
222C22#88611yescharacteristic
223C22#93011yescharacteristic
224C4—42no—
225C4—122no—
226C4—121no—
227C4—121no—
228C4—121no—
229C4—241no—
230C4#88311yescharacteristic
231C4#88511yescharacteristic
232C4#93811yescharacteristic
233C4#93911yescharacteristic
234C3#203811yescharacteristic, complemented, Sylow
235C2—21no—
236C2—21no—
237C2—241no—
238C2#475211yescharacteristic
239C2#475311yescharacteristic
240C2#496111yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#101C2
socleC22×C3#120
prime radicalC2×#235C22
hypercenterC2×#101D3
hypocenterC3#2038
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×C3×#101C2semidirect
3. #5229C2semidirect
6. #5121C2semidirect
7. #4555C2semidirect
32. C2×#101D3semidirect
234. C3#2038semidirect

Maximal subgroups

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

subgroupindex
#58162
C2×C3×#1012
#52292
C2×#8252
#51332
#51212
#45552
#20383