Finite Group Databasebeta

← #72073

Subgroups — #72073

Order 384.

370 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×C3×Q16C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic
3C22×#216C211yesmaximal, characteristic
4#5229C211yesmaximal, characteristic
5C2×#835C211yesmaximal, characteristic
6C2×#816C211yesmaximal, characteristic
7#4550C211yesmaximal, characteristic
8#2016—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C2×C3×Q16—22no—
13C2×C3×Q16—22no—
14C2×C3×Q16—22no—
15C22×C3×C8C2211yescharacteristic
16#944—22no—
17#944—22no—
18C2×#216—22no—
19C2×#216—22no—
20C2×#216—22no—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4—22no—
23C2×C4×Dic3C2211yescharacteristic
24C22×C3⋊C8C2211yescharacteristic
25#835—22no—
26C3⋊#95—22no—
27#816—22no—
28C22×Q16—31no—
29C22×Q16D311yescharacteristic
30C2×C4⋊Q8—31no—
31C2×#103—31no—
32C2×#103—31no—
33#391—31no—
34#336—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—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8D412yes—
46C2×C3×Q8D412yes—
47C22×C3×C4—21no—
48C22×C3×C4—21no—
49C22×C3×C4C2311yescharacteristic
50C22×Dic3—21no—
51C22×Dic3—21no—
52C22×Dic3D412yes—
53C2×Dic6—21no—
54C2×Dic6—21no—
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×Dic6—21no—
58C2×Dic6D412yes—
59C3×Q16—28no—
60C2×C3×C8—21no—
61C2×C3×C8—21no—
62C2×C3×C8D412yes—
63#216—28no—
64C4⋊Dic3—21no—
65C4⋊Dic3—21no—
66Dic3⋊C4—28no—
67C4×Dic3—21no—
68C4×Dic3—21no—
69C2×C3⋊C8—21no—
70C2×C3⋊C8—21no—
71C2×C3⋊C8D412yes—
72C22×Q8—31no—
73C22×Q8C2×D311yescharacteristic
74C22×Q8C2×D311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—62no—
79C2×Q16—62no—
80C2×Q16—62no—
81C22×C8—31no—
82C22×C8C2×D311yescharacteristic
83C4⋊Q8—62no—
84C4⋊Q8—62no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—62no—
87C2×C42—31no—
88#103—62no—
89#103—62no—
90#95—62no—
91C23×C3C2×D411yescharacteristic
92C3×Q8—22no—
93C3×Q8—22no—
94C3×Q8—22no—
95C3×Q8—22no—
96C3×Q8—41no—
97C3×Q8—41no—
98C3×Q8—41no—
99C3×Q8—41no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4#4111yescharacteristic
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—41no—
116C2×Dic3—41no—
117C2×Dic3—41no—
118C2×Dic3—42no—
119C2×Dic3Q1614yescomplemented
120Dic6—22no—
121Dic6—22no—
122Dic6—41no—
123Dic6—41no—
124C3×C8—22no—
125C3⋊C8—22no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—32no—
137C2×Q8—61no—
138C2×Q8—61no—
139C2×Q8—61no—
140C2×Q8—61no—
141C2×Q8—61no—
142C2×Q8#6712yes—
143C2×Q8#6712yes—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—32no—
147C22×C4—61no—
148C22×C4—61no—
149C22×C4C22×D311yescharacteristic
150Q16—28no—
151Q16—68no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—32no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8#6712yes—
158C4⋊C4—68no—
159C4⋊C4—61no—
160C4⋊C4—61no—
161C42—61no—
162C42—61no—
163C22×C3#12011yescharacteristic
164C22×C3#12111yescharacteristic
165C22×C3#12711yescharacteristic
166C22×C3C2×Q1612yes—
167C22×C3#13712yes—
168C3×C4—42no—
169C3×C4—42no—
170C3×C4#12011yescharacteristic
171C3×C4#12011yescharacteristic
172C3×C4#12111yescharacteristic
173C3×C4#12111yescharacteristic
174Dic3—24no—
175Dic3—42no—
176Dic3—81no—
177C23D3×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—62no—
187Q8—62no—
188Q8—121no—
189Q8—121no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—34no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4D3×D411yescharacteristic
205C2×C4D3×D411yescharacteristic
206C2×C4#23711yescharacteristic
207C2×C4C2×#6711yescharacteristic
208C2×C4C2×#6711yescharacteristic
209C2×C4C2×#6711yescharacteristic
210C8—22no—
211C8—62no—
212C2×C3#39211yescharacteristic
213C2×C3#45112yes—
214C2×C3#46212yes—
215C2×C3#49411yescharacteristic
216C2×C3#49711yescharacteristic
217C22D3×Q1612yes—
218C22#91812yes—
219C22#93711yescharacteristic
220C22#93911yescharacteristic
221C22#94011yescharacteristic
222C4—42no—
223C4—42no—
224C4—64no—
225C4—122no—
226C4—241no—
227C4#93711yescharacteristic
228C4#93811yescharacteristic
229C4#93911yescharacteristic
230C4#95311yescharacteristic
231C3#201611yescharacteristic, complemented, Sylow
232C2#518212yes—
233C2#518311yescharacteristic
234C2#518612yes—
235C2#519011yescharacteristic
236C2#522311yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC22×C3×Q16C2
socle, prime radicalC23×C3C2×D4
hypercenterC22×Q16D3
hypocenterC3#2016
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
119. C2×Dic3Q16semidirect
231. C3#2016semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q162
C2×#9442
C22×#2162
#52292
C2×#8352
C2×#8162
#45502
#20163