Finite Group Databasebeta

← #77189

Subgroups — #77189

Order 384.

500 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#944C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic
5#4979C211yesmaximal, characteristic
6#4976C212yesmaximal
7#4975C212yesmaximal
8#4656C211yesmaximal, characteristic
9#4649C211yesmaximal, characteristic
10#4647C212yesmaximal
11#4646C212yesmaximal
12#2567—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18#944—24no—
19C2×C4⋊Dic3C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31#887—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C4⋊Q8—31no—
39C2×C4×Q8—31no—
40#395—31no—
41#395—31no—
42#395—31no—
43#395D311yescharacteristic
44#391—32no—
45#389—32no—
46#389—32no—
47#384—32no—
48C4×C4⋊C4—31no—
49C2×C3×Q8—22no—
50C2×C3×Q8—42no—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×Dic6—22no—
63C2×Dic6—22no—
64C2×Dic6—22no—
65C2×Dic6—42no—
66C2×Dic6Q814yescomplemented
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69C4⋊Dic3—42no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—42no—
75Dic3⋊C4—42no—
76Dic3⋊C4Q818yescomplemented
77C4×Dic3—22no—
78C4×Dic3—22no—
79C4×Dic3—24no—
80C4×Dic3D414yes—
81C22×Q8—31no—
82C22×Q8—31no—
83C22×Q8C2×D311yescharacteristic
84C4⋊Q8—64no—
85C4×Q8—64no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D312yes—
96C2×C42—31no—
97C2×C42—31no—
98C2×C42—31no—
99C2×C42—32no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95C2×D311yescharacteristic
105#95C2×D312yes—
106C23×C3C2411yescharacteristic
107C3×Q8—44no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×Q814yes—
120C2×C3×C4C2×Q812yes—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×Q814yes—
133C2×Dic3C2×Q812yes—
134C2×Dic3C2×Q812yes—
135C2×Dic3C2×Q814yes—
136C2×Dic3#4112yes—
137Dic6—28no—
138Dic6—44no—
139C2×Q8—22no—
140C2×Q8—34no—
141C2×Q8—42no—
142C2×Q8—62no—
143C2×Q8—62no—
144C2×Q8—62no—
145C2×Q8—122no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—31no—
152C22×C4C22×D312yes—
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D312yes—
157C4⋊C4—22no—
158C4⋊C4—38no—
159C4⋊C4—42no—
160C4⋊C4—62no—
161C4⋊C4—64no—
162C4⋊C4—64no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C4⋊C4—122no—
167C42—34no—
168C42—62no—
169C42—62no—
170C42—64no—
171C22×C3C22×D411yescharacteristic
172C22×C3C22×Q811yescharacteristic
173C22×C3C22×Q812yes—
174C22×C3C2×#4111yescharacteristic
175C22×C3H(5,𝔽2)12yes—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—24no—
179C3×C4—41no—
180C3×C4—42no—
181Dic3—22no—
182Dic3—24no—
183Dic3—42no—
184Dic3—41no—
185Dic3C22⋊Q814yes—
186Dic3C4⋊Q814yes—
187C23C23×D311yescharacteristic
188Q8—44no—
189Q8—68no—
190Q8—124no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—34no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—34no—
207C2×C4—32no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4D3×D412yes—
218C2×C4D3×Q814yes—
219C2×C4D3×Q812yes—
220C2×C3C2×C22⋊Q811yescharacteristic
221C2×C3C2×C4⋊Q811yescharacteristic
222C2×C3#53511yescharacteristic
223C2×C3#55412yes—
224C2×C3Q8212yes—
225C22C2×D3×D411yescharacteristic
226C22#100412yes—
227C22C2×D3×Q812yes—
228C22C2×D3×Q811yescharacteristic
229C22D3×#4111yescharacteristic
230C4—24no—
231C4—22no—
232C4—22no—
233C4—34no—
234C4—34no—
235C4—42no—
236C4—41no—
237C4—64no—
238C4—62no—
239C4—122no—
240C4—121no—
241C3#256711yescharacteristic, complemented, Sylow
242C2#559211yescharacteristic
243C2D3×C22⋊Q811yescharacteristic
244C2#563612yes—
245C2#572212yes—
246C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#395C2
hypercenter#395D3
hypocenterC3#2567
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
66. C2×Dic6Q8semidirect
76. Dic3⋊C4Q8semidirect
241. C3#2567semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8872
C3×#3952
#52302
#49792
#49762
#49752
#46562
#46492
#46472
#46462
#25673