Finite Group Databasebeta

← #77100

Subgroups — #77100

Order 384.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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×#394C211yesmaximal, characteristic
2#5222C212yesmaximal
3#4979C211yesmaximal, characteristic, complemented
4#4973C211yesmaximal, characteristic, complemented
5#4953C212yesmaximal, complemented
6#4951C212yesmaximal, complemented
7#4947C212yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#4660C212yesmaximal, complemented
10#2832—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2211yescharacteristic
24C3×#95C2211yescharacteristic
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#397—32no—
34#395—31no—
35#394—32no—
36#394D311yescharacteristic
37#393—32no—
38#393—32no—
39#391—31no—
40#388—32no—
41C24×C3C2312yes—
42C2×C3×D4—42no—
43C2×C3×D4—41no—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×Dic6—22no—
53C2×Dic6—41no—
54C2×Dic6—41no—
55C2×Dic6—42no—
56C2×Dic6—41no—
57C3×C4⋊C4—41no—
58C3×C22⋊C4—24no—
59C3×C22⋊C4—42no—
60#217—42no—
61#217—42no—
62#217—42no—
63#217—42no—
64C4⋊Dic3—42no—
65Dic3⋊C4—42no—
66C4×Dic3—41no—
67C22×Q8—31no—
68C22×Q8—31no—
69C22×D4C2×D311yescharacteristic
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4C2×D312yes—
78C2×C22⋊C4C2×D312yes—
79C2×C42—31no—
80#95—32no—
81#95—32no—
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—31no—
87#95C2×D311yescharacteristic
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3C2411yescharacteristic
96C3×D4—44no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4#4112yes—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119Dic6—44no—
120Dic6—44no—
121C24C22×D312yes—
122C2×Q8—62no—
123C2×Q8—121no—
124C2×Q8—121no—
125C2×Q8—122no—
126C2×Q8—121no—
127C2×D4—41no—
128C2×D4—42no—
129C22×C4—32no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4C22×D312yes—
134C22×C4C22×D311yescharacteristic
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D311yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—122no—
139C4⋊C4—122no—
140C22⋊C4—24no—
141C22⋊C4—42no—
142C22⋊C4—122no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C22⋊C4—122no—
146C42—121no—
147C22×C3—42no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3C2×#4111yescharacteristic
155C22×C3H(5,𝔽2)11yescharacteristic
156C22×C3H(5,𝔽2)12yes—
157C22×C3H(5,𝔽2)11yescharacteristic
158C22×C32−1+412yes—
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—41no—
162C3×C4—41no—
163Dic3—42no—
164Dic3—42no—
165Dic3—42no—
166Dic3—42no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23C23×D311yescharacteristic
175Q8—124no—
176Q8—124no—
177D4—44no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4#23712yes—
200C2×C3—82no—
201C2×C3#53811yescharacteristic
202C2×C3#54212yes—
203C2×C3H(3,𝔽4)11yescharacteristic
204C2×C3#56212yes—
205C2×C3#56311yescharacteristic
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22C2×#23711yescharacteristic
214C22#100412yes—
215C22#100411yescharacteristic
216C22#100911yescharacteristic
217C22#101012yes—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—122no—
223C4—122no—
224C4—122no—
225C4—122no—
226C3#283211yescharacteristic, complemented, Sylow
227C2—82no—
228C2#559411yescharacteristic
229C2#560312yes—
230C2#561711yescharacteristic
231C2#566712yes—
232C2#567211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2832
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #4979C2semidirect
4. #4973C2semidirect
5. #4953C2semidirect
6. #4951C2semidirect
7. #4947C2semidirect
8. #4947C2semidirect
9. #4660C2semidirect
226. C3#2832semidirect

Maximal subgroups

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

subgroupindex
C3×#3942
#52222
#49792
#49732
#49532
#49512
#49472
#46602
#28323