Finite Group Databasebeta

← #80704

Subgroups — #80704

Order 384.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#5600C212yesmaximal, complemented
2C3×#466C211yesmaximal, characteristic
3#5153C212yesmaximal, complemented
4#5116C211yesmaximal, characteristic, complemented
5C2×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#5041C212yesmaximal, complemented
7#4877C212yesmaximal, complemented
8#4767C212yesmaximal, complemented
9#4766C212yesmaximal, complemented
10#3250—31nomaximal, Sylow
11C2×C3×D8C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×#121C2212yes—
14#936C2212yes—
15#935—22no—
16D4×Dic3—22no—
17D4×Dic3C2212yes—
18C2×#214C2212yes—
19#924C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C22×C3⋊C8C2211yescharacteristic
22C2×Dic12C2211yescharacteristic
23#890C2212yes—
24#887C2212yes—
25#879—22no—
26#877—22no—
27C3×#107C2211yescharacteristic
28C3×#102C2212yes—
29#832C2212yes—
30#818C2211yescharacteristic
31#816C2212yes—
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#810C2212yes—
35#808C2212yes—
36#529—32no—
37#479—32no—
38#478—32no—
39#467—31no—
40#466D311yescharacteristic
41#465—32no—
42#464—32no—
43#463—32no—
44C2×C4⋊C8—31no—
45C2×C3×D4—22no—
46C2×C3×D4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×Dic3—22no—
49C22×Dic3D411yescharacteristic
50C22×Dic3D411yescharacteristic
51C2×Dic6C2312yes—
52C3×D8—41no—
53C2×C3×C8—41no—
54C2×C3×C8C2311yescharacteristic
55C2×C3×C8C2311yescharacteristic
56C3×C4⋊C4C2312yes—
57C3×C22⋊C4—22no—
58#217—22no—
59#217—22no—
60#217—22no—
61#217—22no—
62#214—42no—
63C4⋊Dic3—22no—
64C4⋊Dic3C2312yes—
65Dic3⋊C4—22no—
66Dic3⋊C4—22no—
67Dic3⋊C4—22no—
68C4×Dic3—22no—
69C4×Dic3D412yes—
70C4×Dic3C2312yes—
71C2×C3⋊C8—41no—
72C2×C3⋊C8C2312yes—
73Dic12—41no—
74C2×Q16—31no—
75C2×SD16—32no—
76C2×D8C2×D311yescharacteristic
77C22×C8—31no—
78C22×C8C2×D311yescharacteristic
79#126—62no—
80#125—32no—
81#124—32no—
82#123—62no—
83C22⋊Q8—32no—
84#121C2×D312yes—
85C4×Q8—32no—
86C4×D4—32no—
87C4×D4—62no—
88#117—62no—
89C2×C42—31no—
90#107—31no—
91#107C2×D311yescharacteristic
92#106—32no—
93C4⋊C8—32no—
94C4⋊C8—32no—
95#103—32no—
96#103—32no—
97#102—32no—
98#102C2×D312yes—
99C23×C3—22no—
100C23×C3C2×D411yescharacteristic
101C3×D4—22no—
102C3×D4—22no—
103C3×D4—44no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—41no—
113C2×Dic3—44no—
114C2×Dic3C2×D412yes—
115C2×Dic3C2×D412yes—
116Dic6—22no—
117Dic6—22no—
118C3×C8—21no—
119C3×C8—21no—
120C3⋊C8—22no—
121C2×Q8—32no—
122C2×D4—22no—
123C2×D4C22×D312yes—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—62no—
127C22×C4C22×D311yescharacteristic
128Q16—121no—
129SD16—122no—
130D8—41no—
131C2×C8—32no—
132C2×C8—41no—
133C2×C8—121no—
134C2×C8C22×D311yescharacteristic
135C2×C8C22×D311yescharacteristic
136C4⋊C4—32no—
137C4⋊C4—62no—
138C4⋊C4—62no—
139C4⋊C4—62no—
140C4⋊C4—62no—
141C4⋊C4C22×D312yes—
142C22⋊C4—22no—
143C22⋊C4—62no—
144C22⋊C4—62no—
145C22⋊C4—62no—
146C22⋊C4—62no—
147C42—32no—
148C42—32no—
149C42—62no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3#13512yes—
156C22×C3C22×D411yescharacteristic
157C3×C4—21no—
158C3×C4—42no—
159C3×C4H(5,𝔽2)11yescharacteristic
160C3×C42−1+411yescharacteristic
161Dic3—22no—
162Dic3—22no—
163Dic3—42no—
164Dic3—42no—
165C23—22no—
166C23D3×D411yescharacteristic
167Q8—62no—
168Q8—62no—
169D4—22no—
170D4—22no—
171D4—44no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—32no—
176C2×C4—32no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—121no—
181C2×C4—124no—
182C2×C4D3×D411yescharacteristic
183C2×C4C23×D311yescharacteristic
184C8—21no—
185C8—21no—
186C8—62no—
187C2×C3—22no—
188C2×C3—82no—
189C2×C3#53711yescharacteristic
190C2×C3C2×#13511yescharacteristic
191C2×C3#57511yescharacteristic
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#91212yes—
198C22C2×D3×D411yescharacteristic
199C4—21no—
200C4—42no—
201C4—62no—
202C4—62no—
203C4—122no—
204C4—122no—
205C4#100411yescharacteristic
206C4#101011yescharacteristic
207C3#325011yescharacteristic, complemented, Sylow
208C2—22no—
209C2—82no—
210C2#559711yescharacteristic
211C2C2×#91211yescharacteristic
212C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3250
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #5600C2semidirect
3. #5153C2semidirect
4. #5116C2semidirect
5. C2×Dic3⋊C8C2semidirect
6. #5041C2semidirect
7. #4877C2semidirect
8. #4767C2semidirect
9. #4766C2semidirect
207. C3#3250semidirect

Maximal subgroups

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

subgroupindex
#56002
C3×#4662
#51532
#51162
C2×Dic3⋊C82
#50412
#48772
#47672
#47662
#32503