Finite Group Databasebeta

← #13166388

Subgroups — #13166388

Order 1728.

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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×#88988C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3#11723404C211yesmaximal, characteristic
4#11722732C212yesmaximal, complemented
5Dic3⋊#622C212yesmaximal, complemented
6#10593755—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3⋊#622C2211yescharacteristic
9C2×C3⋊#622C2211yescharacteristic
10#88988—22no—
11C2×C3×#622C2212yes—
12C3×#3788—22no—
13C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
14C32×C4⋊Dic3C411yescharacteristic
15#88803C412yes—
16Dic3×C3⋊Dic3C2212yes—
17C2×#3788D311yescharacteristic
18C2×#3787—31no—
19#64101—31no—
20#63841—62no—
21#63779—32no—
22C4⋊#622—32no—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C412yes—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3D412yes—
28C3×C4×C3⋊D3Q812yes—
29C2×C32×Dic3C2×C412yes—
30C3⋊#622—21no—
31C3⋊#622—21no—
32C3×#622—22no—
33#5119—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#622—31no—
37C2×#622—31no—
38C2×#622C2×D312yes—
39#3788—22no—
40#3787—62no—
41C32×C4⋊C4—31no—
42#3752—32no—
43C4⋊(C3⋊Dic3)—31no—
44C4⋊(C3⋊Dic3)—62no—
45C4⋊(C3⋊Dic3)—122no—
46C4×C3⋊Dic3—32no—
47C3×C4⋊Dic3—22no—
48(C2×D3)⋊Dic3—64no—
49Dic32—64no—
50C22×C33C22×C411yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3C4⋊C412yes—
54C33×C4C4⋊C412yes—
55C3⋊(C3⋊Dic3)—22no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3C2×Q811yescharacteristic
58C32×Dic3—22no—
59C2×C3×M16—91no—
60#926—91no—
61C22×C3⋊C8—91no—
62#892—182no—
63Dic3⋊C8—92no—
64Dic3⋊C8—92no—
65C22×C3⋊D3C4×D311yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—32no—
69C2×C32×C4—42no—
70C2×C32×C4C4×D311yescharacteristic
71C2×C3⋊Dic3—32no—
72C2×C3⋊Dic3—64no—
73C2×C3⋊Dic3—124no—
74C2×C3⋊Dic3C22×D311yescharacteristic
75C4×C3⋊D3Dic612yes—
76C4×C3⋊D3#6712yes—
77C2×C3×Dic3—24no—
78C2×C3×Dic3—62no—
79C3×C4×D3—64no—
80C3×C4×D3—64no—
81#622—22no—
82#622—61no—
83#622—61no—
84#424—271noSylow
85C2×C33C2×C4⋊C411yescharacteristic
86C2×C33#13111yescharacteristic
87C2×C33#13111yescharacteristic
88C3×C3⋊D3—22no—
89C22×C3×C4—91no—
90C2×C4×D3—62no—
91C3×M16—182no—
92C2×C3×C8—92no—
93C3×C4⋊C4—62no—
94#217—92no—
95(C2×D3)⋊C4—184no—
96C4⋊Dic3—91no—
97C4⋊Dic3—182no—
98C4⋊Dic3—362no—
99C4⋊Dic3#15711yescharacteristic
100C4×Dic3—92no—
101C4×Dic3—184no—
102C2×C3⋊C8—91no—
103C2×C3⋊C8—91no—
104C2×C3⋊C8—182no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32C2×C4×D311yescharacteristic
108C2×C3⋊D3—21no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3Dic3⋊C412yes—
111C2×C3×D3—64no—
112C2×C3×D3—122no—
113C2×C3×D3—122no—
114C32×C4—24no—
115C32×C4—44no—
116C32×C4—62no—
117C32×C4Dic3⋊C412yes—
118C3⋊Dic3—62no—
119C3⋊Dic3—124no—
120C3⋊Dic3—244no—
121C3⋊Dic3C2×Dic611yescharacteristic
122C3⋊Dic3C2×#6711yescharacteristic
123C3×Dic3—44no—
124C3×Dic3—62no—
125C3×Dic3—62no—
126C2×M16—91no—
127C22×C8—271no—
128#117—271no—
129C4⋊C8—272no—
130C4⋊C8—272no—
131C33#42411yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—62no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4—64no—
137C2×C3×C4—91no—
138C2×C3×C4—92no—
139C2×C3×C4—92no—
140C2×C3×C4C2×#15711yescharacteristic
141C2×Dic3—62no—
142C2×Dic3—92no—
143C2×Dic3—184no—
144C2×Dic3—364no—
145C2×Dic3C2×#15712yes—
146C4×D3—64no—
147C4×D3—64no—
148C3×C8—182no—
149C3⋊C8—181no—
150C3⋊C8—181no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32#90711yescharacteristic
158C2×C32#90711yescharacteristic
159C2×C32C2×Dic3⋊C411yescharacteristic
160C3⋊D3—22no—
161C3×D3—124no—
162C22×C4—91no—
163M16—182no—
164C2×C8—92no—
165C2×C8—271no—
166C2×C8—271no—
167C2×C8—542no—
168C4⋊C4—31no—
169C4⋊C4—271no—
170C22⋊C4—272no—
171C42—272no—
172C22×C3—22no—
173C22×C3—42no—
174C22×C3—92no—
175C22×C3—181no—
176C22×C3—181no—
177C22×C3C22×#15711yescharacteristic
178C2×D3—64no—
179C2×D3—122no—
180C2×D3—122no—
181C3×C4—24no—
182C3×C4—44no—
183C3×C4—91no—
184C3×C4—91no—
185C3×C4—124no—
186C3×C4C4⋊#15712yes—
187Dic3—22no—
188Dic3—62no—
189Dic3—62no—
190Dic3—182no—
191Dic3—364no—
192Dic3—724no—
193C32—22no—
194C32—42no—
195C32#511911yescharacteristic, complemented
196C23—91no—
197C2×C4—32no—
198C2×C4—91no—
199C2×C4—92no—
200C2×C4—92no—
201C2×C4—272no—
202C2×C4D3×#15711yescharacteristic
203C8—182no—
204C8—541no—
205C8—541no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—42no—
211C2×C3—42no—
212C2×C3—182no—
213C2×C3C2×C4⋊#15711yescharacteristic
214C2×C3#6428711yescharacteristic
215C2×C3#6428911yescharacteristic
216D3—124no—
217C22—92no—
218C22—181no—
219C22—181no—
220C22C2×D3×#15711yescharacteristic
221C4—62no—
222C4—91no—
223C4—91no—
224C4—542no—
225C4Dic3⋊#15712yes—
226C3—22no—
227C3—42no—
228C3#1059375511yescharacteristic, complemented
229C2—182no—
230C2#1172425111yescharacteristic
231C2#1172425311yescharacteristic
232C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722732C2semidirect
5. Dic3⋊#622C2semidirect
131. C33#424semidirect
195. C32#5119semidirect
228. C3#10593755semidirect

Maximal subgroups

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

subgroupindex
C2×#889882
C2×C3×#37882
#117234042
#117227322
Dic3⋊#6222
#105937553
#51199