Finite Group Databasebeta

← #13173349

Subgroups — #13173349

Order 1728.

432 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88989C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic
3C3×#64101C211yesmaximal, characteristic
4#11723104C212yesmaximal, complemented
5C4⋊(C3⋊#622)C212yesmaximal, complemented
6#10593755—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3⋊#622C2211yescharacteristic
9C2×C3⋊#622C2212yes—
10C2×C3⋊#622C2211yescharacteristic
11#88989—22no—
12#88988—22no—
13C33×C4⋊C4C411yescharacteristic
14C3×#3752C412yes—
15C3×C4⋊(C3⋊Dic3)C411yescharacteristic
16C3×C4×C3⋊Dic3C2212yes—
17C2×#3788—31no—
18C2×#3787—31no—
19#64101D311yescharacteristic
20C3×#892—62no—
21#63779—32no—
22C4⋊#622—32no—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C33×C4C2×C412yes—
25C2×C33×C4C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C412yes—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3D412yes—
29C3×C4×C3⋊D3Q812yes—
30C3⋊#622—21no—
31C3⋊#622—21no—
32C3⋊#622—22no—
33#4999—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#622—31no—
37C2×#622—31no—
38C2×#622—32no—
39#3788—62no—
40#3787—62no—
41C32×C4⋊C4—22no—
42C32×C4⋊C4—42no—
43C32×C4⋊C4Dic311yescharacteristic
44#3752Dic312yes—
45C4⋊(C3⋊Dic3)Dic311yescharacteristic
46C4×C3⋊Dic3C2×D312yes—
47C3×(C2×D3)⋊C4—64no—
48C3×C4⋊Dic3—62no—
49C3×C4×Dic3—64no—
50C22×C33C22×C411yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3C4⋊C412yes—
54C33×C4—22no—
55C33×C4C4⋊C412yes—
56C3×C3⋊Dic3—22no—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3C2×Q811yescharacteristic
59C3×#117—91no—
60C2×#208—91no—
61C22×C3⋊C8—91no—
62#892—62no—
63C4⋊(C3⋊C8)—92no—
64C4⋊(C3⋊C8)—92no—
65C22×C3⋊D3C2×Dic311yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—24no—
69C2×C32×C4—42no—
70C2×C32×C4—44no—
71C2×C32×C4C2×Dic312yes—
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C3⋊Dic3C2×Dic312yes—
74C2×C3⋊Dic3C22×D311yescharacteristic
75C4×C3⋊D3Dic612yes—
76C4×C3⋊D3D1212yes—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—64no—
79C3×C4×D3—64no—
80C3×C4×D3—64no—
81#622—61no—
82#622—61no—
83#622—62no—
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×C4⋊C4—22no—
92C3×C4⋊C4—42no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4#15711yescharacteristic
95C3×C22⋊C4—92no—
96C3×C42—92no—
97(C2×D3)⋊C4—64no—
98C4⋊Dic3—62no—
99C4×Dic3—64no—
100#208—182no—
101C2×C3⋊C8—91no—
102C2×C3⋊C8—91no—
103C2×C3⋊C8—92no—
104C2×C3⋊C8—182no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32C22×Dic311yescharacteristic
108C2×C3⋊D3—21no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3C4⋊Dic312yes—
111C2×C3×D3—64no—
112C2×C3×D3—122no—
113C2×C3×D3—122no—
114C32×C4—24no—
115C32×C4—22no—
116C32×C4—44no—
117C32×C4—44no—
118C32×C4—84no—
119C32×C4C4⋊Dic312yes—
120C3⋊Dic3—22no—
121C3⋊Dic3C2×Dic611yescharacteristic
122C3⋊Dic3C2×D1211yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—124no—
126C2×M16—271no—
127C22×C8—271no—
128#117—91no—
129C4⋊C8—272no—
130C4⋊C8—272no—
131C33#42411yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—62no—
134C2×C3×C4—22no—
135C2×C3×C4—24no—
136C2×C3×C4—42no—
137C2×C3×C4—44no—
138C2×C3×C4—91no—
139C2×C3×C4—92no—
140C2×C3×C4—92no—
141C2×C3×C4—92no—
142C2×C3×C4C2×#15712yes—
143C2×C3×C4C2×#15711yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—64no—
146C4×D3—64no—
147C4×D3—64no—
148C3⋊C8—181no—
149C3⋊C8—181no—
150C3⋊C8—182no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32C2×C4⋊Dic311yescharacteristic
158C2×C32#94811yescharacteristic
159C2×C32#94811yescharacteristic
160C3⋊D3—22no—
161C3×D3—124no—
162C22×C4—91no—
163M16—542no—
164C2×C8—271no—
165C2×C8—271no—
166C2×C8—272no—
167C2×C8—542no—
168C4⋊C4—91no—
169C4⋊C4C3⋊#15711yescharacteristic
170C22⋊C4—92no—
171C42—92no—
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—22no—
182C3×C4—24no—
183C3×C4—44no—
184C3×C4—44no—
185C3×C4—84no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—182no—
189C3×C4C4⋊#15712yes—
190Dic3—62no—
191Dic3—62no—
192Dic3—124no—
193C32—22no—
194C32—42no—
195C32#499911yescharacteristic, complemented
196C23—91no—
197C2×C4—91no—
198C2×C4—92no—
199C2×C4—92no—
200C2×C4—92no—
201C2×C4C2×C3⋊#15712yes—
202C2×C4C2×C3⋊#15711yescharacteristic
203C8—541no—
204C8—541no—
205C8—542no—
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—
220C22C22×C3⋊#15711yescharacteristic
221C4—22no—
222C4—91no—
223C4—91no—
224C4—182no—
225C4C4⋊(C3⋊#157)12yes—
226C3—22no—
227C3—42no—
228C3#1059375511yescharacteristic, complemented
229C2—182no—
230C2C2×C4⋊(C3⋊#157)11yescharacteristic
231C2#1172451511yescharacteristic
232C2#1172451711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C4⋊C4C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11723104C2semidirect
5. C4⋊(C3⋊#622)C2semidirect
131. C33#424semidirect
195. C32#4999semidirect
228. C3#10593755semidirect

Maximal subgroups

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

subgroupindex
C2×#889892
C2×#889882
C3×#641012
#117231042
C4⋊(C3⋊#622)2
#105937553
#49999