Finite Group Databasebeta

← #13166499

Subgroups — #13166499

Order 1728.

540 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 74 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×#88926C212yesmaximal, complemented
2#11723494C211yesmaximal, characteristic
3C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
4C3×C4×#622C211yesmaximal, characteristic, complemented
5#11722726C212yesmaximal, complemented
6#10593752—31nomaximal
7C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
8C2×#6289C2212yes—
9C2×C3⋊#622C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88926—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C411yescharacteristic
15#88804C412yes—
16C2×#3788—32no—
17#64102—31no—
18#63894—62no—
19#63778—32no—
20C4⋊#622—31no—
21C4×#622D311yescharacteristic, complemented
22C22×C3⋊(C3⋊D3)C2×C412yes—
23C2×C33×C4C2×C411yescharacteristic
24C4⋊(C3⋊(C3⋊D3))—22no—
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3C2×C412yes—
28#6289—22no—
29#6289—22no—
30C3⋊#622—22no—
31C3×#622—21no—
32C3×#622D412yescomplemented
33#4688—91nomaximal
34C2×C4⋊(C3⋊D3)—31no—
35C2×C4⋊(C3⋊D3)—62no—
36C2×C4⋊(C3⋊D3)—122no—
37C2×C4×C3⋊D3—32no—
38C2×#624—64no—
39C2×#622—32no—
40C2×#622C2×D311yescharacteristic
41C2×#622C2×D311yescharacteristic
42#3788—64no—
43C32×C4⋊C4—31no—
44#3752—32no—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C3×C4⋊Dic3—22no—
47C3×C4×Dic3—62no—
48#3722—64no—
49C22×C33C22×C411yescharacteristic
50C2×C3⋊(C3⋊D3)—22no—
51C2×C3⋊(C3⋊D3)—22no—
52C2×C3⋊(C3⋊D3)—22no—
53C33×C4M1612yescomplemented
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3C2×D411yescharacteristic
56C3×C3⋊Dic3#4111yescharacteristic
57C32×Dic3—22no—
58C2×ΓL(1,𝔽25)—92no—
59#893—182no—
60C4×D12—91no—
61C3×C4×C8—91no—
62(C2×D3)⋊C8—92no—
63C4⋊(C3⋊C8)—91no—
64C22×C3⋊D3—32no—
65C22×C3⋊D3—64no—
66C22×C3⋊D3—124no—
67C2×C32×C4—22no—
68C2×C32×C4—32no—
69C2×C32×C4—42no—
70C2×C32×C4C4×D311yescharacteristic
71C2×C3⋊Dic3C4×D311yescharacteristic
72C2×C3⋊Dic3C22×D311yescharacteristic
73C4⋊(C3⋊D3)—62no—
74C4⋊(C3⋊D3)—68no—
75C4⋊(C3⋊D3)—128no—
76C4×C3⋊D3—62no—
77C4×C3⋊D3—62no—
78C2×C3×Dic3—24no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81#624—124no—
82#624—124no—
83#622—21no—
84#622—62no—
85#622D1212yescomplemented
86#436—271noSylow
87C2×C33C4×D411yescharacteristic
88C2×C33C2×M1611yescharacteristic
89C2×C33#13111yescharacteristic
90C3⋊(C3⋊D3)—42no—
91C2×D12—91no—
92C2×D12—182no—
93C2×D12—362no—
94C2×C4×D3—92no—
95C2×C4×D3—184no—
96C2×C3×C8—91no—
97C2×C3×C8—91no—
98C3×C4⋊C4—62no—
99C3×C42—91no—
100(C2×D3)⋊C4—92no—
101(C2×D3)⋊C4—184no—
102C4⋊Dic3#15711yescharacteristic
103C4×Dic3—62no—
104C2×C3⋊C8—92no—
105ΓL(1,𝔽25)—184no—
106C22×C32—22no—
107C22×C32—42no—
108C22×C32C2×C4×D311yescharacteristic
109C2×C3⋊D3—62no—
110C2×C3⋊D3—68no—
111C2×C3⋊D3—62no—
112C2×C3⋊D3—62no—
113C2×C3⋊D3—128no—
114C2×C3⋊D3—124no—
115C2×C3⋊D3—124no—
116C2×C3⋊D3—244no—
117C2×C3⋊D3—244no—
118C32×C4—24no—
119C32×C4—44no—
120C32×C4—62no—
121C32×C4ΓL(1,𝔽25)12yescomplemented
122C3⋊Dic3—21no—
123C3⋊Dic3C2×D1211yescharacteristic
124C3⋊Dic3#23511yescharacteristic
125C3×Dic3—44no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—122no—
129C2×M16—272no—
130C4×D4—271no—
131C4⋊C8—271no—
132C22⋊C8—272no—
133C4×C8—91no—
134C33#43611yescharacteristic, complemented, Sylow
135C22×D3—92no—
136C22×D3—184no—
137C22×D3—364no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—64no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4C2×#15711yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3C2×#15712yes—
147D12—188no—
148D12—182no—
149D12—368no—
150C4×D3—182no—
151C4×D3—182no—
152C4×D3—364no—
153C4×D3—364no—
154C3×C8—92no—
155C3×C8—181no—
156C3⋊C8—182no—
157C2×C32—22no—
158C2×C32—22no—
159C2×C32—22no—
160C2×C32—42no—
161C2×C32—42no—
162C2×C32—42no—
163C2×C32C4×D1211yescharacteristic
164C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
165C2×C32#90111yescharacteristic
166C3⋊D3—122no—
167C3⋊D3—128no—
168C3⋊D3—248no—
169C2×D4—271no—
170C22×C4—272no—
171M16—544no—
172C2×C8—91no—
173C2×C8—91no—
174C2×C8—272no—
175C4⋊C4—31no—
176C22⋊C4—272no—
177C42—91no—
178C22×C3—22no—
179C22×C3—42no—
180C22×C3C22×#15711yescharacteristic
181C2×D3—188no—
182C2×D3—182no—
183C2×D3—182no—
184C2×D3—182no—
185C2×D3—364no—
186C2×D3—364no—
187C2×D3—368no—
188C2×D3—724no—
189C2×D3—724no—
190C3×C4—24no—
191C3×C4—44no—
192C3×C4—91no—
193C3×C4—91no—
194C3×C4—124no—
195C3×C4—181no—
196C3×C4#378812yescomplemented
197Dic3—22no—
198Dic3—62no—
199Dic3—62no—
200Dic3—122no—
201C32—22no—
202C32—42no—
203C32#468811yescharacteristic, complemented
204C23—272no—
205D4—542no—
206C2×C4—32no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—542no—
210C2×C4—542no—
211C2×C4D3×#15711yescharacteristic
212C8—92no—
213C8—181no—
214C8—542no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—42no—
219C2×C3—42no—
220C2×C3—42no—
221C2×C3C2×#378811yescharacteristic
222C2×C3D4×#15711yescharacteristic
223C2×C3#6428911yescharacteristic
224D3—368no—
225D3—362no—
226D3—728no—
227C22—542no—
228C22—542no—
229C22—542no—
230C22C2×D3×#15711yescharacteristic
231C4—62no—
232C4—91no—
233C4—91no—
234C4—181no—
235C4#8892612yescomplemented
236C3—22no—
237C3—42no—
238C3#1059375211yescharacteristic, complemented
239C2—1082no—
240C2#1172425211yescharacteristic
241C2D12×#15711yescharacteristic
242C2C2×#8892611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88926C2semidirect
3. C4⋊(C3⋊#622)C2semidirect
4. C3×C4×#622C2semidirect
5. #11722726C2semidirect
21. C4×#622D3semidirect
32. C3×#622D4semidirect
53. C33×C4M16semidirect
85. #622D12semidirect
121. C32×C4ΓL(1,𝔽25)semidirect
134. C33#436semidirect
196. C3×C4#3788semidirect
203. C32#4688semidirect
235. C4#88926semidirect
238. C3#10593752semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
#117234942
C4⋊(C3⋊#622)2
C3×C4×#6222
#117227262
#105937523
#46889