Finite Group Databasebeta

← #13166508

Subgroups — #13166508

Order 1728.

576 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 84 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
2C4×#6289C211yesmaximal, characteristic
3C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5Dic3⋊#622C212yesmaximal, complemented
6#10593728—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×#6289C411yescharacteristic
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12#88926—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4×Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C2×#3788—32no—
18C42×C3⋊D3—31no—
19C4×#624—62no—
20C4⋊#622—32no—
21C4⋊#622—31no—
22C4⋊#622D311yescharacteristic, complemented
23C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C4×C3⋊(C3⋊D3)—22no—
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30C2×C32×Dic3C2×C411yescharacteristic
31#6289—22no—
32#6289D412yes—
33#6289Q812yes—
34C3⋊#622—22no—
35C3×#622—22no—
36#4837—91nomaximal
37C2×C4×C3⋊D3—31no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3—62no—
41C2×C4×C3⋊D3—122no—
42C2×#624—62no—
43C2×#624—62no—
44C2×#622—32no—
45C2×#622C2×D312yes—
46#3788—64no—
47C32×C42—31no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3C2×D311yescharacteristic
51C3×C4×Dic3—22no—
52C3×C4×Dic3—62no—
53Dic32—62no—
54Dic32—62no—
55C22×C33C22×C411yescharacteristic
56C2×C3⋊(C3⋊D3)—21no—
57C2×C3⋊(C3⋊D3)—21no—
58C2×C3⋊(C3⋊D3)C4⋊C412yes—
59C33×C4M1612yescomplemented
60C3⋊(C3⋊Dic3)—21no—
61C3×C3⋊Dic3—21no—
62C3×C3⋊Dic3C2×D411yescharacteristic
63C3×C3⋊Dic3C2×Q811yescharacteristic
64C32×Dic3C4⋊C412yes—
65C32×Dic3M1612yescomplemented
66C2×ΓL(1,𝔽25)—92no—
67C42×D3—91no—
68C42×D3—182no—
69C3×C4⋊C8—91no—
70Dic3⋊C8—92no—
71C4⋊(C3⋊C8)—91no—
72C22×C3⋊D3—31no—
73C22×C3⋊D3—62no—
74C22×C3⋊D3—122no—
75C2×C32×C4—22no—
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C2×C32×C4—42no—
79C2×C32×C4C4×D311yescharacteristic
80C2×C3⋊Dic3—31no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—122no—
83C2×C3⋊Dic3C4×D311yescharacteristic
84C2×C3⋊Dic3C22×D311yescharacteristic
85C4×C3⋊D3—32no—
86C4×C3⋊D3—32no—
87C4×C3⋊D3—62no—
88C4×C3⋊D3—62no—
89C4×C3⋊D3—68no—
90C4×C3⋊D3—128no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95#624—64no—
96#624—64no—
97#624—68no—
98#622—22no—
99#622—62no—
100#423—271noSylow
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33C2×M1611yescharacteristic
103C2×C33C2×M1611yescharacteristic
104C3⋊(C3⋊D3)—22no—
105C2×C4×D3—91no—
106C2×C4×D3—91no—
107C2×C4×D3—91no—
108C2×C4×D3—182no—
109C2×C4×D3—182no—
110C2×C4×D3—182no—
111C2×C4×D3—362no—
112C2×C3×C8—92no—
113C3×C42—62no—
114C3×C42—91no—
115C4×Dic3—62no—
116C4×Dic3—91no—
117C4×Dic3—91no—
118C4×Dic3—182no—
119C4×Dic3—182no—
120C4×Dic3#15711yescharacteristic
121C2×C3⋊C8—92no—
122ΓL(1,𝔽25)—184no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C2×C4×D311yescharacteristic
126C2×C3⋊D3—32no—
127C2×C3⋊D3—64no—
128C2×C3⋊D3—61no—
129C2×C3⋊D3—61no—
130C2×C3⋊D3—64no—
131C2×C3⋊D3—64no—
132C2×C3⋊D3—124no—
133C2×C3⋊D3—124no—
134C2×C3⋊D3—124no—
135C32×C4—24no—
136C32×C4—32no—
137C32×C4—32no—
138C32×C4—44no—
139C32×C4ΓL(1,𝔽25)12yescomplemented
140C3⋊Dic3—21no—
141C3⋊Dic3—61no—
142C3⋊Dic3—64no—
143C3⋊Dic3—124no—
144C3⋊Dic3D3×D411yescharacteristic
145C3⋊Dic3D3×Q811yescharacteristic
146C3×Dic3—24no—
147C3×Dic3—24no—
148C3×Dic3—62no—
149C3×Dic3—62no—
150C3×Dic3—64no—
151C2×M16—272no—
152C2×C42—271no—
153C4⋊C8—91no—
154C4⋊C8—271no—
155C4⋊C8—272no—
156C33#42311yescharacteristic, complemented, Sylow
157C22×D3—91no—
158C22×D3—182no—
159C22×D3—362no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—62no—
163C2×C3×C4—62no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4C2×#15711yescharacteristic
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—91no—
170C2×Dic3—182no—
171C2×Dic3—362no—
172C2×Dic3C2×#15711yescharacteristic
173C2×Dic3C2×#15711yescharacteristic
174C4×D3—92no—
175C4×D3—92no—
176C4×D3—184no—
177C4×D3—184no—
178C4×D3—188no—
179C4×D3—188no—
180C4×D3—182no—
181C4×D3—182no—
182C4×D3—368no—
183C3×C8—182no—
184C3⋊C8—182no—
185C2×C32—22no—
186C2×C32—22no—
187C2×C32—22no—
188C2×C32—42no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32D3×C4⋊C411yescharacteristic
192C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
193C2×C32D3×M1611yescharacteristic
194C3⋊D3—62no—
195C3⋊D3—68no—
196C3⋊D3—128no—
197C22×C4—271no—
198C22×C4—271no—
199C22×C4—271no—
200M16—544no—
201C2×C8—92no—
202C2×C8—272no—
203C42—31no—
204C42—91no—
205C42—271no—
206C42—271no—
207C22×C3—22no—
208C22×C3—42no—
209C22×C3C22×#15711yescharacteristic
210C2×D3—92no—
211C2×D3—184no—
212C2×D3—184no—
213C2×D3—184no—
214C2×D3—181no—
215C2×D3—181no—
216C2×D3—364no—
217C2×D3—364no—
218C2×D3—364no—
219C3×C4—24no—
220C3×C4—44no—
221C3×C4—64no—
222C3×C4—64no—
223C3×C4—91no—
224C3×C4—91no—
225C3×C4—181no—
226C3×C4#378812yescomplemented
227Dic3—62no—
228Dic3—62no—
229Dic3—64no—
230Dic3—184no—
231Dic3—181no—
232Dic3—364no—
233Dic3#378812yescomplemented
234Dic3C4⋊#15712yes—
235C32—22no—
236C32—42no—
237C32#483711yescharacteristic, complemented
238C23—271no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—272no—
244C2×C4—272no—
245C2×C4—271no—
246C2×C4—542no—
247C2×C4—542no—
248C2×C4D3×#15711yescharacteristic
249C8—182no—
250C8—542no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3C2×#378811yescharacteristic
258C2×C3C2×#378811yescharacteristic
259C2×C3C2×C4⋊#15711yescharacteristic
260D3—188no—
261D3—182no—
262D3—368no—
263C22—272no—
264C22—541no—
265C22—541no—
266C22C2×D3×#15711yescharacteristic
267C4—32no—
268C4—32no—
269C4—91no—
270C4—91no—
271C4—181no—
272C4—541no—
273C4#8892612yescomplemented
274C3—22no—
275C3—42no—
276C3#1059372811yescharacteristic, complemented
277C2—542no—
278C2D3×#378811yescharacteristic
279C2D3×C4⋊#15711yescharacteristic
280C2C2×#8892611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#423
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊#622)C2semidirect
4. C3×C4⋊#622C2semidirect
5. Dic3⋊#622C2semidirect
22. C4⋊#622D3semidirect
59. C33×C4M16semidirect
65. C32×Dic3M16semidirect
139. C32×C4ΓL(1,𝔽25)semidirect
156. C33#423semidirect
226. C3×C4#3788semidirect
233. Dic3#3788semidirect
237. C32#4837semidirect
273. C4#88926semidirect
276. C3#10593728semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
C4×#62892
C4⋊(C3⋊#622)2
C3×C4⋊#6222
Dic3⋊#6222
#105937283
#48379