Finite Group Databasebeta

← #13154806

Subgroups — #13154806

Order 1728.

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

normal subgroups only

This group has 424 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×C4⋊(C3⋊#157)C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#10590182—31nomaximal
5C22×C3⋊#157—21no—
6C22×C3×#157—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C4⋊(C3⋊#157)—21no—
9C4⋊(C3⋊#157)C412yescomplemented
10C3×C4⋊#157—21no—
11C3×C4⋊#157C412yescomplemented
12C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic, complemented
13C2×C32×C3⋊C8C411yescharacteristic, complemented
14C3⋊D3×C3⋊C8—21no—
15C3⋊D3×C3⋊C8—21no—
16C2×C4⋊#157—31no—
17C2×C4⋊#157D311yescharacteristic
18C2×C8×C3⋊D3—31no—
19C2×D3×C3⋊C8—62no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C3⋊#157—41no—
23C2×C3⋊#157—41no—
24C2×C3×#157—22no—
25C2×C3×#157—41no—
26C2×C3×#157—41no—
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3⋊(C3⋊(C3⋊C8))—21no—
34C32×C3⋊C8—21no—
35#4529—91nomaximal
36C22×#157—21no—
37C22×#157—61no—
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C3×C4×D3—62no—
40C4⋊#157—21no—
41C4⋊#157—32no—
42C4⋊#157—61no—
43C4⋊#157Dic312yescomplemented
44C2×C32×C8—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C2×C3⋊(C3⋊C8)—62no—
47C2×C3⋊(C3⋊C8)—122no—
48C8×C3⋊D3—61no—
49C8×C3⋊D3—61no—
50C2×C3×C3⋊C8—22no—
51D3×C3⋊C8—64no—
52D3×C3⋊C8—64no—
53C22×C33C4⋊C411yescharacteristic
54C2×C3×C3⋊D3C22⋊C411yescharacteristic
55C2×C3×C3⋊D3C22⋊C411yescharacteristic
56C2×C3×C3⋊D3D811yescharacteristic
57C2×C3×C3⋊D3SD1611yescharacteristic
58C2×C3×C3⋊D3SD1611yescharacteristic
59C2×C3×C3⋊D3Q1611yescharacteristic
60C3⋊#157—42no—
61C3×#157—42no—
62C33×C4C4211yescharacteristic
63C33×C4C22⋊C411yescharacteristic
64C3×C3⋊Dic3C4⋊C411yescharacteristic
65C3×C3⋊Dic3C4⋊C411yescharacteristic
66C2×C3×C4⋊C4—91no—
67C2×C4⋊Dic3—91no—
68C22×C3⋊C8—91no—
69C2×D3×C8—182no—
70C22×C3⋊D3#6711yescharacteristic
71C22×C3×D3—62no—
72C2×#157—22no—
73C2×#157—41no—
74C2×#157—41no—
75C2×#157—62no—
76C2×#157—121no—
77C2×#157—121no—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4C4×D311yescharacteristic
81C2×C3⋊Dic3Dic611yescharacteristic
82C4×C3⋊D3C4×D311yescharacteristic
83C4×C3⋊D3D1211yescharacteristic
84C4×C3⋊D3C2×Dic311yescharacteristic
85C4×C3⋊D3#6711yescharacteristic
86C2×C3×Dic3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C32×C8—61no—
92C3⋊(C3⋊C8)—61no—
93C3⋊(C3⋊C8)—64no—
94C3⋊(C3⋊C8)—124no—
95C3×C3⋊C8—24no—
96#340—271noSylow
97C2×C33#9511yescharacteristic
98C2×C33#10611yescharacteristic
99C2×C33#10711yescharacteristic
100C3×C3⋊D3#10211yescharacteristic
101C3×C3⋊D3#10211yescharacteristic
102C3×C3⋊D3#10311yescharacteristic
103C3×C3⋊D3#10311yescharacteristic
104C22×C3×C4—91no—
105C22×C3×C4—181no—
106C22×Dic3—181no—
107C2×C4×D3—62no—
108C2×C3×C8—62no—
109C3×C4⋊C4—92no—
110C3×C4⋊C4—181no—
111C4⋊Dic3—92no—
112C4⋊Dic3—181no—
113C2×C3⋊C8—91no—
114C2×C3⋊C8—182no—
115C2×C3⋊C8—181no—
116C2×C3⋊C8—181no—
117C2×C3⋊C8—362no—
118C2×C3⋊C8#15711yescharacteristic, complemented
119D3×C8—184no—
120D3×C8—184no—
121C22×C32—22no—
122C22×C32—42no—
123C22×C32Dic3⋊C411yescharacteristic
124C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
125C2×C3⋊D3#21311yescharacteristic
126C2×C3⋊D3#21411yescharacteristic
127C2×C3⋊D3#21511yescharacteristic
128C2×C3⋊D3#21611yescharacteristic
129C2×C3⋊D3#21711yescharacteristic
130C2×C3×D3—62no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136#157—42no—
137#157—122no—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—42no—
141C32×C4—42no—
142C32×C4C4×Dic311yescharacteristic
143C32×C4(C2×D3)⋊C411yescharacteristic
144C3⋊Dic3Dic3⋊C411yescharacteristic
145C3⋊Dic3C4⋊Dic311yescharacteristic
146C3×Dic3—62no—
147C3×Dic3—62no—
148C22×C8—271no—
149C2×C4⋊C4—91no—
150C2×C4⋊C4—271no—
151C33#34011yescharacteristic, complemented, Sylow
152C23×C3—91no—
153C22×D3—62no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—182no—
162C2×C3×C4—361no—
163C2×C3×C4—361no—
164C2×C3×C4C2×#15711yescharacteristic
165C2×Dic3—62no—
166C2×Dic3—182no—
167C2×Dic3—361no—
168C2×Dic3—361no—
169C4×D3—62no—
170C4×D3—62no—
171C4×D3—62no—
172C4×D3—62no—
173C3×C8—64no—
174C3⋊C8—21no—
175C3⋊C8—184no—
176C3⋊C8—181no—
177C3⋊C8—364no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32—42no—
184C2×C32#80711yescharacteristic
185C2×C32#80811yescharacteristic
186C2×C32C3⋊#9511yescharacteristic
187C3⋊D3#80911yescharacteristic
188C3⋊D3#81011yescharacteristic
189C3⋊D3#83211yescharacteristic
190C3⋊D3#83511yescharacteristic
191C3×D3—62no—
192C3×D3—62no—
193C3×D3—62no—
194C3×D3—62no—
195C22×C4—91no—
196C22×C4—181no—
197C22×C4—541no—
198C2×C8—31no—
199C2×C8—271no—
200C2×C8—541no—
201C2×C8—541no—
202C4⋊C4—92no—
203C4⋊C4—181no—
204C4⋊C4—272no—
205C4⋊C4—541no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—91no—
209C22×C3—91no—
210C22×C3—91no—
211C22×C3—91no—
212C22×C3—91no—
213C22×C3—91no—
214C22×C3C4⋊#15711yescharacteristic
215C2×D3—62no—
216C2×D3—62no—
217C2×D3—62no—
218C2×D3—62no—
219C2×D3—62no—
220C2×D3—62no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—91no—
226C3×C4—91no—
227C3×C4—362no—
228C3×C4C4×#15711yescharacteristic
229C3×C4C22⋊#15711yescharacteristic
230Dic3—62no—
231Dic3—62no—
232Dic3—362no—
233C32—22no—
234C32—42no—
235C32#452911yescharacteristic, complemented
236C23—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—182no—
243C2×C4—361no—
244C2×C4—361no—
245C2×C4—542no—
246C2×C4—1081no—
247C2×C4—1081no—
248C2×C4D3×#15711yescharacteristic
249C8—61no—
250C8—541no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—91no—
258C2×C3—91no—
259C2×C3—91no—
260C2×C3—91no—
261C2×C3#6376811yescharacteristic
262C2×C3#6376911yescharacteristic
263C2×C3#6378111yescharacteristic
264D3—62no—
265D3—62no—
266D3—62no—
267D3—62no—
268C22—91no—
269C22—91no—
270C22—91no—
271C22—91no—
272C22—91no—
273C22—91no—
274C22Dic3⋊#15711yescharacteristic
275C4—91no—
276C4—91no—
277C4—362no—
278C4—1082no—
279C4Dic3×#15711yescharacteristic
280C4(C2×D3)⋊#15711yescharacteristic
281C3—22no—
282C3—42no—
283C3#1059018211yescharacteristic, complemented
284C2—91no—
285C2—91no—
286C2—91no—
287C2—91no—
288C2#1172271411yescharacteristic
289C2#1172271511yescharacteristic
290C2#1172272211yescharacteristic
subgroupG/H
centerC22Dic3⋊#157
commutatorC33×C4C42
Frattini, hypercenterC2×C4D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#340
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
9. C4⋊(C3⋊#157)C4semidirect
11. C3×C4⋊#157C4semidirect
12. C2×C3⋊(C3⋊(C3⋊C8))C4semidirect
13. C2×C32×C3⋊C8C4semidirect
43. C4⋊#157Dic3semidirect
118. C2×C3⋊C8#157semidirect
151. C33#340semidirect
235. C32#4529semidirect
283. C3#10590182semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
C2×C3×C4⋊#1572
C2×C3⋊D3×C3⋊C82
#105901823
#45299