Finite Group Databasebeta

← #13166360

Subgroups — #13166360

Order 1728.

680 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 78 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×C3⋊D3×Dic6C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic
3C3×#64283C211yesmaximal, characteristic
4#11722730C214yesmaximal, complemented
5#10593836—31nomaximal
6C2×#6308C411yescharacteristic
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6C411yescharacteristic
9C2×#6293C2212yes—
10C2×Dic3×C3⋊D3C412yes—
11C3⋊D3×Dic6—21no—
12C3⋊D3×Dic6—21no—
13C3⋊D3×Dic6—21no—
14C3⋊D3×Dic6—21no—
15#88993C2212yes—
16#88989—21no—
17#88988—21no—
18C3×#3792C2212yes—
19C3×#3788—21no—
20C3×#3787—21no—
21C2×Q8×C3⋊D3—31no—
22C2×D3×Dic6—62no—
23#64283—31no—
24#64283D311yescharacteristic
25#63780—34no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C3⋊(C3⋊Dic3)C2×C412yes—
29#6308—21no—
30#6308—21no—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C2×C32×Dic3C2×C412yes—
37C32×Dic6—21no—
38C32×Dic6—21no—
39#6293—22no—
40#6293—22no—
41Dic3×C3⋊D3—22no—
42Dic3×C3⋊D3—22no—
43C3⋊#622—22no—
44C3×#622—22no—
45C2×#824—91nomaximal
46C2×C32×Q8—31no—
47Q8×C3⋊D3—61no—
48Q8×C3⋊D3—61no—
49Q8×C3⋊D3—61no—
50Q8×C3⋊D3—61no—
51C2×C4×C3⋊D3—32no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×#634—31no—
54C2×#634—32no—
55C2×#634—62no—
56C2×#634—122no—
57C2×C3×C4×D3—62no—
58C2×C3×Dic6—22no—
59C2×#627—64no—
60C2×D3×Dic3—64no—
61D3×Dic6—64no—
62D3×Dic6—64no—
63D3×Dic6—64no—
64D3×Dic6—64no—
65#3792—32no—
66#3792C2×D312yes—
67#3788—21no—
68#3788—61no—
69#3787—21no—
70#3787—61no—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3C22⋊C411yescharacteristic
75C33×C4C22⋊C411yescharacteristic
76C33×C4C22⋊C411yescharacteristic
77C3⋊(C3⋊Dic3)—22no—
78C3×C3⋊Dic3C2×D411yescharacteristic
79C3×C3⋊Dic3C2×D411yescharacteristic
80C32×Dic3—22no—
81C2×D3×Q8—182no—
82C22×Dic6—91no—
83C2×C3×M16—91no—
84C2×#208—91no—
85#824—94no—
86C22×C3⋊D3C4×D311yescharacteristic
87C22×C3×D3—62no—
88C32×Q8—61no—
89C32×Q8—61no—
90C2×C32×C4—22no—
91C2×C32×C4—32no—
92C2×C32×C4—42no—
93C2×C32×C4C4×D311yescharacteristic
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—64no—
96C2×C3⋊Dic3—124no—
97C2×C3⋊Dic3C22×D311yescharacteristic
98C4×C3⋊D3—62no—
99C4×C3⋊D3—62no—
100C4×C3⋊D3D1211yescharacteristic
101C4×C3⋊D3D1211yescharacteristic
102C4×C3⋊D3#6711yescharacteristic
103C4×C3⋊D3#6711yescharacteristic
104#634—61no—
105#634—61no—
106#634—62no—
107#634—62no—
108#634—64no—
109#634—64no—
110#634—124no—
111#634—124no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—62no—
114C3×C4×D3—62no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×Dic6—24no—
119C3×Dic6—24no—
120#627—68no—
121#627—68no—
122D3×Dic3—68no—
123D3×Dic3—68no—
124#622—22no—
125#622—62no—
126C2×#101—271noSylow
127C2×C33—21no—
128C2×C33C2×C22⋊C411yescharacteristic
129C3×C3⋊D3—21no—
130C3×C3⋊D3#10112yescomplemented
131C2×C3×Q8—62no—
132C22×C3×C4—91no—
133C22×Dic3—92no—
134D3×Q8—184no—
135D3×Q8—184no—
136D3×Q8—184no—
137D3×Q8—184no—
138C2×C4×D3—62no—
139C2×C4×D3—184no—
140C2×Dic6—91no—
141C2×Dic6—92no—
142C2×Dic6—184no—
143C2×Dic6—182no—
144C2×Dic6—181no—
145C2×Dic6—181no—
146C2×Dic6—181no—
147C2×Dic6—181no—
148C2×Dic6—362no—
149C2×Dic6#15711yescharacteristic
150C3×M16—92no—
151C3×M16—181no—
152C2×C3×C8—181no—
153#208—92no—
154#208—181no—
155C2×C3⋊C8—181no—
156C22×C32—22no—
157C22×C32—42no—
158C22×C32C2×C4×D311yescharacteristic
159C2×C3⋊D3—22no—
160C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
161C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—68no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—62no—
170C32×C4(C2×D3)⋊C411yescharacteristic
171C32×C4(C2×D3)⋊C411yescharacteristic
172C3⋊Dic3—62no—
173C3⋊Dic3—68no—
174C3⋊Dic3—128no—
175C3⋊Dic3C2×D1211yescharacteristic
176C3⋊Dic3C2×#6711yescharacteristic
177C3×Dic3—28no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C22×Q8—271no—
181C2×M16—91no—
182C2×M16—271no—
183#101—274no—
184C33C2×#10111yescharacteristic, complemented, Sylow
185C23×C3—91no—
186C22×D3—62no—
187C3×Q8—64no—
188C3×Q8—64no—
189C2×C3×C4—22no—
190C2×C3×C4—42no—
191C2×C3×C4—64no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4C2×#15711yescharacteristic
198C2×Dic3—62no—
199C2×Dic3—92no—
200C2×Dic3—184no—
201C2×Dic3—182no—
202C2×Dic3—182no—
203C2×Dic3—364no—
204C2×Dic3C2×#15712yes—
205C4×D3—62no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—62no—
209C4×D3—188no—
210C4×D3—188no—
211Dic6—21no—
212Dic6—21no—
213Dic6—182no—
214Dic6—182no—
215Dic6—188no—
216Dic6—188no—
217Dic6—184no—
218Dic6—184no—
219Dic6—181no—
220Dic6—181no—
221Dic6—364no—
222Dic6—364no—
223C3×C8—182no—
224C3⋊C8—182no—
225C2×C32—21no—
226C2×C32—24no—
227C2×C32—22no—
228C2×C32—44no—
229C2×C32—42no—
230C2×C32C2×(C2×D3)⋊C411yescharacteristic
231C3⋊D3—21no—
232C3⋊D3#82412yescomplemented
233C3×D3—64no—
234C3×D3—64no—
235C2×Q8—31no—
236C2×Q8—272no—
237C2×Q8—271no—
238C2×Q8—541no—
239C2×Q8—541no—
240C2×Q8—541no—
241C2×Q8—541no—
242C22×C4—91no—
243C22×C4—272no—
244M16—92no—
245M16—181no—
246M16—272no—
247M16—541no—
248C2×C8—181no—
249C2×C8—541no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—91no—
253C22×C3—91no—
254C22×C3—182no—
255C22×C3C22×#15711yescharacteristic
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—68no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—68no—
264C3×C4—91no—
265C3×C4—91no—
266C3×C4C22⋊#15711yescharacteristic
267C3×C4C22⋊#15711yescharacteristic
268Dic3—22no—
269Dic3—62no—
270Dic3—62no—
271Dic3—188no—
272Dic3—182no—
273Dic3—368no—
274C32—22no—
275C32—42no—
276C32C2×#82411yescharacteristic, complemented
277C23—91no—
278Q8—61no—
279Q8—61no—
280Q8—542no—
281Q8—542no—
282Q8—541no—
283Q8—541no—
284C2×C4—32no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—272no—
291C2×C4—542no—
292C2×C4—542no—
293C2×C4D3×#15711yescharacteristic
294C8—182no—
295C8—542no—
296C2×C3—24no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—44no—
300C2×C3—42no—
301C2×C3—92no—
302C2×C3—181no—
303C2×C3C2×C22⋊#15711yescharacteristic
304D3—64no—
305D3—64no—
306C22—91no—
307C22—91no—
308C22—182no—
309C22C2×D3×#15711yescharacteristic
310C4—62no—
311C4—91no—
312C4—91no—
313C4—542no—
314C4(C2×D3)⋊#15711yescharacteristic
315C4(C2×D3)⋊#15711yescharacteristic
316C3—22no—
317C3—42no—
318C3#1059383611yescharacteristic, complemented
319C2—21no—
320C2—92no—
321C2—181no—
322C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722730C2semidirect
130. C3×C3⋊D3#101semidirect
184. C33C2×#101semidirect
232. C3⋊D3#824semidirect
276. C32C2×#824semidirect
318. C3#10593836semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
#117245112
C3×#642832
#117227302
#105938363
C2×#8249