Finite Group Databasebeta

← #13165372

Subgroups — #13165372

Order 1728.

664 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 112 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×#88934C212yesmaximal
2C2×Dic3×#157C211yesmaximal, characteristic
3Dic3×#624C212yesmaximal
4#11722791C211yesmaximal, characteristic
5C3×#63737C211yesmaximal, characteristic
6C4×#3773—31nomaximal
7C22×C3⋊#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×#6294C2212yes—
10C2×Dic3×C3⋊D3C2211yescharacteristic
11C2×C3×#624C2212yes—
12#88934C418yescomplemented
13Dic3×#157C414yescomplemented
14Dic3×C3⋊Dic3—22no—
15C3×Dic32—22no—
16C2×C4×#157—31no—
17C2×#3773—32no—
18C4×#624—32no—
19C4×D3×Dic3—62no—
20#63737—31no—
21#63737D311yescharacteristic
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3⋊#157C2×C412yes—
24C2×C3⋊#157C2×C414yes—
25C2×C3×#157—22no—
26C2×C3×#157C2×C412yes—
27C2×C3⋊(C3⋊Dic3)D411yescharacteristic
28C2×C3×C3⋊Dic3—22no—
29C2×C32×Dic3—22no—
30C2×C32×Dic3D411yescharacteristic
31#6294—22no—
32#6294C2×C414yes—
33Dic3×C3⋊D3C2×C412yes—
34Dic3×C3⋊D3Q812yes—
35C3×#624—22no—
36C3×#624C2×C414yes—
37#4649—91nomaximal
38C22×#157—31no—
39C22×#157C2×D311yescharacteristic
40C2×C4×C3⋊D3—31no—
41C2×C3×C4×D3—62no—
42C2×#624—32no—
43C2×#624C2×D312yes—
44C2×D3×Dic3—62no—
45C2×D3×Dic3—62no—
46C4×#157—34no—
47#3773—38no—
48C4×C3⋊Dic3—62no—
49C3×C4×Dic3—62no—
50C3×C4×Dic3—62no—
51Dic32—22no—
52Dic32—62no—
53Dic32—62no—
54Dic32—122no—
55C22×C33C2×D411yescharacteristic
56C2×C3×C3⋊D3C22×C411yescharacteristic
57C2×C3×C3⋊D3C22×C412yes—
58C2×C3×C3⋊D3C2×Q811yescharacteristic
59C2×C3×C3⋊D3#4112yes—
60C3⋊#157C4214yescomplemented
61C3×#157—22no—
62C3⋊(C3⋊Dic3)C4⋊C412yescomplemented
63C3×C3⋊Dic3—24no—
64C32×Dic3—24no—
65C32×Dic3C4⋊C412yescomplemented
66C2×Dic3⋊C4—92no—
67C2×C4×Dic3—92no—
68C2×C4×Dic3—91no—
69C42×D3—182no—
70C3×#95—91no—
71C3⋊#95—91no—
72C22×C3⋊D3C22×D311yescharacteristic
73C22×C3×D3—62no—
74C2×#157—22no—
75C2×#157—34no—
76C2×#157—32no—
77C2×#157C4×D312yes—
78C2×C32×C4—31no—
79C2×C32×C4—62no—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—122no—
84C4×C3⋊D3—32no—
85C4×C3⋊D3—32no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C3×C4×D3—64no—
94C3×C4×D3—64no—
95#624—22no—
96#624—34no—
97#624—62no—
98#624C4×D314yes—
99D3×Dic3—64no—
100D3×Dic3—64no—
101D3×Dic3—64no—
102D3×Dic3—64no—
103C4×C4⋊C4—271noSylow
104C2×C33C2×C4⋊C411yescharacteristic
105C2×C33C4×D412yes—
106C3×C3⋊D3C2×C4211yescharacteristic
107C3×C3⋊D3#11711yescharacteristic
108C3×C3⋊D3C4×Q812yes—
109C22×C3×C4—92no—
110C22×C3×C4—91no—
111C22×Dic3—92no—
112C22×Dic3—91no—
113C22×Dic3—91no—
114C2×C4×D3—62no—
115C2×C4×D3—182no—
116C2×C4×D3—182no—
117C3×C42—182no—
118Dic3⋊C4—98no—
119C4×Dic3—62no—
120C4×Dic3—62no—
121C4×Dic3—94no—
122C4×Dic3—182no—
123C4×Dic3—182no—
124C4×Dic3—182no—
125C4×Dic3—362no—
126C22×C32—22no—
127C22×C32—42no—
128C22×C32D3×D411yescharacteristic
129C2×C3⋊D3C2×C4×D311yescharacteristic
130C2×C3⋊D3C2×C4×D312yes—
131C2×C3⋊D3#23712yes—
132C2×C3⋊D3D3×Q811yescharacteristic
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C2×C3×D3—62no—
139#157—22no—
140#157—34no—
141C32×C4—32no—
142C32×C4—64no—
143C3⋊Dic3—24no—
144C3⋊Dic3—32no—
145C3⋊Dic3—64no—
146C3⋊Dic3—124no—
147C3×Dic3—24no—
148C3×Dic3—24no—
149C3×Dic3—24no—
150C3×Dic3—44no—
151C3×Dic3—64no—
152C3×Dic3—64no—
153C3×Dic3—124no—
154C2×C4⋊C4—272no—
155C2×C42—272no—
156C2×C42—271no—
157#95—91no—
158#95—271no—
159C33C4×C4⋊C411yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—62no—
162C2×C3×C4—62no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—94no—
166C2×C3×C4—92no—
167C2×C3×C4—122no—
168C2×C3×C4—182no—
169C2×C3×C4—182no—
170C2×C3×C4—182no—
171C2×Dic3—22no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—94no—
175C2×Dic3—92no—
176C2×Dic3—94no—
177C2×Dic3—91no—
178C2×Dic3—92no—
179C2×Dic3—92no—
180C2×Dic3—122no—
181C2×Dic3—182no—
182C2×Dic3—182no—
183C2×Dic3—362no—
184C2×Dic3GO+(4,𝔽2)11yescharacteristic
185C4×D3—64no—
186C4×D3—64no—
187C4×D3—184no—
188C4×D3—184no—
189C4×D3—184no—
190C4×D3—184no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32#67⋊C412yes—
198C2×C32D3×C4⋊C411yescharacteristic
199C3⋊D3C42×D311yescharacteristic
200C3⋊D3#87711yescharacteristic
201C3⋊D3#88712yes—
202C3×D3—62no—
203C3×D3—62no—
204C3×D3—62no—
205C3×D3—62no—
206C22×C4—92no—
207C22×C4—91no—
208C22×C4—271no—
209C22×C4—272no—
210C22×C4—271no—
211C4⋊C4—278no—
212C42—182no—
213C42—274no—
214C42—542no—
215C22×C3—22no—
216C22×C3—42no—
217C22×C3—92no—
218C22×C3—92no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3C2×GO+(4,𝔽2)11yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—62no—
228C3×C4—64no—
229C3×C4—64no—
230C3×C4—64no—
231C3×C4—124no—
232C3×C4—184no—
233C3×C4—182no—
234Dic3—24no—
235Dic3—64no—
236Dic3—64no—
237Dic3—94no—
238Dic3—92no—
239Dic3—124no—
240Dic3—184no—
241Dic3—364no—
242Dic3#377312yescomplemented
243C32—22no—
244C32—42no—
245C32#464911yescharacteristic, complemented
246C23—91no—
247C2×C4—31no—
248C2×C4—62no—
249C2×C4—94no—
250C2×C4—92no—
251C2×C4—182no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—272no—
255C2×C4—272no—
256C2×C4—274no—
257C2×C4—272no—
258C2×C4—274no—
259C2×C4—271no—
260C2×C4—542no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—22no—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—91no—
268C2×C3—91no—
269C2×C3—92no—
270C2×C3C4×GO+(4,𝔽2)12yes—
271C2×C3C2×#377311yescharacteristic
272D3—62no—
273D3—62no—
274D3—62no—
275D3—62no—
276C22—91no—
277C22—91no—
278C22—92no—
279C22—92no—
280C22D3×GO+(4,𝔽2)11yescharacteristic
281C4—32no—
282C4—64no—
283C4—182no—
284C4—184no—
285C4—274no—
286C4—272no—
287C3—22no—
288C3—42no—
289C3C4×#377311yescharacteristic, complemented
290C2—91no—
291C2—91no—
292C2—92no—
293C2D3×#377311yescharacteristic
294C2(D32⋊D3)⋊C412yes—
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊D3C2×C42
Fitting, socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33C4×C4⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
12. #88934C4semidirect
13. Dic3×#157C4semidirect
60. C3⋊#157C42semidirect
62. C3⋊(C3⋊Dic3)C4⋊C4semidirect
65. C32×Dic3C4⋊C4semidirect
159. C33C4×C4⋊C4semidirect
242. Dic3#3773semidirect
245. C32#4649semidirect
289. C3C4×#3773semidirect

Maximal subgroups

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

subgroupindex
C2×#889342
C2×Dic3×#1572
Dic3×#6242
#117227912
C3×#637372
C4×#37733
#46499