Finite Group Databasebeta

← #13167588

Subgroups — #13167588

Order 1728.

800 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C23×C3×#157C211yesmaximal, characteristic
2C2×#88940C212yesmaximal, complemented
3#11723723C212yesmaximal
4#11722791C212yesmaximal, complemented
5#10593516—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×#157—24no—
8C22×C3×#157C414yescomplemented
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×#6294C2214yes—
12#88940—24no—
13C3×#3757—22no—
14#88814—22no—
15C23×#157D311yescharacteristic
16C2×#3773—32no—
17#63974—32no—
18D3×#217—62no—
19#63737—32no—
20C23×C33D411yescharacteristic
21C22×C3×C3⋊D3—22no—
22C22×C3×C3⋊D3—22no—
23C22×C3×C3⋊D3C2×C412yes—
24C22×C3×C3⋊D3D411yescharacteristic
25C22×C3×C3⋊D3Q812yes—
26C22×C3×C3⋊D3C2311yescharacteristic
27C2×C3×#157—22no—
28C2×C3×#157—28no—
29C2×C3×#157C2×C412yes—
30C2×C3×#157C2×C412yes—
31C2×C3×#157D414yes—
32C2×C3×C3⋊Dic3—24no—
33#6294—24no—
34#6294—24no—
35#5212—91nomaximal
36C23×C3⋊D3C2×D311yescharacteristic
37C23×C3×D3—62no—
38C22×#157—24no—
39C22×#157Dic314yescomplemented
40C22×#157C2×D311yescharacteristic
41C22×#157C2×D311yescharacteristic
42C2×#624—34no—
43C2×D3×Dic3—64no—
44#3773—64no—
45#3757—22no—
46C3×#217—62no—
47C3×#217—62no—
48C3×#217—122no—
49#3722—62no—
50(C2×D3)⋊Dic3—64no—
51C22×C33—22no—
52C22×C33C4⋊C412yescomplemented
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3—24no—
58C2×C3×C3⋊D3C4⋊C412yescomplemented
59C2×C3×C3⋊D3C22×C411yescharacteristic
60C2×C3×C3⋊D3C2×D412yes—
61C2×C3×C3⋊D3C2×Q811yescharacteristic
62C2×C3×C3⋊D3#4112yes—
63C3×#157—22no—
64C3×#157C22⋊C414yescomplemented
65C3×C3⋊Dic3—44no—
66C23×C3×C4—91no—
67C2×#217—92no—
68C2×C4⋊Dic3—92no—
69D3×C22⋊C4—182no—
70C3⋊#95—92no—
71C23×C32—22no—
72C23×C32—42no—
73C23×C32D1211yescharacteristic
74C22×C3⋊D3—22no—
75C22×C3⋊D3—22no—
76C22×C3⋊D3Dic612yes—
77C22×C3⋊D3D1211yescharacteristic
78C22×C3⋊D3C2×Dic312yes—
79C22×C3⋊D3C22×D311yescharacteristic
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—64no—
83C22×C3×D3—64no—
84C22×C3×D3—124no—
85C22×C3×D3—124no—
86C2×#157—22no—
87C2×#157—28no—
88C2×#157C2×Dic312yes—
89C2×#157C2×Dic312yes—
90C2×#157#6714yes—
91C2×C3⋊Dic3—24no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—124no—
95#624—64no—
96#624—64no—
97D3×Dic3—124no—
98D3×Dic3—124no—
99#380—271noSylow
100C2×C33—22no—
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33#12112yes—
103C3×C3⋊D3—22no—
104C3×C3⋊D3C2×C22⋊C411yescharacteristic
105C3×C3⋊D3#11711yescharacteristic
106C3×C3⋊D3C22⋊Q812yes—
107C24×C3—91no—
108C23×D3—62no—
109C22×C3×C4—91no—
110C22×C3×C4—94no—
111C22×C3×C4—91no—
112C22×C3×C4—184no—
113C22×Dic3—94no—
114C2×C4×D3—184no—
115C3×C22⋊C4—182no—
116#217—62no—
117#217—62no—
118#217—122no—
119#217—182no—
120(C2×D3)⋊C4—184no—
121C4⋊Dic3—184no—
122C22×C32—22no—
123C22×C32—24no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—44no—
128C22×C32—84no—
129C22×C32C4⋊Dic312yescomplemented
130C22×C32C2×D1211yescharacteristic
131C2×C3⋊D3—22no—
132C2×C3⋊D3—24no—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—22no—
135C2×C3⋊D3C4⋊Dic312yescomplemented
136C2×C3⋊D3C2×Dic611yescharacteristic
137C2×C3⋊D3#23512yes—
138C2×C3⋊D3C22×Dic311yescharacteristic
139C2×C3⋊D3C2×#6712yes—
140C2×C3×D3—62no—
141C2×C3×D3—64no—
142C2×C3×D3—62no—
143C2×C3×D3—64no—
144C2×C3×D3—64no—
145C2×C3×D3—124no—
146C2×C3×D3—124no—
147C2×C3×D3—128no—
148C2×C3×D3—124no—
149#157—22no—
150#157#21714yescomplemented
151C3⋊Dic3—44no—
152C3×Dic3—124no—
153C3×Dic3—124no—
154C3×Dic3—244no—
155C23×C4—91no—
156C2×C4⋊C4—272no—
157C2×C22⋊C4—272no—
158#95—272no—
159C33#38011yescharacteristic, complemented, Sylow
160C23×C3—22no—
161C23×C3—42no—
162C23×C3—91no—
163C23×C3—91no—
164C23×C3—92no—
165C23×C3—92no—
166C23×C3—182no—
167C23×C3—182no—
168C23×C3GO+(4,𝔽2)11yescharacteristic
169C22×D3—62no—
170C22×D3—64no—
171C22×D3—62no—
172C22×D3—64no—
173C22×D3—124no—
174C22×D3—124no—
175C2×C3×C4—92no—
176C2×C3×C4—94no—
177C2×C3×C4—92no—
178C2×C3×C4—188no—
179C2×C3×C4—182no—
180C2×C3×C4—184no—
181C2×Dic3—64no—
182C2×Dic3—64no—
183C2×Dic3—124no—
184C2×Dic3—184no—
185C2×Dic3—184no—
186C4×D3—364no—
187C4×D3—364no—
188C2×C32—22no—
189C2×C32—24no—
190C2×C32—22no—
191C2×C32—42no—
192C2×C32—44no—
193C2×C32—44no—
194C2×C32—84no—
195C2×C32C2×C4⋊Dic311yescharacteristic
196C2×C32#93012yes—
197C3⋊D3—22no—
198C3⋊D3#92412yes—
199C3⋊D3#92611yescharacteristic
200C3⋊D3C2×#21711yescharacteristic
201C3×D3—62no—
202C3×D3—62no—
203C3×D3—64no—
204C3×D3—124no—
205C24—91no—
206C22×C4—91no—
207C22×C4—94no—
208C22×C4—91no—
209C22×C4—184no—
210C22×C4—274no—
211C4⋊C4—544no—
212C22⋊C4—182no—
213C22⋊C4—542no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—24no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—44no—
220C22×C3—84no—
221C22×C3—91no—
222C22×C3—91no—
223C22×C3—92no—
224C22×C3—92no—
225C22×C3—92no—
226C22×C3—182no—
227C22×C3—182no—
228C22×C3—184no—
229C22×C3—182no—
230C22×C3#377312yescomplemented
231C22×C3C2×GO+(4,𝔽2)11yescharacteristic
232C2×D3—64no—
233C2×D3—64no—
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—64no—
237C2×D3—124no—
238C2×D3—128no—
239C2×D3—124no—
240C2×D3—124no—
241C3×C4—94no—
242C3×C4—182no—
243C3×C4—364no—
244Dic3—124no—
245Dic3—124no—
246Dic3—244no—
247C32—22no—
248C32—42no—
249C32#521211yescharacteristic, complemented
250C23—91no—
251C23—91no—
252C23—92no—
253C23—92no—
254C23—182no—
255C23—182no—
256C23#157⋊D311yescharacteristic
257C2×C4—94no—
258C2×C4—92no—
259C2×C4—92no—
260C2×C4—188no—
261C2×C4—182no—
262C2×C4—184no—
263C2×C4—544no—
264C2×C4—544no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—24no—
268C2×C3—44no—
269C2×C3—42no—
270C2×C3—44no—
271C2×C3—84no—
272C2×C3—91no—
273C2×C3—91no—
274C2×C3—92no—
275C2×C3—182no—
276C2×C3C2×#377311yescharacteristic
277C2×C3#6424212yes—
278D3—64no—
279D3—62no—
280D3—62no—
281D3—124no—
282C22—22no—
283C22—91no—
284C22—91no—
285C22—92no—
286C22—92no—
287C22—92no—
288C22—182no—
289C22—182no—
290C22—184no—
291C22—182no—
292C22#8894012yescomplemented
293C22C2×#157⋊D311yescharacteristic
294C4—94no—
295C4—182no—
296C4—364no—
297C3—22no—
298C3—42no—
299C3#1059351611yescharacteristic, complemented
300C2—22no—
301C2—91no—
302C2—91no—
303C2—92no—
304C2—182no—
305C2C2×#8894011yescharacteristic
306C2#1172435512yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#380
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88940C2semidirect
4. #11722791C2semidirect
8. C22×C3×#157C4semidirect
39. C22×#157Dic3semidirect
52. C22×C33C4⋊C4semidirect
58. C2×C3×C3⋊D3C4⋊C4semidirect
64. C3×#157C22⋊C4semidirect
129. C22×C32C4⋊Dic3semidirect
135. C2×C3⋊D3C4⋊Dic3semidirect
150. #157#217semidirect
159. C33#380semidirect
230. C22×C3#3773semidirect
249. C32#5212semidirect
292. C22#88940semidirect
299. C3#10593516semidirect

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×#889402
#117237232
#117227912
#105935163
#52129