Finite Group Databasebeta

← C2×#11723113 #13173443

Subgroups — C2×#11723113 #13173443

Order 1728.

1112 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×C3×#638C211yesmaximal, characteristic
2C2×#88993C212yesmaximal, complemented
3#11723113C214yesmaximal, direct product factor
4C2×#63788—31nomaximal
5C24×C33C411yescharacteristic
6C23×C3×C3⋊D3C411yescharacteristic
7C2×C3×#638—22no—
8C2×C3×#638—22no—
9C2×C3×#638C412yes—
10C2×C3×#638C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12#88993—22no—
13#88993C2214yescomplemented
14C2×C3⋊#622—22no—
15C22×#638D311yescharacteristic
16C22×C3×#67—62no—
17C2×#3792—32no—
18#63788—34no—
19C23×C33—21no—
20C23×C33—21no—
21C23×C33—42no—
22C23×C33C2×C411yescharacteristic
23C23×C33C2×C412yes—
24C22×C3×C3⋊D3—21no—
25C22×C3×C3⋊D3—21no—
26C22×C3×C3⋊D3—42no—
27C22×C3×C3⋊D3C2×C412yes—
28C3×#638—24no—
29C3×#638—24no—
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3D412yes—
32C2×C3×C3⋊Dic3D412yes—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3⋊#622—24no—
35C2×#833—91nomaximal
36C24×C32—22no—
37C24×C32—42no—
38C24×C32Dic311yescharacteristic
39C23×C3⋊D3Dic311yescharacteristic
40C23×C3×D3—62no—
41C2×#638—22no—
42C2×#638—22no—
43C2×#638Dic312yes—
44C2×#638C2×D312yes—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×#67—64no—
47C2×C3×#67—64no—
48C2×C3×#67—68no—
49C2×C3×#67—68no—
50C22×C3×Dic3—62no—
51#3792—34no—
52#3792—62no—
53C2×#622—62no—
54C22×C33—22no—
55C22×C33—22no—
56C22×C33—42no—
57C22×C33—42no—
58C22×C33—42no—
59C22×C33C22⋊C411yescharacteristic
60C22×C33C22⋊C411yescharacteristic
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—42no—
66C3×C3⋊Dic3C22⋊C412yes—
67C3×C3⋊Dic3C2×D412yes—
68C22×C3×D4—91no—
69C22×#67—62no—
70C2×#208—92no—
71#833—94no—
72C23×C32—21no—
73C23×C32—21no—
74C23×C32—24no—
75C23×C32—24no—
76C23×C32—22no—
77C23×C32—24no—
78C23×C32—44no—
79C23×C32—44no—
80C23×C32—44no—
81C23×C32—42no—
82C23×C32—42no—
83C23×C32—48no—
84C23×C32—416no—
85C23×C32C2×Dic311yescharacteristic
86C23×C32C2×Dic312yes—
87C22×C3⋊D3—21no—
88C22×C3⋊D3—21no—
89C22×C3⋊D3—42no—
90C22×C3⋊D3C2×Dic312yes—
91C22×C3×D3—64no—
92C22×C3×D3—64no—
93C22×C3×D3—64no—
94C22×C3×D3—128no—
95#638—24no—
96#638—24no—
97C2×C3⋊Dic3C2×Dic311yescharacteristic
98C2×C3⋊Dic3#6712yes—
99C2×C3⋊Dic3#6712yes—
100C2×C3⋊Dic3C22×D311yescharacteristic
101C3×#67—616no—
102C3×#67—616no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—62no—
107#622—64no—
108C2×#100—271noSylow
109C2×C33—21no—
110C2×C33—21no—
111C2×C33—42no—
112C2×C33#10012yescomplemented
113C2×C33C2×C22⋊C411yescharacteristic
114C3×C3⋊D3—42no—
115C24×C3—22no—
116C24×C3—42no—
117C24×C3—91no—
118C24×C3#15711yescharacteristic
119C23×D3—62no—
120C2×C3×D4—92no—
121C2×C3×D4—92no—
122C2×C3×D4—182no—
123C2×C3×D4—182no—
124C22×C3×C4—91no—
125C2×#67—64no—
126C2×#67—64no—
127C2×#67—68no—
128C2×#67—68no—
129C22×Dic3—62no—
130#208—94no—
131#208—182no—
132C2×C3⋊C8—182no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—28no—
136C22×C32—28no—
137C22×C32—22no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—48no—
141C22×C32—48no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32—42no—
145C22×C32—42no—
146C22×C32—48no—
147C22×C32—42no—
148C22×C32—48no—
149C22×C32—48no—
150C22×C32—416no—
151C22×C32—416no—
152C22×C32—416no—
153C22×C32—42no—
154C22×C32#21711yescharacteristic
155C22×C32#21711yescharacteristic
156C22×C32C22×Dic311yescharacteristic
157C2×C3⋊D3—22no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3—42no—
161C2×C3×D3—68no—
162C2×C3×D3—128no—
163C2×C3×D3—128no—
164C2×C3×D3—128no—
165C3⋊Dic3#21712yes—
166C3⋊Dic3C2×#6712yes—
167C3×Dic3—64no—
168C3×Dic3—64no—
169C22×D4—91no—
170C2×M16—272no—
171#100—274no—
172C33C2×#10011yescharacteristic, complemented, Sylow
173C23×C3—24no—
174C23×C3—24no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—24no—
178C23×C3—22no—
179C23×C3—44no—
180C23×C3—44no—
181C23×C3—42no—
182C23×C3—48no—
183C23×C3—416no—
184C23×C3—44no—
185C23×C3—42no—
186C23×C3—92no—
187C23×C3—181no—
188C23×C3—181no—
189C23×C3—362no—
190C23×C3C2×#15712yes—
191C23×C3C2×#15711yescharacteristic
192C22×D3—64no—
193C22×D3—64no—
194C22×D3—64no—
195C22×D3—128no—
196C3×D4—184no—
197C3×D4—184no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—92no—
201C2×C3×C4—92no—
202#67—616no—
203#67—616no—
204C2×Dic3—62no—
205C2×Dic3—64no—
206C2×Dic3—64no—
207C2×Dic3—62no—
208C3⋊C8—184no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—22no—
214C2×C32—24no—
215C2×C32—44no—
216C2×C32—44no—
217C2×C32—44no—
218C2×C32—42no—
219C2×C32—48no—
220C2×C32—416no—
221C2×C32—42no—
222C2×C32#83312yescomplemented
223C2×C32C2×#21711yescharacteristic
224C3⋊D3—42no—
225C3×D3—128no—
226C24—91no—
227C24C3⋊#15711yescharacteristic
228C2×D4—92no—
229C2×D4—92no—
230C2×D4—182no—
231C2×D4—182no—
232C22×C4—91no—
233M16—274no—
234M16—542no—
235C2×C8—542no—
236C22×C3—28no—
237C22×C3—22no—
238C22×C3—28no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—48no—
244C22×C3—48no—
245C22×C3—42no—
246C22×C3—48no—
247C22×C3—48no—
248C22×C3—42no—
249C22×C3—48no—
250C22×C3—416no—
251C22×C3—416no—
252C22×C3—416no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—182no—
258C22×C3—362no—
259C22×C3—362no—
260C22×C3—362no—
261C22×C3C22⋊#15711yescharacteristic
262C22×C3C22⋊#15711yescharacteristic
263C22×C3C22×#15711yescharacteristic
264C2×D3—68no—
265C2×D3—128no—
266C2×D3—128no—
267C2×D3—128no—
268C3×C4—92no—
269C3×C4—92no—
270Dic3—64no—
271Dic3—64no—
272C32—22no—
273C32—42no—
274C32C2×#83311yescharacteristic, complemented
275C23—21no—
276C23—21no—
277C23—42no—
278C23—92no—
279C23—181no—
280C23—181no—
281C23—362no—
282C23C2×C3⋊#15712yes—
283C23C2×C3⋊#15711yescharacteristic
284D4—184no—
285D4—184no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—92no—
289C2×C4—92no—
290C8—544no—
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—24no—
296C2×C3—22no—
297C2×C3—44no—
298C2×C3—44no—
299C2×C3—48no—
300C2×C3—416no—
301C2×C3—42no—
302C2×C3—44no—
303C2×C3—42no—
304C2×C3—362no—
305C2×C3#6378812yescomplemented
306C2×C3C2×C22⋊#15711yescharacteristic
307D3—128no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—182no—
314C22—362no—
315C22—362no—
316C22—362no—
317C22#8899411yescharacteristic
318C22#8899411yescharacteristic
319C22C22×C3⋊#15711yescharacteristic
320C4—92no—
321C4—92no—
322C3—22no—
323C3—42no—
324C3C2×#6378811yescharacteristic, complemented
325C2—21no—
326C2—21no—
327C2—42no—
328C2—362no—
329C2#1172311312yesdirect product factor
330C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC24×C33C4
socleC22×C33C22⋊C4
prime radicalC22×C3×#638C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC24C3⋊#157
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11723113C2direct (N × H)
13. #88993C22semidirect
112. C2×C33#100semidirect
172. C33C2×#100semidirect
222. C2×C32#833semidirect
274. C32C2×#833semidirect
305. C2×C3#63788semidirect
324. C3C2×#63788semidirect
329. C2#11723113direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×#889932
#117231132
C2×#637883
C2×#8339