Finite Group Databasebeta

← C2×D3×#3792 #13181000

Subgroups — C2×D3×#3792 #13181000

Order 1728.

1176 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 194 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×D3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3C2×C3×#3792C211yesmaximal, characteristic, complemented
4C2×#88925C212yesmaximal, complemented
5D3×#3792C218yesmaximal, direct product factor
6C2×D3×#622C212yesmaximal, complemented
7C22×#3792—31nomaximal
8C23×C32×D3C411yescharacteristic
9C22×C3⋊(C3⋊Dic3)C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×D3×C3⋊Dic3C411yescharacteristic
12C2×D3×C3⋊Dic3C414yes—
13C2×D3×C3⋊Dic3C411yescharacteristic
14C2×D3×C3⋊Dic3C2211yescharacteristic
15C2×D3×C3⋊Dic3C2214yes—
16C2×D3×C3⋊Dic3C2211yescharacteristic
17#88993C2214yescomplemented
18C2×C3⋊#622C2212yescomplemented
19C3×#3792C2214yescomplemented
20C2×C3×#622C2212yescomplemented
21#88925C2218yescomplemented
22D3×#622C2218yescomplemented
23C23×C3⋊Dic3—31no—
24C22×D3×Dic3—62no—
25C2×#3792—31no—
26C2×#3792—38no—
27C2×#3792—32no—
28C2×#3792D311yescharacteristic, direct product factor
29C22×#622—32no—
30C23×C33C2×C411yescharacteristic
31C22×C32×D3—24no—
32C22×C32×D3C2×C411yescharacteristic
33C22×C32×D3C2×C414yes—
34C22×C32×D3C2×C411yescharacteristic
35C2×C3⋊(C3⋊Dic3)C2×C414yes—
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2311yescharacteristic
41C2×C3×C3⋊Dic3C2312yes—
42D3×C3⋊Dic3C2×C414yes—
43D3×C3⋊Dic3C2×C412yes—
44D3×C3⋊Dic3C2×C412yes—
45D3×C3⋊Dic3C2×C414yes—
46D3×C3⋊Dic3C2312yes—
47D3×C3⋊Dic3C2312yes—
48C3⋊#622C2314yescomplemented
49C3×#622C2314yescomplemented
50C2×D3×M16—91nomaximal
51C24×C32—31no—
52C23×C3×D3—22no—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—31no—
55C22×C3⋊Dic3—34no—
56C22×C3⋊Dic3—31no—
57C22×C3⋊Dic3—31no—
58C22×C3⋊Dic3—34no—
59C22×C3⋊Dic3—31no—
60C22×C3⋊Dic3—62no—
61C22×C3⋊Dic3—122no—
62C22×C3⋊Dic3C2×D311yescharacteristic
63C22×C3×Dic3—62no—
64C2×D3×Dic3—68no—
65C2×D3×Dic3—62no—
66C2×D3×Dic3—62no—
67C2×D3×Dic3—62no—
68C2×D3×Dic3—62no—
69C2×D3×Dic3—68no—
70#3792—34no—
71#3792—38no—
72#3792C2×D314yesdirect product factor
73C2×#622—32no—
74C2×#622—38no—
75C2×#622C2×D312yescomplemented
76C22×C33—22no—
77C22×C33C22×C412yes—
78C22×C33C22×C411yescharacteristic
79C2×C32×D3—22no—
80C2×C32×D3—28no—
81C2×C32×D3M1614yescomplemented
82C2×C32×D3C22×C412yes—
83C2×C32×D3C22×C412yes—
84C3⋊(C3⋊Dic3)C22×C412yes—
85C3⋊(C3⋊Dic3)C22×C412yes—
86C3×C3⋊Dic3C22×C411yescharacteristic
87C3×C3⋊Dic3C22×C412yes—
88C3×C3⋊Dic3C2411yescharacteristic
89C23×Dic3—182no—
90C22×C4×D3—91no—
91C2×C3×M16—91no—
92C2×#208—91no—
93D3×M16—98no—
94C2×ΓL(1,𝔽25)—92no—
95C2×D3×C8—92no—
96C23×C32—22no—
97C23×C32—31no—
98C23×C32—34no—
99C23×C32—31no—
100C23×C32—42no—
101C23×C32—64no—
102C23×C32C4×D311yescharacteristic
103C22×C3×D3—216no—
104C22×C3×D3—28no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—32no—
116C2×C3⋊Dic3—68no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—128no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3—122no—
122C2×C3⋊Dic3C4×D312yes—
123C2×C3⋊Dic3C4×D311yescharacteristic
124C2×C3⋊Dic3C22×D311yescharacteristic
125C2×C3⋊Dic3C22×D312yes—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—64no—
130D3×Dic3—64no—
131D3×Dic3—68no—
132D3×Dic3—64no—
133D3×Dic3—68no—
134D3×Dic3—64no—
135D3×Dic3—64no—
136#622—34no—
137#622C22×D314yescomplemented
138C22×M16—271noSylow
139C2×C33—22no—
140C2×C33C2×M1612yescomplemented
141C2×C33C23×C411yescharacteristic
142C32×D3—22no—
143C32×D3C2×M1614yescomplemented
144C24×C3—62no—
145C23×D3#15711yescharacteristic
146C22×C3×C4—91no—
147C22×Dic3—62no—
148C22×Dic3—91no—
149C22×Dic3—182no—
150C22×Dic3—188no—
151C22×Dic3—182no—
152C22×Dic3—188no—
153C22×Dic3—182no—
154C22×Dic3—182no—
155C22×Dic3—182no—
156C22×Dic3—362no—
157C2×C4×D3—91no—
158C2×C4×D3—91no—
159C2×C4×D3—94no—
160C2×C4×D3—94no—
161C2×C4×D3—91no—
162C2×C4×D3—91no—
163C3×M16—94no—
164C2×C3×C8—92no—
165#208—94no—
166C2×C3⋊C8—92no—
167ΓL(1,𝔽25)—98no—
168D3×C8—98no—
169C22×C32—22no—
170C22×C32—28no—
171C22×C32—24no—
172C22×C32—22no—
173C22×C32—34no—
174C22×C32—32no—
175C22×C32—32no—
176C22×C32—48no—
177C22×C32—44no—
178C22×C32—42no—
179C22×C32—68no—
180C22×C32—62no—
181C22×C32C2×C4×D312yes—
182C22×C32C2×C4×D311yescharacteristic
183C2×C3×D3—28no—
184C2×C3×D3—232no—
185C2×C3×D3—24no—
186C2×C3×D3—24no—
187C2×C3×D3—28no—
188C3⋊Dic3—32no—
189C3⋊Dic3—32no—
190C3⋊Dic3—64no—
191C3⋊Dic3—64no—
192C3⋊Dic3—124no—
193C3⋊Dic3—124no—
194C3⋊Dic3C2×C4×D311yescharacteristic
195C3⋊Dic3C2×C4×D312yes—
196C3⋊Dic3C23×D311yescharacteristic
197C3×Dic3—62no—
198C3×Dic3—64no—
199C3×Dic3—62no—
200C23×C4—271no—
201C2×M16—91no—
202C2×M16—272no—
203C2×M16—278no—
204C2×M16—271no—
205C22×C8—272no—
206C33C22×M1611yescharacteristic, complemented, Sylow
207C23×C3—22no—
208C23×C3—42no—
209C23×C3—62no—
210C23×C3—68no—
211C23×C3—62no—
212C23×C3—616no—
213C23×C3C2×#15711yescharacteristic
214C22×D3—24no—
215C22×D3C2×#15714yes—
216C22×D3C2×#15711yescharacteristic
217C22×D3C2×#15711yescharacteristic
218C2×C3×C4—91no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4—91no—
222C2×Dic3—64no—
223C2×Dic3—62no—
224C2×Dic3—64no—
225C2×Dic3—62no—
226C2×Dic3—94no—
227C2×Dic3—91no—
228C2×Dic3—91no—
229C2×Dic3—184no—
230C2×Dic3—184no—
231C2×Dic3—184no—
232C2×Dic3—188no—
233C2×Dic3—188no—
234C2×Dic3—184no—
235C2×Dic3—188no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238C2×Dic3—368no—
239C2×Dic3—362no—
240C2×Dic3—362no—
241C4×D3—92no—
242C4×D3—92no—
243C4×D3—92no—
244C4×D3—94no—
245C4×D3—92no—
246C4×D3—94no—
247C3×C8—94no—
248C3⋊C8—94no—
249C2×C32—22no—
250C2×C32—28no—
251C2×C32—24no—
252C2×C32—22no—
253C2×C32—34no—
254C2×C32—48no—
255C2×C32—44no—
256C2×C32—42no—
257C2×C32—62no—
258C2×C32D3×M1612yescomplemented
259C2×C32C22×C4×D311yescharacteristic
260C3×D3—28no—
261C3×D3—28no—
262C24—31no—
263C22×C4—91no—
264C22×C4—271no—
265C22×C4—274no—
266C22×C4—271no—
267C22×C4—274no—
268C22×C4—271no—
269C22×C4—271no—
270C22×C4—271no—
271M16—94no—
272M16—278no—
273M16—274no—
274C2×C8—92no—
275C2×C8—278no—
276C2×C8—272no—
277C22×C3—28no—
278C22×C3—22no—
279C22×C3—24no—
280C22×C3—22no—
281C22×C3—48no—
282C22×C3—44no—
283C22×C3—42no—
284C22×C3—64no—
285C22×C3—64no—
286C22×C3—68no—
287C22×C3—632no—
288C22×C3—68no—
289C22×C3C22×#15712yes—
290C22×C3C22×#15711yescharacteristic
291C2×D3—22no—
292C2×D3—28no—
293C2×D3#379214yesdirect product factor
294C2×D3C22×#15712yes—
295C2×D3C22×#15712yes—
296C3×C4—91no—
297C3×C4—91no—
298C3×C4—92no—
299Dic3—64no—
300Dic3—62no—
301Dic3—62no—
302Dic3—92no—
303Dic3—92no—
304Dic3—184no—
305Dic3—184no—
306Dic3—364no—
307Dic3—364no—
308C32—22no—
309C32—42no—
310C32C2×D3×M1611yescharacteristic, complemented
311C23—31no—
312C23—34no—
313C23—31no—
314C23—64no—
315C23D3×#15711yescharacteristic
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—272no—
321C2×C4—272no—
322C2×C4—272no—
323C2×C4—274no—
324C2×C4—274no—
325C2×C4—272no—
326C2×C4—274no—
327C2×C4—271no—
328C2×C4—271no—
329C8—94no—
330C8—274no—
331C2×C3—28no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—24no—
335C2×C3—48no—
336C2×C3—42no—
337C2×C3—44no—
338C2×C3—68no—
339C2×C3—68no—
340C2×C3C2×#379212yescomplemented
341C2×C3C23×#15711yescharacteristic
342D3—22no—
343D3C2×#379214yesdirect product factor
344C22—22no—
345C22—32no—
346C22—34no—
347C22—32no—
348C22—68no—
349C22—62no—
350C22C2×D3×#15712yes—
351C22C2×D3×#15711yescharacteristic
352C4—91no—
353C4—92no—
354C4—91no—
355C4—272no—
356C4—272no—
357C3—22no—
358C3—42no—
359C3C22×#379211yescharacteristic, complemented
360C2—22no—
361C2—34no—
362C2—62no—
363C2D3×#379212yesdirect product factor
364C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC23×C32×D3C4
cosocleC3×C3⋊Dic3C24
hypercenterC23D3×#157
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. C2×C3×#3792C2semidirect
4. C2×#88925C2semidirect
5. D3×#3792C2direct (N × H)
6. C2×D3×#622C2semidirect
17. #88993C22semidirect
18. C2×C3⋊#622C22semidirect
19. C3×#3792C22semidirect
20. C2×C3×#622C22semidirect
21. #88925C22semidirect
22. D3×#622C22semidirect
28. C2×#3792D3direct (N × H)
48. C3⋊#622C23semidirect
49. C3×#622C23semidirect
72. #3792C2×D3direct (N × H)
75. C2×#622C2×D3semidirect
81. C2×C32×D3M16semidirect
137. #622C22×D3semidirect
140. C2×C33C2×M16semidirect
143. C32×D3C2×M16semidirect
206. C33C22×M16semidirect
258. C2×C32D3×M16semidirect
293. C2×D3#3792direct (N × H)
310. C32C2×D3×M16semidirect
340. C2×C3C2×#3792semidirect
343. D3C2×#3792direct (N × H)
359. C3C22×#3792semidirect
363. C2D3×#3792direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C2×#889932
C2×C3×#37922
C2×#889252
D3×#37922
C2×D3×#6222
C22×#37923
C2×D3×M169