Finite Group Databasebeta

← C2×#11724255 #13180964

Subgroups — C2×#11724255 #13180964

Order 1728.

1176 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 186 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×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4C2×#88925C212yesmaximal, complemented
5C2×#88924C212yesmaximal, complemented
6#11724255C218yesmaximal, direct product factor
7C2×#64283—31nomaximal
8C22×D3×C3⋊D3C411yescharacteristic
9C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3C411yescharacteristic
12C2×#6289C2211yescharacteristic
13C2×Dic3×C3⋊D3C411yescharacteristic
14C2×D3×C3⋊Dic3C2211yescharacteristic
15C4×D3×C3⋊D3C414yes—
16C4×D3×C3⋊D3C2214yes—
17C2×C3⋊#622C2212yescomplemented
18#88989C2214yescomplemented
19C2×C3×#622C2212yescomplemented
20C3×#3788C2214yescomplemented
21#88925C2218yescomplemented
22#88924C2218yescomplemented
23C22×C4×C3⋊D3—31no—
24C2×C4×D32—62no—
25C2×#3792—32no—
26#64283—38no—
27C2×#3788—31no—
28C2×#3788D311yescharacteristic, complemented
29C2×#3787—32no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C22×C32×D3C2×C411yescharacteristic
33C2×D3×C3⋊D3—24no—
34C2×D3×C3⋊D3C2×C414yes—
35C2×C33×C4C2×C411yescharacteristic
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C4×C3⋊(C3⋊D3)C2×C414yes—
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C3×C4×C3⋊D3C2×C412yes—
40C3×C4×C3⋊D3C2312yes—
41C2×C32×Dic3C2×C411yescharacteristic
42C32×C4×D3C2×C414yes—
43#6289C2×C412yes—
44#6289C2312yes—
45Dic3×C3⋊D3C2×C414yes—
46D3×C3⋊Dic3C2×C412yes—
47D3×C3⋊Dic3C2312yes—
48C3⋊#622C2314yescomplemented
49C3×#622C2314yescomplemented
50C2×D3×M16—91nomaximal
51C23×C3⋊D3—31no—
52C22×D32—62no—
53C22×C32×C4—31no—
54C22×C3⋊Dic3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—31no—
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3—62no—
61C2×C4×C3⋊D3—122no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3—62no—
65C2×#624—62no—
66C2×D3×Dic3—62no—
67C2×D3×Dic3—62no—
68C4×D32—68no—
69C4×D32—68no—
70#3792—38no—
71C2×#622—32no—
72C2×#622C2×D312yescomplemented
73#3788—34no—
74#3788C2×D314yescomplemented
75#3787—38no—
76C22×C33C22×C411yescharacteristic
77C2×C3⋊(C3⋊D3)M1614yescomplemented
78C2×C3⋊(C3⋊D3)C22×C412yes—
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3C22×C412yes—
81C2×C32×D3—22no—
82C2×C32×D3C22×C412yes—
83D3×C3⋊D3—28no—
84C33×C4C22×C412yes—
85C3⋊(C3⋊Dic3)C22×C412yes—
86C3×C3⋊Dic3C22×C411yescharacteristic
87C3×C3⋊Dic3C2411yescharacteristic
88C32×Dic3C22×C412yes—
89C22×C4×D3—91no—
90C22×C4×D3—182no—
91C2×C3×M16—91no—
92C2×#208—91no—
93D3×M16—98no—
94C2×ΓL(1,𝔽25)—92no—
95C2×D3×C8—92no—
96C23×C32—31no—
97C22×C3⋊D3—31no—
98C22×C3⋊D3—34no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—64no—
101C22×C3⋊D3—122no—
102C22×C3⋊D3C4×D311yescharacteristic
103C22×C3×D3—22no—
104C22×C3×D3—62no—
105C2×D32—68no—
106C2×D32—616no—
107C2×C32×C4—22no—
108C2×C32×C4—31no—
109C2×C32×C4—34no—
110C2×C32×C4—42no—
111C2×C32×C4C4×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—122no—
117C2×C3⋊Dic3C22×D311yescharacteristic
118C4×C3⋊D3—34no—
119C4×C3⋊D3—34no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—32no—
122C4×C3⋊D3—68no—
123C4×C3⋊D3—128no—
124C4×C3⋊D3C4×D312yes—
125C4×C3⋊D3C22×D312yes—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—62no—
128C3×C4×D3—28no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131#624—64no—
132#624—64no—
133D3×Dic3—64no—
134D3×Dic3—64no—
135D3×Dic3—68no—
136#622—34no—
137#622C22×D314yescomplemented
138C22×M16—271noSylow
139C2×C33C2×M1612yescomplemented
140C2×C33C23×C411yescharacteristic
141C3⋊(C3⋊D3)C2×M1614yescomplemented
142C3×C3⋊D3—22no—
143C32×D3—22no—
144C23×D3—91no—
145C23×D3—182no—
146C22×C3×C4—62no—
147C22×C3×C4—91no—
148C22×Dic3—91no—
149C22×Dic3—182no—
150C2×C4×D3—62no—
151C2×C4×D3—91no—
152C2×C4×D3—91no—
153C2×C4×D3—91no—
154C2×C4×D3—94no—
155C2×C4×D3—94no—
156C2×C4×D3—188no—
157C2×C4×D3—188no—
158C2×C4×D3—182no—
159C2×C4×D3—182no—
160C2×C4×D3—182no—
161C2×C4×D3—362no—
162C2×C4×D3#15711yescharacteristic
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—32no—
171C22×C32—42no—
172C22×C32—62no—
173C22×C32C2×C4×D311yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3—34no—
176C2×C3⋊D3—32no—
177C2×C3⋊D3—68no—
178C2×C3⋊D3—64no—
179C2×C3⋊D3—68no—
180C2×C3⋊D3—128no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3C2×C4×D312yes—
183C2×C3×D3—28no—
184C2×C3×D3—24no—
185C2×C3×D3—64no—
186C2×C3×D3—68no—
187D32—632no—
188C32×C4—24no—
189C32×C4—32no—
190C32×C4—44no—
191C32×C4C2×C4×D312yes—
192C3⋊Dic3—32no—
193C3⋊Dic3—64no—
194C3⋊Dic3—124no—
195C3⋊Dic3C2×C4×D311yescharacteristic
196C3⋊Dic3C23×D311yescharacteristic
197C3×Dic3—24no—
198C3×Dic3—62no—
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—62no—
208C23×C3—91no—
209C22×D3—62no—
210C22×D3—91no—
211C22×D3—94no—
212C22×D3—188no—
213C22×D3—182no—
214C22×D3—184no—
215C22×D3—1816no—
216C22×D3—362no—
217C22×D3C2×#15711yescharacteristic
218C2×C3×C4—22no—
219C2×C3×C4—42no—
220C2×C3×C4—68no—
221C2×C3×C4—62no—
222C2×C3×C4—91no—
223C2×C3×C4—92no—
224C2×C3×C4—92no—
225C2×C3×C4C2×#15711yescharacteristic
226C2×Dic3—62no—
227C2×Dic3—91no—
228C2×Dic3—94no—
229C2×Dic3—184no—
230C2×Dic3—184no—
231C2×Dic3—182no—
232C2×Dic3—362no—
233C2×Dic3C2×#15711yescharacteristic
234C4×D3—64no—
235C4×D3—64no—
236C4×D3—92no—
237C4×D3—92no—
238C4×D3—94no—
239C4×D3—92no—
240C4×D3—92no—
241C4×D3—184no—
242C4×D3—184no—
243C4×D3—188no—
244C4×D3—188no—
245C4×D3—368no—
246C4×D3C2×#15714yes—
247C3×C8—94no—
248C3⋊C8—94no—
249C2×C32—24no—
250C2×C32—22no—
251C2×C32—44no—
252C2×C32—42no—
253C2×C32—62no—
254C2×C32D3×M1612yescomplemented
255C2×C32C22×C4×D311yescharacteristic
256C3⋊D3—22no—
257C3⋊D3—34no—
258C3⋊D3—68no—
259C3⋊D3—128no—
260C3×D3—28no—
261C3×D3—68no—
262C24—271no—
263C22×C4—31no—
264C22×C4—91no—
265C22×C4—271no—
266C22×C4—274no—
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—22no—
278C22×C3—42no—
279C22×C3—64no—
280C22×C3—68no—
281C22×C3—92no—
282C22×C3—182no—
283C22×C3C22×#15711yescharacteristic
284C2×D3—22no—
285C2×D3—64no—
286C2×D3—68no—
287C2×D3—94no—
288C2×D3—92no—
289C2×D3—184no—
290C2×D3—188no—
291C2×D3—188no—
292C2×D3—1832no—
293C2×D3—364no—
294C2×D3—368no—
295C2×D3C22×#15712yes—
296C3×C4—24no—
297C3×C4—44no—
298C3×C4—64no—
299C3×C4—91no—
300C3×C4—91no—
301C3×C4C22×#15712yes—
302Dic3—62no—
303Dic3—62no—
304Dic3—92no—
305Dic3—184no—
306Dic3—364no—
307Dic3C22×#15712yes—
308C32—22no—
309C32—42no—
310C32C2×D3×M1611yescharacteristic, complemented
311C23—31no—
312C23—91no—
313C23—274no—
314C23—271no—
315C23—544no—
316C2×C4—31no—
317C2×C4—34no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—272no—
322C2×C4—272no—
323C2×C4—274no—
324C2×C4—272no—
325C2×C4—272no—
326C2×C4—271no—
327C2×C4—274no—
328C2×C4D3×#15711yescharacteristic
329C8—94no—
330C8—274no—
331C2×C3—22no—
332C2×C3—24no—
333C2×C3—42no—
334C2×C3—44no—
335C2×C3—68no—
336C2×C3—182no—
337C2×C3#6428312yescomplemented
338C2×C3C23×#15711yescharacteristic
339D3—22no—
340D3—68no—
341D3—94no—
342D3—188no—
343D3—368no—
344C22—32no—
345C22—62no—
346C22—92no—
347C22—182no—
348C22—272no—
349C22—274no—
350C22—548no—
351C22C2×D3×#15711yescharacteristic
352C4—32no—
353C4—91no—
354C4—91no—
355C4—272no—
356C4C2×D3×#15712yes—
357C3—22no—
358C3—42no—
359C3C2×#6428311yescharacteristic, complemented
360C2—62no—
361C2—182no—
362C2—274no—
363C2#1172425512yesdirect product factor
364C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C4
cosocleC3×C3⋊Dic3C24
hypercenterC2×C4D3×#157
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. C2×C3×#3788C2semidirect
4. C2×#88925C2semidirect
5. C2×#88924C2semidirect
6. #11724255C2direct (N × H)
17. C2×C3⋊#622C22semidirect
18. #88989C22semidirect
19. C2×C3×#622C22semidirect
20. C3×#3788C22semidirect
21. #88925C22semidirect
22. #88924C22semidirect
28. C2×#3788D3semidirect
48. C3⋊#622C23semidirect
49. C3×#622C23semidirect
72. C2×#622C2×D3semidirect
74. #3788C2×D3semidirect
77. C2×C3⋊(C3⋊D3)M16semidirect
137. #622C22×D3semidirect
139. C2×C33C2×M16semidirect
141. C3⋊(C3⋊D3)C2×M16semidirect
206. C33C22×M16semidirect
254. C2×C32D3×M16semidirect
310. C32C2×D3×M16semidirect
337. C2×C3#64283semidirect
359. C3C2×#64283semidirect
363. C2#11724255direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×#889892
C2×C3×#37882
C2×#889252
C2×#889242
#117242552
C2×#642833
C2×D3×M169