Finite Group Databasebeta

← C2×#11724269 #13181004

Subgroups — C2×#11724269 #13181004

Order 1728.

1336 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1336 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
1C22×#6289C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3C2×C3×#3792C211yesmaximal, characteristic, complemented
4C2×#88926C212yesmaximal, complemented
5#11724269C218yesmaximal, direct product factor
6C2×#88924C212yesmaximal, complemented
7C2×#64283—31nomaximal
8C23×C3⋊(C3⋊D3)C411yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C411yescharacteristic
11C2×#6289C411yescharacteristic
12C2×#6289C414yes—
13C2×#6289C411yescharacteristic
14C2×#6289C2211yescharacteristic
15C2×#6289C2214yes—
16C2×#6289C2211yescharacteristic
17#88993C2214yescomplemented
18C2×C3⋊#622C2212yescomplemented
19C3×#3792C2214yescomplemented
20C2×C3×#622C2212yescomplemented
21#88926C2218yescomplemented
22#88924C2218yescomplemented
23C22×C4×C3⋊D3—31no—
24C22×#624—62no—
25C2×#3792—31no—
26C2×#3792D311yescharacteristic, complemented
27#64283—38no—
28C2×#3788—32no—
29C2×#3787—32no—
30C23×C33C2×C411yescharacteristic
31C22×C3⋊(C3⋊D3)—24no—
32C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
33C22×C3⋊(C3⋊D3)C2×C414yes—
34C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C2×C3×C3⋊Dic3C2312yes—
39C2×C32×Dic3C2×C414yes—
40C2×C32×Dic3C2×C411yescharacteristic
41C2×C32×Dic3C2×C411yescharacteristic
42#6289C2×C414yes—
43#6289C2×C412yes—
44#6289C2×C412yes—
45#6289C2×C414yes—
46#6289C2312yes—
47#6289C2312yes—
48C3⋊#622C2314yescomplemented
49C3×#622C2314yescomplemented
50C2×D3×M16—91nomaximal
51C23×C3⋊D3—31no—
52C23×C3⋊D3—62no—
53C23×C3⋊D3—122no—
54C22×C32×C4—31no—
55C22×C3⋊Dic3C2×D311yescharacteristic
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3—31no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—34no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—62no—
64C2×#624—62no—
65C2×#624—68no—
66C2×#624—62no—
67C2×#624—62no—
68C2×#624—62no—
69C2×#624—68no—
70#3792—34no—
71#3792C2×D314yescomplemented
72C2×#622—32no—
73C2×#622C2×D312yescomplemented
74#3788—38no—
75#3787—38no—
76C22×C33—22no—
77C22×C33C22×C412yes—
78C22×C33C22×C411yescharacteristic
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)—28no—
81C2×C3⋊(C3⋊D3)M1614yescomplemented
82C2×C3⋊(C3⋊D3)C22×C412yes—
83C2×C3⋊(C3⋊D3)C22×C412yes—
84C3×C3⋊Dic3C22×C411yescharacteristic
85C3×C3⋊Dic3C22×C412yes—
86C3×C3⋊Dic3C2411yescharacteristic
87C32×Dic3C22×C412yes—
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—22no—
97C23×C32—42no—
98C23×C32C4×D311yescharacteristic
99C22×C3⋊D3—31no—
100C22×C3⋊D3—34no—
101C22×C3⋊D3—31no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—68no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—64no—
106C22×C3⋊D3—616no—
107C22×C3⋊D3—1216no—
108C22×C3⋊D3—128no—
109C22×C3⋊D3—122no—
110C22×C3⋊D3—122no—
111C2×C32×C4—34no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C3⋊Dic3C4×D312yes—
115C2×C3⋊Dic3C4×D311yescharacteristic
116C2×C3⋊Dic3C22×D311yescharacteristic
117C2×C3⋊Dic3C22×D312yes—
118C4×C3⋊D3—32no—
119C4×C3⋊D3—34no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—34no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C2×C3×Dic3—28no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—64no—
131#624—64no—
132#624—68no—
133#624—64no—
134#624—68no—
135#624—64no—
136#624—64no—
137#622—34no—
138#622C22×D314yescomplemented
139C22×M16—271noSylow
140C2×C33—22no—
141C2×C33C2×M1612yescomplemented
142C2×C33C23×C411yescharacteristic
143C3⋊(C3⋊D3)—22no—
144C3⋊(C3⋊D3)C2×M1614yescomplemented
145C23×D3—91no—
146C23×D3—182no—
147C23×D3—362no—
148C22×C3×C4—62no—
149C22×C3×C4—91no—
150C22×Dic3—62no—
151C22×Dic3#15711yescharacteristic
152C2×C4×D3—91no—
153C2×C4×D3—91no—
154C2×C4×D3—94no—
155C2×C4×D3—91no—
156C2×C4×D3—94no—
157C2×C4×D3—91no—
158C2×C4×D3—182no—
159C2×C4×D3—182no—
160C2×C4×D3—188no—
161C2×C4×D3—188no—
162C2×C4×D3—182no—
163C2×C4×D3—182no—
164C3×M16—94no—
165C2×C3×C8—92no—
166#208—94no—
167C2×C3⋊C8—92no—
168ΓL(1,𝔽25)—98no—
169D3×C8—98no—
170C22×C32—22no—
171C22×C32—28no—
172C22×C32—24no—
173C22×C32—22no—
174C22×C32—48no—
175C22×C32—44no—
176C22×C32—42no—
177C22×C32C2×C4×D312yes—
178C22×C32C2×C4×D311yescharacteristic
179C2×C3⋊D3—34no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—32no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—64no—
185C2×C3⋊D3—68no—
186C2×C3⋊D3—62no—
187C2×C3⋊D3—632no—
188C2×C3⋊D3—68no—
189C2×C3⋊D3—128no—
190C2×C3⋊D3—1232no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—124no—
193C2×C3⋊D3—124no—
194C32×C4—32no—
195C32×C4—32no—
196C3⋊Dic3C2×C4×D311yescharacteristic
197C3⋊Dic3C2×C4×D312yes—
198C3⋊Dic3C23×D311yescharacteristic
199C3×Dic3—24no—
200C3×Dic3—24no—
201C3×Dic3—62no—
202C3×Dic3—64no—
203C3×Dic3—62no—
204C23×C4—271no—
205C2×M16—91no—
206C2×M16—272no—
207C2×M16—278no—
208C2×M16—271no—
209C22×C8—272no—
210C33C22×M1611yescharacteristic, complemented, Sylow
211C23×C3—22no—
212C23×C3—42no—
213C23×C3C2×#15711yescharacteristic
214C22×D3—91no—
215C22×D3—94no—
216C22×D3—91no—
217C22×D3—182no—
218C22×D3—188no—
219C22×D3—182no—
220C22×D3—184no—
221C22×D3—1816no—
222C22×D3—362no—
223C22×D3—368no—
224C22×D3—362no—
225C22×D3—3616no—
226C2×C3×C4—68no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—91no—
230C2×C3×C4—92no—
231C2×C3×C4—92no—
232C2×C3×C4—91no—
233C2×Dic3—64no—
234C2×Dic3—62no—
235C2×Dic3—64no—
236C2×Dic3—62no—
237C2×Dic3C2×#15714yes—
238C2×Dic3C2×#15711yescharacteristic
239C2×Dic3C2×#15711yescharacteristic
240C4×D3—92no—
241C4×D3—92no—
242C4×D3—92no—
243C4×D3—94no—
244C4×D3—92no—
245C4×D3—94no—
246C4×D3—184no—
247C4×D3—184no—
248C4×D3—184no—
249C4×D3—188no—
250C4×D3—184no—
251C4×D3—188no—
252C3×C8—94no—
253C3⋊C8—94no—
254C2×C32—22no—
255C2×C32—28no—
256C2×C32—24no—
257C2×C32—22no—
258C2×C32—48no—
259C2×C32—44no—
260C2×C32—42no—
261C2×C32D3×M1612yescomplemented
262C2×C32C22×C4×D311yescharacteristic
263C3⋊D3—34no—
264C3⋊D3—68no—
265C3⋊D3—62no—
266C3⋊D3—68no—
267C3⋊D3—128no—
268C3⋊D3—128no—
269C24—271no—
270C22×C4—31no—
271C22×C4—91no—
272C22×C4—271no—
273C22×C4—274no—
274C22×C4—271no—
275C22×C4—274no—
276C22×C4—271no—
277C22×C4—271no—
278M16—94no—
279M16—278no—
280M16—274no—
281C2×C8—92no—
282C2×C8—278no—
283C2×C8—272no—
284C22×C3—28no—
285C22×C3—22no—
286C22×C3—24no—
287C22×C3—22no—
288C22×C3—48no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3C22×#15712yes—
292C22×C3C22×#15711yescharacteristic
293C2×D3—94no—
294C2×D3—92no—
295C2×D3—92no—
296C2×D3—184no—
297C2×D3—184no—
298C2×D3—188no—
299C2×D3—188no—
300C2×D3—1832no—
301C2×D3—188no—
302C2×D3—182no—
303C2×D3—364no—
304C2×D3—368no—
305C2×D3—364no—
306C2×D3—3632no—
307C2×D3—368no—
308C3×C4—64no—
309C3×C4—64no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—92no—
313Dic3—64no—
314Dic3—62no—
315Dic3—62no—
316Dic3C22×#15712yes—
317Dic3C22×#15712yes—
318C32—22no—
319C32—42no—
320C32C2×D3×M1611yescharacteristic, complemented
321C23—271no—
322C23—274no—
323C23—271no—
324C23—544no—
325C23D3×#15711yescharacteristic
326C2×C4—34no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—92no—
330C2×C4—92no—
331C2×C4—91no—
332C2×C4—91no—
333C2×C4—272no—
334C2×C4—272no—
335C2×C4—272no—
336C2×C4—274no—
337C2×C4—274no—
338C2×C4—272no—
339C8—94no—
340C8—274no—
341C2×C3—28no—
342C2×C3—22no—
343C2×C3—22no—
344C2×C3—24no—
345C2×C3—48no—
346C2×C3—42no—
347C2×C3—44no—
348C2×C3#6428312yescomplemented
349C2×C3C23×#15711yescharacteristic
350D3—94no—
351D3—188no—
352D3—188no—
353D3—182no—
354D3—368no—
355D3—368no—
356C22—22no—
357C22—272no—
358C22—274no—
359C22—272no—
360C22—548no—
361C22—542no—
362C22C2×D3×#15712yes—
363C22C2×D3×#15711yescharacteristic
364C4—32no—
365C4—32no—
366C4—91no—
367C4—92no—
368C4—91no—
369C3—22no—
370C3—42no—
371C3C2×#6428311yescharacteristic, complemented
372C2—22no—
373C2—274no—
374C2—542no—
375C2#1172426912yesdirect product factor
376C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC23×C3⋊(C3⋊D3)C4
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×#88926C2semidirect
5. #11724269C2direct (N × H)
6. C2×#88924C2semidirect
17. #88993C22semidirect
18. C2×C3⋊#622C22semidirect
19. C3×#3792C22semidirect
20. C2×C3×#622C22semidirect
21. #88926C22semidirect
22. #88924C22semidirect
26. C2×#3792D3semidirect
48. C3⋊#622C23semidirect
49. C3×#622C23semidirect
71. #3792C2×D3semidirect
73. C2×#622C2×D3semidirect
81. C2×C3⋊(C3⋊D3)M16semidirect
138. #622C22×D3semidirect
141. C2×C33C2×M16semidirect
144. C3⋊(C3⋊D3)C2×M16semidirect
210. C33C22×M16semidirect
261. C2×C32D3×M16semidirect
320. C32C2×D3×M16semidirect
348. C2×C3#64283semidirect
371. C3C2×#64283semidirect
375. C2#11724269direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#889932
C2×C3×#37922
C2×#889262
#117242692
C2×#889242
C2×#642833
C2×D3×M169