Finite Group Databasebeta

← C2×#11724214 #13180875

Subgroups — C2×#11724214 #13180875

Order 1728.

1146 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 86 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×C3⋊S4C212yesmaximal, complemented
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11724214C214yesmaximal, direct product factor
4C2×#64207—32nomaximal
5C2×#63980—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3×C3⋊S4—24no—
8C2×C3×C3⋊S4C2214yescomplemented
9C2×#6295—41nomaximal
10A4×C3⋊Dic3C2212yescomplemented
11C22×C3⋊S4D312yescomplemented
12C22×C3×S4—32no—
13C22×C3×S4—62no—
14C23×C3⋊Dic3D311yescharacteristic, complemented
15C22×C3×#67—32no—
16C22×#625—31no—
17C2×Dic3×A4—32no—
18C2×Dic3×A4—61no—
19#64207—38no—
20#63980—34no—
21#63980—34no—
22C2×#3724—31no—
23C2×(C2×D3)⋊Dic3—32no—
24C22×C3×C3⋊D3—42no—
25C2×C32×A4D412yescomplemented
26C2×C32×A4C2311yescharacteristic
27C3×C3⋊S4—24no—
28C2×C3×C3⋊Dic3—41no—
29#6295—44no—
30C2×C4⋊S4—91no—
31C2×#939—92no—
32C24×C32C2×D311yescharacteristic
33C23×C3×D3—32no—
34C22×C3×A4—22no—
35C22×C3×A4—21no—
36C22×C3×A4—41no—
37C22×C3×A4C2×D312yes—
38C2×C3⋊S4—24no—
39C2×C3⋊S4C2×D314yescomplemented
40C2×C3×S4—34no—
41C2×C3×S4—64no—
42C2×C3×S4—68no—
43C2×C3×S4—64no—
44C22×C3⋊Dic3—31no—
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—32no—
47C22×C3⋊Dic3C2×D312yescomplemented
48C2×C3×#67—34no—
49C2×C3×#67—34no—
50C2×C3×#67—64no—
51C2×C3×#67—64no—
52C22×C3×Dic3—32no—
53C2×#626—122no—
54C2×#625—34no—
55C2×#625—61no—
56C2×#625—62no—
57C2×#625—61no—
58Dic3×A4—34no—
59Dic3×A4—62no—
60#3724—34no—
61(C2×D3)⋊Dic3—38no—
62C22×C33—41no—
63C2×C3×C3⋊D3—44no—
64C2×C3×C3⋊D3—84no—
65C32×A4C2×D411yescharacteristic, complemented
66C3×C3⋊Dic3—42no—
67C22×S4—32no—
68C22×S4—62no—
69C22×C3×D4—92no—
70C22×#67—32no—
71C22×#67—92no—
72C23×Dic3—32no—
73C23×Dic3—61no—
74C2×C4×A4—91no—
75C4⋊S4—94no—
76C2×#217—92no—
77#939—98no—
78#939—98no—
79C2×(C2×D3)⋊C4—92no—
80C2×Dic3⋊C4—92no—
81C23×C32—31no—
82C23×C32—31no—
83C23×C32—32no—
84C23×C32D1212yescomplemented
85C23×C32C22×D311yescharacteristic
86C22×C3⋊D3—42no—
87C22×C3×D3—34no—
88C22×C3×D3—64no—
89C22×C3×D3—62no—
90C22×C3×D3—64no—
91C22×C3×D3—62no—
92C22×C3×D3—122no—
93C22×C3×D3—242no—
94C2×C3×A4—24no—
95C2×C3×A4—22no—
96C2×C3×A4—22no—
97C2×C3×A4—21no—
98C2×C3×A4—42no—
99C2×C3×A4—41no—
100C2×C3×A4#6714yescomplemented
101C2×C3×A4C22×D312yes—
102C3⋊S4—24no—
103C3×S4—68no—
104C3×S4—64no—
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—32no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3S411yescharacteristic, complemented
111C3×#67—68no—
112C3×#67—64no—
113C3×#67—64no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—122no—
117C2×C3×Dic3—241no—
118#626—128no—
119#625—64no—
120#625—64no—
121C2×#121—271noSylow
122C2×C33—42no—
123C2×C33—41no—
124C3×C3⋊D3—84no—
125C24×C3—21no—
126C24×C3—92no—
127C24×C3D3212yes—
128C23×D3—32no—
129C22×A4—22no—
130C22×A4—41no—
131C22×A4D3211yescharacteristic
132C2×S4—34no—
133C2×S4—64no—
134C2×S4—68no—
135C2×S4—64no—
136C2×C3×D4—94no—
137C2×C3×D4—94no—
138C2×C3×D4—184no—
139C2×C3×D4—184no—
140C22×C3×C4—92no—
141C2×#67—34no—
142C2×#67—34no—
143C2×#67—64no—
144C2×#67—64no—
145C2×#67—98no—
146C2×#67—182no—
147C2×#67—184no—
148C2×#67—182no—
149C22×Dic3—32no—
150C22×Dic3—34no—
151C22×Dic3—62no—
152C22×Dic3—92no—
153C22×Dic3—92no—
154C22×Dic3—94no—
155C22×Dic3—181no—
156C22×Dic3—181no—
157C22×Dic3—182no—
158C2×D12—361no—
159C4×A4—92no—
160#217—98no—
161(C2×D3)⋊C4—98no—
162Dic3⋊C4—98no—
163C22×C32—32no—
164C22×C32—31no—
165C22×C32—32no—
166C22×C32—32no—
167C22×C32—31no—
168C22×C32—31no—
169C22×C32—42no—
170C22×C32—61no—
171C22×C32—81no—
172C22×C32—82no—
173C22×C32—161no—
174C22×C32C2×D1211yescharacteristic, complemented
175C22×C32C2×S411yescharacteristic
176C2×C3⋊D3—44no—
177C2×C3⋊D3—84no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—64no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C2×C3×D3—124no—
184C2×C3×D3—244no—
185C2×C3×D3—244no—
186C2×C3×D3—248no—
187C3×A4—22no—
188C3×A4—21no—
189C3×A4—41no—
190C3×A4C2×#6712yescomplemented
191C3⋊Dic3—32no—
192C3⋊Dic3C2×S412yescomplemented
193C3×Dic3—64no—
194C3×Dic3—124no—
195C3×Dic3—242no—
196C22×D4—92no—
197C22×D4—271no—
198C23×C4—91no—
199#121—274no—
200#121—274no—
201C2×C4⋊C4—271no—
202C2×C22⋊C4—272no—
203C33—41noSylow
204C23×C3—22no—
205C23×C3—21no—
206C23×C3—32no—
207C23×C3—32no—
208C23×C3—34no—
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—62no—
212C23×C3—94no—
213C23×C3—182no—
214C23×C3—184no—
215C23×C3—184no—
216C23×C3—182no—
217C23×C3#62614yescomplemented
218C23×C3C2×D3212yes—
219C22×D3—34no—
220C22×D3—64no—
221C22×D3—62no—
222C22×D3—64no—
223C22×D3—62no—
224C22×D3—122no—
225C22×D3—242no—
226C2×A4—24no—
227C2×A4—22no—
228C2×A4—42no—
229C2×A4—41no—
230C2×A4#62512yescomplemented
231C2×A4C2×D3211yescharacteristic
232S4—68no—
233S4—64no—
234C3×D4—188no—
235C3×D4—184no—
236C3×D4—184no—
237C2×C3×C4—94no—
238C2×C3×C4—184no—
239C2×C3×C4—361no—
240#67—68no—
241#67—64no—
242#67—64no—
243#67—188no—
244#67—188no—
245C2×Dic3—32no—
246C2×Dic3—34no—
247C2×Dic3—61no—
248C2×Dic3—64no—
249C2×Dic3—94no—
250C2×Dic3—94no—
251C2×Dic3—92no—
252C2×Dic3—94no—
253C2×Dic3—182no—
254C2×Dic3—182no—
255C2×Dic3—181no—
256C2×Dic3—182no—
257C2×Dic3—182no—
258C2×Dic3—182no—
259D12—364no—
260C2×C32—31no—
261C2×C32—31no—
262C2×C32—32no—
263C2×C32—42no—
264C2×C32—44no—
265C2×C32—81no—
266C2×C32—82no—
267C2×C32—82no—
268C2×C32—84no—
269C2×C32—162no—
270C2×C32—161no—
271C2×C32C4⋊S412yescomplemented
272C2×C32C22×S411yescharacteristic
273C3⋊D3—84no—
274C3×D3—124no—
275C3×D3—244no—
276C3×D3—248no—
277C24—92no—
278C24#111311yescharacteristic
279C2×D4—94no—
280C2×D4—94no—
281C2×D4—184no—
282C2×D4—184no—
283C2×D4—274no—
284C2×D4—541no—
285C2×D4—542no—
286C2×D4—541no—
287C22×C4—92no—
288C22×C4—92no—
289C22×C4—271no—
290C22×C4—271no—
291C22×C4—272no—
292C4⋊C4—274no—
293C22⋊C4—278no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—34no—
297C22×C3—32no—
298C22×C3—32no—
299C22×C3—32no—
300C22×C3—34no—
301C22×C3—34no—
302C22×C3—41no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—62no—
310C22×C3—62no—
311C22×C3—82no—
312C22×C3—161no—
313C22×C3—184no—
314C22×C3—184no—
315C22×C3—184no—
316C22×C3—364no—
317C22×C3—364no—
318C22×C3C2×#62612yescomplemented
319C22×C3D3×S412yes—
320C2×D3—64no—
321C2×D3—64no—
322C2×D3—64no—
323C2×D3—124no—
324C2×D3—124no—
325C2×D3—124no—
326C2×D3—244no—
327C2×D3—244no—
328C2×D3—248no—
329A4—22no—
330A4—41no—
331A4C2×#62511yescharacteristic, complemented
332C3×C4—184no—
333C3×C4—362no—
334Dic3—34no—
335Dic3—62no—
336Dic3—64no—
337Dic3—94no—
338Dic3—182no—
339C32—42no—
340C32—81no—
341C32—82no—
342C32—161no—
343C32C2×C4⋊S411yescharacteristic, complemented
344C23—31no—
345C23—31no—
346C23—32no—
347C23—94no—
348C23—184no—
349C23—182no—
350C23—184no—
351C23—182no—
352C23#629512yescomplemented
353C23C2×#111311yescharacteristic
354D4—188no—
355D4—184no—
356D4—184no—
357D4—544no—
358D4—544no—
359C2×C4—91no—
360C2×C4—94no—
361C2×C4—184no—
362C2×C4—272no—
363C2×C4—272no—
364C2×C4—271no—
365C2×C4—272no—
366C2×C4—541no—
367C2×C3—22no—
368C2×C3—21no—
369C2×C3—32no—
370C2×C3—32no—
371C2×C3—34no—
372C2×C3—42no—
373C2×C3—41no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—62no—
377C2×C3—84no—
378C2×C3—82no—
379C2×C3—162no—
380C2×C3—161no—
381C2×C3—364no—
382C2×C3#6420714yescomplemented
383C2×C3C2×D3×S412yes—
384D3—124no—
385D3—244no—
386D3—248no—
387C22—32no—
388C22—31no—
389C22—32no—
390C22—32no—
391C22—31no—
392C22—31no—
393C22—61no—
394C22—184no—
395C22—184no—
396C22—184no—
397C22—364no—
398C22—364no—
399C22C2×#629511yescharacteristic, complemented
400C22#8910111yescharacteristic
401C4—92no—
402C4—184no—
403C4—272no—
404C3—21no—
405C3—41no—
406C3—82no—
407C3—161no—
408C3C2×#6420712yescomplemented
409C2—31no—
410C2—31no—
411C2—32no—
412C2—364no—
413C2#1172421412yesdirect product factor
414C2C2×#8910111yescharacteristic
subgroupG/H
center, hypercenterC22#89101
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×#89101
Fitting, socleC24×C32C2×D3
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊S4C2semidirect
2. C2×A4×C3⋊Dic3C2semidirect
3. #11724214C2direct (N × H)
8. C2×C3×C3⋊S4C22semidirect
10. A4×C3⋊Dic3C22semidirect
11. C22×C3⋊S4D3semidirect
14. C23×C3⋊Dic3D3semidirect
25. C2×C32×A4D4semidirect
39. C2×C3⋊S4C2×D3semidirect
47. C22×C3⋊Dic3C2×D3semidirect
65. C32×A4C2×D4semidirect
84. C23×C32D12semidirect
100. C2×C3×A4#67semidirect
110. C2×C3⋊Dic3S4semidirect
174. C22×C32C2×D12semidirect
190. C3×A4C2×#67semidirect
192. C3⋊Dic3C2×S4semidirect
217. C23×C3#626semidirect
230. C2×A4#625semidirect
271. C2×C32C4⋊S4semidirect
318. C22×C3C2×#626semidirect
331. A4C2×#625semidirect
343. C32C2×C4⋊S4semidirect
352. C23#6295semidirect
382. C2×C3#64207semidirect
399. C22C2×#6295semidirect
408. C3C2×#64207semidirect
413. C2#11724214direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊S42
C2×A4×C3⋊Dic32
#117242142
C2×#642073
C2×#639803
C2×#62954