Finite Group Databasebeta

← C2×#11724246 #13180952

Subgroups — C2×#11724246 #13180952

Order 1728.

1240 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1240 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
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×C4×#157C211yesmaximal, characteristic, complemented
4C2×Dic3⋊#157C212yesmaximal, complemented
5C2×(C2×D3)⋊#157C212yesmaximal
6#11724246C218yesmaximal, direct product factor
7C2×#64286—31nomaximal
8C22×D3×C3⋊D3C2211yescharacteristic
9C22×C3⋊#157C2212yes—
10C22×C3×#157C2212yes—
11C2×C4×C3⋊(C3⋊D3)C411yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×C4×D3C411yescharacteristic, complemented
14C2×#6289C411yescharacteristic, complemented
15C2×Dic3×C3⋊D3C2211yescharacteristic
16C2×D3×C3⋊Dic3C411yescharacteristic, complemented
17C4×D3×C3⋊D3C414yescomplemented
18C4×D3×C3⋊D3C2214yes—
19C4×C3⋊#157C2214yescomplemented
20C3×C4×#157C2214yescomplemented
21Dic3⋊#157C2218yescomplemented
22(C2×D3)⋊#157C2218yes—
23C22×C4×C3⋊D3—31no—
24C2×C4×D32—62no—
25C2×C22⋊#157—32no—
26#64286—38no—
27C2×C4⋊#157—32no—
28C2×C4×#157—31no—
29C2×C4×#157D311yescharacteristic, complemented
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3C2×C411yescharacteristic
33C2×D3×C3⋊D3—24no—
34C2×D3×C3⋊D3C2×C412yes—
35C2×D3×C3⋊D3C2312yes—
36C2×C3⋊#157—24no—
37C2×C3⋊#157C2314yes—
38C2×C3×#157—24no—
39C2×C3×#157C2314yes—
40C2×C33×C4C2×C411yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C4×C3⋊(C3⋊D3)C2×C414yescomplemented
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C3×C4×C3⋊D3C2×C412yes—
45C3×C4×C3⋊D3C2312yes—
46C2×C32×Dic3C2×C411yescharacteristic
47C32×C4×D3C2×C414yescomplemented
48#6289C2×C414yescomplemented
49Dic3×C3⋊D3C2×C412yes—
50Dic3×C3⋊D3C2312yes—
51D3×C3⋊Dic3C2×C414yescomplemented
52C2×#872—91nomaximal
53C23×C3⋊D3—31no—
54C22×D32—62no—
55C22×#157—32no—
56C22×#157C2×D312yes—
57C22×C32×C4—31no—
58C22×C3⋊Dic3—31no—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—62no—
65C2×C4×C3⋊D3—122no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—62no—
69C2×#624—62no—
70C2×D3×Dic3—62no—
71C2×D3×Dic3—62no—
72C4×D32—68no—
73C4×D32—68no—
74C22⋊#157—38no—
75C4⋊#157—38no—
76C4×#157—34no—
77C4×#157C2×D314yescomplemented
78C22×C33C22×C411yescharacteristic
79C2×C3⋊(C3⋊D3)—22no—
80C2×C3⋊(C3⋊D3)C22×C412yes—
81C2×C3×C3⋊D3C22×C411yescharacteristic
82C2×C3×C3⋊D3#4114yes—
83C2×C3×C3⋊D3C2411yescharacteristic
84C2×C32×D3—22no—
85C2×C32×D3C22×C412yes—
86D3×C3⋊D3—24no—
87D3×C3⋊D3—24no—
88C3⋊#157—24no—
89C3×#157—24no—
90C33×C4C22×C412yes—
91C3⋊(C3⋊Dic3)C22×C412yes—
92C3×C3⋊Dic3C22×C412yes—
93C32×Dic3C22×C412yes—
94C22×C4×D3—91no—
95C22×C4×D3—182no—
96C2×C3×C42—91no—
97C2×(C2×D3)⋊C4—92no—
98C2×Dic3⋊C4—92no—
99C2×C4×Dic3—91no—
100#872—98no—
101C23×C32—31no—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—32no—
104C22×C3⋊D3—32no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—122no—
108C22×C3⋊D3C22×D311yescharacteristic
109C22×C3×D3—22no—
110C22×C3×D3—62no—
111C2×D32—64no—
112C2×D32—64no—
113C2×D32—616no—
114C2×#157—24no—
115C2×#157—34no—
116C2×#157—64no—
117C2×#157C22×D314yes—
118C2×C32×C4—22no—
119C2×C32×C4—31no—
120C2×C32×C4—34no—
121C2×C32×C4—42no—
122C2×C32×C4C4×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3C4×D311yescharacteristic
128C4×C3⋊D3—32no—
129C4×C3⋊D3—32no—
130C4×C3⋊D3—34no—
131C4×C3⋊D3—34no—
132C4×C3⋊D3—68no—
133C4×C3⋊D3—128no—
134C4×C3⋊D3C4×D312yes—
135C4×C3⋊D3C22×D312yes—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—62no—
138C3×C4×D3—28no—
139C3×C4×D3—64no—
140C3×C4×D3—64no—
141#624—68no—
142D3×Dic3—68no—
143D3×Dic3—64no—
144D3×Dic3—64no—
145C2×#117—271noSylow
146C2×C33#11712yescomplemented
147C2×C33C23×C411yescharacteristic
148C3⋊(C3⋊D3)—22no—
149C3×C3⋊D3#11712yescomplemented
150C3×C3⋊D3C2×#4112yes—
151C32×D3—22no—
152C23×D3—91no—
153C23×D3—182no—
154C22×C3×C4—62no—
155C22×C3×C4—91no—
156C22×C3×C4—92no—
157C22×Dic3—91no—
158C22×Dic3—92no—
159C22×Dic3—182no—
160C2×C4×D3—62no—
161C2×C4×D3—91no—
162C2×C4×D3—91no—
163C2×C4×D3—91no—
164C2×C4×D3—94no—
165C2×C4×D3—94no—
166C2×C4×D3—188no—
167C2×C4×D3—188no—
168C2×C4×D3—182no—
169C2×C4×D3—182no—
170C2×C4×D3—182no—
171C2×C4×D3—362no—
172C2×C4×D3#15711yescharacteristic, complemented
173C3×C42—94no—
174(C2×D3)⋊C4—98no—
175Dic3⋊C4—98no—
176C4×Dic3—94no—
177C22×C32—22no—
178C22×C32—32no—
179C22×C32—42no—
180C22×C32—62no—
181C22×C32C2×C4×D311yescharacteristic
182C2×C3⋊D3—32no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—64no—
186C2×C3⋊D3—64no—
187C2×C3⋊D3—68no—
188C2×C3⋊D3—128no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3C2×C4×D311yescharacteristic
191C2×C3⋊D3#23514yes—
192C2×C3⋊D3C23×D311yescharacteristic
193C2×C3×D3—28no—
194C2×C3×D3—24no—
195C2×C3×D3—62no—
196C2×C3×D3—68no—
197C2×C3×D3—62no—
198D32—616no—
199D32—616no—
200#157—24no—
201#157—64no—
202C32×C4—24no—
203C32×C4—32no—
204C32×C4—44no—
205C32×C4C2×C4×D312yes—
206C3⋊Dic3—32no—
207C3⋊Dic3—64no—
208C3⋊Dic3—124no—
209C3⋊Dic3C2×C4×D312yes—
210C3×Dic3—24no—
211C3×Dic3—64no—
212C23×C4—271no—
213#117—278no—
214C2×C4⋊C4—272no—
215C2×C22⋊C4—272no—
216C2×C42—91no—
217C2×C42—271no—
218C33C2×#11711yescharacteristic, complemented, Sylow
219C23×C3—62no—
220C23×C3—91no—
221C22×D3—62no—
222C22×D3—91no—
223C22×D3—92no—
224C22×D3—92no—
225C22×D3—184no—
226C22×D3—184no—
227C22×D3—182no—
228C22×D3—184no—
229C22×D3—1816no—
230C22×D3—362no—
231C22×D3C2×#15711yescharacteristic
232C2×C3×C4—22no—
233C2×C3×C4—42no—
234C2×C3×C4—68no—
235C2×C3×C4—62no—
236C2×C3×C4—91no—
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4—94no—
240C2×C3×C4—184no—
241C2×C3×C4C2×#15711yescharacteristic
242C2×Dic3—62no—
243C2×Dic3—91no—
244C2×Dic3—92no—
245C2×Dic3—92no—
246C2×Dic3—94no—
247C2×Dic3—188no—
248C2×Dic3—182no—
249C2×Dic3—184no—
250C2×Dic3—362no—
251C2×Dic3C2×#15711yescharacteristic
252C4×D3—64no—
253C4×D3—64no—
254C4×D3—94no—
255C4×D3—94no—
256C4×D3—94no—
257C4×D3—188no—
258C4×D3—188no—
259C4×D3—184no—
260C4×D3—184no—
261C4×D3—368no—
262C4×D3C2×#15714yescomplemented
263C2×C32—24no—
264C2×C32—22no—
265C2×C32—44no—
266C2×C32—42no—
267C2×C32—62no—
268C2×C32#87212yescomplemented
269C2×C32C22×C4×D311yescharacteristic
270C3⋊D3—62no—
271C3⋊D3—68no—
272C3⋊D3—128no—
273C3⋊D3#87212yescomplemented
274C3⋊D3C2×#23512yes—
275C3×D3—28no—
276C3×D3—64no—
277C3×D3—64no—
278C24—271no—
279C22×C4—31no—
280C22×C4—91no—
281C22×C4—92no—
282C22×C4—274no—
283C22×C4—271no—
284C22×C4—274no—
285C22×C4—271no—
286C22×C4—271no—
287C22×C4—271no—
288C22×C4—272no—
289C4⋊C4—278no—
290C22⋊C4—278no—
291C42—94no—
292C42—274no—
293C22×C3—22no—
294C22×C3—42no—
295C22×C3—64no—
296C22×C3—68no—
297C22×C3—94no—
298C22×C3—91no—
299C22×C3—91no—
300C22×C3C22×#15711yescharacteristic
301C2×D3—22no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—68no—
305C2×D3—92no—
306C2×D3—184no—
307C2×D3—184no—
308C2×D3—184no—
309C2×D3—1816no—
310C2×D3—1816no—
311C2×D3—188no—
312C2×D3—182no—
313C2×D3—364no—
314C2×D3—368no—
315C2×D3C22×#15712yes—
316C3×C4—24no—
317C3×C4—44no—
318C3×C4—64no—
319C3×C4—92no—
320C3×C4—184no—
321C3×C4C22×#15712yes—
322Dic3—64no—
323Dic3—92no—
324Dic3—184no—
325Dic3—184no—
326Dic3—364no—
327Dic3C22×#15712yes—
328C32—22no—
329C32—42no—
330C32C2×#87211yescharacteristic, complemented
331C23—31no—
332C23—91no—
333C23—272no—
334C23—272no—
335C23—271no—
336C23—544no—
337C2×C4—31no—
338C2×C4—34no—
339C2×C4—91no—
340C2×C4—92no—
341C2×C4—92no—
342C2×C4—94no—
343C2×C4—184no—
344C2×C4—274no—
345C2×C4—272no—
346C2×C4—272no—
347C2×C4—274no—
348C2×C4—271no—
349C2×C4—274no—
350C2×C4—274no—
351C2×C4—544no—
352C2×C4D3×#15711yescharacteristic
353C2×C3—22no—
354C2×C3—24no—
355C2×C3—42no—
356C2×C3—44no—
357C2×C3—68no—
358C2×C3—92no—
359C2×C3—92no—
360C2×C3#6428612yescomplemented
361C2×C3C23×#15711yescharacteristic
362D3—22no—
363D3—64no—
364D3—64no—
365D3—188no—
366D3—182no—
367D3—368no—
368C22—32no—
369C22—62no—
370C22—91no—
371C22—91no—
372C22—94no—
373C22—272no—
374C22—544no—
375C22—544no—
376C22—542no—
377C22C2×D3×#15711yescharacteristic
378C4—32no—
379C4—92no—
380C4—184no—
381C4—272no—
382C4—544no—
383C4C2×D3×#15712yes—
384C3—22no—
385C3—42no—
386C3C2×#6428611yescharacteristic, complemented
387C2—62no—
388C2—92no—
389C2—92no—
390C2—542no—
391C2#1172424612yesdirect product factor
392C2C22×D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. C2×C3×C4×#157C2semidirect
4. C2×Dic3⋊#157C2semidirect
6. #11724246C2direct (N × H)
11. C2×C4×C3⋊(C3⋊D3)C4semidirect
13. C2×C32×C4×D3C4semidirect
14. C2×#6289C4semidirect
16. C2×D3×C3⋊Dic3C4semidirect
17. C4×D3×C3⋊D3C4semidirect
19. C4×C3⋊#157C22semidirect
20. C3×C4×#157C22semidirect
21. Dic3⋊#157C22semidirect
29. C2×C4×#157D3semidirect
42. C4×C3⋊(C3⋊D3)C2×C4semidirect
47. C32×C4×D3C2×C4semidirect
48. #6289C2×C4semidirect
51. D3×C3⋊Dic3C2×C4semidirect
77. C4×#157C2×D3semidirect
146. C2×C33#117semidirect
149. C3×C3⋊D3#117semidirect
172. C2×C4×D3#157semidirect
218. C33C2×#117semidirect
262. C4×D3C2×#157semidirect
268. C2×C32#872semidirect
273. C3⋊D3#872semidirect
330. C32C2×#872semidirect
360. C2×C3#64286semidirect
386. C3C2×#64286semidirect
391. C2#11724246direct (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×C4×C3⋊#1572
C2×C3×C4×#1572
C2×Dic3⋊#1572
C2×(C2×D3)⋊#1572
#117242462
C2×#642863
C2×#8729