Finite Group Databasebeta

← C2×#11724247 #13180949

Subgroups — C2×#11724247 #13180949

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⋊#157)C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#157C212yesmaximal
5C2×Dic3×#157C212yesmaximal, complemented
6#11724247C218yesmaximal, 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⋊#157)C2214yescomplemented
20C3×C4⋊#157C2214yescomplemented
21(C2×D3)⋊#157C2218yes—
22Dic3×#157C2218yescomplemented
23C22×C4×C3⋊D3—31no—
24C2×C4×D32—62no—
25C2×C22⋊#157—32no—
26#64286—38no—
27C2×C4⋊#157—31no—
28C2×C4⋊#157D311yescharacteristic, complemented
29C2×C4×#157—32no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3C2×C411yescharacteristic
33C2×D3×C3⋊D3—22no—
34C2×D3×C3⋊D3—22no—
35C2×D3×C3⋊D3C2×C412yes—
36C2×D3×C3⋊D3C2312yes—
37C2×C3⋊#157—24no—
38C2×C3⋊#157C2314yes—
39C2×C3×#157—24no—
40C2×C3×#157C2314yes—
41C2×C33×C4C2×C411yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
43C4×C3⋊(C3⋊D3)C2×C414yescomplemented
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4×C3⋊D3C2×C412yes—
46C3×C4×C3⋊D3C2312yes—
47C2×C32×Dic3C2×C411yescharacteristic
48C32×C4×D3C2×C414yescomplemented
49#6289C2×C414yescomplemented
50Dic3×C3⋊D3C2×C412yes—
51Dic3×C3⋊D3C2312yes—
52D3×C3⋊Dic3C2×C414yescomplemented
53C2×#892—91nomaximal
54C23×C3⋊D3—31no—
55C22×D32—62no—
56C22×#157—32no—
57C22×#157C2×D312yes—
58C22×C32×C4—31no—
59C22×C3⋊Dic3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—34no—
65C2×C4×C3⋊D3—62no—
66C2×C4×C3⋊D3—122no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3—62no—
70C2×#624—62no—
71C2×D3×Dic3—62no—
72C2×D3×Dic3—62no—
73C4×D32—68no—
74C4×D32—68no—
75C22⋊#157—38no—
76C4⋊#157—34no—
77C4⋊#157C2×D314yescomplemented
78C4×#157—38no—
79C22×C33C22×C411yescharacteristic
80C2×C3⋊(C3⋊D3)—22no—
81C2×C3⋊(C3⋊D3)C22×C412yes—
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C3×C3⋊D3#4112yes—
84C2×C3×C3⋊D3#4112yes—
85C2×C3×C3⋊D3C2411yescharacteristic
86C2×C32×D3—22no—
87C2×C32×D3C22×C412yes—
88D3×C3⋊D3—24no—
89D3×C3⋊D3—22no—
90D3×C3⋊D3—22no—
91C3⋊#157—24no—
92C3×#157—24no—
93C33×C4C22×C412yes—
94C3⋊(C3⋊Dic3)C22×C412yes—
95C3×C3⋊Dic3C22×C412yes—
96C32×Dic3C22×C412yes—
97C22×C4×D3—91no—
98C22×C4×D3—182no—
99C2×C3×C4⋊C4—91no—
100C2×(C2×D3)⋊C4—92no—
101C2×C4⋊Dic3—91no—
102C2×C4×Dic3—92no—
103#892—98no—
104C23×C32—31no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—32no—
107C22×C3⋊D3—32no—
108C22×C3⋊D3—62no—
109C22×C3⋊D3—62no—
110C22×C3⋊D3—62no—
111C22×C3⋊D3—122no—
112C22×C3⋊D3C22×D311yescharacteristic
113C22×C3×D3—22no—
114C22×C3×D3—62no—
115C2×D32—64no—
116C2×D32—64no—
117C2×D32—68no—
118C2×D32—68no—
119C2×#157—24no—
120C2×#157—34no—
121C2×#157—64no—
122C2×#157C22×D314yes—
123C2×C32×C4—22no—
124C2×C32×C4—31no—
125C2×C32×C4—34no—
126C2×C32×C4—42no—
127C2×C32×C4C4×D311yescharacteristic
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—62no—
131C2×C3⋊Dic3—122no—
132C2×C3⋊Dic3C4×D311yescharacteristic
133C4×C3⋊D3—32no—
134C4×C3⋊D3—32no—
135C4×C3⋊D3—34no—
136C4×C3⋊D3—34no—
137C4×C3⋊D3—68no—
138C4×C3⋊D3—128no—
139C4×C3⋊D3C4×D312yes—
140C4×C3⋊D3C22×D312yes—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—62no—
143C3×C4×D3—28no—
144C3×C4×D3—64no—
145C3×C4×D3—64no—
146#624—68no—
147D3×Dic3—68no—
148D3×Dic3—64no—
149D3×Dic3—64no—
150C2×#117—271noSylow
151C2×C33#11712yescomplemented
152C2×C33C23×C411yescharacteristic
153C3⋊(C3⋊D3)—22no—
154C3×C3⋊D3#11712yescomplemented
155C3×C3⋊D3C2×#4111yescharacteristic
156C3×C3⋊D3C2×#4111yescharacteristic
157C32×D3—22no—
158C23×D3—91no—
159C23×D3—182no—
160C22×C3×C4—62no—
161C22×C3×C4—91no—
162C22×C3×C4—92no—
163C22×Dic3—91no—
164C22×Dic3—92no—
165C22×Dic3—182no—
166C2×C4×D3—62no—
167C2×C4×D3—91no—
168C2×C4×D3—91no—
169C2×C4×D3—91no—
170C2×C4×D3—94no—
171C2×C4×D3—94no—
172C2×C4×D3—188no—
173C2×C4×D3—188no—
174C2×C4×D3—182no—
175C2×C4×D3—182no—
176C2×C4×D3—182no—
177C2×C4×D3—362no—
178C2×C4×D3#15711yescharacteristic, complemented
179C3×C4⋊C4—94no—
180(C2×D3)⋊C4—98no—
181C4⋊Dic3—94no—
182C4×Dic3—98no—
183C22×C32—22no—
184C22×C32—32no—
185C22×C32—42no—
186C22×C32—62no—
187C22×C32C2×C4×D311yescharacteristic
188C2×C3⋊D3—32no—
189C2×C3⋊D3—64no—
190C2×C3⋊D3—62no—
191C2×C3⋊D3—64no—
192C2×C3⋊D3—62no—
193C2×C3⋊D3—62no—
194C2×C3⋊D3—68no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3C2×C4×D311yescharacteristic
198C2×C3⋊D3#23712yes—
199C2×C3⋊D3#23912yes—
200C2×C3⋊D3C23×D311yescharacteristic
201C2×C3×D3—28no—
202C2×C3×D3—24no—
203C2×C3×D3—62no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—62no—
207D32—616no—
208D32—68no—
209D32—68no—
210#157—24no—
211#157—64no—
212C32×C4—24no—
213C32×C4—32no—
214C32×C4—44no—
215C32×C4C2×C4×D312yes—
216C3⋊Dic3—32no—
217C3⋊Dic3—64no—
218C3⋊Dic3—124no—
219C3⋊Dic3C2×C4×D312yes—
220C3×Dic3—24no—
221C3×Dic3—64no—
222C23×C4—271no—
223#117—278no—
224C2×C4⋊C4—91no—
225C2×C4⋊C4—271no—
226C2×C22⋊C4—272no—
227C2×C42—272no—
228C33C2×#11711yescharacteristic, complemented, Sylow
229C23×C3—62no—
230C23×C3—91no—
231C22×D3—62no—
232C22×D3—91no—
233C22×D3—92no—
234C22×D3—92no—
235C22×D3—184no—
236C22×D3—184no—
237C22×D3—182no—
238C22×D3—182no—
239C22×D3—182no—
240C22×D3—188no—
241C22×D3—188no—
242C22×D3—362no—
243C22×D3C2×#15711yescharacteristic
244C2×C3×C4—22no—
245C2×C3×C4—42no—
246C2×C3×C4—68no—
247C2×C3×C4—62no—
248C2×C3×C4—91no—
249C2×C3×C4—92no—
250C2×C3×C4—92no—
251C2×C3×C4—94no—
252C2×C3×C4—184no—
253C2×C3×C4C2×#15711yescharacteristic
254C2×Dic3—62no—
255C2×Dic3—91no—
256C2×Dic3—92no—
257C2×Dic3—92no—
258C2×Dic3—94no—
259C2×Dic3—188no—
260C2×Dic3—182no—
261C2×Dic3—184no—
262C2×Dic3—362no—
263C2×Dic3C2×#15711yescharacteristic
264C4×D3—64no—
265C4×D3—64no—
266C4×D3—94no—
267C4×D3—94no—
268C4×D3—94no—
269C4×D3—188no—
270C4×D3—188no—
271C4×D3—184no—
272C4×D3—184no—
273C4×D3—368no—
274C4×D3C2×#15714yescomplemented
275C2×C32—24no—
276C2×C32—22no—
277C2×C32—44no—
278C2×C32—42no—
279C2×C32—62no—
280C2×C32#89212yescomplemented
281C2×C32C22×C4×D311yescharacteristic
282C3⋊D3—62no—
283C3⋊D3—68no—
284C3⋊D3—128no—
285C3⋊D3#89212yescomplemented
286C3⋊D3C2×#23711yescharacteristic
287C3⋊D3C2×#23911yescharacteristic
288C3×D3—28no—
289C3×D3—64no—
290C3×D3—62no—
291C3×D3—62no—
292C24—271no—
293C22×C4—31no—
294C22×C4—91no—
295C22×C4—92no—
296C22×C4—274no—
297C22×C4—271no—
298C22×C4—274no—
299C22×C4—271no—
300C22×C4—271no—
301C22×C4—271no—
302C22×C4—272no—
303C4⋊C4—94no—
304C4⋊C4—274no—
305C22⋊C4—278no—
306C42—278no—
307C22×C3—22no—
308C22×C3—42no—
309C22×C3—64no—
310C22×C3—68no—
311C22×C3—92no—
312C22×C3—92no—
313C22×C3—91no—
314C22×C3—91no—
315C22×C3C22×#15711yescharacteristic
316C2×D3—22no—
317C2×D3—62no—
318C2×D3—62no—
319C2×D3—64no—
320C2×D3—64no—
321C2×D3—92no—
322C2×D3—184no—
323C2×D3—182no—
324C2×D3—182no—
325C2×D3—184no—
326C2×D3—1816no—
327C2×D3—188no—
328C2×D3—188no—
329C2×D3—188no—
330C2×D3—182no—
331C2×D3—364no—
332C2×D3—368no—
333C2×D3C22×#15712yes—
334C3×C4—24no—
335C3×C4—44no—
336C3×C4—64no—
337C3×C4—92no—
338C3×C4—184no—
339C3×C4C22×#15712yes—
340Dic3—64no—
341Dic3—92no—
342Dic3—184no—
343Dic3—184no—
344Dic3—364no—
345Dic3C22×#15712yes—
346C32—22no—
347C32—42no—
348C32C2×#89211yescharacteristic, complemented
349C23—31no—
350C23—91no—
351C23—272no—
352C23—272no—
353C23—271no—
354C23—542no—
355C23—542no—
356C2×C4—31no—
357C2×C4—34no—
358C2×C4—91no—
359C2×C4—92no—
360C2×C4—92no—
361C2×C4—94no—
362C2×C4—184no—
363C2×C4—274no—
364C2×C4—272no—
365C2×C4—272no—
366C2×C4—274no—
367C2×C4—271no—
368C2×C4—274no—
369C2×C4—274no—
370C2×C4—544no—
371C2×C4D3×#15711yescharacteristic
372C2×C3—22no—
373C2×C3—24no—
374C2×C3—42no—
375C2×C3—44no—
376C2×C3—68no—
377C2×C3—91no—
378C2×C3—91no—
379C2×C3—92no—
380C2×C3#6428612yescomplemented
381C2×C3C23×#15711yescharacteristic
382D3—22no—
383D3—64no—
384D3—62no—
385D3—62no—
386D3—188no—
387D3—182no—
388D3—368no—
389C22—32no—
390C22—62no—
391C22—91no—
392C22—91no—
393C22—92no—
394C22—92no—
395C22—272no—
396C22—542no—
397C22—542no—
398C22—544no—
399C22—542no—
400C22C2×D3×#15711yescharacteristic
401C4—32no—
402C4—92no—
403C4—184no—
404C4—272no—
405C4—544no—
406C4C2×D3×#15712yes—
407C3—22no—
408C3—42no—
409C3C2×#6428611yescharacteristic, complemented
410C2—62no—
411C2—91no—
412C2—91no—
413C2—92no—
414C2—542no—
415C2#1172424712yesdirect product factor
416C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. C2×C3×C4⋊#157C2semidirect
5. C2×Dic3×#157C2semidirect
6. #11724247C2direct (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⋊#157)C22semidirect
20. C3×C4⋊#157C22semidirect
22. Dic3×#157C22semidirect
28. C2×C4⋊#157D3semidirect
43. C4×C3⋊(C3⋊D3)C2×C4semidirect
48. C32×C4×D3C2×C4semidirect
49. #6289C2×C4semidirect
52. D3×C3⋊Dic3C2×C4semidirect
77. C4⋊#157C2×D3semidirect
151. C2×C33#117semidirect
154. C3×C3⋊D3#117semidirect
178. C2×C4×D3#157semidirect
228. C33C2×#117semidirect
274. C4×D3C2×#157semidirect
280. C2×C32#892semidirect
285. C3⋊D3#892semidirect
348. C32C2×#892semidirect
380. C2×C3#64286semidirect
409. C3C2×#64286semidirect
415. C2#11724247direct (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⋊#157)2
C2×C3×C4⋊#1572
C2×(C2×D3)⋊#1572
C2×Dic3×#1572
#117242472
C2×#642863
C2×#8929