Finite Group Databasebeta

← C3×#71689 #11775320

Subgroups — C3×#71689 #11775320

Order 1152.

709 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C3×#5211C211yesmaximal, characteristic
3C3×#5207C211yesmaximal, characteristic
4C3×#4538C214yesmaximal, complemented
5C3×#1511—31nomaximal
6#71689C311yesmaximal, characteristic, direct product factor
7C22×C32×Q8C411yescharacteristic, complemented
8C23×C32×C4C2211yescharacteristic
9C32×C22⋊Q8—21no—
10C32×C22⋊Q8—21no—
11C32×C22⋊Q8C412yescomplemented
12C32×C22⋊Q8C2212yes—
13C2×C32×C4⋊C4—21no—
14C2×C32×C4⋊C4C411yescharacteristic, complemented
15C2×C32×C22⋊C4—21no—
16C2×C3×#217—21no—
17C2×C3×#208—21no—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2212yes—
20C3×#830—21no—
21C3×#830C2212yes—
22C2×C3×C22⋊Q8—21no—
23C2×C3×C22⋊Q8D311yescharacteristic
24C2×C3×C22⋊Q8C2×C311yescharacteristic
25C3×#407—31no—
26C3×#381—31no—
27#5211C2×C311yescharacteristic
28#5207C2×C311yescharacteristic
29C3×#328—34no—
30#4538C2×C314yescomplemented
31C24×C32D411yescharacteristic
32C2×C32×Q8—21no—
33C2×C32×Q8—42no—
34C2×C32×Q8C2×C412yes—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4—42no—
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4C2311yescharacteristic
44C22×C3×Dic3—22no—
45C32×C4⋊C4—22no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—22no—
48C32×C4⋊C4C2×C412yes—
49C32×C22⋊C4—22no—
50C32×C22⋊C4—22no—
51C3×#217—42no—
52C3×#208—42no—
53C2×C3×C3⋊C8—22no—
54#1511—31noSylow
55C22×C3×Q8—21no—
56C22×C3×Q8Dic311yescharacteristic, complemented
57C22×C3×Q8C3×C411yescharacteristic, complemented
58C23×C3×C4—21no—
59C23×C3×C4C2×D311yescharacteristic
60C23×C3×C4C22×C311yescharacteristic
61C2×C3×M16—61no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—21no—
65C3×C22⋊Q8—21no—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8Dic312yescomplemented
71C3×C22⋊Q8C3×C412yescomplemented
72C3×C22⋊Q8C2×D312yes—
73C3×C22⋊Q8C22×C312yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4Dic311yescharacteristic, complemented
79C2×C3×C4⋊C4C3×C411yescharacteristic, complemented
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—61no—
84C2×#217—21no—
85C2×#208—21no—
86C3×C22⋊C8—32no—
87C3×C22⋊C8—61no—
88C3×#95—32no—
89C3×#95—61no—
90C3⋊#95—21no—
91C3⋊#95C22×C312yes—
92#830—21no—
93#830C22×C312yes—
94C23×C32—22no—
95C23×C32—22no—
96C23×C32—41no—
97C23×C32C22⋊C411yescharacteristic
98C23×C32C22⋊C411yescharacteristic
99C23×C32C2×D411yescharacteristic
100C32×Q8—42no—
101C32×Q8—42no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—42no—
111C2×C32×C4—42no—
112C2×C32×C4C22⋊C411yescharacteristic
113C2×C32×C4C22⋊C411yescharacteristic
114C2×C32×C4C22×C411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C3×Dic3—42no—
117C2×C3×Dic3—42no—
118C2×C3×Dic3—42no—
119C3×C3⋊C8—42no—
120C2×C22⋊Q8C3×D311yescharacteristic
121#407—31no—
122#381—31no—
123#328—34no—
124C24×C3—21no—
125C24×C3#6711yescharacteristic
126C24×C3C3×D411yescharacteristic
127C2×C3×Q8—22no—
128C2×C3×Q8—21no—
129C2×C3×Q8—21no—
130C2×C3×Q8—22no—
131C2×C3×Q8—44no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8C2×Dic312yes—
135C2×C3×Q8C2×C3×C412yes—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—44no—
151C22×C3×C4—62no—
152C22×C3×C4C2×Dic311yescharacteristic
153C22×C3×C4C2×Dic311yescharacteristic
154C22×C3×C4#6711yescharacteristic
155C22×C3×C4#6711yescharacteristic
156C22×C3×C4#6711yescharacteristic
157C22×C3×C4C2×C3×C411yescharacteristic
158C22×C3×C4C2×C3×C411yescharacteristic
159C22×C3×C4C3×D411yescharacteristic
160C22×C3×C4C3×D411yescharacteristic
161C22×C3×C4C3×D411yescharacteristic
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C23×C311yescharacteristic
164C22×Dic3—22no—
165C3×M16—122no—
166C2×C3×C8—62no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4C2×Dic312yes—
178C3×C4⋊C4C2×C3×C412yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—122no—
186#217—42no—
187#208—42no—
188C2×C3⋊C8—22no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—44no—
197C22×C32#9912yes—
198C22×C32C2×C22⋊C411yescharacteristic
199C22×C32#13712yes—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—42no—
205C32×C4—42no—
206C3×Dic3—82no—
207C22×Q8C3×Dic311yescharacteristic, complemented
208C23×C4C2×C3×D311yescharacteristic
209C2×M16—61no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8C3×Dic312yescomplemented
213C22⋊Q8C2×C3×D312yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C3×Dic311yescharacteristic, complemented
216C2×C22⋊C4—21no—
217C2×C22⋊C4—61no—
218C22⋊C8—32no—
219C22⋊C8—61no—
220#95—32no—
221#95—61no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—42no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3#21711yescharacteristic
235C23×C3#21711yescharacteristic
236C23×C3C3×C22⋊C411yescharacteristic
237C23×C3C3×C22⋊C411yescharacteristic
238C23×C3C2×#6711yescharacteristic
239C23×C3C2×C3×D411yescharacteristic
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—44no—
243C3×Q8—44no—
244C3×Q8—42no—
245C3×Q8—42no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—122no—
281C2×C3×C4—122no—
282C2×C3×C4—122no—
283C2×C3×C4#21711yescharacteristic
284C2×C3×C4#21711yescharacteristic
285C2×C3×C4C3×C22⋊C411yescharacteristic
286C2×C3×C4C3×C22⋊C411yescharacteristic
287C2×C3×C4C22×Dic311yescharacteristic
288C2×C3×C4C2×#6711yescharacteristic
289C2×C3×C4C22×C3×C411yescharacteristic
290C2×C3×C4C2×C3×D411yescharacteristic
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C3×C8—122no—
295C3⋊C8—42no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—41no—
299C2×C32C2×#9911yescharacteristic
300C2×C32#41911yescharacteristic
301C2×C32#42111yescharacteristic
302C24C3×#6711yescharacteristic
303C2×Q8—21no—
304C2×Q8—42no—
305C2×Q8C2×C3×Dic312yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—62no—
310C22×C4C2×C3×Dic311yescharacteristic
311C22×C4C2×C3×Dic311yescharacteristic
312C22×C4C3×#6711yescharacteristic
313C22×C4C3×#6711yescharacteristic
314C22×C4C3×#6711yescharacteristic
315C22×C4C22×C3×D311yescharacteristic
316M16—122no—
317C2×C8—62no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4C2×C3×Dic312yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—122no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—48no—
352C22×C3#83412yes—
353C22×C3C3×#9912yes—
354C22×C3#94212yes—
355C22×C3C2×#21711yescharacteristic
356C22×C3C2×C3×C22⋊C411yescharacteristic
357C22×C3C3×#13712yes—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—44no—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—242no—
377Dic3—82no—
378C32#151111yescharacteristic, complemented, Sylow
379C23—22no—
380C23—22no—
381C23—41no—
382C23C3×#21711yescharacteristic
383C23C3×#21711yescharacteristic
384C23C2×C3×#6711yescharacteristic
385Q8—42no—
386Q8—42no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4C3×#21711yescharacteristic
401C2×C4C3×#21711yescharacteristic
402C2×C4C22×C3×Dic311yescharacteristic
403C2×C4C2×C3×#6711yescharacteristic
404C8—122no—
405C2×C3—24no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—42no—
414C2×C3—42no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3C2×#83411yescharacteristic
418C2×C3#522511yescharacteristic
419C2×C3#523611yescharacteristic
420C2×C3C2×C3×#9911yescharacteristic
421C2×C3C3×#41911yescharacteristic
422C2×C3C3×#42111yescharacteristic
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22C3×#83412yes—
432C22C3×#94212yes—
433C22C2×C3×#21711yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—42no—
439C4—42no—
440C4—242no—
441C3—21no—
442C3#7168911yescharacteristic, direct product factor
443C3C3×#151111yescharacteristic, complemented
444C2—22no—
445C2—22no—
446C2—41no—
447C2C2×C3×#83411yescharacteristic
448C2C3×#522511yescharacteristic
449C2C3×#523611yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC22×C32C2×C22⋊C4
prime radicalC24×C32D4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C3×#1511
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. C3×#4538C2semidirect
6. #71689C3direct (N × H)
7. C22×C32×Q8C4semidirect
11. C32×C22⋊Q8C4semidirect
14. C2×C32×C4⋊C4C4semidirect
30. #4538C2×C3semidirect
56. C22×C3×Q8Dic3semidirect
57. C22×C3×Q8C3×C4semidirect
70. C3×C22⋊Q8Dic3semidirect
71. C3×C22⋊Q8C3×C4semidirect
78. C2×C3×C4⋊C4Dic3semidirect
79. C2×C3×C4⋊C4C3×C4semidirect
207. C22×Q8C3×Dic3semidirect
212. C22⋊Q8C3×Dic3semidirect
215. C2×C4⋊C4C3×Dic3semidirect
378. C32#1511semidirect
442. C3#71689direct (N × H)
443. C3C3×#1511semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C3×#52112
C3×#52072
C3×#45382
C3×#15113
#716893