Finite Group Databasebeta

← C3×#71688 #11775228

Subgroups — C3×#71688 #11775228

Order 1152.

749 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C3×C4×#217C211yesmaximal, characteristic
3C3×#5207C211yesmaximal, characteristic
4C3×#4538C214yesmaximal, complemented
5C3×#1508—31nomaximal
6#71688C311yesmaximal, 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×C4×Dic3—21no—
18C2×C3×#208—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×C4×C22⋊C4—31no—
27C4×#217C2×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—24no—
52C3×C4×Dic3—42no—
53C3×#208—42no—
54C2×C3×C3⋊C8—22no—
55#1508—31noSylow
56C22×C3×Q8—21no—
57C22×C3×Q8Dic311yescharacteristic, complemented
58C22×C3×Q8C3×C411yescharacteristic, complemented
59C23×C3×C4—21no—
60C23×C3×C4C2×D311yescharacteristic
61C23×C3×C4C22×C311yescharacteristic
62C2×C3×M16—61no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—21no—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8—21no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8Dic312yescomplemented
72C3×C22⋊Q8C3×C412yescomplemented
73C3×C22⋊Q8C2×D312yes—
74C3×C22⋊Q8C22×C312yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4Dic311yescharacteristic, complemented
80C2×C3×C4⋊C4C3×C411yescharacteristic, complemented
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—61no—
85C2×C3×C42—61no—
86C2×#217—21no—
87C2×C4×Dic3—21no—
88C2×#208—21no—
89C3×C22⋊C8—32no—
90C3×C22⋊C8—61no—
91C3×#95—32no—
92C3⋊#95C22×C312yes—
93#830—21no—
94#830C22×C312yes—
95C23×C32—22no—
96C23×C32—22no—
97C23×C32—41no—
98C23×C32C22⋊C411yescharacteristic
99C23×C32C22⋊C411yescharacteristic
100C23×C32C2×D411yescharacteristic
101C32×Q8—42no—
102C32×Q8—42no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4C22⋊C411yescharacteristic
114C2×C32×C4C22⋊C411yescharacteristic
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C3×Dic3—24no—
118C2×C3×Dic3—42no—
119C2×C3×Dic3—42no—
120C3×C3⋊C8—42no—
121C2×C22⋊Q8C3×D311yescharacteristic
122#407—31no—
123C4×C22⋊C4—31no—
124#328—34no—
125C24×C3—21no—
126C24×C3#6711yescharacteristic
127C24×C3C3×D411yescharacteristic
128C2×C3×Q8—22no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—22no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8—44no—
135C2×C3×Q8C2×Dic312yes—
136C2×C3×Q8C2×C3×C412yes—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
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—21no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—44no—
152C22×C3×C4—62no—
153C22×C3×C4C2×Dic311yescharacteristic
154C22×C3×C4C2×Dic311yescharacteristic
155C22×C3×C4#6711yescharacteristic
156C22×C3×C4#6711yescharacteristic
157C22×C3×C4#6711yescharacteristic
158C22×C3×C4C2×C3×C411yescharacteristic
159C22×C3×C4C2×C3×C411yescharacteristic
160C22×C3×C4C3×D411yescharacteristic
161C22×C3×C4C3×D411yescharacteristic
162C22×C3×C4C3×D411yescharacteristic
163C22×C3×C4C22×D311yescharacteristic
164C22×C3×C4C23×C311yescharacteristic
165C22×Dic3—22no—
166C3×M16—122no—
167C2×C3×C8—62no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4C2×Dic312yes—
179C3×C4⋊C4C2×C3×C412yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—64no—
187C3×C42—122no—
188#217—24no—
189C4×Dic3—42no—
190#208—42no—
191C2×C3⋊C8—22no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—44no—
200C22×C32#10412yes—
201C22×C32C2×C22⋊C411yescharacteristic
202C22×C32#13712yes—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—42no—
207C32×C4—42no—
208C32×C4#9912yes—
209C3×Dic3—44no—
210C22×Q8C3×Dic311yescharacteristic, complemented
211C23×C4C2×C3×D311yescharacteristic
212C2×M16—61no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8C3×Dic312yescomplemented
216C22⋊Q8C2×C3×D312yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C3×Dic311yescharacteristic, complemented
219C2×C22⋊C4—21no—
220C2×C22⋊C4—61no—
221C2×C42—61no—
222C22⋊C8—32no—
223C22⋊C8—61no—
224#95—32no—
225C23×C3—24no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3#21711yescharacteristic
238C23×C3#21711yescharacteristic
239C23×C3C3×C22⋊C411yescharacteristic
240C23×C3C3×C22⋊C411yescharacteristic
241C23×C3C2×#6711yescharacteristic
242C23×C3C2×C3×D411yescharacteristic
243C3×Q8—42no—
244C3×Q8—42no—
245C3×Q8—44no—
246C3×Q8—44no—
247C3×Q8—42no—
248C3×Q8—42no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—44no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—64no—
284C2×C3×C4—122no—
285C2×C3×C4—122no—
286C2×C3×C4#21711yescharacteristic
287C2×C3×C4#21711yescharacteristic
288C2×C3×C4C3×C22⋊C411yescharacteristic
289C2×C3×C4C3×C22⋊C411yescharacteristic
290C2×C3×C4C22×Dic311yescharacteristic
291C2×C3×C4C2×#6711yescharacteristic
292C2×C3×C4C22×C3×C411yescharacteristic
293C2×C3×C4C2×C3×D411yescharacteristic
294C2×Dic3—24no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C3×C8—122no—
298C3⋊C8—42no—
299C2×C32—22no—
300C2×C32—22no—
301C2×C32—41no—
302C2×C32C2×#9911yescharacteristic
303C2×C32#41911yescharacteristic
304C2×C32C2×#10411yescharacteristic
305C24C3×#6711yescharacteristic
306C2×Q8—21no—
307C2×Q8—42no—
308C2×Q8C2×C3×Dic312yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—62no—
313C22×C4C2×C3×Dic311yescharacteristic
314C22×C4C2×C3×Dic311yescharacteristic
315C22×C4C3×#6711yescharacteristic
316C22×C4C3×#6711yescharacteristic
317C22×C4C3×#6711yescharacteristic
318C22×C4C22×C3×D311yescharacteristic
319M16—122no—
320C2×C8—62no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4C2×C3×Dic312yes—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—64no—
328C42—122no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—48no—
356C22×C3#83712yes—
357C22×C3C3×#10412yes—
358C22×C3#94212yes—
359C22×C3C2×#21711yescharacteristic
360C22×C3C2×C3×C22⋊C411yescharacteristic
361C22×C3C3×#13712yes—
362C3×C4—22no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—44no—
373C3×C4—44no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—124no—
379C3×C4#83412yes—
380C3×C4C3×#9912yes—
381Dic3—44no—
382C32#150811yescharacteristic, complemented, Sylow
383C23—22no—
384C23—22no—
385C23—41no—
386C23C3×#21711yescharacteristic
387C23C3×#21711yescharacteristic
388C23C2×C3×#6711yescharacteristic
389Q8—42no—
390Q8—42no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—64no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4C3×#21711yescharacteristic
405C2×C4C3×#21711yescharacteristic
406C2×C4C22×C3×Dic311yescharacteristic
407C2×C4C2×C3×#6711yescharacteristic
408C8—122no—
409C2×C3—24no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—42no—
418C2×C3—42no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3C2×#83411yescharacteristic
422C2×C3#522511yescharacteristic
423C2×C3C2×#83711yescharacteristic
424C2×C3C2×C3×#9911yescharacteristic
425C2×C3C3×#41911yescharacteristic
426C2×C3C2×C3×#10411yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22C3×#83712yes—
436C22C3×#94212yes—
437C22C2×C3×#21711yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—124no—
444C4C3×#83412yes—
445C3—21no—
446C3#7168811yescharacteristic, direct product factor
447C3C3×#150811yescharacteristic, complemented
448C2—22no—
449C2—22no—
450C2—41no—
451C2C2×C3×#83411yescharacteristic
452C2C3×#522511yescharacteristic
453C2C2×C3×#83711yescharacteristic
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×#1508
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. #71688C3direct (N × H)
7. C22×C32×Q8C4semidirect
11. C32×C22⋊Q8C4semidirect
14. C2×C32×C4⋊C4C4semidirect
30. #4538C2×C3semidirect
57. C22×C3×Q8Dic3semidirect
58. C22×C3×Q8C3×C4semidirect
71. C3×C22⋊Q8Dic3semidirect
72. C3×C22⋊Q8C3×C4semidirect
79. C2×C3×C4⋊C4Dic3semidirect
80. C2×C3×C4⋊C4C3×C4semidirect
210. C22×Q8C3×Dic3semidirect
215. C22⋊Q8C3×Dic3semidirect
218. C2×C4⋊C4C3×Dic3semidirect
382. C32#1508semidirect
446. C3#71688direct (N × H)
447. C3C3×#1508semidirect

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×C4×#2172
C3×#52072
C3×#45382
C3×#15083
#716883