Finite Group Databasebeta

← C3×#68310 #11775166

Subgroups — C3×#68310 #11775166

Order 1152.

496 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5622C211yesmaximal, characteristic
2C3×#4719C211yesmaximal, characteristic
3C3×#4648C211yesmaximal, characteristic
4C32×#328C211yesmaximal, characteristic
5C3×#4538C211yesmaximal, characteristic
6C3×#4467C211yesmaximal, characteristic
7C3×#4452C211yesmaximal, characteristic
8C3×#1506—31nomaximal
9#68310C311yesmaximal, characteristic, direct product factor
10C32×C22⋊Q8C2211yescharacteristic
11C3×Q8×Dic3C411yescharacteristic, complemented
12C3×#944C411yescharacteristic, complemented
13C3×#924C2211yescharacteristic
14C2×C3×C4×Dic3—21no—
15C2×C3×C4×Dic3C2211yescharacteristic
16C3×#889—21no—
17C3×#888C411yescharacteristic, complemented
18C3×#887—21no—
19C3×#887C411yescharacteristic, complemented
20C3×#878—21no—
21C3×#877—21no—
22C32×C22⋊C8C2211yescharacteristic
23C32×#95C2211yescharacteristic
24C3×C3⋊#95—21no—
25C3×C3⋊#95C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#815—21no—
28C3×Dic3⋊C8—21no—
29C3×#533—31no—
30#5622C2×C311yescharacteristic
31C3×#432—31no—
32C3×#376—31no—
33#4719C2×C311yescharacteristic
34#4648C2×C311yescharacteristic
35C3×#328—21no—
36C3×#328—31no—
37C3×#328—31no—
38C3×#328—31no—
39C3×#328D311yescharacteristic
40C3×#328C2×C311yescharacteristic
41#4538C2×C311yescharacteristic
42#4467C2×C311yescharacteristic
43#4452C2×C311yescharacteristic
44C2×C32×Q8C2×C411yescharacteristic
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C22×C3×Dic3—21no—
48C22×C3×Dic3D411yescharacteristic
49C22×C3×Dic3D411yescharacteristic
50C2×C3×Dic6—21no—
51C2×C3×Dic6C2×C411yescharacteristic
52C2×C32×C8—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×C411yescharacteristic
55C32×C22⋊C4—21no—
56C3×#217—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3C2×C411yescharacteristic
59C3×Dic3⋊C4—21no—
60C3×Dic3⋊C4—21no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×Dic3⋊C4—21no—
64C3×C4×Dic3—24no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3—21no—
67C3×C4×Dic3C2×C411yescharacteristic
68C3×C4×Dic3C2×C411yescharacteristic
69C3×C4×Dic3D411yescharacteristic
70C3×C4×Dic3D411yescharacteristic
71C2×C3×C3⋊C8—21no—
72#1506—31noSylow
73C3×C4⋊Q8—31no—
74C3×C4⋊Q8—31no—
75C3×#125—61no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—31no—
78C3×C22⋊Q8—61no—
79C3×C22⋊Q8C2×D311yescharacteristic
80C3×C22⋊Q8C22×C311yescharacteristic
81C3×C4×Q8—31no—
82C3×C4×Q8—31no—
83C3×C4×Q8—61no—
84C3×#117—61no—
85C2×C3×C42—31no—
86C2×C3×C42—61no—
87Q8×Dic3C3×C411yescharacteristic, complemented
88#944C3×C411yescharacteristic, complemented
89#924C22×C311yescharacteristic
90C2×C4×Dic3—21no—
91C2×C4×Dic3C22×C311yescharacteristic
92#889—21no—
93#888C3×C411yescharacteristic, complemented
94#887—21no—
95#887C3×C411yescharacteristic, complemented
96#878—21no—
97#877—21no—
98C3×C4⋊C8—61no—
99C3×C22⋊C8—21no—
100C3×C22⋊C8—31no—
101C3×C22⋊C8C2×D311yescharacteristic
102C3×C22⋊C8C22×C311yescharacteristic
103C3×#97—61no—
104C3×#95—21no—
105C3×#95—31no—
106C3×#95—61no—
107C3×#95C2×D311yescharacteristic
108C3×#95C22×C311yescharacteristic
109C3⋊#95—21no—
110C3⋊#95C22×C311yescharacteristic
111#830C22×C311yescharacteristic
112#815—21no—
113Dic3⋊C8—21no—
114C23×C32C2×D411yescharacteristic
115C32×Q8—41no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4C22×C411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3C22⋊C411yescharacteristic
133C2×C3×Dic3C22⋊C411yescharacteristic
134C2×C3×Dic3C22⋊C411yescharacteristic
135C2×C3×Dic3C22⋊C411yescharacteristic
136C3×Dic6—42no—
137C3×Dic6—41no—
138C32×C8—41no—
139C3×C3⋊C8—41no—
140#533—31no—
141#432—31no—
142#376—31no—
143#328—31no—
144#328—31no—
145#328—31no—
146#328C3×D311yescharacteristic
147C2×C3×Q8—21no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8C4×D311yescharacteristic
151C2×C3×Q8C2×C3×C411yescharacteristic
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—41no—
158C22×C3×C4—61no—
159C22×C3×C4C22×D311yescharacteristic
160C22×C3×C4C23×C311yescharacteristic
161C22×Dic3—21no—
162C22×Dic3C3×D411yescharacteristic
163C22×Dic3C3×D411yescharacteristic
164C2×Dic6—21no—
165C2×Dic6C2×C3×C411yescharacteristic
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—41no—
169C2×C3×C8—61no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—61no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4C4×D311yescharacteristic
182C3×C4⋊C4C2×C3×C411yescharacteristic
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—61no—
187C3×C42—31no—
188C3×C42—31no—
189C3×C42—31no—
190C3×C42—31no—
191C3×C42—61no—
192C3×C42—61no—
193C3×C42—64no—
194#217—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3C2×C3×C411yescharacteristic
197Dic3⋊C4—21no—
198Dic3⋊C4—21no—
199Dic3⋊C4—21no—
200Dic3⋊C4—21no—
201Dic3⋊C4—21no—
202C4×Dic3—24no—
203C4×Dic3—21no—
204C4×Dic3—21no—
205C4×Dic3C2×C3×C411yescharacteristic
206C4×Dic3C2×C3×C411yescharacteristic
207C4×Dic3C3×D411yescharacteristic
208C4×Dic3C3×D411yescharacteristic
209C2×C3⋊C8—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32C2×C22⋊C411yescharacteristic
213C22×C32#13711yescharacteristic
214C22×C32#13711yescharacteristic
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—41no—
220C3×Dic3—21no—
221C3×Dic3—21no—
222C3×Dic3—21no—
223C3×Dic3—42no—
224C3×Dic3—41no—
225C3×Dic3—41no—
226C3×Dic3#10412yes—
227C4⋊Q8—31no—
228C4⋊Q8—31no—
229#125—61no—
230C22⋊Q8—31no—
231C22⋊Q8—61no—
232C22⋊Q8C2×C3×D311yescharacteristic
233C4×Q8—31no—
234C4×Q8—31no—
235C4×Q8—61no—
236#117—61no—
237C2×C42—31no—
238C2×C42—61no—
239C4⋊C8—61no—
240C22⋊C8—31no—
241C22⋊C8C2×C3×D311yescharacteristic
242#97—61no—
243#95—31no—
244#95—61no—
245#95C2×C3×D311yescharacteristic
246C23×C3—21no—
247C23×C3D3×D411yescharacteristic
248C23×C3C2×C3×D411yescharacteristic
249C3×Q8—41no—
250C3×Q8—41no—
251C3×Q8—81no—
252C3×Q8—121no—
253C3×Q8—122no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—62no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—62no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4C2×C4×D311yescharacteristic
287C2×C3×C4D3×D411yescharacteristic
288C2×C3×C4C22×C3×C411yescharacteristic
289C2×C3×C4C2×C3×D411yescharacteristic
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3C3×C22⋊C411yescharacteristic
299C2×Dic3C3×C22⋊C411yescharacteristic
300C2×Dic3C3×C22⋊C411yescharacteristic
301C2×Dic3C3×C22⋊C411yescharacteristic
302Dic6—42no—
303Dic6—41no—
304C3×C8—41no—
305C3×C8—41no—
306C3×C8—81no—
307C3×C8—121no—
308C3⋊C8—41no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#41011yescharacteristic
312C2×C32#41911yescharacteristic
313C2×C32C2×#10411yescharacteristic
314C2×Q8—31no—
315C2×Q8—61no—
316C2×Q8C3×C4×D311yescharacteristic
317C22×C4—21no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—61no—
321C22×C4C22×C3×D311yescharacteristic
322C2×C8—21no—
323C2×C8—61no—
324C4⋊C4—21no—
325C4⋊C4—31no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4C3×C4×D311yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—61no—
335C42—31no—
336C42—31no—
337C42—31no—
338C42—31no—
339C42—61no—
340C42—61no—
341C42—64no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3D3×C22⋊C411yescharacteristic
352C22×C3#91511yescharacteristic
353C22×C3#91811yescharacteristic
354C22×C3C2×C3×C22⋊C411yescharacteristic
355C22×C3C3×#13711yescharacteristic
356C22×C3C3×#13711yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—32no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—81no—
374C3×C4—81no—
375C3×C4—82no—
376C3×C4—122no—
377C3×C4—121no—
378C3×C4—121no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—42no—
383Dic3—41no—
384Dic3—41no—
385Dic3C3×#10412yes—
386C32#150611yescharacteristic, complemented, Sylow
387C23C3×D3×D411yescharacteristic
388Q8—41no—
389Q8—121no—
390Q8—122no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—31no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—62no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4C2×C3×C4×D311yescharacteristic
410C2×C4C3×D3×D411yescharacteristic
411C8—41no—
412C8—121no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3#474011yescharacteristic
423C2×C3#480211yescharacteristic
424C2×C3D3×#10411yescharacteristic
425C2×C3C3×#41011yescharacteristic
426C2×C3C3×#41911yescharacteristic
427C2×C3C2×C3×#10411yescharacteristic
428C22—21no—
429C22—21no—
430C22C3×D3×C22⋊C411yescharacteristic
431C22C3×#91511yescharacteristic
432C22C3×#91811yescharacteristic
433C4—21no—
434C4—21no—
435C4—32no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—61no—
440C4—61no—
441C4—61no—
442C4—122no—
443C4—121no—
444C4—121no—
445C3—21no—
446C3#6831011yescharacteristic, direct product factor
447C3C3×#150611yescharacteristic, complemented
448C2—21no—
449C2—21no—
450C2C3×#474011yescharacteristic
451C2C3×#480211yescharacteristic
452C2C3×D3×#10411yescharacteristic
subgroupG/H
centerC22×C3D3×C22⋊C4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#328C2
socleC22×C32C2×C22⋊C4
prime radicalC23×C32C2×D4
cosocleC22×C3×C4C23×C3
hypercenterC3×#328D3
hypocenterC3C3×#1506
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#328

Complements

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

NH ≅ G/Ntype
9. #68310C3direct (N × H)
11. C3×Q8×Dic3C4semidirect
12. C3×#944C4semidirect
17. C3×#888C4semidirect
19. C3×#887C4semidirect
87. Q8×Dic3C3×C4semidirect
88. #944C3×C4semidirect
93. #888C3×C4semidirect
95. #887C3×C4semidirect
386. C32#1506semidirect
446. C3#68310direct (N × H)
447. C3C3×#1506semidirect

Maximal subgroups

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

subgroupindex
C3×#56222
C3×#47192
C3×#46482
C32×#3282
C3×#45382
C3×#44672
C3×#44522
C3×#15063
#683103