Finite Group Databasebeta

← C3×#71842 #11793079

Subgroups — C3×#71842 #11793079

Order 1152.

599 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#533C211yesmaximal, characteristic, complemented
2C3×#5006C211yesmaximal, characteristic
3C3×#5005C211yesmaximal, characteristic
4C2×C3×#805C211yesmaximal, characteristic
5C3×#4533C211yesmaximal, characteristic, complemented
6C3×#4531C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#2031—31nomaximal
9#71842C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C32×C4⋊Q8—21no—
12C32×#125—21no—
13C32×C22⋊Q8—22no—
14C32×C4×Q8—22no—
15C32×#117C2211yescharacteristic
16C32×#117C2211yescharacteristic
17C2×C32×C42C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C3×#806—22no—
22C3×#805—21no—
23C3×#805—21no—
24C3×#805—22no—
25C3×#533—21no—
26C3×#533D311yescharacteristic, complemented
27C3×#533C2×C311yescharacteristic, complemented
28C3×#421—31no—
29C2×C3×#104—31no—
30C3×#410—31no—
31#5006C2×C311yescharacteristic
32#5005C2×C311yescharacteristic
33C2×#805C2×C311yescharacteristic
34C3×#344—31no—
35C3×#339—31no—
36C3×H(3,ℤ/4)—31no—
37#4533C2×C311yescharacteristic, complemented
38#4531C2×C311yescharacteristic, complemented
39#4523C2×C311yescharacteristic, complemented
40C2×C32×Q8—22no—
41C22×C32×C4—21no—
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67—21no—
44C3×#235—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C2×C3×D12Q811yescharacteristic, complemented
49C2×C3×C4×D3—21no—
50C2×C3×Dic6Q811yescharacteristic, complemented
51C32×C4⋊C4—24no—
52C32×C4⋊C4—22no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C22⋊C4—21no—
56C32×C22⋊C4D412yes—
57C32×C42—21no—
58C32×C42—21no—
59C32×C42D411yescharacteristic
60C32×C42D411yescharacteristic
61C32×C42D412yes—
62C3×#217—21no—
63C3×C4⋊Dic3—21no—
64C3×C4×Dic3—21no—
65C3×#208—21no—
66C3×#208—21no—
67C3×#208—21no—
68C3×#208—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#2031—31noSylow
72C2×C3×#41—31no—
73C2×#235C22×C311yescharacteristic
74C2×C3×M16—31no—
75C2×C3×M16—31no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8—21no—
79C3×#125—22no—
80C3×#125—21no—
81C3×#125—21no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C4×Q8—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×#117—21no—
89C3×#117—21no—
90C3×#117—31no—
91C3×#117C2×D311yescharacteristic
92C3×#117C2×D311yescharacteristic
93C3×#117C22×C311yescharacteristic
94C3×#117C22×C311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2×D311yescharacteristic
97C2×C3×C42C22×C311yescharacteristic
98#926C22×C311yescharacteristic
99C2×#208C22×C311yescharacteristic
100C2×#208C22×C311yescharacteristic
101C3×#104—61no—
102C3×#104—61no—
103C3×#104—62no—
104C3×#99—62no—
105#806—22no—
106#805—21no—
107#805—21no—
108#805—22no—
109C23×C32C2×D411yescharacteristic
110C22×C3×D3—21no—
111C32×Q8—44no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—42no—
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×Q811yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C3×#67—41no—
124C3×#67—41no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—41no—
127C3×D12—21no—
128C3×D12—21no—
129C3×C4×D3—41no—
130C3×C4×D3—41no—
131C3×Dic6—21no—
132C3×Dic6—21no—
133C3×C3⋊C8—41no—
134C3×C3⋊C8—41no—
135#533C3×D311yescharacteristic, complemented
136#421—31no—
137C2×#104—31no—
138#410—31no—
139#344—31no—
140#339—31no—
141H(3,ℤ/4)—31no—
142C3×#41—61no—
143C3×#41—61no—
144C3×#41—61no—
145C3×#41—61no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—31no—
150C2×C3×D4—31no—
151C2×C3×D4—61no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—61no—
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C23×C311yescharacteristic
159C2×#67—21no—
160#235—21no—
161#235—21no—
162#235—21no—
163#235—21no—
164C2×D12C3×Q811yescharacteristic, complemented
165C2×C4×D3—21no—
166C2×Dic6C3×Q811yescharacteristic, complemented
167C3×M16—61no—
168C3×M16—61no—
169C3×M16—61no—
170C3×M16—61no—
171C2×C3×C8—61no—
172C2×C3×C8—61no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—61no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—61no—
191C3×C22⋊C4#6712yes—
192C3×C22⋊C4C3×D412yes—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42—22no—
202C3×C42—61no—
203C3×C42#6711yescharacteristic
204C3×C42#6711yescharacteristic
205C3×C42#6712yes—
206C3×C42C3×D411yescharacteristic
207C3×C42C3×D411yescharacteristic
208C3×C42C3×D412yes—
209#217—21no—
210C4⋊Dic3—21no—
211C4×Dic3—21no—
212#208—21no—
213#208—21no—
214#208—21no—
215#208—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C22×C32—41no—
219C22×C32#12011yescharacteristic
220C22×C32C22⋊Q811yescharacteristic
221C22×C32#12411yescharacteristic
222C2×C3×D3—41no—
223C2×C3×D3—81no—
224C32×C4—22no—
225C32×C4—41no—
226C32×C4—42no—
227C32×C4—42no—
228C32×C4C22⋊Q811yescharacteristic
229C32×C4C22⋊Q811yescharacteristic
230C32×C4#12411yescharacteristic
231C32×C4#12411yescharacteristic
232C3×Dic3—41no—
233C3×Dic3—81no—
234C2×#41—31no—
235C2×M16—31no—
236C2×M16—31no—
237C4⋊Q8—21no—
238#125—21no—
239C22⋊Q8—22no—
240C4×Q8—22no—
241#117—31no—
242#117C2×C3×D311yescharacteristic
243#117C2×C3×D311yescharacteristic
244C2×C42C2×C3×D311yescharacteristic
245#104—61no—
246#104—61no—
247#104—62no—
248#99—62no—
249C23×C3—21no—
250C23×C3—61no—
251C23×C3C2×#6711yescharacteristic
252C23×C3C2×C3×D411yescharacteristic
253C22×D3—21no—
254C3×Q8—48no—
255C3×Q8—44no—
256C3×Q8—44no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—121no—
262C3×D4—121no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—44no—
284C2×C3×C4—61no—
285C2×C3×C4—121no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4D3×Q811yescharacteristic
289C2×C3×C4#23911yescharacteristic
290C2×C3×C4#23911yescharacteristic
291C2×C3×C4#23911yescharacteristic
292C2×C3×C4C2×#6711yescharacteristic
293C2×C3×C4C2×#6711yescharacteristic
294C2×C3×C4C2×C3×D411yescharacteristic
295C2×C3×C4C2×C3×D411yescharacteristic
296C2×C3×C4C2×C3×Q811yescharacteristic
297C2×C3×C4C3×#4111yescharacteristic
298C2×C3×C4C3×#4111yescharacteristic
299C2×C3×C4C3×#4111yescharacteristic
300#67—41no—
301#67—41no—
302C2×Dic3—21no—
303C2×Dic3—41no—
304D12—21no—
305D12—21no—
306C4×D3—41no—
307C4×D3—41no—
308Dic6—21no—
309Dic6—21no—
310C3×C8—121no—
311C3×C8—121no—
312C3⋊C8—41no—
313C3⋊C8—41no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32#39311yescharacteristic
318C3×D3—81no—
319#41—61no—
320#41—61no—
321#41—61no—
322#41—61no—
323C2×Q8—22no—
324C2×Q8—31no—
325C2×D4—31no—
326C2×D4—61no—
327C22×C4—21no—
328C22×C4—61no—
329C22×C4C22×C3×D311yescharacteristic
330M16—61no—
331M16—61no—
332M16—61no—
333M16—61no—
334C2×C8—61no—
335C2×C8—61no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—61no—
341C22⋊C4—21no—
342C22⋊C4—61no—
343C22⋊C4C3×#6712yes—
344C42—21no—
345C42—21no—
346C42—61no—
347C42C3×#6711yescharacteristic
348C42C3×#6711yescharacteristic
349C42C3×#6712yes—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3—121no—
357C22×C3—241no—
358C22×C3#94611yescharacteristic
359C22×C3#94711yescharacteristic
360C22×C3#95311yescharacteristic
361C22×C3C3×#12011yescharacteristic
362C22×C3C3×C22⋊Q811yescharacteristic
363C22×C3C3×#12411yescharacteristic
364C2×D3—41no—
365C2×D3—81no—
366C3×C4—24no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—44no—
377C3×C4—44no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—121no—
383C3×C4—241no—
384C3×C4#94611yescharacteristic
385C3×C4#94611yescharacteristic
386C3×C4#94711yescharacteristic
387C3×C4#94711yescharacteristic
388C3×C4C3×C22⋊Q811yescharacteristic
389C3×C4C3×C22⋊Q811yescharacteristic
390C3×C4C3×#12411yescharacteristic
391C3×C4C3×#12411yescharacteristic
392Dic3—41no—
393Dic3—81no—
394C32#203111yescharacteristic, complemented, Sylow
395C23—61no—
396C23C2×C3×#6711yescharacteristic
397Q8—44no—
398Q8—61no—
399Q8—61no—
400D4—61no—
401D4—61no—
402D4—121no—
403D4—121no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—61no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4C3×D3×Q811yescharacteristic
414C2×C4C3×#23911yescharacteristic
415C2×C4C3×#23911yescharacteristic
416C2×C4C3×#23911yescharacteristic
417C2×C4C2×C3×#6711yescharacteristic
418C2×C4C2×C3×#6711yescharacteristic
419C8—121no—
420C8—121no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—241no—
432C2×C3#523111yescharacteristic
433C2×C3C3×#39311yescharacteristic
434D3—81no—
435C22—41no—
436C22—121no—
437C22—241no—
438C22C3×#94611yescharacteristic
439C22C3×#94711yescharacteristic
440C22C3×#95311yescharacteristic
441C4—22no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—121no—
446C4—241no—
447C4C3×#94611yescharacteristic
448C4C3×#94611yescharacteristic
449C4C3×#94711yescharacteristic
450C4C3×#94711yescharacteristic
451C3—21no—
452C3#7184211yescharacteristic, direct product factor
453C3C3×#203111yescharacteristic, complemented
454C2—21no—
455C2—21no—
456C2—21no—
457C2—241no—
458C2C3×#523111yescharacteristic
subgroupG/H
centerC3×C4#947
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#533C2
socleC2×C32#393
prime radicalC2×C3×#235C22
hypercenterC3×#533D3
hypocenterC3C3×#2031
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#533

Complements

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

NH ≅ G/Ntype
1. C32×#533C2semidirect
5. C3×#4533C2semidirect
6. C3×#4531C2semidirect
7. C3×#4523C2semidirect
9. #71842C3direct (N × H)
26. C3×#533D3semidirect
27. C3×#533C2×C3semidirect
37. #4533C2×C3semidirect
38. #4531C2×C3semidirect
39. #4523C2×C3semidirect
48. C2×C3×D12Q8semidirect
50. C2×C3×Dic6Q8semidirect
135. #533C3×D3semidirect
164. C2×D12C3×Q8semidirect
166. C2×Dic6C3×Q8semidirect
394. C32#2031semidirect
452. C3#71842direct (N × H)
453. C3C3×#2031semidirect

Maximal subgroups

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

subgroupindex
C32×#5332
C3×#50062
C3×#50052
C2×C3×#8052
C3×#45332
C3×#45312
C3×#45232
C3×#20313
#718423