Finite Group Databasebeta

← C3×#71120 #11781663

Subgroups — C3×#71120 #11781663

Order 1152.

384 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#475C211yesmaximal, characteristic, complemented
2C3×#5085C211yesmaximal, characteristic
3C3×#5027C211yesmaximal, characteristic
4C3×#5024C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6C3×#4485C211yesmaximal, characteristic
7C3×#4479C211yesmaximal, characteristic, complemented
8C3×#1686—31nomaximal
9#71120C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4×Q8C2211yescharacteristic
12C2×C3×#215—21no—
13C2×C3×#215—21no—
14C2×C3×#215—21no—
15C3×#874C2211yescharacteristic
16C32×#106—21no—
17C32×C4⋊C8C2211yescharacteristic
18C32×#103—21no—
19C3×#809—21no—
20C3×C4⋊(C3⋊C8)—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×#803—21no—
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#492—31no—
26C3×#475—21no—
27C3×#475D311yescharacteristic, complemented
28C3×#475C2×C311yescharacteristic, complemented
29C3×#460—31no—
30C3×#446—31no—
31#5085C2×C311yescharacteristic
32#5027C2×C311yescharacteristic
33#5024C2×C311yescharacteristic, complemented
34C3×#335—31no—
35C3×#332—31no—
36C3×#331—31no—
37#4547C2×C311yescharacteristic, complemented
38#4485C2×C311yescharacteristic
39#4479C2×C311yescharacteristic, complemented
40C2×C32×Q8—21no—
41C2×C32×Q8D411yescharacteristic
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4D411yescharacteristic
50C32×C42—21no—
51C32×C42C2311yescharacteristic
52C3×#215—42no—
53C3×#215—42no—
54C3×#215—42no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57C2×C3×C3⋊C8D411yescharacteristic
58C2×C3×C3⋊C8D411yescharacteristic
59#1686—31noSylow
60C2×C3×SD16—61no—
61C2×C3×SD16—61no—
62C2×C3×SD16—61no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×C4⋊Q8C22×C311yescharacteristic
66C3×#127—31no—
67C3×C4×Q8—21no—
68C3×C4×Q8C2×D311yescharacteristic
69C3×C4×Q8C22×C311yescharacteristic
70C2×#215—21no—
71C2×#215—21no—
72C2×#215—21no—
73#874C22×C311yescharacteristic
74C3×#106—22no—
75C3×#106—21no—
76C3×#106—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—31no—
79C3×C4⋊C8—31no—
80C3×C4⋊C8—61no—
81C3×C4⋊C8C2×D311yescharacteristic
82C3×C4⋊C8C22×C311yescharacteristic
83C3×#103—22no—
84C3×#103—21no—
85C3×#103—21no—
86C3×#102—61no—
87C3×#97—61no—
88C3×C4×C8—31no—
89#809—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)C22×C311yescharacteristic
92C4⋊(C3⋊C8)C22×C311yescharacteristic
93#803—21no—
94C4×C3⋊C8C22×C311yescharacteristic
95C22×C3×D3—41no—
96C32×Q8—21no—
97C32×Q8—21no—
98C32×Q8—41no—
99C32×Q8—41no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4#4111yescharacteristic
106C3×D12—41no—
107C3×D12—41no—
108C3×D12—41no—
109C3×D12—41no—
110C3×D12—81no—
111C32×C8—41no—
112C3×C3⋊C8—21no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115C3×C3⋊C8SD1612yescomplemented
116#492—31no—
117#475C3×D311yescharacteristic, complemented
118#460—31no—
119#446—31no—
120#335—31no—
121#332—31no—
122#331—31no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—41no—
127C2×C3×Q8#6711yescharacteristic
128C2×C3×Q8C3×D411yescharacteristic
129C2×C3×D4—61no—
130C2×C3×D4—61no—
131C2×C3×D4—61no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C3×SD16—122no—
136C3×SD16—122no—
137C3×SD16—122no—
138C2×C3×C8—22no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C2×C3×C8—61no—
144C2×C3×C8—61no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4#6711yescharacteristic
156C3×C4⋊C4C3×D411yescharacteristic
157C3×C42—21no—
158C3×C42—21no—
159C3×C42—21no—
160C3×C42—41no—
161C3×C42C22×D311yescharacteristic
162C3×C42C23×C311yescharacteristic
163#215—42no—
164#215—42no—
165#215—42no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8C3×D411yescharacteristic
169C2×C3⋊C8C3×D411yescharacteristic
170C22×C32#12111yescharacteristic
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—21no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—81no—
178C32×C4C2×SD1611yescharacteristic
179C32×C4AGL(1,ℤ/8)11yescharacteristic
180C32×C4AGL(1,ℤ/8)11yescharacteristic
181C32×C4#13711yescharacteristic
182C2×SD16—61no—
183C2×SD16—61no—
184C2×SD16—61no—
185C4⋊Q8C2×C3×D311yescharacteristic
186#127—31no—
187C4×Q8C2×C3×D311yescharacteristic
188#106—21no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—61no—
192C4⋊C8C2×C3×D311yescharacteristic
193#103—21no—
194#102—61no—
195#97—61no—
196C4×C8—31no—
197C23×C3—121no—
198C22×D3—41no—
199C3×Q8—22no—
200C3×Q8—21no—
201C3×Q8—21no—
202C3×Q8—21no—
203C3×Q8—21no—
204C3×Q8—41no—
205C3×Q8—41no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×D4—121no—
212C3×D4—121no—
213C3×D4—121no—
214C3×D4—121no—
215C3×D4—241no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4D3×D411yescharacteristic
229C2×C3×C4#23711yescharacteristic
230C2×C3×C4C2×#6711yescharacteristic
231C2×C3×C4C2×C3×D411yescharacteristic
232C2×C3×C4C2×C3×D411yescharacteristic
233C2×C3×C4C3×#4111yescharacteristic
234D12—41no—
235D12—41no—
236D12—41no—
237D12—41no—
238D12—81no—
239C3×C8—32no—
240C3×C8—42no—
241C3×C8—41no—
242C3×C8—41no—
243C3×C8—61no—
244C3×C8—121no—
245C3×C8—121no—
246C3⋊C8—21no—
247C3⋊C8—41no—
248C3⋊C8—41no—
249C3⋊C8C3×SD1612yescomplemented
250C2×C32#46011yescharacteristic
251C2×C32#46811yescharacteristic
252C2×C32#47111yescharacteristic
253C3×D3—161no—
254C2×Q8—21no—
255C2×Q8C3×#6711yescharacteristic
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259SD16—122no—
260SD16—122no—
261SD16—122no—
262C2×C8—21no—
263C2×C8—31no—
264C2×C8—31no—
265C2×C8—61no—
266C2×C8—61no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C3×#6711yescharacteristic
271C42—21no—
272C42C22×C3×D311yescharacteristic
273C22×C3—21no—
274C22×C3—241no—
275C22×C3—241no—
276C22×C3—241no—
277C22×C3#93911yescharacteristic
278C22×C3C3×#12111yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—81no—
295C3×C4—82no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4D3×SD1611yescharacteristic
299C3×C4#91411yescharacteristic
300C3×C4#94211yescharacteristic
301C3×C4#94911yescharacteristic
302C3×C4C2×C3×SD1611yescharacteristic
303C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
304C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
305C3×C4C3×#13711yescharacteristic
306C32#168611yescharacteristic, complemented, Sylow
307C23—121no—
308Q8—21no—
309Q8—21no—
310Q8—41no—
311Q8—41no—
312D4—121no—
313D4—121no—
314D4—121no—
315D4—121no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4C3×D3×D411yescharacteristic
321C2×C4C3×#23711yescharacteristic
322C2×C4C2×C3×#6711yescharacteristic
323C8—32no—
324C8—41no—
325C8—61no—
326C8—121no—
327C8—121no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—481no—
332C2×C3#504911yescharacteristic
333C2×C3#516311yescharacteristic
334C2×C3#520311yescharacteristic
335C2×C3C3×#46011yescharacteristic
336C2×C3C3×#46811yescharacteristic
337C2×C3C3×#47111yescharacteristic
338D3—161no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22C3×#93911yescharacteristic
343C4—21no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4C3×D3×SD1611yescharacteristic
348C4C3×#91411yescharacteristic
349C4C3×#94211yescharacteristic
350C4C3×#94911yescharacteristic
351C3—21no—
352C3#7112011yescharacteristic, direct product factor
353C3C3×#168611yescharacteristic, complemented
354C2—481no—
355C2C3×#504911yescharacteristic
356C2C3×#516311yescharacteristic
357C2C3×#520311yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#475C2
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC3×#475D3
hypocenterC3C3×#1686
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#475

Complements

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

NH ≅ G/Ntype
1. C32×#475C2semidirect
4. C3×#5024C2semidirect
5. C3×#4547C2semidirect
7. C3×#4479C2semidirect
9. #71120C3direct (N × H)
27. C3×#475D3semidirect
28. C3×#475C2×C3semidirect
33. #5024C2×C3semidirect
37. #4547C2×C3semidirect
39. #4479C2×C3semidirect
115. C3×C3⋊C8SD16semidirect
117. #475C3×D3semidirect
249. C3⋊C8C3×SD16semidirect
306. C32#1686semidirect
352. C3#71120direct (N × H)
353. C3C3×#1686semidirect

Maximal subgroups

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

subgroupindex
C32×#4752
C3×#50852
C3×#50272
C3×#50242
C3×#45472
C3×#44852
C3×#44792
C3×#16863
#711203