Finite Group Databasebeta

← C3×#71202 #11782048

Subgroups — C3×#71202 #11782048

Order 1152.

399 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#486C211yesmaximal, characteristic, complemented
2C3×#5084C211yesmaximal, characteristic, complemented
3C3×#5072C211yesmaximal, characteristic
4C3×#4688C211yesmaximal, characteristic, complemented
5C3×#4546C211yesmaximal, characteristic, complemented
6C3×#4544C211yesmaximal, characteristic, complemented
7C3×#4458C211yesmaximal, characteristic, complemented
8C3×#1697—31nomaximal
9#71202C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×#127C2211yescharacteristic
12C2×C3×#213—21no—
13C2×C3×ΓL(1,𝔽25)—21no—
14C3×C4×D12C2211yescharacteristic
15C32×#102—21no—
16C32×#102—21no—
17C32×C4×C8C2211yescharacteristic
18C3×#832—21no—
19C3×(C2×D3)⋊C8—21no—
20C3×#809—21no—
21C3×#808—21no—
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25C3×#486—21no—
26C3×#486D311yescharacteristic, complemented
27C3×#486C2×C311yescharacteristic, complemented
28C3×#476—31no—
29C3×#459—31no—
30C3×#436—31no—
31#5084C2×C311yescharacteristic, complemented
32#5072C2×C311yescharacteristic
33#4688C2×C311yescharacteristic, complemented
34C3×#335—31no—
35C3×#333—31no—
36C3×#331—31no—
37#4546C2×C311yescharacteristic, complemented
38#4544C2×C311yescharacteristic, complemented
39#4458C2×C311yescharacteristic, complemented
40C2×C32×Q8—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C32×C8D411yescharacteristic
47C2×C32×C8D411yescharacteristic
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C42C2311yescharacteristic
51C3×#213—42no—
52C3×(C2×D3)⋊C4—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57C3×ΓL(1,𝔽25)—42no—
58#1697—31noSylow
59C2×C3×D8—61no—
60C2×C3×M16—61no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C22×C311yescharacteristic
64C3×#127—21no—
65C3×#127C2×D311yescharacteristic
66C3×#127C22×C311yescharacteristic
67C3×C4×D4—31no—
68C2×#213—21no—
69C2×ΓL(1,𝔽25)—21no—
70C4×D12C22×C311yescharacteristic
71C3×#106—61no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8—31no—
75C3×#102—22no—
76C3×#102—22no—
77C3×#102—21no—
78C3×#102—21no—
79C3×#102—21no—
80C3×#102—21no—
81C3×#102—61no—
82C3×#102—61no—
83C3×C22⋊C8—61no—
84C3×C4×C8—21no—
85C3×C4×C8C2×D311yescharacteristic
86C3×C4×C8C22×C311yescharacteristic
87#832—21no—
88(C2×D3)⋊C8—21no—
89#809—21no—
90#808—21no—
91C4⋊(C3⋊C8)C22×C311yescharacteristic
92C4⋊(C3⋊C8)C22×C311yescharacteristic
93C4⋊(C3⋊C8)C22×C311yescharacteristic
94C23×C32—41no—
95C22×C3×D3—21no—
96C32×Q8—81no—
97C32×D4—41no—
98C32×D4—41no—
99C32×D4—41no—
100C32×D4—41no—
101C32×D4—81no—
102C2×C32×C4—41no—
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4#4111yescharacteristic
106C2×C3×Dic3—21no—
107C3×D12—21no—
108C3×D12—21no—
109C3×C4×D3—41no—
110C3×C4×D3—41no—
111C32×C8—21no—
112C32×C8—21no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115C3×C3⋊C8—41no—
116#486C3×D311yescharacteristic, complemented
117#476—31no—
118#459—31no—
119#436—31no—
120#335—31no—
121#333—31no—
122#331—31no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—41no—
126C2×C3×D4—22no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—31no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C22×C3×C4—61no—
137C2×D12C3×D411yescharacteristic
138C2×C4×D3—21no—
139C3×D8—122no—
140C3×M16—122no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—61no—
144C2×C3×C8—61no—
145C2×C3×C8—61no—
146C2×C3×C8#6711yescharacteristic
147C2×C3×C8#6711yescharacteristic
148C2×C3×C8C3×D411yescharacteristic
149C2×C3×C8C3×D411yescharacteristic
150C3×C4⋊C4—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—61no—
158C3×C42—21no—
159C3×C42C22×D311yescharacteristic
160C3×C42C23×C311yescharacteristic
161#213—42no—
162(C2×D3)⋊C4—21no—
163C4⋊Dic3C3×D411yescharacteristic
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167ΓL(1,𝔽25)—42no—
168C22×C32—81no—
169C22×C32—81no—
170C22×C32—81no—
171C22×C32#12111yescharacteristic
172C2×C3×D3—41no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4—21no—
176C32×C4—81no—
177C32×C4#13511yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C32×C4AGL(1,ℤ/8)11yescharacteristic
180C32×C4#13711yescharacteristic
181C3×Dic3—41no—
182C2×D8—61no—
183C2×M16—61no—
184C4⋊Q8C2×C3×D311yescharacteristic
185#127C2×C3×D311yescharacteristic
186C4×D4—31no—
187#106—61no—
188C4⋊C8—31no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191#102—21no—
192#102—21no—
193#102—61no—
194#102—61no—
195C22⋊C8—61no—
196C4×C8C2×C3×D311yescharacteristic
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—61no—
201C22×D3—21no—
202C3×Q8—81no—
203C3×Q8—81no—
204C3×Q8—82no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—41no—
208C3×D4—41no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—61no—
216C3×D4—61no—
217C3×D4—81no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—82no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—61no—
229C2×C3×C4—121no—
230C2×C3×C4—121no—
231C2×C3×C4D3×D411yescharacteristic
232C2×C3×C4#23711yescharacteristic
233C2×C3×C4C2×#6711yescharacteristic
234C2×C3×C4C2×C3×D411yescharacteristic
235C2×C3×C4C2×C3×D411yescharacteristic
236C2×C3×C4C3×#4111yescharacteristic
237C2×Dic3—21no—
238D12—21no—
239D12—21no—
240C4×D3—41no—
241C4×D3—41no—
242C3×C8—21no—
243C3×C8—21no—
244C3×C8—21no—
245C3×C8—21no—
246C3×C8—41no—
247C3×C8—41no—
248C3×C8—121no—
249C3×C8—121no—
250C3×C8—121no—
251C3⋊C8—41no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C2×C32—161no—
255C2×C32#46311yescharacteristic
256C2×C32#46811yescharacteristic
257C2×C32#47211yescharacteristic
258C3×D3—81no—
259C2×Q8—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—31no—
264C22×C4—61no—
265D8—122no—
266M16—122no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8C3×#6711yescharacteristic
271C2×C8C3×#6711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—31no—
275C22⋊C4—61no—
276C42C22×C3×D311yescharacteristic
277C22×C3—21no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—82no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—121no—
290C22×C3#93811yescharacteristic
291C22×C3C3×#12111yescharacteristic
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—82no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4—121no—
306C3×C4#91411yescharacteristic
307C3×C4#91511yescharacteristic
308C3×C4#94911yescharacteristic
309C3×C4#95011yescharacteristic
310C3×C4C3×#13511yescharacteristic
311C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
312C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
313C3×C4C3×#13711yescharacteristic
314Dic3—41no—
315C32#169711yescharacteristic, complemented, Sylow
316C23—41no—
317C23—61no—
318Q8—81no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—61no—
324D4—61no—
325D4—81no—
326C2×C4—41no—
327C2×C4—61no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4C3×D3×D411yescharacteristic
331C2×C4C3×#23711yescharacteristic
332C2×C4C2×C3×#6711yescharacteristic
333C8—21no—
334C8—21no—
335C8—121no—
336C8—121no—
337C8—121no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—162no—
342C2×C3—161no—
343C2×C3—161no—
344C2×C3—241no—
345C2×C3#505711yescharacteristic
346C2×C3#516011yescharacteristic
347C2×C3#517411yescharacteristic
348C2×C3C3×#46311yescharacteristic
349C2×C3C3×#46811yescharacteristic
350C2×C3C3×#47211yescharacteristic
351D3—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22C3×#93811yescharacteristic
359C4—21no—
360C4—81no—
361C4—121no—
362C4C3×#91411yescharacteristic
363C4C3×#91511yescharacteristic
364C4C3×#94911yescharacteristic
365C4C3×#95011yescharacteristic
366C3—21no—
367C3#7120211yescharacteristic, direct product factor
368C3C3×#169711yescharacteristic, complemented
369C2—161no—
370C2—241no—
371C2C3×#505711yescharacteristic
372C2C3×#516011yescharacteristic
373C2C3×#517411yescharacteristic
subgroupG/H
centerC22×C3#938
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#486C2
socleC22×C32#121
prime radicalC3×#5072C2
hypercenterC3×#486D3
hypocenterC3C3×#1697
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#486

Complements

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

NH ≅ G/Ntype
1. C32×#486C2semidirect
2. C3×#5084C2semidirect
4. C3×#4688C2semidirect
5. C3×#4546C2semidirect
6. C3×#4544C2semidirect
7. C3×#4458C2semidirect
9. #71202C3direct (N × H)
26. C3×#486D3semidirect
27. C3×#486C2×C3semidirect
31. #5084C2×C3semidirect
33. #4688C2×C3semidirect
37. #4546C2×C3semidirect
38. #4544C2×C3semidirect
39. #4458C2×C3semidirect
116. #486C3×D3semidirect
315. C32#1697semidirect
367. C3#71202direct (N × H)
368. C3C3×#1697semidirect

Maximal subgroups

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

subgroupindex
C32×#4862
C3×#50842
C3×#50722
C3×#46882
C3×#45462
C3×#45442
C3×#44582
C3×#16973
#712023