Finite Group Databasebeta

← C3×#71156 #11779409

Subgroups — C3×#71156 #11779409

Order 1152.

476 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5579C211yesmaximal, characteristic
2C32×#477C211yesmaximal, characteristic, complemented
3C3×#5092C211yesmaximal, characteristic, complemented
4C3×#5084C211yesmaximal, characteristic
5C3×#4535C211yesmaximal, characteristic, complemented
6C3×#4483C211yesmaximal, characteristic
7C3×#4481C211yesmaximal, characteristic, complemented
8C3×#1632—31nomaximal
9#71156C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4×Q8C2211yescharacteristic
12C3×#946—21no—
13C2×C3×#216—22no—
14C3×#896—22no—
15C3×D3×C4⋊C4—21no—
16C3×C4×D12—21no—
17C3×C4×D12C2211yescharacteristic
18C3×#870—21no—
19C3×#869C2211yescharacteristic
20C32×#107—21no—
21C32×C4⋊C8C2211yescharacteristic
22C32×#103—21no—
23C3×#809—21no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27C3×#554—31no—
28#5579C2×C311yescharacteristic
29C3×#494—31no—
30C3×#477—21no—
31C3×#477D311yescharacteristic, complemented
32C3×#477C2×C311yescharacteristic, complemented
33C3×#476—31no—
34#5092C2×C311yescharacteristic, complemented
35#5084C2×C311yescharacteristic
36C3×#333—31no—
37C3×Q8⋊C8—31no—
38C3×D4⋊C8—31no—
39#4535C2×C311yescharacteristic, complemented
40#4483C2×C311yescharacteristic
41#4481C2×C311yescharacteristic, complemented
42C2×C32×Q8—21no—
43C2×C32×Q8D411yescharacteristic
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—22no—
47C2×C3×Dic6D412yes—
48C2×C32×C8—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4D411yescharacteristic
53C32×C42—21no—
54C32×C42C2311yescharacteristic
55C3×#216—44no—
56C3×(C2×D3)⋊C4—22no—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—22no—
62C3×Dic3⋊C4—22no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65#1632—31noSylow
66C2×C3×Q16—62no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—31no—
69C3×C4⋊Q8C2×D311yescharacteristic
70C3×C4⋊Q8C22×C311yescharacteristic
71C3×#125—61no—
72C3×C22⋊Q8—62no—
73C3×C22⋊Q8—61no—
74C3×C4×Q8—21no—
75C3×C4×Q8C2×D311yescharacteristic
76C3×C4×Q8C22×C311yescharacteristic
77C3×C4×D4—31no—
78C3×C4×D4—61no—
79C2×C3×C4⋊C4—61no—
80#946—21no—
81C2×#216—22no—
82#896—22no—
83D3×C4⋊C4—21no—
84C4×D12—21no—
85C4×D12C22×C311yescharacteristic
86#870—21no—
87#869C22×C311yescharacteristic
88C3×#107—22no—
89C3×#107—21no—
90C3×#107—21no—
91C3×#106—61no—
92C3×C4⋊C8—21no—
93C3×C4⋊C8—31no—
94C3×C4⋊C8C2×D311yescharacteristic
95C3×C4⋊C8C22×C311yescharacteristic
96C3×#103—22no—
97C3×#103—21no—
98C3×#103—21no—
99C3×#102—61no—
100C3×C4×C8—31no—
101#809—21no—
102#808—21no—
103C4⋊(C3⋊C8)C22×C311yescharacteristic
104C4×C3⋊C8C22×C311yescharacteristic
105C22×C3×D3—21no—
106C32×Q8—21no—
107C32×Q8—21no—
108C32×Q8—42no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3×Dic3—21no—
116C2×C3×Dic3—22no—
117C3×D12—21no—
118C3×D12Q1612yescomplemented
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×C4×D3—82no—
122C3×Dic6—42no—
123C32×C8—41no—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—41no—
126#554—31no—
127#494—31no—
128#477C3×D311yescharacteristic, complemented
129#476—31no—
130#333—31no—
131Q8⋊C8—31no—
132D4⋊C8—31no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—32no—
137C2×C3×Q8—41no—
138C2×C3×Q8#6711yescharacteristic
139C2×C3×Q8C3×D411yescharacteristic
140C2×C3×D4—31no—
141C22×C3×C4—62no—
142C22×C3×C4—61no—
143C2×D12C3×D411yescharacteristic
144C2×C4×D3—21no—
145C2×C4×D3—22no—
146C2×Dic6C3×D412yes—
147C3×Q16—124no—
148C2×C3×C8—22no—
149C2×C3×C8—21no—
150C2×C3×C8—21no—
151C2×C3×C8—61no—
152C2×C3×C8—61no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—62no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4#6711yescharacteristic
169C3×C4⋊C4C3×D411yescharacteristic
170C3×C22⋊C4—62no—
171C3×C22⋊C4—61no—
172C3×C42—21no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—41no—
176C3×C42C22×D311yescharacteristic
177C3×C42C23×C311yescharacteristic
178#216—44no—
179(C2×D3)⋊C4—22no—
180(C2×D3)⋊C4—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—21no—
183C4⋊Dic3C3×D411yescharacteristic
184Dic3⋊C4—22no—
185Dic3⋊C4—22no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C22×C32#12011yescharacteristic
189C2×C3×D3—22no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C32×C4—21no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—81no—
196C32×C4C2×Q1611yescharacteristic
197C32×C4#13511yescharacteristic
198C32×C4#13711yescharacteristic
199C32×C4#13711yescharacteristic
200C3×Dic3—41no—
201C3×Dic3—42no—
202C2×Q16—62no—
203C4⋊Q8—31no—
204C4⋊Q8C2×C3×D311yescharacteristic
205#125—61no—
206C22⋊Q8—62no—
207C22⋊Q8—61no—
208C4×Q8C2×C3×D311yescharacteristic
209C4×D4—31no—
210C4×D4—61no—
211C2×C4⋊C4—61no—
212#107—21no—
213#106—61no—
214C4⋊C8—31no—
215C4⋊C8C2×C3×D311yescharacteristic
216#103—21no—
217#102—61no—
218C4×C8—31no—
219C23×C3—61no—
220C22×D3—21no—
221C3×Q8—22no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—41no—
227C3×Q8—42no—
228C3×Q8—42no—
229C3×Q8—82no—
230C3×Q8—122no—
231C3×D4—32no—
232C3×D4—61no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—61no—
246C2×C3×C4—62no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4—242no—
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4C2×#6711yescharacteristic
253C2×C3×C4C2×C3×D411yescharacteristic
254C2×C3×C4C2×C3×D411yescharacteristic
255C2×C3×C4C2×C3×D411yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—22no—
258D12—21no—
259D12C3×Q1612yescomplemented
260C4×D3—41no—
261C4×D3—41no—
262C4×D3—82no—
263Dic6—42no—
264C3×C8—42no—
265C3×C8—41no—
266C3×C8—41no—
267C3×C8—62no—
268C3×C8—121no—
269C3⋊C8—22no—
270C3⋊C8—41no—
271C2×C32#45111yescharacteristic
272C2×C32#45211yescharacteristic
273C2×C32#45311yescharacteristic
274C3×D3—42no—
275C2×Q8—21no—
276C2×Q8—32no—
277C2×Q8C3×#6711yescharacteristic
278C2×D4—31no—
279C22×C4—62no—
280C22×C4—61no—
281Q16—124no—
282C2×C8—21no—
283C2×C8—61no—
284C2×C8—61no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—31no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4C3×#6711yescharacteristic
294C22⋊C4—62no—
295C22⋊C4—61no—
296C42—21no—
297C42C22×C3×D311yescharacteristic
298C22×C3—21no—
299C22×C3—62no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3#93711yescharacteristic
303C22×C3C3×#12011yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—42no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—81no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—122no—
324C3×C4—121no—
325C3×C4D3×Q1611yescharacteristic
326C3×C4#91811yescharacteristic
327C3×C4#94211yescharacteristic
328C3×C4#95011yescharacteristic
329C3×C4C2×C3×Q1611yescharacteristic
330C3×C4C3×#13511yescharacteristic
331C3×C4C3×#13711yescharacteristic
332C3×C4C3×#13711yescharacteristic
333Dic3—41no—
334Dic3—42no—
335C32#163211yescharacteristic, complemented, Sylow
336C23—61no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—122no—
341D4—32no—
342D4—61no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—61no—
347C2×C4—62no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—242no—
351C2×C4C3×D3×D411yescharacteristic
352C2×C4C3×D3×D411yescharacteristic
353C2×C4C2×C3×#6711yescharacteristic
354C8—41no—
355C8—62no—
356C8—121no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3#504711yescharacteristic
362C2×C3#518611yescharacteristic
363C2×C3#519811yescharacteristic
364C2×C3C3×#45111yescharacteristic
365C2×C3C3×#45211yescharacteristic
366C2×C3C3×#45311yescharacteristic
367D3—42no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22C3×#93711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—122no—
377C4—121no—
378C4C3×D3×Q1611yescharacteristic
379C4C3×#91811yescharacteristic
380C4C3×#94211yescharacteristic
381C4C3×#95011yescharacteristic
382C3—21no—
383C3#7115611yescharacteristic, direct product factor
384C3C3×#163211yescharacteristic, complemented
385C2—122no—
386C2C3×#504711yescharacteristic
387C2C3×#518611yescharacteristic
388C2C3×#519811yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#477C2
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC3×#477D3
hypocenterC3C3×#1632
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#477

Complements

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

NH ≅ G/Ntype
2. C32×#477C2semidirect
3. C3×#5092C2semidirect
5. C3×#4535C2semidirect
7. C3×#4481C2semidirect
9. #71156C3direct (N × H)
31. C3×#477D3semidirect
32. C3×#477C2×C3semidirect
34. #5092C2×C3semidirect
39. #4535C2×C3semidirect
41. #4481C2×C3semidirect
118. C3×D12Q16semidirect
128. #477C3×D3semidirect
259. D12C3×Q16semidirect
335. C32#1632semidirect
383. C3#71156direct (N × H)
384. C3C3×#1632semidirect

Maximal subgroups

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

subgroupindex
C3×#55792
C32×#4772
C3×#50922
C3×#50842
C3×#45352
C3×#44832
C3×#44812
C3×#16323
#711563