Finite Group Databasebeta

← C3×#68496 #11781023

Subgroups — C3×#68496 #11781023

Order 1152.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5084C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic, complemented
3C3×#4841C211yesmaximal, characteristic
4C3×#4834C211yesmaximal, characteristic
5C32×#333C211yesmaximal, characteristic, complemented
6C3×#4486C211yesmaximal, characteristic, complemented
7C3×#4480C211yesmaximal, characteristic, complemented
8C3×#1671—31nomaximal
9#68496C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#204—21no—
12C2×C3×D3×C8—21no—
13C3×C4×D12C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C32×C4⋊C8C2211yescharacteristic
16C32×C4⋊C8C2211yescharacteristic
17C3×(C2×D3)⋊C8—21no—
18C3×#816—21no—
19C3×#809—21no—
20C3×#808—21no—
21C3×#808—21no—
22C3×#808—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#498—31no—
26C3×#476—31no—
27C3×#461—31no—
28C3×C8×D4—31no—
29#5084C2×C311yescharacteristic
30#5080C2×C311yescharacteristic, complemented
31#4841C2×C311yescharacteristic
32#4834C2×C311yescharacteristic
33C3×#334—31no—
34C3×#333—21no—
35C3×#333D311yescharacteristic, complemented
36C3×#333C2×C311yescharacteristic, complemented
37C3×#332—31no—
38#4486C2×C311yescharacteristic, complemented
39#4480C2×C311yescharacteristic, complemented
40C2×C32×Q8—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×#204—42no—
57C3×D3×C8—42no—
58#1671—31noSylow
59C2×C3×SD16—61no—
60C22×C3×C8—61no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—31no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C22×C311yescharacteristic
65C3×C4×D4—31no—
66C2×#204—21no—
67C2×D3×C8—21no—
68C4×D12C22×C311yescharacteristic
69#869C22×C311yescharacteristic
70C3×#106—61no—
71C3×#106—61no—
72C3×#106—61no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—31no—
76C3×C4⋊C8C2×D311yescharacteristic
77C3×C4⋊C8C2×D311yescharacteristic
78C3×C4⋊C8C22×C311yescharacteristic
79C3×C4⋊C8C22×C311yescharacteristic
80C3×#103—61no—
81C3×#102—61no—
82C3×C22⋊C8—61no—
83C3×C4×C8—31no—
84(C2×D3)⋊C8—21no—
85#816—21no—
86#809—21no—
87#808—21no—
88#808—21no—
89#808—21no—
90C4⋊(C3⋊C8)C22×C311yescharacteristic
91C4×C3⋊C8C22×C311yescharacteristic
92C22×C3×D3—21no—
93C32×Q8—81no—
94C2×C32×C4—41no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4#4111yescharacteristic
98C2×C3×Dic3—21no—
99C2×C3×Dic3—41no—
100C3×D12—21no—
101C3×D12SD1612yes—
102C3×C4×D3—41no—
103C3×C4×D3—41no—
104C3×Dic6—41no—
105C3×Dic6—41no—
106C32×C8—41no—
107C32×C8—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yescomplemented
111#498—31no—
112#476—31no—
113#461—31no—
114C8×D4—31no—
115#334—31no—
116#333C3×D311yescharacteristic, complemented
117#332—31no—
118C2×C3×Q8—22no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—61no—
122C2×C3×D4—31no—
123C22×C3×C4—61no—
124C2×D12C3×D411yescharacteristic
125C2×C4×D3—21no—
126C2×Dic6—21no—
127C3×SD16—122no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—61no—
137C2×C3×C8—122no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—61no—
147C3×C22⋊C4—61no—
148C3×C42—21no—
149C3×C42C22×D311yescharacteristic
150C3×C42C23×C311yescharacteristic
151(C2×D3)⋊C4—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3C3×D411yescharacteristic
155C2×C3⋊C8—21no—
156C2×C3⋊C8C3×D411yescharacteristic
157C2×C3⋊C8C3×D411yescharacteristic
158#204—42no—
159D3×C8—42no—
160C22×C32#12111yescharacteristic
161C2×C3×D3—22no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4—81no—
166C32×C4C2×SD1611yescharacteristic
167C32×C4C2×SD1611yescharacteristic
168C32×C4#13511yescharacteristic
169C32×C4#13711yescharacteristic
170C3×Dic3—41no—
171C3×Dic3—81no—
172C2×SD16—61no—
173C22×C8—61no—
174C4⋊Q8—31no—
175C4⋊Q8C2×C3×D311yescharacteristic
176C4×D4—31no—
177#106—61no—
178#106—61no—
179#106—61no—
180C4⋊C8—31no—
181C4⋊C8C2×C3×D311yescharacteristic
182C4⋊C8C2×C3×D311yescharacteristic
183#103—61no—
184#102—61no—
185C22⋊C8—61no—
186C4×C8—31no—
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—81no—
190C3×Q8—81no—
191C3×Q8—82no—
192C3×Q8—121no—
193C3×Q8—121no—
194C3×D4—32no—
195C3×D4—61no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—61no—
203C2×C3×C4—121no—
204C2×C3×C4—121no—
205C2×C3×C4—121no—
206C2×C3×C4#23511yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4C2×C3×D411yescharacteristic
210C2×C3×C4C2×C3×D411yescharacteristic
211C2×C3×C4C3×#4111yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—41no—
214D12—21no—
215D12C3×SD1612yes—
216C4×D3—41no—
217C4×D3—41no—
218Dic6—41no—
219Dic6—41no—
220C3×C8—32no—
221C3×C8—41no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—41no—
225C3×C8—61no—
226C3×C8—81no—
227C3×C8—81no—
228C3×C8—121no—
229C3⋊C8—21no—
230C3⋊C8—41no—
231C3⋊C8C3×SD1612yescomplemented
232C2×C32#46111yescharacteristic
233C2×C32#46511yescharacteristic
234C2×C32#47111yescharacteristic
235C3×D3—42no—
236C2×Q8—21no—
237C2×Q8—61no—
238C2×D4—31no—
239C22×C4—61no—
240SD16—122no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—61no—
246C2×C8—122no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C22⋊C4—61no—
253C42C22×C3×D311yescharacteristic
254C22×C3—21no—
255C22×C3—62no—
256C22×C3—121no—
257C22×C3—121no—
258C22×C3#88411yescharacteristic
259C22×C3C3×#12111yescharacteristic
260C2×D3—22no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—82no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—121no—
274C3×C4—241no—
275C3×C4D3×SD1611yescharacteristic
276C3×C4D3×SD1611yescharacteristic
277C3×C4#91811yescharacteristic
278C3×C4#91911yescharacteristic
279C3×C4C2×C3×SD1611yescharacteristic
280C3×C4C2×C3×SD1611yescharacteristic
281C3×C4C3×#13511yescharacteristic
282C3×C4C3×#13711yescharacteristic
283Dic3—41no—
284Dic3—81no—
285C32#167111yescharacteristic, complemented, Sylow
286C23—61no—
287Q8—81no—
288Q8—121no—
289Q8—121no—
290D4—32no—
291D4—61no—
292C2×C4—41no—
293C2×C4—61no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4C3×#23511yescharacteristic
298C2×C4C3×D3×D411yescharacteristic
299C2×C4C3×D3×D411yescharacteristic
300C8—32no—
301C8—41no—
302C8—41no—
303C8—61no—
304C8—121no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—122no—
309C2×C3#474811yescharacteristic
310C2×C3#480711yescharacteristic
311C2×C3#481811yescharacteristic
312C2×C3C3×#46111yescharacteristic
313C2×C3C3×#46511yescharacteristic
314C2×C3C3×#47111yescharacteristic
315D3—42no—
316C22—62no—
317C22—121no—
318C22—121no—
319C22C3×#88411yescharacteristic
320C4—21no—
321C4—81no—
322C4—121no—
323C4—241no—
324C4C3×D3×SD1611yescharacteristic
325C4C3×D3×SD1611yescharacteristic
326C4C3×#91811yescharacteristic
327C4C3×#91911yescharacteristic
328C3—21no—
329C3#6849611yescharacteristic, direct product factor
330C3C3×#167111yescharacteristic, complemented
331C2—122no—
332C2C3×#474811yescharacteristic
333C2C3×#480711yescharacteristic
334C2C3×#481811yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#333C2
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC3×#333D3
hypocenterC3C3×#1671
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#333

Complements

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

NH ≅ G/Ntype
2. C3×#5080C2semidirect
5. C32×#333C2semidirect
6. C3×#4486C2semidirect
7. C3×#4480C2semidirect
9. #68496C3direct (N × H)
30. #5080C2×C3semidirect
35. C3×#333D3semidirect
36. C3×#333C2×C3semidirect
38. #4486C2×C3semidirect
39. #4480C2×C3semidirect
110. C3×C3⋊C8SD16semidirect
116. #333C3×D3semidirect
231. C3⋊C8C3×SD16semidirect
285. C32#1671semidirect
329. C3#68496direct (N × H)
330. C3C3×#1671semidirect

Maximal subgroups

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

subgroupindex
C3×#50842
C3×#50802
C3×#48412
C3×#48342
C32×#3332
C3×#44862
C3×#44802
C3×#16713
#684963