Finite Group Databasebeta

← C3×#68565 #11777872

Subgroups — C3×#68565 #11777872

Order 1152.

351 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#4838C211yesmaximal, characteristic
2C3×#4831C211yesmaximal, characteristic, complemented
3C3×C4×#204C211yesmaximal, characteristic
4C32×#335C211yesmaximal, characteristic, complemented
5C3×#4484C211yesmaximal, characteristic, complemented
6C3×#4483C211yesmaximal, characteristic
7C3×#4455C211yesmaximal, characteristic, complemented
8C3×#1592—31nomaximal
9#68565C311yesmaximal, characteristic, direct product factor
10C2×C3×#204C411yescharacteristic
11C2×C3×ΓL(1,𝔽25)—21no—
12C3×C4×D12C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C32×C4⋊C8C2211yescharacteristic
16C32×C4×C8C2211yescharacteristic
17C3×#821C411yescharacteristic
18C3×(C2×D3)⋊C8—21no—
19C3×#817C411yescharacteristic
20C3×#816C411yescharacteristic
21C3×#815—21no—
22C3×Dic3⋊C8—21no—
23C3×C4×C3⋊C8C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#446—31no—
26C3×C4×SD16—31no—
27C3×#436—31no—
28#4838C2×C311yescharacteristic
29#4831C2×C311yescharacteristic, complemented
30C4×#204C2×C311yescharacteristic
31C3×#335—21no—
32C3×#335D311yescharacteristic, complemented
33C3×#335C2×C311yescharacteristic, complemented
34C3×Q8⋊C8—31no—
35C3×D4⋊C8—31no—
36C3×#322—31no—
37#4484C2×C311yescharacteristic, complemented
38#4483C2×C311yescharacteristic
39#4455C2×C311yescharacteristic, complemented
40C2×C3×D12C2×C411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C2×C32×C8—21no—
44C2×C32×C8—21no—
45C2×C32×C8C2×C411yescharacteristic
46C2×C32×C8C2×C411yescharacteristic
47C32×C42C2311yescharacteristic
48C3×(C2×D3)⋊C4—21no—
49C3×C4⋊Dic3C2×C411yescharacteristic
50C3×C4⋊Dic3C2×C411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×#204—22no—
57C3×ΓL(1,𝔽25)—42no—
58#1592—31noSylow
59C2×C3×SD16—31no—
60C2×C3×M16—61no—
61C3×C4×Q8—31no—
62C3×C4×D4—31no—
63C2×#204C3×C411yescharacteristic
64C2×ΓL(1,𝔽25)—21no—
65C4×D12C22×C311yescharacteristic
66C4×Dic6C22×C311yescharacteristic
67C3×#106—31no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—61no—
71C3×C4⋊C8C2×D311yescharacteristic
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4⋊C8C22×C311yescharacteristic
74C3×C4⋊C8C22×C311yescharacteristic
75C3×#103—31no—
76C3×#102—31no—
77C3×C22⋊C8—61no—
78C3×#97—61no—
79C3×C4×C8—21no—
80C3×C4×C8—31no—
81C3×C4×C8—31no—
82C3×C4×C8C2×D311yescharacteristic
83C3×C4×C8C22×C311yescharacteristic
84#821C3×C411yescharacteristic
85(C2×D3)⋊C8—21no—
86#817C3×C411yescharacteristic
87#816C3×C411yescharacteristic
88#815—21no—
89Dic3⋊C8—21no—
90C4×C3⋊C8C22×C311yescharacteristic
91C4×C3⋊C8C22×C311yescharacteristic
92C22×C3×D3—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C3×D12—21no—
100C3×D12—21no—
101C3×C4×D3—41no—
102C3×C4×D3—41no—
103C3×Dic6—21no—
104C3×Dic6—21no—
105C32×C8—21no—
106C32×C8—41no—
107C32×C8—41no—
108C32×C8M1612yescomplemented
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112#446—31no—
113C4×SD16—31no—
114#436—31no—
115#335C3×D311yescharacteristic, complemented
116Q8⋊C8—31no—
117D4⋊C8—31no—
118#322—31no—
119C2×C3×Q8—31no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12C2×C3×C411yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6C2×C3×C411yescharacteristic
125C3×SD16—62no—
126C3×M16—122no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—61no—
138C2×C3×C8C4×D311yescharacteristic
139C2×C3×C8C4×D311yescharacteristic
140C2×C3×C8C2×C3×C411yescharacteristic
141C2×C3×C8C2×C3×C411yescharacteristic
142C3×C4⋊C4—31no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C42—21no—
147C3×C42—61no—
148C3×C42C22×D311yescharacteristic
149C3×C42C23×C311yescharacteristic
150(C2×D3)⋊C4—21no—
151C4⋊Dic3C2×C3×C411yescharacteristic
152C4⋊Dic3C2×C3×C411yescharacteristic
153Dic3⋊C4—21no—
154C4×Dic3—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8C3×D411yescharacteristic
157C2×C3⋊C8C3×D411yescharacteristic
158#204—22no—
159ΓL(1,𝔽25)—42no—
160C22×C32C4×D411yescharacteristic
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4C2×M1611yescharacteristic
166C32×C4#13111yescharacteristic
167C32×C4AGL(1,ℤ/8)11yescharacteristic
168C32×C4#13711yescharacteristic
169C3×Dic3—41no—
170C3×Dic3—41no—
171C3×Dic3—41no—
172C2×SD16—31no—
173C2×M16—61no—
174C4×Q8—31no—
175C4×D4—31no—
176#106—31no—
177C4⋊C8—61no—
178C4⋊C8C2×C3×D311yescharacteristic
179C4⋊C8C2×C3×D311yescharacteristic
180#103—31no—
181#102—31no—
182C22⋊C8—61no—
183#97—61no—
184C4×C8—31no—
185C4×C8—31no—
186C4×C8C2×C3×D311yescharacteristic
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—61no—
190C3×Q8—61no—
191C3×D4—61no—
192C3×D4—61no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4C2×C4×D311yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4#23911yescharacteristic
204C2×C3×C4C22×C3×C411yescharacteristic
205C2×C3×C4C2×C3×D411yescharacteristic
206C2×C3×C4C3×#4111yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210D12—21no—
211D12—21no—
212C4×D3—41no—
213C4×D3—41no—
214Dic6—21no—
215Dic6—21no—
216C3×C8—21no—
217C3×C8—21no—
218C3×C8—22no—
219C3×C8—41no—
220C3×C8—41no—
221C3×C8—41no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—61no—
225C3×C8—61no—
226C3×C8—81no—
227C3×C8—81no—
228C3×C8—121no—
229C3×C8ΓL(1,𝔽25)12yescomplemented
230C3×C8C3×M1612yescomplemented
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—41no—
234C2×C32#43611yescharacteristic
235C2×C32#44011yescharacteristic
236C2×C32#44411yescharacteristic
237C3×D3—81no—
238C2×Q8—31no—
239C2×D4—31no—
240C22×C4—61no—
241SD16—62no—
242M16—122no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—61no—
248C2×C8C3×C4×D311yescharacteristic
249C2×C8C3×C4×D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C22⋊C4—61no—
254C42—61no—
255C42C22×C3×D311yescharacteristic
256C22×C3—21no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3#89311yescharacteristic
261C22×C3C3×C4×D411yescharacteristic
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—121no—
275C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
276C3×C4#90711yescharacteristic
277C3×C4#91111yescharacteristic
278C3×C4#91811yescharacteristic
279C3×C4C2×C3×M1611yescharacteristic
280C3×C4C3×#13111yescharacteristic
281C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
282C3×C4C3×#13711yescharacteristic
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286C32#159211yescharacteristic, complemented, Sylow
287C23—61no—
288Q8—61no—
289Q8—61no—
290D4—61no—
291D4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4C2×C3×C4×D311yescharacteristic
298C2×C4C3×D3×D411yescharacteristic
299C2×C4C3×#23911yescharacteristic
300C8—21no—
301C8—41no—
302C8—41no—
303C8—61no—
304C8—61no—
305C8—121no—
306C8C3×ΓL(1,𝔽25)12yescomplemented
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—241no—
311C2×C3#483811yescharacteristic
312C2×C3#488911yescharacteristic
313C2×C3#489411yescharacteristic
314C2×C3C3×#43611yescharacteristic
315C2×C3C3×#44011yescharacteristic
316C2×C3C3×#44411yescharacteristic
317D3—81no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22C3×#89311yescharacteristic
322C4—21no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4C2×C3×ΓL(1,𝔽25)11yescharacteristic
327C4C3×#90711yescharacteristic
328C4C3×#91111yescharacteristic
329C4C3×#91811yescharacteristic
330C3—21no—
331C3#6856511yescharacteristic, direct product factor
332C3C3×#159211yescharacteristic, complemented
333C2—241no—
334C2C3×#483811yescharacteristic
335C2C3×#488911yescharacteristic
336C2C3×#489411yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC42C22×C3×D3
FittingC32×#335C2
socleC22×C32C4×D4
prime radicalC2×C3×D12C2×C4
cosocleC3×C42C23×C3
hypercenterC3×#335D3
hypocenterC3C3×#1592
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×#335

Complements

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

NH ≅ G/Ntype
2. C3×#4831C2semidirect
4. C32×#335C2semidirect
5. C3×#4484C2semidirect
7. C3×#4455C2semidirect
9. #68565C3direct (N × H)
29. #4831C2×C3semidirect
32. C3×#335D3semidirect
33. C3×#335C2×C3semidirect
37. #4484C2×C3semidirect
39. #4455C2×C3semidirect
108. C32×C8M16semidirect
115. #335C3×D3semidirect
229. C3×C8ΓL(1,𝔽25)semidirect
230. C3×C8C3×M16semidirect
286. C32#1592semidirect
306. C8C3×ΓL(1,𝔽25)semidirect
331. C3#68565direct (N × H)
332. C3C3×#1592semidirect

Maximal subgroups

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

subgroupindex
C3×#48382
C3×#48312
C3×C4×#2042
C32×#3352
C3×#44842
C3×#44832
C3×#44552
C3×#15923
#685653