Finite Group Databasebeta

← C3×#83688 #11895143

Subgroups — C3×#83688 #11895143

Order 1152.

840 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C3×Q8×D12C211yesmaximal, characteristic
3C3×#5083C213yesmaximal
4C3×#5079C213yesmaximal, complemented
5C3×C4×#215C213yesmaximal
6C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
7C3×#5020C213yesmaximal, complemented
8C3×#3359—31nomaximal
9#83688C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×Q8—21no—
11C32×C4⋊Q8C2213yes—
12C32×#125—23no—
13C32×C4×Q8—21no—
14C32×C4×Q8C2213yes—
15C32×C4×Q8C2211yescharacteristic
16C2×C3×#215C2213yes—
17C3×#897—23no—
18C3×C4×D12C2213yes—
19C3×#869C2213yes—
20C3×#835C2216yes—
21C3×#835C2213yes—
22C3×#809C2211yescharacteristic
23C3×#808C2213yes—
24C3×C4⋊(C3⋊C8)C2213yes—
25C3×C4×C3⋊C8C2213yes—
26C3×#557—21no—
27C3×#557D311yescharacteristic, complemented
28C3×#557C2×C311yescharacteristic, complemented
29C3×D4×Q8—31no—
30Q8×D12C2×C311yescharacteristic
31C3×#487—33no—
32C3×#475—33no—
33C3×#461—33no—
34C3×C8×Q8—31no—
35C3×C4×SD16—33no—
36#5083C2×C313yes—
37#5079C2×C313yescomplemented
38C4×#215C2×C313yes—
39Q8×C3⋊C8C2×C311yescharacteristic, complemented
40#5020C2×C313yescomplemented
41C2×C32×Q8D411yescharacteristic
42C2×C32×Q8C2313yes—
43C3×D3×Q8—44no—
44C2×C3×D12C2311yescharacteristic
45C2×C3×C4×D3—23no—
46C2×C3×Dic6—23no—
47C32×C4⋊C4—23no—
48C32×C4⋊C4—23no—
49C32×C4⋊C4—23no—
50C32×C4⋊C4D413yes—
51C32×C4⋊C4C2311yescharacteristic
52C32×C42—23no—
53C32×C42C2313yes—
54C3×#215—26no—
55C3×(C2×D3)⋊C4—23no—
56C3×C4⋊Dic3—23no—
57C3×C4⋊Dic3C2313yes—
58C2×C3×C3⋊C8C2313yes—
59C2×C3×C3⋊C8C2311yescharacteristic
60#3359—31noSylow
61C22×C3×Q8—61no—
62C2×D3×Q8—21no—
63C2×C3×SD16—33no—
64C3×C4⋊Q8—23no—
65C3×C4⋊Q8—33no—
66C3×C4⋊Q8C2×D313yes—
67C3×C4⋊Q8C22×C313yes—
68C3×#125—26no—
69C3×#125—23no—
70C3×#125—23no—
71C3×C22⋊Q8—63no—
72C3×C4×Q8—22no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×C4×Q8—23no—
76C3×C4×Q8—21no—
77C3×C4×Q8C2×D313yes—
78C3×C4×Q8C2×D311yescharacteristic
79C3×C4×Q8C22×C313yes—
80C3×C4×Q8C22×C311yescharacteristic
81C3×C4×D4—33no—
82C2×#215C22×C313yes—
83#897—23no—
84C4×D12C22×C313yes—
85#869C22×C313yes—
86C3×#106—33no—
87C3×C4⋊C8—33no—
88C3×#103—33no—
89C3×#103—36no—
90C3×#102—31no—
91C3×C4×C8—33no—
92#835C22×C316yes—
93#835C22×C313yes—
94#809C22×C311yescharacteristic
95#808C22×C313yes—
96C4⋊(C3⋊C8)C22×C313yes—
97C4×C3⋊C8C22×C313yes—
98C22×C3×D3—21no—
99C32×Q8—23no—
100C32×Q8—23no—
101C32×Q8SD1614yescomplemented
102C2×C32×C4—21no—
103C2×C32×C4—23no—
104C2×C32×C4C2×D413yes—
105C2×C32×C4C2×D413yes—
106C2×C32×C4C2411yescharacteristic
107C2×C3×Dic3—23no—
108C3×D12—21no—
109C3×D12#4112yes—
110C3×C4×D3—46no—
111C3×Dic6—46no—
112C3×C3⋊C8—23no—
113C3×C3⋊C8—21no—
114#557C3×D311yescharacteristic, complemented
115D4×Q8—31no—
116#487—33no—
117#475—33no—
118#461—33no—
119C8×Q8—31no—
120C4×SD16—33no—
121C2×C3×Q8—21no—
122C2×C3×Q8—23no—
123C2×C3×Q8—63no—
124C2×C3×Q8—124no—
125C2×C3×Q8#6711yescharacteristic
126C2×C3×Q8C3×D411yescharacteristic
127C2×C3×Q8C22×D313yes—
128C2×C3×Q8C23×C313yes—
129C2×C3×D4—31no—
130C22×C3×C4—63no—
131D3×Q8—44no—
132C2×D12C23×C311yescharacteristic
133C2×C4×D3—23no—
134C2×Dic6—23no—
135C3×SD16—66no—
136C2×C3×C8—31no—
137C2×C3×C8—33no—
138C3×C4⋊C4—26no—
139C3×C4⋊C4—26no—
140C3×C4⋊C4—26no—
141C3×C4⋊C4—23no—
142C3×C4⋊C4—23no—
143C3×C4⋊C4—23no—
144C3×C4⋊C4—23no—
145C3×C4⋊C4—23no—
146C3×C4⋊C4—23no—
147C3×C4⋊C4—23no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—33no—
150C3×C4⋊C4—63no—
151C3×C4⋊C4#6713yes—
152C3×C4⋊C4C3×D413yes—
153C3×C4⋊C4C22×D311yescharacteristic
154C3×C4⋊C4C23×C311yescharacteristic
155C3×C22⋊C4—63no—
156C3×C42—26no—
157C3×C42—23no—
158C3×C42—23no—
159C3×C42—23no—
160C3×C42C22×D313yes—
161C3×C42C23×C313yes—
162#215—26no—
163(C2×D3)⋊C4—23no—
164C4⋊Dic3—23no—
165C4⋊Dic3C23×C313yes—
166C2×C3⋊C8C23×C311yescharacteristic
167C2×C3⋊C8C23×C313yes—
168C22×C32C22×D411yescharacteristic
169C2×C3×D3—22no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C32×C4—23no—
173C32×C4—43no—
174C32×C4—41no—
175C32×C4C2×SD1616yes—
176C32×C4C2×#4111yescharacteristic
177C32×C42−1+411yescharacteristic
178C3×Dic3—43no—
179C22×Q8—61no—
180C2×SD16—33no—
181C4⋊Q8—33no—
182C4⋊Q8C2×C3×D313yes—
183#125—23no—
184C22⋊Q8—63no—
185C4×Q8—21no—
186C4×Q8C2×C3×D313yes—
187C4×Q8C2×C3×D311yescharacteristic
188C4×D4—33no—
189#106—33no—
190C4⋊C8—33no—
191#103—33no—
192#103—36no—
193#102—31no—
194C4×C8—33no—
195C23×C3—61no—
196C22×D3—21no—
197C3×Q8—23no—
198C3×Q8—23no—
199C3×Q8—23no—
200C3×Q8—23no—
201C3×Q8—24no—
202C3×Q8—43no—
203C3×Q8—43no—
204C3×Q8—126no—
205C3×Q8#21514yescomplemented
206C3×Q8C3×SD1614yescomplemented
207C3×D4—32no—
208C3×D4—61no—
209C2×C3×C4—22no—
210C2×C3×C4—26no—
211C2×C3×C4—21no—
212C2×C3×C4—23no—
213C2×C3×C4—21no—
214C2×C3×C4—23no—
215C2×C3×C4—21no—
216C2×C3×C4—23no—
217C2×C3×C4—23no—
218C2×C3×C4—63no—
219C2×C3×C4—126no—
220C2×C3×C4C2×#6713yes—
221C2×C3×C4C2×#6713yes—
222C2×C3×C4C2×C3×D413yes—
223C2×C3×C4C2×C3×D413yes—
224C2×C3×C4C23×D311yescharacteristic
225C2×C3×C4C24×C311yescharacteristic
226C2×Dic3—23no—
227D12—21no—
228D12C3×#4112yes—
229C4×D3—46no—
230Dic6—46no—
231C3×C8—63no—
232C3×C8—61no—
233C3⋊C8—23no—
234C3⋊C8—21no—
235C2×C32#54811yescharacteristic
236C2×C32C22×SD1611yescharacteristic
237C2×C32#57811yescharacteristic
238C3×D3—42no—
239C2×Q8—63no—
240C2×Q8—124no—
241C2×Q8C3×#6711yescharacteristic
242C2×Q8C22×C3×D313yes—
243C2×D4—31no—
244C22×C4—63no—
245SD16—66no—
246C2×C8—31no—
247C2×C8—33no—
248C4⋊C4—23no—
249C4⋊C4—23no—
250C4⋊C4—23no—
251C4⋊C4—33no—
252C4⋊C4—63no—
253C4⋊C4C3×#6713yes—
254C4⋊C4C22×C3×D311yescharacteristic
255C22⋊C4—63no—
256C42—23no—
257C42C22×C3×D313yes—
258C22×C3—21no—
259C22×C3—62no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3C22×#6711yescharacteristic
263C22×C3C22×C3×D411yescharacteristic
264C2×D3—22no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—23no—
268C3×C4—23no—
269C3×C4—26no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—43no—
273C3×C4—46no—
274C3×C4—42no—
275C3×C4—41no—
276C3×C4—43no—
277C3×C4—41no—
278C3×C4—43no—
279C3×C4—123no—
280C3×C4C2×#21516yes—
281C3×C4C2×C3×SD1616yes—
282C3×C4D3×#4111yescharacteristic
283C3×C4#101011yescharacteristic
284C3×C4C2×C3×#4111yescharacteristic
285C3×C4C3×2−1+411yescharacteristic
286Dic3—43no—
287C32#335911yescharacteristic, complemented, Sylow
288C23—61no—
289Q8—23no—
290Q8—23no—
291Q8—126no—
292Q8C3×#21514yescomplemented
293D4—32no—
294D4—61no—
295C2×C4—21no—
296C2×C4—23no—
297C2×C4—63no—
298C2×C4—126no—
299C2×C4C2×C3×#6713yes—
300C2×C4C2×C3×#6713yes—
301C2×C4C23×C3×D311yescharacteristic
302C8—63no—
303C8—61no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—122no—
308C2×C3C22×#21511yescharacteristic
309C2×C3#583111yescharacteristic
310C2×C3#583811yescharacteristic
311C2×C3C3×#54811yescharacteristic
312C2×C3C22×C3×SD1611yescharacteristic
313C2×C3C3×#57811yescharacteristic
314D3—42no—
315C22—62no—
316C22—121no—
317C22—121no—
318C22C22×C3×#6711yescharacteristic
319C4—23no—
320C4—43no—
321C4—41no—
322C4—123no—
323C4C2×C3×#21516yes—
324C4C3×D3×#4111yescharacteristic
325C4C3×#101011yescharacteristic
326C3—21no—
327C3#8368811yescharacteristic, direct product factor
328C3C3×#335911yescharacteristic, complemented
329C2—122no—
330C2C22×C3×#21511yescharacteristic
331C2C3×#583111yescharacteristic
332C2C3×#583811yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#557C2
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC3×#557D3
hypocenterC3C3×#3359
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
4. C3×#5079C2semidirect
6. C3×Q8×C3⋊C8C2semidirect
7. C3×#5020C2semidirect
9. #83688C3direct (N × H)
27. C3×#557D3semidirect
28. C3×#557C2×C3semidirect
37. #5079C2×C3semidirect
39. Q8×C3⋊C8C2×C3semidirect
40. #5020C2×C3semidirect
101. C32×Q8SD16semidirect
114. #557C3×D3semidirect
205. C3×Q8#215semidirect
206. C3×Q8C3×SD16semidirect
287. C32#3359semidirect
292. Q8C3×#215semidirect
327. C3#83688direct (N × H)
328. C3C3×#3359semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
C3×Q8×D122
C3×#50832
C3×#50792
C3×C4×#2152
C3×Q8×C3⋊C82
C3×#50202
C3×#33593
#836883