Finite Group Databasebeta

← C3×#68443 #11781106

Subgroups — C3×#68443 #11781106

Order 1152.

376 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5083C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3C3×#4841C211yesmaximal, characteristic
4C3×#4834C211yesmaximal, characteristic
5C32×#332C211yesmaximal, characteristic, complemented
6C3×#4486C211yesmaximal, characteristic, complemented
7C3×#4479C211yesmaximal, characteristic, complemented
8C3×#1673—31nomaximal
9#68443C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#215—21no—
12C2×C3×#204—21no—
13C2×C3×D3×C8—21no—
14C3×C4×D12C2211yescharacteristic
15C3×#869C2211yescharacteristic
16C32×C4⋊C8C2211yescharacteristic
17C32×C4⋊C8C2211yescharacteristic
18C3×#835—21no—
19C3×(C2×D3)⋊C8—21no—
20C3×#816—21no—
21C3×#807—21no—
22C3×#807—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#500—31no—
26C3×#461—31no—
27C3×#461—31no—
28C3×C8×D4—31no—
29#5083C2×C311yescharacteristic
30#5078C2×C311yescharacteristic, complemented
31#4841C2×C311yescharacteristic
32#4834C2×C311yescharacteristic
33C3×#335—31no—
34C3×#332—21no—
35C3×#332—31no—
36C3×#332D311yescharacteristic, complemented
37C3×#332C2×C311yescharacteristic, complemented
38#4486C2×C311yescharacteristic, complemented
39#4479C2×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×#215—42no—
50C3×(C2×D3)⋊C4—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57C3×#204—42no—
58C3×D3×C8—42no—
59#1673—31noSylow
60C2×C3×SD16—61no—
61C2×C3×SD16—61no—
62C22×C3×C8—61no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—31no—
65C3×C4⋊Q8C2×D311yescharacteristic
66C3×C4⋊Q8C22×C311yescharacteristic
67C3×C4×D4—31no—
68C2×#215—21no—
69C2×#204—21no—
70C2×D3×C8—21no—
71C4×D12C22×C311yescharacteristic
72#869C22×C311yescharacteristic
73C3×#107—61no—
74C3×#107—61no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8—31no—
78C3×C4⋊C8C2×D311yescharacteristic
79C3×C4⋊C8C2×D311yescharacteristic
80C3×C4⋊C8C22×C311yescharacteristic
81C3×C4⋊C8C22×C311yescharacteristic
82C3×#103—61no—
83C3×#103—61no—
84C3×C22⋊C8—61no—
85C3×C4×C8—31no—
86#835—21no—
87(C2×D3)⋊C8—21no—
88#816—21no—
89#807—21no—
90#807—21no—
91C4⋊(C3⋊C8)C22×C311yescharacteristic
92C4×C3⋊C8C22×C311yescharacteristic
93C22×C3×D3—21no—
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C3×D12—21no—
103C3×D12D812yes—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C32×C8—41no—
109C32×C8—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8SD1612yescomplemented
113#500—31no—
114#461—31no—
115#461—31no—
116C8×D4—31no—
117#335—31no—
118#332—31no—
119#332C3×D311yescharacteristic, complemented
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—41no—
123C2×C3×Q8—61no—
124C2×C3×D4—31no—
125C22×C3×C4—61no—
126C2×D12C3×D411yescharacteristic
127C2×C4×D3—21no—
128C2×Dic6—21no—
129C3×SD16—122no—
130C3×SD16—122no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—61no—
140C2×C3×C8—122no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—61no—
150C3×C22⋊C4—61no—
151C3×C42—21no—
152C3×C42C22×D311yescharacteristic
153C3×C42C23×C311yescharacteristic
154#215—42no—
155(C2×D3)⋊C4—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3C3×D411yescharacteristic
159C2×C3⋊C8—21no—
160C2×C3⋊C8C3×D411yescharacteristic
161C2×C3⋊C8C3×D411yescharacteristic
162#204—42no—
163D3×C8—42no—
164C22×C32#12111yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—21no—
169C32×C4—81no—
170C32×C4C2×D811yescharacteristic
171C32×C4C2×SD1611yescharacteristic
172C32×C4#13511yescharacteristic
173C32×C4#13711yescharacteristic
174C3×Dic3—41no—
175C3×Dic3—81no—
176C2×SD16—61no—
177C2×SD16—61no—
178C22×C8—61no—
179C4⋊Q8—31no—
180C4⋊Q8C2×C3×D311yescharacteristic
181C4×D4—31no—
182#107—61no—
183#107—61no—
184C4⋊C8—31no—
185C4⋊C8C2×C3×D311yescharacteristic
186C4⋊C8C2×C3×D311yescharacteristic
187#103—61no—
188#103—61no—
189C22⋊C8—61no—
190C4×C8—31no—
191C23×C3—61no—
192C22×D3—21no—
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—81no—
198C3×Q8—81no—
199C3×Q8—121no—
200C3×Q8—121no—
201C3×D4—32no—
202C3×D4—61no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—61no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4#23511yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4C2×C3×D411yescharacteristic
217C2×C3×C4C2×C3×D411yescharacteristic
218C2×C3×C4C3×#4111yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—41no—
221D12—21no—
222D12C3×D812yes—
223C4×D3—41no—
224C4×D3—41no—
225Dic6—41no—
226Dic6—41no—
227C3×C8—32no—
228C3×C8—41no—
229C3×C8—41no—
230C3×C8—41no—
231C3×C8—41no—
232C3×C8—61no—
233C3×C8—81no—
234C3×C8—81no—
235C3×C8—121no—
236C3⋊C8—21no—
237C3⋊C8—41no—
238C3⋊C8C3×SD1612yescomplemented
239C2×C32#46111yescharacteristic
240C2×C32#46611yescharacteristic
241C2×C32#47311yescharacteristic
242C3×D3—42no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×D4—31no—
246C22×C4—61no—
247SD16—122no—
248SD16—122no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—61no—
254C2×C8—122no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C22⋊C4—61no—
261C42C22×C3×D311yescharacteristic
262C22×C3—21no—
263C22×C3—62no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3#88411yescharacteristic
267C22×C3C3×#12111yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—121no—
282C3×C4—241no—
283C3×C4D3×D811yescharacteristic
284C3×C4D3×SD1611yescharacteristic
285C3×C4#91611yescharacteristic
286C3×C4#91811yescharacteristic
287C3×C4C2×C3×D811yescharacteristic
288C3×C4C2×C3×SD1611yescharacteristic
289C3×C4C3×#13511yescharacteristic
290C3×C4C3×#13711yescharacteristic
291Dic3—41no—
292Dic3—81no—
293C32#167311yescharacteristic, complemented, Sylow
294C23—61no—
295Q8—41no—
296Q8—41no—
297Q8—121no—
298Q8—121no—
299D4—32no—
300D4—61no—
301C2×C4—41no—
302C2×C4—61no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4C3×#23511yescharacteristic
307C2×C4C3×D3×D411yescharacteristic
308C2×C4C3×D3×D411yescharacteristic
309C8—32no—
310C8—41no—
311C8—41no—
312C8—61no—
313C8—121no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—122no—
318C2×C3#475511yescharacteristic
319C2×C3#477511yescharacteristic
320C2×C3#481811yescharacteristic
321C2×C3C3×#46111yescharacteristic
322C2×C3C3×#46611yescharacteristic
323C2×C3C3×#47311yescharacteristic
324D3—42no—
325C22—62no—
326C22—121no—
327C22—121no—
328C22C3×#88411yescharacteristic
329C4—21no—
330C4—81no—
331C4—121no—
332C4—241no—
333C4C3×D3×D811yescharacteristic
334C4C3×D3×SD1611yescharacteristic
335C4C3×#91611yescharacteristic
336C4C3×#91811yescharacteristic
337C3—21no—
338C3#6844311yescharacteristic, direct product factor
339C3C3×#167311yescharacteristic, complemented
340C2—122no—
341C2C3×#475511yescharacteristic
342C2C3×#477511yescharacteristic
343C2C3×#481811yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#332C2
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC3×#332D3
hypocenterC3C3×#1673
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#332

Complements

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

NH ≅ G/Ntype
2. C3×#5078C2semidirect
5. C32×#332C2semidirect
6. C3×#4486C2semidirect
7. C3×#4479C2semidirect
9. #68443C3direct (N × H)
30. #5078C2×C3semidirect
36. C3×#332D3semidirect
37. C3×#332C2×C3semidirect
38. #4486C2×C3semidirect
39. #4479C2×C3semidirect
112. C3×C3⋊C8SD16semidirect
119. #332C3×D3semidirect
238. C3⋊C8C3×SD16semidirect
293. C32#1673semidirect
338. C3#68443direct (N × H)
339. C3C3×#1673semidirect

Maximal subgroups

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

subgroupindex
C3×#50832
C3×#50782
C3×#48412
C3×#48342
C32×#3322
C3×#44862
C3×#44792
C3×#16733
#684433