Finite Group Databasebeta

← C3×#71663 #11788810

Subgroups — C3×#71663 #11788810

Order 1152.

761 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic, complemented
2C3×#5816C211yesmaximal, characteristic
3C2×C3×#836C211yesmaximal, characteristic, complemented
4C3×#5005C212yesmaximal
5C3×#4531C212yesmaximal, complemented
6C3×#1912—31nomaximal
7#71663C311yesmaximal, characteristic, direct product factor
8C22×C32×Q8C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10C32×C4×Q8—24no—
11C32×#117—22no—
12C32×#117C2212yes—
13C3×#947C412yescomplemented
14C3×#946—22no—
15C3×#944C412yescomplemented
16C3×#929—22no—
17C3×#926C2211yescharacteristic
18C2×C3×#208C2212yes—
19C3×#836—22no—
20C3×#805—24no—
21C3×#536—31no—
22C3×#519—21no—
23C3×#519D311yescharacteristic, complemented
24C3×#519C2×C311yescharacteristic, complemented
25#5816C2×C311yescharacteristic
26C3×#421—32no—
27C2×C3×#101—31no—
28C2×#836C2×C311yescharacteristic, complemented
29#5005C2×C312yes—
30C3×H(3,ℤ/4)—32no—
31#4531C2×C312yescomplemented
32C2×C32×Q8—22no—
33C2×C32×Q8—22no—
34C2×C32×Q8C2×C412yes—
35C2×C32×Q8D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C2×C3×#67—21no—
39C3×#235—22no—
40C3×#235—22no—
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6C2×C411yescharacteristic
44C32×C4⋊C4—24no—
45C32×C4⋊C4—22no—
46C32×C22⋊C4—22no—
47C32×C42—24no—
48C32×C42D414yes—
49C3×#217—21no—
50C3×(C2×D3)⋊C4—24no—
51C3×C4⋊Dic3—21no—
52C3×Dic3⋊C4—24no—
53C3×C4×Dic3C2×C412yes—
54C3×#208—22no—
55C3×#208—22no—
56C2×C3×C3⋊C8—22no—
57#1912—31noSylow
58C2×C3×#41—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2×D311yescharacteristic
61C22×C3×Q8C22×C311yescharacteristic
62C2×#235C22×C311yescharacteristic
63C2×C3×M16—32no—
64C3×C4⋊Q8—32no—
65C3×#124—32no—
66C3×#123—62no—
67C3×C22⋊Q8—62no—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×#117—24no—
72C3×#117—22no—
73C3×#117—22no—
74C3×#117—22no—
75C3×#117—31no—
76C3×#117C2×D312yes—
77C3×#117C22×C312yes—
78#947C3×C412yescomplemented
79#946—22no—
80#944C3×C412yescomplemented
81#929—22no—
82#926C22×C311yescharacteristic
83C2×#208C22×C312yes—
84C3×#104—64no—
85C3×#101—62no—
86#836—22no—
87#805—24no—
88C23×C32C2×D411yescharacteristic
89C22×C3×D3—21no—
90C32×Q8—24no—
91C32×Q8—24no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—24no—
95C2×C32×C4C22⋊C414yes—
96C2×C32×C4C22×C411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D412yes—
99C2×C32×C4#4112yes—
100C3×#67—41no—
101C3×#67—41no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—21no—
104C3×D12—22no—
105C3×C4×D3—42no—
106C3×Dic6—22no—
107C3×C3⋊C8—42no—
108#536—31no—
109#519C3×D311yescharacteristic, complemented
110#421—32no—
111C2×#101—31no—
112H(3,ℤ/4)—32no—
113C3×#41—62no—
114C3×#41—62no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—31no—
124C2×C3×Q8C4×D312yes—
125C2×C3×Q8D1212yes—
126C2×C3×Q8C2×C3×C412yes—
127C2×C3×Q8C3×D412yes—
128C2×C3×D4—31no—
129C2×C3×D4—61no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—61no—
133C22×C3×C4#6712yes—
134C22×C3×C4C3×D412yes—
135C22×C3×C4C22×D311yescharacteristic
136C22×C3×C4C23×C311yescharacteristic
137C2×#67—21no—
138#235—22no—
139#235—22no—
140C2×D12C2×C3×C411yescharacteristic
141C2×C4×D3—21no—
142C2×Dic6C2×C3×C411yescharacteristic
143C3×M16—62no—
144C3×M16—62no—
145C2×C3×C8—62no—
146C3×C4⋊C4—28no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—64no—
153C3×C4⋊C4—61no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—61no—
158C3×C22⋊C4—64no—
159C3×C42—28no—
160C3×C42—24no—
161C3×C42—24no—
162C3×C42—24no—
163C3×C42—32no—
164C3×C42#6714yes—
165C3×C42C3×D414yes—
166#217—21no—
167(C2×D3)⋊C4—24no—
168C4⋊Dic3—21no—
169Dic3⋊C4—24no—
170C4×Dic3C2×C3×C412yes—
171#208—22no—
172#208—22no—
173C2×C3⋊C8—22no—
174C22×C32—41no—
175C22×C32C2×C22⋊C411yescharacteristic
176C22×C32#12011yescharacteristic
177C22×C32#12311yescharacteristic
178C2×C3×D3—42no—
179C2×C3×D3—41no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4C4×D412yes—
183C32×C4#12112yes—
184C3×Dic3—42no—
185C3×Dic3—41no—
186C2×#41—31no—
187C22×Q8C2×C3×D311yescharacteristic
188C2×M16—32no—
189C4⋊Q8—32no—
190#124—32no—
191#123—62no—
192C22⋊Q8—62no—
193C4×Q8—24no—
194#117—22no—
195#117—31no—
196#117C2×C3×D312yes—
197#104—64no—
198#101—62no—
199C23×C3—21no—
200C23×C3—61no—
201C23×C3C2×#6711yescharacteristic
202C23×C3C2×C3×D411yescharacteristic
203C22×D3—21no—
204C3×Q8—28no—
205C3×Q8—28no—
206C3×Q8—24no—
207C3×Q8—24no—
208C3×Q8—24no—
209C3×Q8—24no—
210C3×Q8—62no—
211C3×D4—62no—
212C3×D4—121no—
213C3×D4—121no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—28no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—62no—
229C2×C3×C4—61no—
230C2×C3×C4—122no—
231C2×C3×C4(C2×D3)⋊C414yes—
232C2×C3×C4C3×C22⋊C414yes—
233C2×C3×C4C2×C4×D311yescharacteristic
234C2×C3×C4C2×D1211yescharacteristic
235C2×C3×C4#23512yes—
236C2×C3×C4C2×#6712yes—
237C2×C3×C4C22×C3×C411yescharacteristic
238C2×C3×C4C2×C3×D411yescharacteristic
239C2×C3×C4C2×C3×D412yes—
240C2×C3×C4C3×#4112yes—
241#67—41no—
242#67—41no—
243C2×Dic3—22no—
244C2×Dic3—21no—
245D12—22no—
246C4×D3—42no—
247Dic6—22no—
248C3×C8—122no—
249C3⋊C8—42no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32#38611yescharacteristic
254C3×D3—81no—
255#41—62no—
256#41—62no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—31no—
260C2×Q8C3×C4×D312yes—
261C2×Q8C3×D1212yes—
262C2×D4—31no—
263C2×D4—61no—
264C22×C4—61no—
265C22×C4C3×#6712yes—
266C22×C4C22×C3×D311yescharacteristic
267M16—62no—
268M16—62no—
269C2×C8—62no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—64no—
273C4⋊C4—61no—
274C22⋊C4—22no—
275C22⋊C4—61no—
276C22⋊C4—64no—
277C42—24no—
278C42—32no—
279C42C3×#6714yes—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—42no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3C2×(C2×D3)⋊C411yescharacteristic
289C22×C3#92911yescharacteristic
290C22×C3#95311yescharacteristic
291C22×C3C2×C3×C22⋊C411yescharacteristic
292C22×C3C3×#12011yescharacteristic
293C22×C3C3×#12311yescharacteristic
294C2×D3—42no—
295C2×D3—41no—
296C3×C4—28no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—48no—
302C3×C4—44no—
303C3×C4—44no—
304C3×C4—122no—
305C3×C4—121no—
306C3×C4C4×#6712yes—
307C3×C4#93012yes—
308C3×C4C3×C4×D412yes—
309C3×C4C3×#12112yes—
310Dic3—42no—
311Dic3—41no—
312C32#191211yescharacteristic, complemented, Sylow
313C23—61no—
314C23C2×C3×#6711yescharacteristic
315Q8—24no—
316Q8—24no—
317Q8—62no—
318D4—62no—
319D4—121no—
320D4—121no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—62no—
325C2×C4—61no—
326C2×C4—122no—
327C2×C4C3×(C2×D3)⋊C414yes—
328C2×C4C2×C3×C4×D311yescharacteristic
329C2×C4C2×C3×D1211yescharacteristic
330C2×C4C3×#23512yes—
331C2×C4C2×C3×#6712yes—
332C8—122no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—241no—
344C2×C3#521311yescharacteristic
345C2×C3C3×#38611yescharacteristic
346D3—81no—
347C22—41no—
348C22—122no—
349C22—121no—
350C22C2×C3×(C2×D3)⋊C411yescharacteristic
351C22C3×#92911yescharacteristic
352C22C3×#95311yescharacteristic
353C4—24no—
354C4—44no—
355C4—122no—
356C4—121no—
357C4C3×C4×#6712yes—
358C4C3×#93012yes—
359C3—21no—
360C3#7166311yescharacteristic, direct product factor
361C3C3×#191211yescharacteristic, complemented
362C2—21no—
363C2—21no—
364C2—21no—
365C2—241no—
366C2C3×#521311yescharacteristic
subgroupG/H
centerC2×C3#5213
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#519C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#519D3
hypocenterC3C3×#1912
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#519

Complements

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

NH ≅ G/Ntype
1. C32×#519C2semidirect
3. C2×C3×#836C2semidirect
5. C3×#4531C2semidirect
7. #71663C3direct (N × H)
13. C3×#947C4semidirect
15. C3×#944C4semidirect
23. C3×#519D3semidirect
24. C3×#519C2×C3semidirect
28. C2×#836C2×C3semidirect
31. #4531C2×C3semidirect
78. #947C3×C4semidirect
80. #944C3×C4semidirect
109. #519C3×D3semidirect
312. C32#1912semidirect
360. C3#71663direct (N × H)
361. C3C3×#1912semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
C3×#58162
C2×C3×#8362
C3×#50052
C3×#45312
C3×#19123
#716633