Finite Group Databasebeta

← C3×#72139 #11784618

Subgroups — C3×#72139 #11784618

Order 1152.

1669 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic
2C2×C3×#834C214yesmaximal, complemented
3C3×#5211C212yesmaximal, complemented
4C3×#1782—31nomaximal
5#72139C311yesmaximal, characteristic, direct product factor
6C2×C32×#41—28no—
7C2×C32×#41—24no—
8C22×C32×Q8C411yescharacteristic, complemented
9C22×C32×D4C411yescharacteristic, complemented
10C22×C32×D4C2212yes—
11C23×C32×C4C412yescomplemented
12C23×C32×C4C2211yescharacteristic
13C2×C3×#217C2214yes—
14C3×#834—28no—
15C3×C3⋊#95—24no—
16C22×C3×#41—21no—
17C22×C3×#41D311yescharacteristic
18C22×C3×#41C2×C311yescharacteristic
19C2×C3×#99—34no—
20C3×#381—32no—
21C2×#834C2×C314yescomplemented
22#5211C2×C312yescomplemented
23C24×C32C2×C412yes—
24C24×C32C2311yescharacteristic
25C32×#41—232no—
26C2×C32×Q8—24no—
27C2×C32×Q8—22no—
28C2×C32×D4—24no—
29C2×C32×D4—24no—
30C2×C32×D4—24no—
31C2×C32×D4—22no—
32C2×C32×D4D418yes—
33C22×C32×C4—24no—
34C22×C32×C4—24no—
35C22×C32×C4—24no—
36C22×C32×C4—24no—
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4C2×C412yes—
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4D414yes—
41C22×C3×Dic3—24no—
42C3×#217—28no—
43#1782—31noSylow
44C2×C3×#41—216no—
45C2×C3×#41—28no—
46C2×C3×#41—28no—
47C2×C3×#41—28no—
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C22×C3×Q8—21no—
51C22×C3×Q8Dic311yescharacteristic, complemented
52C22×C3×Q8C3×C411yescharacteristic, complemented
53C22×C3×D4—21no—
54C22×C3×D4—22no—
55C22×C3×D4Dic311yescharacteristic, complemented
56C22×C3×D4C3×C411yescharacteristic, complemented
57C22×C3×D4C2×D312yes—
58C22×C3×D4C22×C312yes—
59C23×C3×C4—22no—
60C23×C3×C4—21no—
61C23×C3×C4Dic312yescomplemented
62C23×C3×C4C3×C412yescomplemented
63C23×C3×C4C2×D311yescharacteristic
64C23×C3×C4C22×C311yescharacteristic
65C2×C3×C22⋊C4—34no—
66C2×#217C22×C314yes—
67C3×#99—68no—
68C3×#95—64no—
69#834—28no—
70C3⋊#95—24no—
71C23×C32—22no—
72C23×C32—22no—
73C23×C32—42no—
74C23×C32—44no—
75C23×C32C22⋊C414yes—
76C23×C32C22×C411yescharacteristic
77C23×C32C2×D412yes—
78C23×C32C2×D414yes—
79C32×Q8—28no—
80C32×D4—28no—
81C32×D4—216no—
82C2×C32×C4—216no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—28no—
88C2×C32×C4C22⋊C412yes—
89C2×C32×C4C22⋊C414yes—
90C2×C32×C4C22⋊C412yes—
91C2×C3×Dic3—48no—
92C2×C3×Dic3—44no—
93C22×#41C3×D311yescharacteristic
94C2×#99—34no—
95#381—32no—
96C24×C3—22no—
97C24×C3—21no—
98C24×C3C2×Dic312yes—
99C24×C3C2×C3×C412yes—
100C24×C3C22×D311yescharacteristic
101C24×C3C23×C311yescharacteristic
102C3×#41—264no—
103C3×#41—232no—
104C3×#41—232no—
105C2×C3×Q8—28no—
106C2×C3×Q8—24no—
107C2×C3×Q8—24no—
108C2×C3×Q8—22no—
109C2×C3×Q8—24no—
110C2×C3×Q8—22no—
111C2×C3×D4—28no—
112C2×C3×D4—28no—
113C2×C3×D4—28no—
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—28no—
124C2×C3×D4#6718yes—
125C2×C3×D4C3×D418yes—
126C22×C3×C4—28no—
127C22×C3×C4—28no—
128C22×C3×C4—28no—
129C22×C3×C4—28no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—24no—
142C22×C3×C4—64no—
143C22×C3×C4C2×Dic311yescharacteristic
144C22×C3×C4C2×Dic312yes—
145C22×C3×C4C2×Dic311yescharacteristic
146C22×C3×C4#6714yes—
147C22×C3×C4C2×C3×C411yescharacteristic
148C22×C3×C4C2×C3×C412yes—
149C22×C3×C4C2×C3×C411yescharacteristic
150C22×C3×C4C3×D414yes—
151C22×Dic3—24no—
152C3×C22⋊C4—68no—
153#217—28no—
154C22×C32—22no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32—44no—
161C22×C32C2×C22⋊C412yes—
162C22×C32C2×C22⋊C411yescharacteristic
163C22×C32#12014yes—
164C32×C4—22no—
165C32×C4—24no—
166C32×C4—22no—
167C3×Dic3—84no—
168C2×#41—28no—
169C2×#41—24no—
170C22×Q8C3×Dic311yescharacteristic, complemented
171C22×D4C3×Dic311yescharacteristic, complemented
172C22×D4C2×C3×D312yes—
173C23×C4C3×Dic312yescomplemented
174C23×C4C2×C3×D311yescharacteristic
175C2×C22⋊C4—34no—
176#99—68no—
177#95—64no—
178C23×C3—24no—
179C23×C3—24no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—24no—
185C23×C3—21no—
186C23×C3—22no—
187C23×C3—24no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—44no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—48no—
194C23×C3#21714yes—
195C23×C3C3×C22⋊C414yes—
196C23×C3C22×Dic311yescharacteristic
197C23×C3C2×#6712yes—
198C23×C3C2×#6714yes—
199C23×C3C22×C3×C411yescharacteristic
200C23×C3C2×C3×D412yes—
201C23×C3C2×C3×D414yes—
202C3×Q8—216no—
203C3×Q8—28no—
204C3×Q8—28no—
205C3×D4—216no—
206C3×D4—28no—
207C3×D4—28no—
208C3×D4—232no—
209C3×D4—216no—
210C3×D4—216no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—232no—
216C2×C3×C4—216no—
217C2×C3×C4—216no—
218C2×C3×C4—216no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—28no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—28no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—128no—
233C2×C3×C4—124no—
234C2×C3×C4#21714yes—
235C2×C3×C4#21712yes—
236C2×C3×C4#21712yes—
237C2×C3×C4C3×C22⋊C412yes—
238C2×C3×C4C3×C22⋊C414yes—
239C2×C3×C4C3×C22⋊C412yes—
240C2×Dic3—48no—
241C2×Dic3—44no—
242C2×C32—22no—
243C2×C32—24no—
244C2×C32—44no—
245C2×C32#37911yescharacteristic
246C2×C32#41012yes—
247C24C2×C3×Dic312yes—
248C24C22×C3×D311yescharacteristic
249#41—232no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4C3×#6718yes—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—64no—
262C22×C4C2×C3×Dic311yescharacteristic
263C22×C4C2×C3×Dic312yes—
264C22×C4C2×C3×Dic311yescharacteristic
265C22×C4C3×#6714yes—
266C22⋊C4—68no—
267C22×C3—24no—
268C22×C3—24no—
269C22×C3—22no—
270C22×C3—24no—
271C22×C3—22no—
272C22×C3—28no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—24no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3—44no—
281C22×C3—44no—
282C22×C3—44no—
283C22×C3—48no—
284C22×C3—48no—
285C22×C3—48no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—48no—
289C22×C3—44no—
290C22×C3—44no—
291C22×C3C2×#21712yes—
292C22×C3C2×#21711yescharacteristic
293C22×C3#95314yes—
294C22×C3C2×C3×C22⋊C412yes—
295C22×C3C2×C3×C22⋊C411yescharacteristic
296C22×C3C3×#12014yes—
297C3×C4—24no—
298C3×C4—28no—
299C3×C4—24no—
300C3×C4—22no—
301C3×C4—24no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—24no—
305C3×C4—22no—
306C3×C4—244no—
307Dic3—84no—
308C32#178211yescharacteristic, complemented, Sylow
309C23—22no—
310C23—22no—
311C23—42no—
312C23—44no—
313C23C3×#21714yes—
314C23C22×C3×Dic311yescharacteristic
315C23C2×C3×#6712yes—
316C23C2×C3×#6714yes—
317Q8—28no—
318D4—28no—
319D4—216no—
320C2×C4—216no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—28no—
326C2×C4—128no—
327C2×C4—124no—
328C2×C4C3×#21712yes—
329C2×C4C3×#21714yes—
330C2×C4C3×#21712yes—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—22no—
334C2×C3—24no—
335C2×C3—22no—
336C2×C3—22no—
337C2×C3—21no—
338C2×C3—44no—
339C2×C3—48no—
340C2×C3—44no—
341C2×C3—44no—
342C2×C3#524012yes—
343C2×C3#524711yescharacteristic
344C2×C3C3×#37911yescharacteristic
345C2×C3C3×#41012yes—
346C22—24no—
347C22—22no—
348C22—24no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22C2×C3×#21712yes—
354C22C2×C3×#21711yescharacteristic
355C22C3×#95314yes—
356C4—22no—
357C4—24no—
358C4—22no—
359C4—244no—
360C3—21no—
361C3#7213911yescharacteristic, direct product factor
362C3C3×#178211yescharacteristic, complemented
363C2—24no—
364C2—22no—
365C2—44no—
366C2C3×#524012yes—
367C2C3×#524711yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC24C22×C3×D3
Fitting, prime radicalC22×C32×#41C2
socleC22×C32C2×C22⋊C4
cosocleC24×C3C23×C3
hypercenterC22×C3×#41D3
hypocenterC3C3×#1782
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#834C2semidirect
3. C3×#5211C2semidirect
5. #72139C3direct (N × H)
8. C22×C32×Q8C4semidirect
9. C22×C32×D4C4semidirect
11. C23×C32×C4C4semidirect
21. C2×#834C2×C3semidirect
22. #5211C2×C3semidirect
51. C22×C3×Q8Dic3semidirect
52. C22×C3×Q8C3×C4semidirect
55. C22×C3×D4Dic3semidirect
56. C22×C3×D4C3×C4semidirect
61. C23×C3×C4Dic3semidirect
62. C23×C3×C4C3×C4semidirect
170. C22×Q8C3×Dic3semidirect
171. C22×D4C3×Dic3semidirect
173. C23×C4C3×Dic3semidirect
308. C32#1782semidirect
361. C3#72139direct (N × H)
362. C3C3×#1782semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#412
C2×C3×#8342
C3×#52112
C3×#17823
#721393