Finite Group Databasebeta

← C3×#70656 #11796846

Subgroups — C3×#70656 #11796846

Order 1152.

505 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#904C211yesmaximal, characteristic
2C32×#427C211yesmaximal, characteristic, complemented
3C2×C3×#812C211yesmaximal, characteristic, complemented
4C3×#4488C214yesmaximal, complemented
5C3×#2145—31nomaximal
6#70656C311yesmaximal, characteristic, direct product factor
7C2×C3×#235—21no—
8C22×C32×C8C2211yescharacteristic
9C32×#117—21no—
10C2×C3×Dic12C411yescharacteristic, complemented
11C3×#904—21no—
12C3×#904—21no—
13C3×#904C412yescomplemented
14C3×#904C2212yes—
15C2×C3×D24C411yescharacteristic, complemented
16C2×C3×#204—21no—
17C32×#107—21no—
18C32×#106C2212yescomplemented
19C3×#812—21no—
20C3×#812C2212yescomplemented
21C2×C3×C3⋊C16—21no—
22C2×C3×#135—31no—
23C2×#904C2×C311yescharacteristic
24C2×C3×M32—31no—
25C3×#427—21no—
26C3×#427D311yescharacteristic, complemented
27C3×#427C2×C311yescharacteristic, complemented
28C2×#812C2×C311yescharacteristic, complemented
29C3×#360—34no—
30#4488C2×C314yescomplemented
31C22×C32×C4D411yescharacteristic
32C2×C3×#67—41no—
33C3×#235—22no—
34C3×#235—22no—
35C3×#235—41no—
36C3×#235—41no—
37C2×C3×D12—21no—
38C2×C3×C4×D3—41no—
39C2×C3×Dic6—21no—
40C2×C32×C8C2×C411yescharacteristic
41C2×C32×C8C2×C411yescharacteristic
42C2×C32×C8D411yescharacteristic
43C2×C32×C8D411yescharacteristic
44C2×C32×C8D411yescharacteristic
45C2×C32×C8C2311yescharacteristic
46C32×C4⋊C4—22no—
47C32×C22⋊C4—41no—
48C32×C42—41no—
49C3×Dic12—21no—
50C3×Dic12C2×C412yes—
51C3×D24—21no—
52C3×D24C2×C412yes—
53C3×#204—22no—
54C3×#204—22no—
55C3×C3⋊C16—22no—
56#2145—31noSylow
57C2×C3×#41—61no—
58C2×#235—21no—
59C3×#135—32no—
60C3×#135—32no—
61C3×#135—61no—
62C3×#135—61no—
63C2×C3×Q16—31no—
64C2×C3×SD16—61no—
65C2×C3×D8—31no—
66C22×C3×C8—21no—
67C22×C3×C8C2×D311yescharacteristic
68C22×C3×C8C22×C311yescharacteristic
69C3×#117—22no—
70C3×#117—21no—
71C3×#117—21no—
72C2×Dic12C3×C411yescharacteristic, complemented
73#904—21no—
74#904—21no—
75#904C3×C412yescomplemented
76#904C22×C312yes—
77C2×D24C3×C411yescharacteristic, complemented
78C2×#204—21no—
79C3×M32—32no—
80C3×M32—61no—
81C2×C3×C16—61no—
82C3×#107—22no—
83C3×#107—21no—
84C3×#107—21no—
85C3×#106—22no—
86C3×#106C2×D312yescomplemented
87C3×#106C22×C312yescomplemented
88#812—21no—
89#812C22×C312yescomplemented
90C2×C3⋊C16—21no—
91C23×C32D811yescharacteristic
92C22×C3×D3—41no—
93C2×C32×C4—42no—
94C2×C32×C4C22⋊C411yescharacteristic
95C2×C32×C4C22⋊C411yescharacteristic
96C2×C32×C4D811yescharacteristic
97C2×C32×C4SD1611yescharacteristic
98C2×C32×C4SD1611yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C3×#67—42no—
101C3×#67—42no—
102C2×C3×Dic3—41no—
103C3×D12—22no—
104C3×D12—41no—
105C3×C4×D3—42no—
106C3×C4×D3—42no—
107C3×Dic6—22no—
108C3×Dic6—41no—
109C32×C8C22⋊C411yescharacteristic
110C32×C8C22⋊C411yescharacteristic
111C32×C8C22×C411yescharacteristic
112C32×C8C2×D411yescharacteristic
113C2×#135—31no—
114C2×M32—31no—
115#427C3×D311yescharacteristic, complemented
116#360—34no—
117C3×#41—62no—
118C3×#41—62no—
119C3×#41—121no—
120C3×#41—121no—
121C2×C3×Q8—61no—
122C2×C3×D4—61no—
123C2×C3×D4—121no—
124C22×C3×C4—21no—
125C22×C3×C4—121no—
126C22×C3×C4#6711yescharacteristic
127C22×C3×C4C3×D411yescharacteristic
128C2×#67—41no—
129#235—22no—
130#235—22no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C3×Q16—32no—
137C3×Q16—61no—
138C3×SD16—62no—
139C3×SD16—62no—
140C3×D8—32no—
141C3×D8—61no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8C4×D311yescharacteristic
149C2×C3×C8C4×D311yescharacteristic
150C2×C3×C8D1211yescharacteristic
151C2×C3×C8D1211yescharacteristic
152C2×C3×C8#6711yescharacteristic
153C2×C3×C8C2×C3×C411yescharacteristic
154C2×C3×C8C2×C3×C411yescharacteristic
155C2×C3×C8C3×D411yescharacteristic
156C2×C3×C8C3×D411yescharacteristic
157C2×C3×C8C3×D411yescharacteristic
158C2×C3×C8C22×D311yescharacteristic
159C2×C3×C8C23×C311yescharacteristic
160C3×C4⋊C4—24no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C22⋊C4—42no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C42—42no—
167C3×C42—41no—
168C3×C42—41no—
169Dic12—21no—
170Dic12C2×C3×C412yes—
171D24—21no—
172D24C2×C3×C412yes—
173#204—22no—
174#204—22no—
175C3×C16—62no—
176C3⋊C16—22no—
177C22×C32—41no—
178C22×C32#10211yescharacteristic
179C22×C32#10211yescharacteristic
180C22×C32C2×D811yescharacteristic
181C2×C3×D3—42no—
182C2×C3×D3—82no—
183C32×C4—82no—
184C32×C4#10211yescharacteristic
185C32×C4#10211yescharacteristic
186C32×C4C2×C22⋊C411yescharacteristic
187C32×C4C2×SD1611yescharacteristic
188C3×Dic3—42no—
189C2×#41—61no—
190#135—32no—
191#135—32no—
192#135—61no—
193#135—61no—
194C2×Q16—31no—
195C2×SD16—61no—
196C2×D8—31no—
197C22×C8C2×C3×D311yescharacteristic
198#117—21no—
199M32—32no—
200M32—61no—
201C2×C16—61no—
202#107—21no—
203#106C2×C3×D312yescomplemented
204C23×C3—21no—
205C23×C3—121no—
206C23×C3#21311yescharacteristic
207C23×C3C3×D811yescharacteristic
208C22×D3—41no—
209C3×Q8—62no—
210C3×Q8—121no—
211C3×D4—62no—
212C3×D4—122no—
213C3×D4—122no—
214C3×D4—121no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—121no—
225C2×C3×C4—122no—
226C2×C3×C4—122no—
227C2×C3×C4(C2×D3)⋊C411yescharacteristic
228C2×C3×C4(C2×D3)⋊C411yescharacteristic
229C2×C3×C4#21311yescharacteristic
230C2×C3×C4#21511yescharacteristic
231C2×C3×C4#21511yescharacteristic
232C2×C3×C4C3×C22⋊C411yescharacteristic
233C2×C3×C4C3×C22⋊C411yescharacteristic
234C2×C3×C4C3×D811yescharacteristic
235C2×C3×C4C3×SD1611yescharacteristic
236C2×C3×C4C3×SD1611yescharacteristic
237C2×C3×C4C2×#6711yescharacteristic
238C2×C3×C4C2×C3×D411yescharacteristic
239#67—42no—
240#67—42no—
241C2×Dic3—41no—
242D12—22no—
243D12—41no—
244C4×D3—42no—
245C4×D3—42no—
246Dic6—22no—
247Dic6—41no—
248C3×C8—21no—
249C3×C8—21no—
250C3×C8—21no—
251C3×C8—21no—
252C3×C8(C2×D3)⋊C411yescharacteristic
253C3×C8(C2×D3)⋊C411yescharacteristic
254C3×C8C3×C22⋊C411yescharacteristic
255C3×C8C3×C22⋊C411yescharacteristic
256C3×C8C2×C4×D311yescharacteristic
257C3×C8C2×D1211yescharacteristic
258C3×C8C22×C3×C411yescharacteristic
259C3×C8C2×C3×D411yescharacteristic
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32C2×#10211yescharacteristic
264C3×D3—82no—
265#41—62no—
266#41—62no—
267#41—121no—
268#41—121no—
269C2×Q8—61no—
270C2×D4—61no—
271C2×D4—121no—
272C22×C4—121no—
273C22×C4C3×#6711yescharacteristic
274Q16—32no—
275Q16—61no—
276SD16—62no—
277SD16—62no—
278D8—32no—
279D8—61no—
280C2×C8C3×C4×D311yescharacteristic
281C2×C8C3×C4×D311yescharacteristic
282C2×C8C3×D1211yescharacteristic
283C2×C8C3×D1211yescharacteristic
284C2×C8C3×#6711yescharacteristic
285C2×C8C22×C3×D311yescharacteristic
286C4⋊C4—22no—
287C22⋊C4—41no—
288C42—41no—
289C16—62no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—122no—
297C22×C3—242no—
298C22×C3#80911yescharacteristic
299C22×C3#80911yescharacteristic
300C22×C3C3×#10211yescharacteristic
301C22×C3C3×#10211yescharacteristic
302C22×C3C2×#21311yescharacteristic
303C22×C3C2×C3×D811yescharacteristic
304C2×D3—42no—
305C2×D3—82no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—84no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4—122no—
314C3×C4#80911yescharacteristic
315C3×C4#80911yescharacteristic
316C3×C4C3×#10211yescharacteristic
317C3×C4C3×#10211yescharacteristic
318C3×C4C2×(C2×D3)⋊C411yescharacteristic
319C3×C4C2×#21511yescharacteristic
320C3×C4C2×C3×C22⋊C411yescharacteristic
321C3×C4C2×C3×SD1611yescharacteristic
322Dic3—42no—
323C32#214511yescharacteristic, complemented, Sylow
324C23—121no—
325C23C3×#21311yescharacteristic
326Q8—62no—
327Q8—121no—
328D4—62no—
329D4—122no—
330D4—122no—
331D4—121no—
332C2×C4—42no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4C3×(C2×D3)⋊C411yescharacteristic
337C2×C4C3×(C2×D3)⋊C411yescharacteristic
338C2×C4C3×#21311yescharacteristic
339C2×C4C3×#21511yescharacteristic
340C2×C4C3×#21511yescharacteristic
341C2×C4C2×C3×#6711yescharacteristic
342C8C3×(C2×D3)⋊C411yescharacteristic
343C8C3×(C2×D3)⋊C411yescharacteristic
344C8C2×C3×C4×D311yescharacteristic
345C8C2×C3×D1211yescharacteristic
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—242no—
357C2×C3C2×#80911yescharacteristic
358C2×C3C2×C3×#10211yescharacteristic
359D3—82no—
360C22—41no—
361C22—122no—
362C22—242no—
363C22C3×#80911yescharacteristic
364C22C3×#80911yescharacteristic
365C22C2×C3×#21311yescharacteristic
366C4—82no—
367C4—122no—
368C4C3×#80911yescharacteristic
369C4C3×#80911yescharacteristic
370C4C2×C3×(C2×D3)⋊C411yescharacteristic
371C4C2×C3×#21511yescharacteristic
372C3—21no—
373C3#7065611yescharacteristic, direct product factor
374C3C3×#214511yescharacteristic, complemented
375C2—21no—
376C2—21no—
377C2—21no—
378C2—242no—
379C2C2×C3×#80911yescharacteristic
subgroupG/H
centerC3×C4#809
commutatorC3×C8C22×C3×C4
FrattiniC2×C8C22×C3×D3
FittingC32×#427C2
socleC2×C32C2×#102
prime radicalC2×C3×#904C2
cosocleC2×C3×C8C23×C3
hypercenterC3×#427D3
hypocenterC3C3×#2145
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C3×#427

Complements

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

NH ≅ G/Ntype
2. C32×#427C2semidirect
3. C2×C3×#812C2semidirect
4. C3×#4488C2semidirect
6. #70656C3direct (N × H)
10. C2×C3×Dic12C4semidirect
13. C3×#904C4semidirect
15. C2×C3×D24C4semidirect
18. C32×#106C22semidirect
20. C3×#812C22semidirect
26. C3×#427D3semidirect
27. C3×#427C2×C3semidirect
28. C2×#812C2×C3semidirect
30. #4488C2×C3semidirect
72. C2×Dic12C3×C4semidirect
75. #904C3×C4semidirect
77. C2×D24C3×C4semidirect
86. C3×#106C2×D3semidirect
87. C3×#106C22×C3semidirect
89. #812C22×C3semidirect
115. #427C3×D3semidirect
203. #106C2×C3×D3semidirect
323. C32#2145semidirect
373. C3#70656direct (N × H)
374. C3C3×#2145semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9042
C32×#4272
C2×C3×#8122
C3×#44882
C3×#21453
#706563