Finite Group Databasebeta

← C3×#69556 #11785232

Subgroups — C3×#69556 #11785232

Order 1152.

789 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×#382C211yesmaximal, characteristic, complemented
3C2×C3×#809C212yesmaximal
4C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#4529C212yesmaximal, complemented
6C3×#1804—31nomaximal
7#69556C311yesmaximal, characteristic, direct product factor
8C22×C3×D12C2211yescharacteristic
9C22×C3×C4×D3—21no—
10C2×C32×C4⋊C4C2212yes—
11C2×C32×C42C2211yescharacteristic
12C2×C3×(C2×D3)⋊C4—21no—
13C2×C3×C4⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊C8C2212yes—
15C3×C4×D12—22no—
16C3×C4×D12C414yescomplemented
17C32×#95—21no—
18C32×#95—21no—
19C3×#809—24no—
20C3×C4⋊(C3⋊C8)—22no—
21C2×C3×C4×D4—31no—
22C2×C4×D12C2×C311yescharacteristic
23C2×C3×C4⋊C8—31no—
24C2×C3×#102—32no—
25C3×#382—21no—
26C3×#382D311yescharacteristic, complemented
27C3×#382C2×C311yescharacteristic, complemented
28C2×#809C2×C312yes—
29C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
30C3×#340—32no—
31#4529C2×C312yescomplemented
32C23×C3×D3—21no—
33C22×C32×C4—22no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C22×C3×Dic3—21no—
38C2×C3×D12—21no—
39C2×C3×D12—21no—
40C2×C3×D12—21no—
41C2×C3×D12C2×C412yes—
42C2×C3×D12D412yes—
43C2×C3×D12Q812yes—
44C2×C3×C4×D3—24no—
45C2×C3×C4×D3—44no—
46C32×C4⋊C4—22no—
47C32×C4⋊C4—22no—
48C32×C42C2×C412yes—
49C32×C42D412yes—
50C3×(C2×D3)⋊C4—24no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3C2×C412yes—
53C2×C3×C3⋊C8—22no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—22no—
56#1804—31noSylow
57C22×C3×D4—31no—
58C23×C3×C4—61no—
59C22×D12C22×C311yescharacteristic
60C22×C4×D3—21no—
61C22×C3×C8—32no—
62C3×C4×D4—34no—
63C3×C4×D4—62no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4C2×D312yes—
67C2×C3×C4⋊C4C22×C312yes—
68C2×C3×C22⋊C4—61no—
69C2×C3×C42—21no—
70C2×C3×C42C2×D311yescharacteristic
71C2×C3×C42C22×C311yescharacteristic
72C2×(C2×D3)⋊C4—21no—
73C2×C4⋊Dic3C22×C311yescharacteristic
74C22×C3⋊C8C22×C312yes—
75C4×D12—22no—
76C4×D12C3×C414yescomplemented
77C3×C4⋊C8—62no—
78C3×#102—64no—
79C3×#95—22no—
80C3×#95—22no—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3×#95—21no—
85#809—24no—
86C4⋊(C3⋊C8)—22no—
87C23×C32C2×D411yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3—42no—
92C22×C3×D3—42no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—42no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4C22⋊C412yes—
99C2×C32×C4C22⋊C412yes—
100C2×C32×C4C22×C411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4C2×D411yescharacteristic
103C2×C32×C4C2×Q811yescharacteristic
104C2×C32×C4#4111yescharacteristic
105C2×C32×C4#4111yescharacteristic
106C2×C3×Dic3—22no—
107C2×C3×Dic3—42no—
108C3×D12—22no—
109C3×D12—22no—
110C3×D12—22no—
111C3×D12C4⋊C414yes—
112C3×C4×D3—48no—
113C3×C3⋊C8—42no—
114C2×C4×D4—31no—
115C2×C4⋊C8—31no—
116C2×#102—32no—
117#382C3×D311yescharacteristic, complemented
118#340—32no—
119C24×C3—61no—
120C23×D3—21no—
121C2×C3×D4—32no—
122C2×C3×D4—32no—
123C2×C3×D4—32no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C2×C3×D4—61no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—61no—
134C22×C3×C4—64no—
135C22×C3×C4—124no—
136C22×C3×C4#6711yescharacteristic
137C22×C3×C4#6711yescharacteristic
138C22×C3×C4C3×D411yescharacteristic
139C22×C3×C4C3×D411yescharacteristic
140C22×C3×C4C22×D311yescharacteristic
141C22×C3×C4C23×C311yescharacteristic
142C22×Dic3—21no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12C2×C3×C412yes—
147C2×D12C3×D412yes—
148C2×D12C3×Q812yes—
149C2×C4×D3—24no—
150C2×C4×D3—44no—
151C2×C3×C8—62no—
152C2×C3×C8—62no—
153C2×C3×C8—62no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—32no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—61no—
162C3×C22⋊C4—64no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42C4×D312yes—
166C3×C42D1212yes—
167C3×C42C2×C3×C412yes—
168C3×C42C3×D412yes—
169(C2×D3)⋊C4—24no—
170C4⋊Dic3—21no—
171C4⋊Dic3C2×C3×C412yes—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C22×C32C2×C22⋊C411yescharacteristic
176C22×C32#12111yescharacteristic
177C22×C32C22⋊Q811yescharacteristic
178C22×C32#13512yes—
179C22×C32AGL(1,ℤ/8)12yes—
180C2×C3×D3—24no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—44no—
184C2×C3×D3—44no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—82no—
188C32×C4C2×C4⋊C411yescharacteristic
189C32×C4#11711yescharacteristic
190C32×C4#12111yescharacteristic
191C32×C4C22⋊Q811yescharacteristic
192C3×Dic3—42no—
193C22×D4—31no—
194C23×C4—61no—
195C22×C8—32no—
196C4×D4—34no—
197C4×D4—62no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4C2×C3×D312yes—
200C2×C22⋊C4—61no—
201C2×C42C2×C3×D311yescharacteristic
202C4⋊C8—62no—
203#102—64no—
204#95—21no—
205#95—21no—
206C23×C3—21no—
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—62no—
210C23×C3—122no—
211C23×C3—122no—
212C23×C3C2×#6711yescharacteristic
213C23×C3C2×C3×D411yescharacteristic
214C22×D3—22no—
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—42no—
218C22×D3—42no—
219C3×D4—34no—
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—62no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—62no—
247C2×C3×C4—128no—
248C2×C3×C4—122no—
249C2×C3×C4(C2×D3)⋊C412yes—
250C2×C3×C4(C2×D3)⋊C412yes—
251C2×C3×C4C3×C22⋊C412yes—
252C2×C3×C4C3×C22⋊C412yes—
253C2×C3×C4C2×C4×D311yescharacteristic
254C2×C3×C4C2×D1211yescharacteristic
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4#23711yescharacteristic
257C2×C3×C4D3×Q811yescharacteristic
258C2×C3×C4#23911yescharacteristic
259C2×C3×C4C22×C3×C411yescharacteristic
260C2×C3×C4C2×C3×D411yescharacteristic
261C2×C3×C4C2×C3×D411yescharacteristic
262C2×C3×C4C2×C3×Q811yescharacteristic
263C2×C3×C4C3×#4111yescharacteristic
264C2×C3×C4C3×#4111yescharacteristic
265C2×Dic3—22no—
266C2×Dic3—42no—
267D12—22no—
268D12—22no—
269D12—22no—
270D12C3×C4⋊C414yes—
271C4×D3—48no—
272C3×C8—122no—
273C3⋊C8—42no—
274C2×C32#38011yescharacteristic
275C2×C32#41611yescharacteristic
276C2×C32#41811yescharacteristic
277C2×C32#46312yes—
278C2×C32#47812yes—
279C3×D3—44no—
280C24—61no—
281C2×D4—32no—
282C2×D4—32no—
283C2×D4—32no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C22×C4—22no—
288C22×C4—61no—
289C22×C4—64no—
290C22×C4—124no—
291C22×C4C3×#6711yescharacteristic
292C22×C4C3×#6711yescharacteristic
293C22×C4C22×C3×D311yescharacteristic
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—32no—
300C4⋊C4—61no—
301C22⋊C4—64no—
302C42C3×C4×D312yes—
303C42C3×D1212yes—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—64no—
310C22×C3—122no—
311C22×C3—122no—
312C22×C3—124no—
313C22×C3—124no—
314C22×C3C2×(C2×D3)⋊C411yescharacteristic
315C22×C3#93811yescharacteristic
316C22×C3#94611yescharacteristic
317C22×C3#94912yes—
318C22×C3#95012yes—
319C22×C3C2×C3×C22⋊C411yescharacteristic
320C22×C3C3×#12111yescharacteristic
321C22×C3C3×C22⋊Q811yescharacteristic
322C22×C3C3×#13512yes—
323C22×C3C3×AGL(1,ℤ/8)12yes—
324C2×D3—24no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—44no—
328C2×D3—44no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—84no—
340C3×C4—82no—
341C3×C4—82no—
342C3×C4—122no—
343C3×C4D3×C4⋊C411yescharacteristic
344C3×C4#89211yescharacteristic
345C3×C4#89511yescharacteristic
346C3×C4#89711yescharacteristic
347C3×C4C2×C3×C4⋊C411yescharacteristic
348C3×C4C3×#11711yescharacteristic
349C3×C4C3×#12111yescharacteristic
350C3×C4C3×C22⋊Q811yescharacteristic
351Dic3—42no—
352C32#180411yescharacteristic, complemented, Sylow
353C23—62no—
354C23—62no—
355C23—62no—
356C23—122no—
357C23—122no—
358C23C2×C3×#6711yescharacteristic
359D4—34no—
360D4—62no—
361D4—62no—
362D4—62no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—62no—
369C2×C4—128no—
370C2×C4—122no—
371C2×C4C3×(C2×D3)⋊C412yes—
372C2×C4C3×(C2×D3)⋊C412yes—
373C2×C4C2×C3×C4×D311yescharacteristic
374C2×C4C2×C3×D1211yescharacteristic
375C2×C4C3×D3×D411yescharacteristic
376C2×C4C3×#23711yescharacteristic
377C2×C4C3×D3×Q811yescharacteristic
378C2×C4C3×#23911yescharacteristic
379C8—122no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—124no—
386C2×C3#498711yescharacteristic
387C2×C3#500111yescharacteristic
388C2×C3#500211yescharacteristic
389C2×C3#505712yes—
390C2×C3#506612yes—
391C2×C3C3×#38011yescharacteristic
392C2×C3C3×#41611yescharacteristic
393C2×C3C3×#41811yescharacteristic
394C2×C3C3×#46312yes—
395C2×C3C3×#47812yes—
396D3—44no—
397C22—64no—
398C22—122no—
399C22—122no—
400C22—124no—
401C22—124no—
402C22C2×C3×(C2×D3)⋊C411yescharacteristic
403C22C3×#93811yescharacteristic
404C22C3×#94611yescharacteristic
405C22C3×#94912yes—
406C22C3×#95012yes—
407C4—22no—
408C4—22no—
409C4—82no—
410C4—122no—
411C4C3×D3×C4⋊C411yescharacteristic
412C4C3×#89211yescharacteristic
413C4C3×#89511yescharacteristic
414C4C3×#89711yescharacteristic
415C3—21no—
416C3#6955611yescharacteristic, direct product factor
417C3C3×#180411yescharacteristic, complemented
418C2—124no—
419C2C3×#498711yescharacteristic
420C2C3×#500111yescharacteristic
421C2C3×#500211yescharacteristic
422C2C3×#505712yes—
423C2C3×#506612yes—
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#382C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#382D3
hypocenterC3C3×#1804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C3×#382

Complements

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

NH ≅ G/Ntype
2. C32×#382C2semidirect
4. C2×C3×C4⋊(C3⋊C8)C2semidirect
5. C3×#4529C2semidirect
7. #69556C3direct (N × H)
16. C3×C4×D12C4semidirect
26. C3×#382D3semidirect
27. C3×#382C2×C3semidirect
29. C2×C4⋊(C3⋊C8)C2×C3semidirect
31. #4529C2×C3semidirect
76. C4×D12C3×C4semidirect
117. #382C3×D3semidirect
352. C32#1804semidirect
416. C3#69556direct (N × H)
417. C3C3×#1804semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C32×#3822
C2×C3×#8092
C2×C3×C4⋊(C3⋊C8)2
C3×#45292
C3×#18043
#695563