Finite Group Databasebeta

← C3×#69445 #11775225

Subgroups — C3×#69445 #11775225

Order 1152.

717 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#924C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic
3C3×#5207C211yesmaximal, characteristic
4C3×#4452C214yesmaximal, complemented
5C3×#1508—31nomaximal
6#69445C311yesmaximal, characteristic, direct product factor
7C23×C32×C4C2211yescharacteristic
8C22×C3×Dic6C411yescharacteristic, complemented
9C2×C32×C22⋊C4—21no—
10C2×C32×C42—21no—
11C2×C3×#217—21no—
12C2×C3×C4⋊Dic3C411yescharacteristic, complemented
13C3×#924—21no—
14C3×#924—21no—
15C3×#924C412yescomplemented
16C3×#924C2212yes—
17C2×C3×Dic3⋊C4—21no—
18C2×C3×#208—21no—
19C32×#95C2212yes—
20C3×#830—21no—
21C3×#830C2212yes—
22C2×C3×C22⋊Q8—31no—
23C2×#924C2×C311yescharacteristic
24C3×#407—31no—
25C3×C4×C22⋊C4—21no—
26C3×C4×C22⋊C4D311yescharacteristic
27C3×C4×C22⋊C4C2×C311yescharacteristic
28#5207C2×C311yescharacteristic
29C3×#328—34no—
30#4452C2×C314yescomplemented
31C24×C32D411yescharacteristic
32C22×C32×C4—22no—
33C22×C32×C4—42no—
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4C2×C411yescharacteristic
36C22×C32×C4D411yescharacteristic
37C22×C32×C4D411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—42no—
44C2×C3×Dic6C2×C412yes—
45C32×C22⋊C4—24no—
46C32×C42—42no—
47C3×#217—22no—
48C3×#217—22no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3C2×C412yes—
51C3×Dic3⋊C4—22no—
52C3×Dic3⋊C4—22no—
53C3×#208—42no—
54C2×C3×C3⋊C8—22no—
55#1508—31noSylow
56C22×C3×Q8—31no—
57C23×C3×C4—21no—
58C23×C3×C4C2×D311yescharacteristic
59C23×C3×C4C22×C311yescharacteristic
60C22×Dic6C3×C411yescharacteristic, complemented
61C2×C3×M16—61no—
62C3×C22⋊Q8—32no—
63C3×C22⋊Q8—32no—
64C3×C22⋊Q8—61no—
65C3×C22⋊Q8—61no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C4⋊C4—61no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—61no—
72C2×C3×C42—22no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×#217—21no—
76C2×C4⋊Dic3C3×C411yescharacteristic, complemented
77#924—21no—
78#924—21no—
79#924C3×C412yescomplemented
80#924C22×C312yes—
81C2×Dic3⋊C4—21no—
82C2×#208—21no—
83C3×C22⋊C8—32no—
84C3×C22⋊C8—61no—
85C3×#95—22no—
86C3×#95C2×D312yes—
87C3×#95C22×C312yes—
88#830—21no—
89#830C22×C312yes—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—41no—
93C23×C32C22⋊C411yescharacteristic
94C23×C32C22⋊C411yescharacteristic
95C23×C32C2×D411yescharacteristic
96C2×C32×C4—22no—
97C2×C32×C4—24no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4C22⋊C411yescharacteristic
106C2×C32×C4C22⋊C411yescharacteristic
107C2×C32×C4C22×C411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—42no—
113C3×Dic6—42no—
114C3×Dic6—42no—
115C3×C3⋊C8—42no—
116C2×C22⋊Q8—31no—
117#407—31no—
118C4×C22⋊C4C3×D311yescharacteristic
119#328—34no—
120C24×C3—21no—
121C24×C3#6711yescharacteristic
122C24×C3C3×D411yescharacteristic
123C2×C3×Q8—32no—
124C2×C3×Q8—61no—
125C2×C3×Q8—122no—
126C22×C3×C4—24no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4—44no—
138C22×C3×C4—61no—
139C22×C3×C4—61no—
140C22×C3×C4C4×D311yescharacteristic
141C22×C3×C4C4×D311yescharacteristic
142C22×C3×C4D1211yescharacteristic
143C22×C3×C4D1211yescharacteristic
144C22×C3×C4#6711yescharacteristic
145C22×C3×C4C2×C3×C411yescharacteristic
146C22×C3×C4C2×C3×C411yescharacteristic
147C22×C3×C4C3×D411yescharacteristic
148C22×C3×C4C3×D411yescharacteristic
149C22×C3×C4C3×D411yescharacteristic
150C22×C3×C4C22×D311yescharacteristic
151C22×C3×C4C23×C311yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—21no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C2×Dic6C2×C3×C412yes—
157C3×M16—122no—
158C2×C3×C8—62no—
159C3×C4⋊C4—32no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—62no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—24no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—62no—
167C3×C22⋊C4—62no—
168C3×C42—44no—
169C3×C42—42no—
170C3×C42—42no—
171#217—22no—
172#217—22no—
173C4⋊Dic3—21no—
174C4⋊Dic3C2×C3×C412yes—
175Dic3⋊C4—22no—
176Dic3⋊C4—22no—
177#208—42no—
178C2×C3⋊C8—22no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—44no—
187C22×C32#10412yes—
188C22×C32C2×C22⋊C411yescharacteristic
189C22×C32#13712yes—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—44no—
194C32×C4#9912yes—
195C3×Dic3—42no—
196C3×Dic3—42no—
197C22×Q8—31no—
198C23×C4C2×C3×D311yescharacteristic
199C2×M16—61no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—61no—
203C22⋊Q8—61no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—61no—
208C2×C42—21no—
209C22⋊C8—32no—
210C22⋊C8—61no—
211#95C2×C3×D312yes—
212C23×C3—24no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3(C2×D3)⋊C411yescharacteristic
225C23×C3(C2×D3)⋊C411yescharacteristic
226C23×C3C3×C22⋊C411yescharacteristic
227C23×C3C3×C22⋊C411yescharacteristic
228C23×C3C2×#6711yescharacteristic
229C23×C3C2×C3×D411yescharacteristic
230C3×Q8—122no—
231C3×Q8—122no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—44no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—122no—
266C2×C3×C4—122no—
267C2×C3×C4(C2×D3)⋊C411yescharacteristic
268C2×C3×C4(C2×D3)⋊C411yescharacteristic
269C2×C3×C4C3×C22⋊C411yescharacteristic
270C2×C3×C4C3×C22⋊C411yescharacteristic
271C2×C3×C4C2×C4×D311yescharacteristic
272C2×C3×C4C2×D1211yescharacteristic
273C2×C3×C4C22×C3×C411yescharacteristic
274C2×C3×C4C2×C3×D411yescharacteristic
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279Dic6—42no—
280Dic6—42no—
281C3×C8—122no—
282C3⋊C8—42no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—41no—
286C2×C32C2×#9911yescharacteristic
287C2×C32#41911yescharacteristic
288C2×C32C2×#10411yescharacteristic
289C24C3×#6711yescharacteristic
290C2×Q8—32no—
291C2×Q8—61no—
292C2×Q8—122no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4C3×C4×D311yescharacteristic
298C22×C4C3×C4×D311yescharacteristic
299C22×C4C3×D1211yescharacteristic
300C22×C4C3×D1211yescharacteristic
301C22×C4C3×#6711yescharacteristic
302C22×C4C22×C3×D311yescharacteristic
303M16—122no—
304C2×C8—62no—
305C4⋊C4—32no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C22⋊C4—24no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C42—42no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—48no—
340C22×C3#80512yes—
341C22×C3C3×#10412yes—
342C22×C3C2×(C2×D3)⋊C411yescharacteristic
343C22×C3#95112yes—
344C22×C3C2×C3×C22⋊C411yescharacteristic
345C22×C3C3×#13712yes—
346C3×C4—22no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—22no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—48no—
357C3×C4—44no—
358C3×C4—44no—
359C3×C4—122no—
360C3×C4—122no—
361C3×C4#80612yes—
362C3×C4C3×#9912yes—
363Dic3—42no—
364Dic3—42no—
365C32#150811yescharacteristic, complemented, Sylow
366C23—22no—
367C23—22no—
368C23—41no—
369C23C3×(C2×D3)⋊C411yescharacteristic
370C23C3×(C2×D3)⋊C411yescharacteristic
371C23C2×C3×#6711yescharacteristic
372Q8—122no—
373Q8—122no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4C3×(C2×D3)⋊C411yescharacteristic
388C2×C4C3×(C2×D3)⋊C411yescharacteristic
389C2×C4C2×C3×C4×D311yescharacteristic
390C2×C4C2×C3×D1211yescharacteristic
391C8—122no—
392C2×C3—24no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—42no—
401C2×C3—42no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3C2×#80511yescharacteristic
405C2×C3C2×#80611yescharacteristic
406C2×C3#500411yescharacteristic
407C2×C3C2×C3×#9911yescharacteristic
408C2×C3C3×#41911yescharacteristic
409C2×C3C2×C3×#10411yescharacteristic
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22C3×#80512yes—
419C22C2×C3×(C2×D3)⋊C411yescharacteristic
420C22C3×#95112yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—122no—
426C4—122no—
427C4C3×#80612yes—
428C3—21no—
429C3#6944511yescharacteristic, direct product factor
430C3C3×#150811yescharacteristic, complemented
431C2—22no—
432C2—22no—
433C2—41no—
434C2C2×C3×#80511yescharacteristic
435C2C2×C3×#80611yescharacteristic
436C2C3×#500411yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×C4×C22⋊C4C2
socleC22×C32C2×C22⋊C4
prime radicalC24×C32D4
cosocleC22×C3×C4C23×C3
hypercenterC3×C4×C22⋊C4D3
hypocenterC3C3×#1508
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#4452C2semidirect
6. #69445C3direct (N × H)
8. C22×C3×Dic6C4semidirect
12. C2×C3×C4⋊Dic3C4semidirect
15. C3×#924C4semidirect
30. #4452C2×C3semidirect
60. C22×Dic6C3×C4semidirect
76. C2×C4⋊Dic3C3×C4semidirect
79. #924C3×C4semidirect
365. C32#1508semidirect
429. C3#69445direct (N × H)
430. C3C3×#1508semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
C32×C4×C22⋊C42
C3×#52072
C3×#44522
C3×#15083
#694453