Finite Group Databasebeta

← C3×#69448 #11783707

Subgroups — C3×#69448 #11783707

Order 1152.

825 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#208C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic
4C2×C3×#803C211yesmaximal, characteristic, complemented
5C2×C3×C4×C3⋊C8C211yesmaximal, characteristic, complemented
6C3×#4524C211yesmaximal, characteristic, complemented
7C3×#4524C211yesmaximal, characteristic, complemented
8C3×#1753—31nomaximal
9#69448C311yesmaximal, characteristic, direct product factor
10C23×C32×C4C2211yescharacteristic
11C2×C32×C22⋊C4C412yes—
12C2×C32×C42C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×#208—22no—
15C2×C3×#208—22no—
16C2×C3×#208C414yescomplemented
17C22×C3×C3⋊C8C2211yescharacteristic
18C22×C3×C3⋊C8C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C32×#95C412yes—
22C3×#830C414yescomplemented
23C3×#803—22no—
24C3×C4×C3⋊C8—22no—
25C22×C3×M16—31no—
26C22×#208C2×C311yescharacteristic
27C2×C3×C22⋊C8—31no—
28C2×C3×#97—31no—
29C2×C3×C4×C8—31no—
30C3×C4×C22⋊C4—21no—
31C3×C4×C22⋊C4D311yescharacteristic
32C3×C4×C22⋊C4C2×C311yescharacteristic
33C2×#830C2×C311yescharacteristic
34C2×#803C2×C311yescharacteristic, complemented
35C2×C4×C3⋊C8C2×C311yescharacteristic, complemented
36C3×#336—31no—
37C3×#336—31no—
38#4524C2×C311yescharacteristic, complemented
39#4524C2×C311yescharacteristic, complemented
40C24×C32C2×C411yescharacteristic
41C22×C32×C4—21no—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4C2×C412yes—
46C22×C32×C4C2×C411yescharacteristic
47C22×C32×C4C2×C412yes—
48C22×C32×C4C2×C412yes—
49C22×C32×C4C2×C412yes—
50C22×C32×C4C2311yescharacteristic
51C32×C22⋊C4—24no—
52C32×C42—21no—
53C32×C42—21no—
54C32×C42—21no—
55C32×C42—21no—
56C3×#208—28no—
57C2×C3×C3⋊C8—21no—
58C2×C3×C3⋊C8—21no—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8C2×C412yes—
64C2×C3×C3⋊C8C2×C412yes—
65C2×C3×C3⋊C8C2×C412yes—
66C2×C3×C3⋊C8C2×C412yes—
67C2×C3×C3⋊C8D412yes—
68C2×C3×C3⋊C8D412yes—
69#1753—31noSylow
70C23×C3×C4—21no—
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C22×C311yescharacteristic
73C2×C3×M16—34no—
74C2×C3×M16—62no—
75C2×C3×M16—62no—
76C22×C3×C8—31no—
77C22×C3×C8—31no—
78C22×C3×C8—31no—
79C22×C3×C8—31no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4Dic312yes—
82C2×C3×C22⋊C4C3×C412yes—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2×D311yescharacteristic
86C2×C3×C42C2×D311yescharacteristic
87C2×C3×C42C22×C311yescharacteristic
88C2×C3×C42C22×C311yescharacteristic
89C2×#208—22no—
90C2×#208—22no—
91C2×#208C3×C414yescomplemented
92C22×C3⋊C8C22×C311yescharacteristic
93C22×C3⋊C8C22×C311yescharacteristic
94C22×C3⋊C8C22×C311yescharacteristic
95C22×C3⋊C8C22×C311yescharacteristic
96C3×C22⋊C8—34no—
97C3×#97—62no—
98C3×C4×C8—62no—
99C3×#95—22no—
100C3×#95Dic312yes—
101C3×#95C3×C412yes—
102#830C3×C414yescomplemented
103#803—22no—
104C4×C3⋊C8—22no—
105C23×C32—22no—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32C4212yes—
110C23×C32C22×C411yescharacteristic
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4C4212yes—
123C2×C32×C4M1614yescomplemented
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C22×C411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4#4111yescharacteristic
129C2×C32×C4#4111yescharacteristic
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8C22⋊C414yescomplemented
134C22×M16—31no—
135C2×C22⋊C8—31no—
136C2×#97—31no—
137C2×C4×C8—31no—
138C4×C22⋊C4C3×D311yescharacteristic
139#336—31no—
140#336—31no—
141C24×C3—21no—
142C24×C3C2×Dic311yescharacteristic
143C24×C3C2×C3×C411yescharacteristic
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4C4×D312yes—
163C22×C3×C4C4×D312yes—
164C22×C3×C4C2×Dic312yes—
165C22×C3×C4C2×Dic312yes—
166C22×C3×C4C2×Dic311yescharacteristic
167C22×C3×C4C2×C3×C412yes—
168C22×C3×C4C2×C3×C412yes—
169C22×C3×C4C2×C3×C412yes—
170C22×C3×C4C2×C3×C411yescharacteristic
171C22×C3×C4C2×C3×C412yes—
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C23×C311yescharacteristic
174C3×M16—68no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—32no—
178C2×C3×C8—32no—
179C2×C3×C8—32no—
180C2×C3×C8—32no—
181C2×C3×C8—61no—
182C2×C3×C8—61no—
183C2×C3×C8—61no—
184C2×C3×C8—61no—
185C2×C3×C8—61no—
186C2×C3×C8—61no—
187C3×C22⋊C4—28no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—41no—
199C3×C42—41no—
200C3×C42—41no—
201C3×C42—41no—
202#208—28no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C2×C3×C412yes—
210C2×C3⋊C8C2×C3×C412yes—
211C2×C3⋊C8C2×C3×C412yes—
212C2×C3⋊C8C2×C3×C412yes—
213C2×C3⋊C8C3×D412yes—
214C2×C3⋊C8C3×D412yes—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32C2×C4211yescharacteristic
221C22×C32C4×D411yescharacteristic
222C22×C32C4×D411yescharacteristic
223C22×C32C2×M1612yes—
224C22×C32#13112yes—
225C32×C4—22no—
226C32×C4—24no—
227C32×C4—42no—
228C32×C4C2×C22⋊C411yescharacteristic
229C32×C4#11711yescharacteristic
230C32×C4C4×D411yescharacteristic
231C32×C4C4×D411yescharacteristic
232C23×C4C2×C3×D311yescharacteristic
233C2×M16—34no—
234C2×M16—62no—
235C2×M16—62no—
236C22×C8—31no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8—31no—
240C2×C22⋊C4C3×Dic312yes—
241C2×C42C2×C3×D311yescharacteristic
242C2×C42C2×C3×D311yescharacteristic
243C22⋊C8—34no—
244#97—62no—
245C4×C8—62no—
246#95C3×Dic312yes—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—22no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3C4×Dic312yes—
262C23×C3C3×C4212yes—
263C23×C3C22×Dic311yescharacteristic
264C23×C3C22×C3×C411yescharacteristic
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4#20814yescomplemented
307C2×C3×C4C4×Dic312yes—
308C2×C3×C4C3×C4212yes—
309C2×C3×C4C3×M1614yescomplemented
310C2×C3×C4C2×C4×D311yescharacteristic
311C2×C3×C4C2×C4×D311yescharacteristic
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4#23711yescharacteristic
315C2×C3×C4#23711yescharacteristic
316C2×C3×C4C22×C3×C411yescharacteristic
317C2×C3×C4C22×C3×C411yescharacteristic
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C3×#4111yescharacteristic
321C2×C3×C4C3×#4111yescharacteristic
322C3×C8—34no—
323C3×C8—62no—
324C3×C8—62no—
325C3×C8—62no—
326C3⋊C8—22no—
327C3⋊C8—22no—
328C3⋊C8—22no—
329C3⋊C8C3×C22⋊C414yescomplemented
330C2×C32—42no—
331C2×C32C4×C22⋊C411yescharacteristic
332C2×C32C4×M1611yescharacteristic
333C2×C32#40511yescharacteristic
334C2×C32#43512yes—
335C2×C32#43612yes—
336C24C2×C3×Dic311yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4C3×C4×D312yes—
342C22×C4C3×C4×D312yes—
343C22×C4C2×C3×Dic312yes—
344C22×C4C2×C3×Dic312yes—
345C22×C4C2×C3×Dic311yescharacteristic
346C22×C4C22×C3×D311yescharacteristic
347M16—68no—
348C2×C8—32no—
349C2×C8—32no—
350C2×C8—32no—
351C2×C8—32no—
352C2×C8—32no—
353C2×C8—32no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C22⋊C4—24no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C22×C3—24no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3C2×#20812yes—
386C22×C3C2×C4×Dic311yescharacteristic
387C22×C3D4×Dic311yescharacteristic
388C22×C3D4×Dic311yescharacteristic
389C22×C3#94812yes—
390C22×C3C2×C3×C4211yescharacteristic
391C22×C3C3×C4×D411yescharacteristic
392C22×C3C3×C4×D411yescharacteristic
393C22×C3C2×C3×M1612yes—
394C22×C3C3×#13112yes—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—24no—
398C3×C4—24no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—42no—
404C3×C4—44no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4#87711yescharacteristic
409C3×C4D3×C22⋊C411yescharacteristic
410C3×C4#67⋊C411yescharacteristic
411C3×C4#67⋊C411yescharacteristic
412C3×C4C2×C3×C22⋊C411yescharacteristic
413C3×C4C3×#11711yescharacteristic
414C3×C4C3×C4×D411yescharacteristic
415C3×C4C3×C4×D411yescharacteristic
416C32#175311yescharacteristic, complemented, Sylow
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23C3×C4×Dic312yes—
422C23C22×C3×Dic311yescharacteristic
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4C3×#20814yescomplemented
435C2×C4C3×C4×Dic312yes—
436C2×C4C2×C3×C4×D311yescharacteristic
437C2×C4C2×C3×C4×D311yescharacteristic
438C2×C4C3×D3×D411yescharacteristic
439C2×C4C3×D3×D411yescharacteristic
440C2×C4C3×#23711yescharacteristic
441C2×C4C3×#23711yescharacteristic
442C8—34no—
443C8—62no—
444C8—62no—
445C8—62no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—22no—
451C2×C3—44no—
452C2×C3—42no—
453C2×C3—42no—
454C2×C3C4×#20811yescharacteristic
455C2×C3Dic3×C22⋊C411yescharacteristic
456C2×C3#499711yescharacteristic
457C2×C3#501112yes—
458C2×C3#501212yes—
459C2×C3C3×C4×C22⋊C411yescharacteristic
460C2×C3C3×C4×M1611yescharacteristic
461C2×C3C3×#40511yescharacteristic
462C2×C3C3×#43512yes—
463C2×C3C3×#43612yes—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22C2×C3×#20812yes—
470C22C2×C3×C4×Dic311yescharacteristic
471C22C3×D4×Dic311yescharacteristic
472C22C3×D4×Dic311yescharacteristic
473C22C3×#94812yes—
474C4—22no—
475C4—24no—
476C4—42no—
477C4C3×#87711yescharacteristic
478C4C3×D3×C22⋊C411yescharacteristic
479C4C3×#67⋊C411yescharacteristic
480C4C3×#67⋊C411yescharacteristic
481C3—21no—
482C3#6944811yescharacteristic, direct product factor
483C3C3×#175311yescharacteristic, complemented
484C2—42no—
485C2C3×C4×#20811yescharacteristic
486C2C3×Dic3×C22⋊C411yescharacteristic
487C2C3×#499711yescharacteristic
488C2C3×#501112yes—
489C2C3×#501212yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C2×C3×C42
FrattiniC22×C4C22×C3×D3
FittingC32×C4×C22⋊C4C2
socleC23×C32C22×C4
prime radicalC24×C32C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC3×C4×C22⋊C4D3
hypocenterC3C3×#1753
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C3×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×#803C2semidirect
5. C2×C3×C4×C3⋊C8C2semidirect
6. C3×#4524C2semidirect
7. C3×#4524C2semidirect
9. #69448C3direct (N × H)
16. C2×C3×#208C4semidirect
22. C3×#830C4semidirect
34. C2×#803C2×C3semidirect
35. C2×C4×C3⋊C8C2×C3semidirect
38. #4524C2×C3semidirect
39. #4524C2×C3semidirect
91. C2×#208C3×C4semidirect
102. #830C3×C4semidirect
123. C2×C32×C4M16semidirect
133. C3×C3⋊C8C22⋊C4semidirect
306. C2×C3×C4#208semidirect
309. C2×C3×C4C3×M16semidirect
329. C3⋊C8C3×C22⋊C4semidirect
416. C32#1753semidirect
434. C2×C4C3×#208semidirect
482. C3#69448direct (N × H)
483. C3C3×#1753semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2082
C32×C4×C22⋊C42
C2×C3×#8302
C2×C3×#8032
C2×C3×C4×C3⋊C82
C3×#45242
C3×#17533
#694483