Finite Group Databasebeta

← C3×#60498 #10679105

Subgroups — C3×#60498 #10679105

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic, complemented
2C3×C4×2−1+4C211yesmaximal, characteristic
3C3×#3083C216yesmaximal, complemented
4C3×#3082C216yesmaximal, complemented
5C3×#3079C213yesmaximal, complemented
6C3×#3078C213yesmaximal, complemented
7C3×#3076C213yesmaximal, complemented
8C2×C3×#477C213yesmaximal, complemented
9C3×#2913C212yesmaximal, complemented
10C3×#2906C211yesmaximal, characteristic, complemented
11C3×#2898C212yesmaximal
12#60498C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C2×C3×#131C2212yes—
15C3×Q82—21no—
16C3×#557—21no—
17C3×#554—23no—
18C3×D4×Q8—23no—
19C3×#544C2216yes—
20C3×#533C2213yes—
21C2×C3×C4⋊Q8C2213yes—
22C3×#520—23no—
23C3×#520C2213yes—
24C3×#520C2211yescharacteristic
25C3×#520C2212yes—
26C3×#519—21no—
27C3×#519C2213yes—
28C3×C4×#41—23no—
29C3×C4×#41C2213yes—
30C3×C4×#41C2211yescharacteristic
31C2×C3×C4×Q8—21no—
32C2×C3×C4×Q8C2213yes—
33C3×#479C2216yes—
34C3×#479C22112yes—
35C3×#478C2216yes—
36C3×#477C2213yes—
37C3×#477C2216yes—
38C3×#477C22112yes—
39C3×#476C2213yes—
40C3×#475C2213yes—
41C3×#475C2216yes—
42C3×#474C2211yescharacteristic
43C3×#474C2216yes—
44C3×#428C2216yes—
45C3×#428C2216yes—
46C3×#427C2216yes—
47C2×C3×#107C2216yes—
48C3×#424C2216yes—
49C3×#423C2213yes—
50C2×C3×C4⋊C8C2213yes—
51C3×#419C2216yes—
52C3×#417C2216yes—
53C3×#416C2216yes—
54C2×C3×#103C2216yes—
55#3503C2×C311yescharacteristic, complemented
56C4×2−1+4C2×C311yescharacteristic
57#3083C2×C316yescomplemented
58#3082C2×C316yescomplemented
59#3079C2×C313yescomplemented
60#3078C2×C313yescomplemented
61#3076C2×C313yescomplemented
62C2×#477C2×C313yescomplemented
63#2913C2×C312yescomplemented
64#2906C2×C311yescharacteristic, complemented
65#2898C2×C312yes—
66C3×2−1+4—24no—
67C3×2−1+4Q818yescomplemented
68C2×C3×#41—23no—
69C2×C3×#41C2313yes—
70C2×C3×#41C2311yescharacteristic
71C22×C3×Q8—21no—
72C22×C3×Q8—23no—
73C22×C3×Q8C2313yes—
74C3×#131—28no—
75C2×C3×M16C2316yes—
76C22×C3×C8C2316yes—
77C3×C4⋊Q8—26no—
78C3×C4⋊Q8—23no—
79C3×C4⋊Q8—23no—
80C3×C4⋊Q8C2311yescharacteristic
81C3×C4⋊Q8C2316yes—
82C3×C4⋊Q8C2313yes—
83C3×#125—26no—
84C3×#125C2316yes—
85C3×C22⋊Q8—23no—
86C3×C22⋊Q8—23no—
87C3×C22⋊Q8—212no—
88C3×C4×Q8—22no—
89C3×C4×Q8—21no—
90C3×C4×Q8—23no—
91C3×C4×Q8—23no—
92C3×C4×Q8—23no—
93C3×C4×Q8—23no—
94C3×C4×Q8—23no—
95C3×C4×Q8—23no—
96C3×C4×Q8D411yescharacteristic
97C3×C4×Q8D411yescharacteristic
98C3×C4×Q8C2313yes—
99C3×C4×Q8C2316yes—
100C3×C4×Q8C2313yes—
101C3×C4×D4—23no—
102C3×C4×D4—26no—
103C3×C4×D4—23no—
104C3×C4×D4—23no—
105C3×C4×D4—26no—
106C3×C4×D4D413yes—
107C3×C4×D4D413yes—
108C3×C4×D4C2311yescharacteristic
109C3×C4×D4C2313yes—
110C3×#117—23no—
111C3×#117—26no—
112C3×#117—23no—
113C3×#117C2316yes—
114C3×#117C2313yes—
115C3×#117C2313yes—
116C2×C3×C4⋊C4—23no—
117C2×C3×C4⋊C4—23no—
118C2×C3×C4⋊C4—23no—
119C2×C3×C4⋊C4C2316yes—
120C2×C3×C4⋊C4C2313yes—
121C2×C3×C42—23no—
122C2×C3×C42—23no—
123C2×C3×C42C2313yes—
124C3×#107C2316yes—
125C3×#107C23112yes—
126C3×#107C2312yes—
127C3×#106C2316yes—
128C3×#106C2316yes—
129C3×C4⋊C8C2311yescharacteristic
130C3×C4⋊C8C2313yes—
131C3×C4⋊C8C2316yes—
132C3×C4⋊C8C2316yes—
133C3×#103C2316yes—
134C3×#103C23112yes—
135C3×#103C2316yes—
136C3×#102C2312yes—
137C3×#102C2316yes—
138C2×2−1+4C22×C311yescharacteristic
139C2×#131C22×C312yes—
140Q82—21no—
141#557—21no—
142#554—23no—
143D4×Q8—23no—
144#544C22×C316yes—
145#533C22×C313yes—
146C2×C4⋊Q8C22×C313yes—
147#520—23no—
148#520C22×C313yes—
149#520C22×C312yes—
150#520C22×C311yescharacteristic
151#519—21no—
152#519C22×C313yes—
153C4×#41—23no—
154C4×#41C22×C313yes—
155C4×#41C22×C311yescharacteristic
156C2×C4×Q8—21no—
157C2×C4×Q8C22×C313yes—
158#479C22×C316yes—
159#479C22×C3112yes—
160#478C22×C316yes—
161#477C22×C313yes—
162#477C22×C316yes—
163#477C22×C3112yes—
164#476C22×C313yes—
165#475C22×C313yes—
166#475C22×C316yes—
167#474C22×C311yescharacteristic
168#474C22×C316yes—
169#428C22×C316yes—
170#428C22×C316yes—
171#427C22×C316yes—
172C2×#107C22×C316yes—
173#424C22×C316yes—
174#423C22×C313yes—
175C2×C4⋊C8C22×C313yes—
176#419C22×C316yes—
177#417C22×C316yes—
178#416C22×C316yes—
179C2×#103C22×C316yes—
180C3×#41—224no—
181C3×#41—26no—
182C3×#41C2×Q814yes—
183C3×#41C2×Q8112yescomplemented
184C3×#41#4114yes—
185C2×C3×Q8—28no—
186C2×C3×Q8—23no—
187C2×C3×Q8—21no—
188C2×C3×Q8—23no—
189C2×C3×Q8—26no—
190C2×C3×Q8—26no—
191C2×C3×Q8—86no—
192C2×C3×Q8C2×D411yescharacteristic
193C2×C3×Q8C2×Q8112yescomplemented
194C2×C3×Q8C2413yes—
195C2×C3×D4—23no—
196C2×C3×D4C2×D413yes—
197C2×C3×D4C2411yescharacteristic
198C22×C3×C4—23no—
199C22×C3×C4—23no—
200C22×C3×C4—23no—
201C22×C3×C4—26no—
202C22×C3×C4—21no—
203C22×C3×C4—23no—
204C22×C3×C4C2×D413yes—
205C22×C3×C4C2×D413yes—
206C22×C3×C4C2413yes—
207C3×M16—212no—
208C2×C3×C8—212no—
209C2×C3×C8C2416yes—
210C2×C3×C8C2412yes—
211C3×C4⋊C4—26no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4—23no—
215C3×C4⋊C4—26no—
216C3×C4⋊C4—26no—
217C3×C4⋊C4—23no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—26no—
220C3×C4⋊C4—26no—
221C3×C4⋊C4C2×D413yes—
222C3×C4⋊C4C2×D413yes—
223C3×C4⋊C4C2412yes—
224C3×C4⋊C4C2416yes—
225C3×C4⋊C4C2411yescharacteristic
226C3×C4⋊C4C2413yes—
227C3×C22⋊C4—23no—
228C3×C22⋊C4—26no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—23no—
231C3×C22⋊C4C2×D416yes—
232C3×C42—23no—
233C3×C42—23no—
234C3×C42—26no—
235C3×C42—23no—
236C3×C42—26no—
237C3×C42C2×D413yes—
238C3×C42C2×D413yes—
239C3×C42C2411yescharacteristic
240C3×C42C2413yes—
2412−1+4—24no—
2422−1+4C3×Q818yescomplemented
243C2×#41—23no—
244C2×#41C23×C313yes—
245C2×#41C23×C311yescharacteristic
246C22×Q8—21no—
247C22×Q8—23no—
248C22×Q8C23×C313yes—
249#131—28no—
250C2×M16C23×C316yes—
251C22×C8C23×C316yes—
252C4⋊Q8—26no—
253C4⋊Q8—23no—
254C4⋊Q8—23no—
255C4⋊Q8C23×C313yes—
256C4⋊Q8C23×C316yes—
257C4⋊Q8C23×C311yescharacteristic
258#125—26no—
259#125C23×C316yes—
260C22⋊Q8—23no—
261C22⋊Q8—23no—
262C22⋊Q8—212no—
263C4×Q8—21no—
264C4×Q8—22no—
265C4×Q8—23no—
266C4×Q8—23no—
267C4×Q8—23no—
268C4×Q8—23no—
269C4×Q8—23no—
270C4×Q8—23no—
271C4×Q8C3×D411yescharacteristic
272C4×Q8C3×D411yescharacteristic
273C4×Q8C23×C313yes—
274C4×Q8C23×C316yes—
275C4×Q8C23×C313yes—
276C4×D4—23no—
277C4×D4—26no—
278C4×D4—23no—
279C4×D4—23no—
280C4×D4—26no—
281C4×D4C3×D413yes—
282C4×D4C3×D413yes—
283C4×D4C23×C311yescharacteristic
284C4×D4C23×C313yes—
285#117—23no—
286#117—26no—
287#117—23no—
288#117C23×C313yes—
289#117C23×C316yes—
290#117C23×C313yes—
291C2×C4⋊C4—23no—
292C2×C4⋊C4—23no—
293C2×C4⋊C4—23no—
294C2×C4⋊C4C23×C313yes—
295C2×C4⋊C4C23×C316yes—
296C2×C42—23no—
297C2×C42—23no—
298C2×C42C23×C313yes—
299#107C23×C316yes—
300#107C23×C3112yes—
301#107C23×C312yes—
302#106C23×C316yes—
303#106C23×C316yes—
304C4⋊C8C23×C311yescharacteristic
305C4⋊C8C23×C313yes—
306C4⋊C8C23×C316yes—
307C4⋊C8C23×C316yes—
308#103C23×C316yes—
309#103C23×C3112yes—
310#103C23×C316yes—
311#102C23×C312yes—
312#102C23×C316yes—
313C23×C3—21no—
314C23×C3C22×D413yes—
315C3×Q8—212no—
316C3×Q8—23no—
317C3×Q8—46no—
318C3×Q8—42no—
319C3×Q8—86no—
320C3×Q8C22⋊Q814yes—
321C3×Q8C22×Q816yes—
322C3×D4—212no—
323C3×D4—21no—
324C3×D4C22⋊Q8112yes—
325C3×D4C22×Q812yes—
326C2×C3×C4—212no—
327C2×C3×C4—23no—
328C2×C3×C4—212no—
329C2×C3×C4—26no—
330C2×C3×C4—21no—
331C2×C3×C4—23no—
332C2×C3×C4—22no—
333C2×C3×C4—26no—
334C2×C3×C4—43no—
335C2×C3×C4—42no—
336C2×C3×C4—83no—
337C2×C3×C4—83no—
338C2×C3×C4—86no—
339C2×C3×C4C22×D413yes—
340C2×C3×C4C22×D413yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D413yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×Q816yes—
345C2×C3×C4C2×#4116yes—
346C2×C3×C4C2511yescharacteristic
347C3×C8—22no—
348C3×C8—26no—
349#41—224no—
350#41—26no—
351#41C2×C3×Q814yes—
352#41C2×C3×Q8112yescomplemented
353#41C3×#4114yes—
354C2×Q8—28no—
355C2×Q8—23no—
356C2×Q8—21no—
357C2×Q8—23no—
358C2×Q8—26no—
359C2×Q8—26no—
360C2×Q8—86no—
361C2×Q8C2×C3×D411yescharacteristic
362C2×Q8C2×C3×Q8112yescomplemented
363C2×Q8C24×C313yes—
364C2×D4—23no—
365C2×D4C2×C3×D413yes—
366C2×D4C24×C311yescharacteristic
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—26no—
371C22×C4—21no—
372C22×C4—23no—
373C22×C4C2×C3×D413yes—
374C22×C4C2×C3×D413yes—
375C22×C4C24×C313yes—
376M16—212no—
377C2×C8—212no—
378C2×C8C24×C316yes—
379C2×C8C24×C312yes—
380C4⋊C4—26no—
381C4⋊C4—23no—
382C4⋊C4—23no—
383C4⋊C4—23no—
384C4⋊C4—26no—
385C4⋊C4—26no—
386C4⋊C4—23no—
387C4⋊C4—21no—
388C4⋊C4—26no—
389C4⋊C4—26no—
390C4⋊C4C2×C3×D413yes—
391C4⋊C4C2×C3×D413yes—
392C4⋊C4C24×C312yes—
393C4⋊C4C24×C316yes—
394C4⋊C4C24×C311yescharacteristic
395C4⋊C4C24×C313yes—
396C22⋊C4—23no—
397C22⋊C4—26no—
398C22⋊C4—21no—
399C22⋊C4—23no—
400C22⋊C4C2×C3×D416yes—
401C42—23no—
402C42—23no—
403C42—26no—
404C42—23no—
405C42—26no—
406C42C2×C3×D413yes—
407C42C2×C3×D413yes—
408C42C24×C311yescharacteristic
409C42C24×C313yes—
410C22×C3—23no—
411C22×C3—23no—
412C22×C3—22no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3C2×C22⋊Q816yes—
416C22×C3C23×D411yescharacteristic
417C3×C4—21no—
418C3×C4—23no—
419C3×C4—26no—
420C3×C4—23no—
421C3×C4—43no—
422C3×C4—46no—
423C3×C4—41no—
424C3×C4—42no—
425C3×C4C2×C22⋊Q816yes—
426C3×C4#57812yes—
427C3×C4C23×Q811yescharacteristic
428C3×C4C22×#4111yescharacteristic
429C23—21no—
430C23C22×C3×D413yes—
431Q8—23no—
432Q8—212no—
433Q8—46no—
434Q8—42no—
435Q8—86no—
436Q8C3×C22⋊Q814yes—
437Q8C22×C3×Q816yes—
438D4—212no—
439D4—21no—
440D4C3×C22⋊Q8112yes—
441D4C22×C3×Q812yes—
442C2×C4—23no—
443C2×C4—212no—
444C2×C4—26no—
445C2×C4—212no—
446C2×C4—21no—
447C2×C4—23no—
448C2×C4—22no—
449C2×C4—26no—
450C2×C4—43no—
451C2×C4—42no—
452C2×C4—83no—
453C2×C4—86no—
454C2×C4—83no—
455C2×C4C22×C3×D413yes—
456C2×C4C22×C3×D413yes—
457C2×C4C22×C3×D411yescharacteristic
458C2×C4C22×C3×D411yescharacteristic
459C2×C4C22×C3×D413yes—
460C2×C4C22×C3×Q816yes—
461C2×C4C2×C3×#4116yes—
462C2×C4C25×C311yescharacteristic
463C8—22no—
464C8—26no—
465C2×C3—26no—
466C2×C3—42no—
467C2×C3C22×C22⋊Q811yescharacteristic
468C2×C3C2×#57811yescharacteristic
469C2×C3#358511yescharacteristic
470C22—23no—
471C22—23no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22C2×C3×C22⋊Q816yes—
476C22C23×C3×D411yescharacteristic
477C4—26no—
478C4—23no—
479C4—21no—
480C4—23no—
481C4—41no—
482C4—42no—
483C4—43no—
484C4—46no—
485C4C2×C3×C22⋊Q816yes—
486C4C3×#57812yes—
487C4C23×C3×Q811yescharacteristic
488C4C22×C3×#4111yescharacteristic
489C3#6049811yescharacteristic, direct product factor, Sylow
490C2—26no—
491C2—42no—
492C2C22×C3×C22⋊Q811yescharacteristic
493C2C2×C3×#57811yescharacteristic
494C2C3×#358511yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3503C2semidirect
3. C3×#3083C2semidirect
4. C3×#3082C2semidirect
5. C3×#3079C2semidirect
6. C3×#3078C2semidirect
7. C3×#3076C2semidirect
8. C2×C3×#477C2semidirect
9. C3×#2913C2semidirect
10. C3×#2906C2semidirect
12. #60498C3direct (N × H)
55. #3503C2×C3semidirect
57. #3083C2×C3semidirect
58. #3082C2×C3semidirect
59. #3079C2×C3semidirect
60. #3078C2×C3semidirect
61. #3076C2×C3semidirect
62. C2×#477C2×C3semidirect
63. #2913C2×C3semidirect
64. #2906C2×C3semidirect
67. C3×2−1+4Q8semidirect
183. C3×#41C2×Q8semidirect
193. C2×C3×Q8C2×Q8semidirect
242. 2−1+4C3×Q8semidirect
352. #41C2×C3×Q8semidirect
362. C2×Q8C2×C3×Q8semidirect
489. C3#60498direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×C4×2−1+42
C3×#30832
C3×#30822
C3×#30792
C3×#30782
C3×#30762
C2×C3×#4772
C3×#29132
C3×#29062
C3×#28982
#604983