Finite Group Databasebeta

← C3×#25677 #10644284

Subgroups — C3×#25677 #10644284

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C3×#2851C211yesmaximal, characteristic
5C3×#2036C212yesmaximal
6C3×#2035C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic
12#25677C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#533—24no—
14C3×#533—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#501—24no—
26C3×#491—24no—
27C2×C3×#107C2212yes—
28C2×C3×#106C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2211yescharacteristic
35C2×C3×#97C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—21no—
39C3×#387—21no—
40C3×#385—21no—
41C3×#385—21no—
42C3×#382C2211yescharacteristic
43C3×#380C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#533C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#491C2×C311yescharacteristic, complemented
51#2851C2×C311yescharacteristic
52#2036C2×C312yes—
53#2035C2×C312yes—
54#1993C2×C311yescharacteristic, complemented
55#1960C2×C311yescharacteristic, complemented
56#1865C2×C312yescomplemented
57#1863C2×C312yescomplemented
58#1749C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×C4⋊Q8D414yes—
70C3×#125—28no—
71C3×#125—22no—
72C3×#125—22no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×#117—28no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—22no—
98C2×C3×C42—22no—
99C2×C3×C42—22no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C3×#107—24no—
105C3×#106—24no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#103—24no—
109C3×#103—24no—
110C3×C22⋊C8—24no—
111C3×#97—22no—
112C3×#97—22no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119#533—24no—
120#533—24no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#125—21no—
124C2×#125C22×C311yescharacteristic
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C22×C312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8C22×C312yes—
129C2×#117—21no—
130C22×C42C22×C311yescharacteristic
131#501—24no—
132#491—24no—
133C2×#107C22×C312yes—
134C2×#106C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#103C22×C312yes—
139C2×C22⋊C8C22×C312yes—
140C2×#97C22×C311yescharacteristic
141C2×#97C22×C311yescharacteristic
142#389C22×C311yescharacteristic
143#389C22×C311yescharacteristic
144#387—21no—
145#387—21no—
146#385—21no—
147#385—21no—
148#382C22×C311yescharacteristic
149#380C22×C311yescharacteristic
150#380C22×C311yescharacteristic
151#340C22×C312yes—
152#340C22×C312yes—
153#336C22×C312yes—
154C24×C3C2×D411yescharacteristic
155C2×C3×Q8—24no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—41no—
179C22×C3×C4—41no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8#4114yes—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D414yes—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C42—28no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42C2×D412yes—
218C3×C42C2×D412yes—
219C22×Q8—21no—
220C22×Q8C23×C312yes—
221C23×C4C3×D411yescharacteristic
222C23×C4C3×D411yescharacteristic
223C23×C4C23×C311yescharacteristic
224C22×C8C23×C312yes—
225C22×C8C23×C312yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8C3×D414yes—
229C4⋊Q8C3×D414yes—
230#125—28no—
231#125—22no—
232#125—22no—
233C22⋊Q8—28no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—24no—
239#117—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C3×D411yescharacteristic
261C2×C42C3×D411yescharacteristic
262C2×C42C23×C311yescharacteristic
263C2×C42C23×C311yescharacteristic
264#107—24no—
265#106—24no—
266#103—24no—
267#103—24no—
268#103—24no—
269#103—24no—
270C22⋊C8—24no—
271#97—22no—
272#97—22no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3#13712yes—
286C23×C3C22×D411yescharacteristic
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—48no—
290C3×Q8—42no—
291C3×Q8—42no—
292C2×C3×C4—28no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—44no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12012yes—
328C2×C3×C4#13712yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C42−1+412yes—
333C3×C8—24no—
334C3×C8—42no—
335C24C2×C3×D411yescharacteristic
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8C2×C3×D414yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C24×C311yescharacteristic
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8C3×#4114yes—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4C2×C3×D414yes—
390C22⋊C4—28no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C42—28no—
394C42—22no—
395C42—22no—
396C42—21no—
397C42—21no—
398C42C2×C3×D412yes—
399C42C2×C3×D412yes—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#53211yescharacteristic
409C22×C3#53611yescharacteristic
410C22×C3C2×#13711yescharacteristic
411C22×C3C2×#13711yescharacteristic
412C22×C3#57511yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—44no—
419C3×C4—41no—
420C3×C4—44no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4#54712yes—
426C3×C4#54812yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C3×#13712yes—
434C23C22×C3×D411yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—48no—
438Q8—42no—
439Q8—42no—
440C2×C4—28no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4C3×#12012yes—
475C2×C4C3×#12012yes—
476C2×C4C3×#13712yes—
477C2×C4C22×C3×D411yescharacteristic
478C2×C4C22×C3×D411yescharacteristic
479C2×C4C2×C3×#4112yes—
480C2×C4C3×2−1+412yes—
481C8—24no—
482C8—42no—
483C2×C3—42no—
484C2×C3#243211yescharacteristic
485C2×C3#300411yescharacteristic
486C2×C3#300511yescharacteristic
487C2×C3#317211yescharacteristic
488C2×C3#318111yescharacteristic
489C2×C3#323611yescharacteristic
490C2×C3#323711yescharacteristic
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22C2×C3×#12011yescharacteristic
499C22C3×#53211yescharacteristic
500C22C3×#53611yescharacteristic
501C22C2×C3×#13711yescharacteristic
502C22C2×C3×#13711yescharacteristic
503C22C3×#57511yescharacteristic
504C22C3×#57511yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—44no—
510C4—41no—
511C4—44no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4C3×#54712yes—
517C4C3×#54812yes—
518C3#2567711yescharacteristic, direct product factor, Sylow
519C2—42no—
520C2C3×#243211yescharacteristic
521C2C3×#300411yescharacteristic
522C2C3×#300511yescharacteristic
523C2C3×#317211yescharacteristic
524C2C3×#318111yescharacteristic
525C2C3×#323611yescharacteristic
526C2C3×#323711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#491C2semidirect
7. C3×#1993C2semidirect
8. C3×#1960C2semidirect
9. C3×#1865C2semidirect
10. C3×#1863C2semidirect
12. #25677C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#491C2×C3semidirect
54. #1993C2×C3semidirect
55. #1960C2×C3semidirect
56. #1865C2×C3semidirect
57. #1863C2×C3semidirect
518. C3#25677direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#5012
C2×C3×#4912
C3×#28512
C3×#20362
C3×#20352
C3×#19932
C3×#19602
C3×#18652
C3×#18632
C3×#17492
#256773