Finite Group Databasebeta

← C3×#23593 #10642200

Subgroups — C3×#23593 #10642200

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×C4×#137C212yesmaximal, complemented
3C3×#2326C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic, complemented
6C3×#2037C212yesmaximal, complemented
7C3×#2028C211yesmaximal, characteristic, complemented
8C3×#2007C211yesmaximal, characteristic, complemented
9C3×#2004C212yesmaximal, complemented
10C3×#1759C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#23593C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#540—22no—
17C3×#538—22no—
18C3×#537C2211yescharacteristic
19C3×#535C2212yes—
20C3×#532—22no—
21C3×#529—22no—
22C2×C3×#124C2211yescharacteristic
23C3×#525—22no—
24C3×C4×#41C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#441—22no—
28C3×#440—22no—
29C3×C4×Q16—22no—
30C3×C4×SD16—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#417C2212yes—
36C2×C3×#100C2212yes—
37C3×C4×M16C2212yes—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—22no—
41C3×#385—22no—
42C3×#384C2211yescharacteristic
43C3×#376C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic
49C4×#137C2×C312yescomplemented
50#2326C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic, complemented
53#2037C2×C312yescomplemented
54#2028C2×C311yescharacteristic, complemented
55#2007C2×C311yescharacteristic, complemented
56#2004C2×C312yescomplemented
57#1759C2×C312yescomplemented
58C4×#100C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D412yes—
64C3×#137—24no—
65C3×#137—24no—
66C2×C3×Q16—22no—
67C2×C3×SD16—22no—
68C2×C3×M16C2312yes—
69C2×C3×M16C2312yes—
70C3×C4⋊Q8—22no—
71C3×#127—22no—
72C3×#126—24no—
73C3×#125—22no—
74C3×#124—24no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#123—24no—
79C3×#123—24no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×#121—22no—
84C3×#121—24no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#120—24no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8D412yes—
93C3×C4×Q8D412yes—
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D412yes—
100C3×C4×D4D412yes—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117C2312yes—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4D412yes—
111C2×C3×C42—22no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#107—22no—
117C3×#106—22no—
118C3×#104—24no—
119C3×#104—24no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#100—24no—
125C3×#97—22no—
126C3×C4×C8—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—22no—
130C2×#137C22×C312yes—
131#544C22×C311yescharacteristic
132#542—22no—
133#540—22no—
134#538—22no—
135#537C22×C311yescharacteristic
136#535C22×C312yes—
137#532—22no—
138#529—22no—
139C2×#124C22×C311yescharacteristic
140#525—22no—
141C4×#41C22×C312yes—
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144#441—22no—
145#440—22no—
146C4×Q16—22no—
147C4×SD16—22no—
148#428C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#419C22×C312yes—
152#417C22×C312yes—
153C2×#100C22×C312yes—
154C4×M16C22×C312yes—
155#391C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#388—22no—
158#385—22no—
159#384C22×C311yescharacteristic
160#376C22×C311yescharacteristic
161#343C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C24×C3C2×D412yes—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8#4112yescomplemented
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—82no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×Q16—28no—
201C3×SD16—28no—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16#4114yescomplemented
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—82no—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—82no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—21no—
233C3×C42—22no—
234C3×C42—42no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237C3×C42#4112yescomplemented
238C2×#41—22no—
239C2×#41C23×C312yes—
240C22×Q8C23×C311yescharacteristic
241C22×D4C23×C311yescharacteristic
242C23×C4C3×D412yes—
243#137—24no—
244#137—24no—
245C2×Q16—22no—
246C2×SD16—22no—
247C2×M16C23×C312yes—
248C2×M16C23×C312yes—
249C4⋊Q8—22no—
250#127—22no—
251#126—24no—
252#125—22no—
253#124—24no—
254#124—22no—
255#124—22no—
256#124—22no—
257#123—24no—
258#123—24no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—22no—
263#121—24no—
264#121—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#120—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8C3×D412yes—
272C4×Q8C3×D412yes—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4C3×D412yes—
279C4×D4C3×D412yes—
280#117—22no—
281#117—22no—
282#117C23×C312yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4C3×D412yes—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—22no—
296#106—22no—
297#104—24no—
298#104—24no—
299#103—22no—
300#103—22no—
301#103—22no—
302#102—22no—
303#100—24no—
304#97—22no—
305C4×C8—22no—
306#95—21no—
307#95—21no—
308#95—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3C22×D411yescharacteristic
316C3×Q8—24no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—42no—
320C3×Q8—82no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—84no—
328C3×D4—82no—
329C3×D4—84no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—82no—
353C2×C3×C4—82no—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12012yes—
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C22×D412yes—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4112yes—
360C2×C3×C4C2×#4111yescharacteristic
361C3×C8—24no—
362C3×C8—42no—
363C24C2×C3×D412yes—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—84no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8C2×C3×D412yes—
374C2×Q8C3×#4112yescomplemented
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4C2×C3×D412yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—82no—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C24×C311yescharacteristic
398Q16—28no—
399SD16—28no—
400M16—22no—
401M16—22no—
402M16—22no—
403M16C3×#4114yescomplemented
404C2×C8—22no—
405C2×C8—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—82no—
416C4⋊C4C2×C3×D412yes—
417C4⋊C4C2×C3×D412yes—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—82no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—21no—
431C42—22no—
432C42—42no—
433C42C2×C3×D412yes—
434C42C2×C3×D412yes—
435C42C3×#4112yescomplemented
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#52511yescharacteristic
448C22×C3#53211yescharacteristic
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4#52512yes—
461C3×C4#53212yes—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23C22×C3×D411yescharacteristic
469Q8—24no—
470Q8—22no—
471Q8—22no—
472Q8—42no—
473Q8—82no—
474D4—22no—
475D4—22no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—82no—
481D4—84no—
482D4—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4C3×#12011yescharacteristic
508C2×C4C3×#12012yes—
509C2×C4C3×#12011yescharacteristic
510C2×C4C22×C3×D412yes—
511C2×C4C2×C3×#4111yescharacteristic
512C2×C4C2×C3×#4112yes—
513C2×C4C2×C3×#4111yescharacteristic
514C8—24no—
515C8—42no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—82no—
519C2×C3—82no—
520C2×C3#238811yescharacteristic
521C2×C3#301911yescharacteristic
522C2×C3#302211yescharacteristic
523C22—21no—
524C22—21no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22C2×C3×#12011yescharacteristic
534C22C3×#52511yescharacteristic
535C22C3×#53211yescharacteristic
536C4—21no—
537C4—21no—
538C4—22no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4C3×#52512yes—
548C4C3×#53212yes—
549C3#2359311yescharacteristic, direct product factor, Sylow
550C2—21no—
551C2—21no—
552C2—82no—
553C2—82no—
554C2C3×#238811yescharacteristic
555C2C3×#301911yescharacteristic
556C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×#137C2semidirect
3. C3×#2326C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2037C2semidirect
7. C3×#2028C2semidirect
8. C3×#2007C2semidirect
9. C3×#2004C2semidirect
10. C3×#1759C2semidirect
11. C3×C4×#100C2semidirect
12. #23593C3direct (N × H)
49. C4×#137C2×C3semidirect
50. #2326C2×C3semidirect
51. #2060C2×C3semidirect
52. #2039C2×C3semidirect
53. #2037C2×C3semidirect
54. #2028C2×C3semidirect
55. #2007C2×C3semidirect
56. #2004C2×C3semidirect
57. #1759C2×C3semidirect
58. C4×#100C2×C3semidirect
176. C2×C3×Q8#41semidirect
205. C3×M16#41semidirect
237. C3×C42#41semidirect
374. C2×Q8C3×#41semidirect
403. M16C3×#41semidirect
435. C42C3×#41semidirect
549. C3#23593direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×C4×#1372
C3×#23262
C3×#20602
C3×#20392
C3×#20372
C3×#20282
C3×#20072
C3×#20042
C3×#17592
C3×C4×#1002
#235933