Finite Group Databasebeta

← C3×#25603 #10644210

Subgroups — C3×#25603 #10644210

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic
5C3×#2001C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1878C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25603C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#533—24no—
16C3×#533—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—21no—
20C2×C3×#124C2211yescharacteristic
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8—21no—
26C2×C3×C4×Q8C2212yes—
27C2×C3×#117—21no—
28C22×C3×C42C2211yescharacteristic
29C3×#497—24no—
30C3×#489—24no—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C2×C3×#97C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—21no—
41C3×#388—21no—
42C3×#386—21no—
43C3×#386—21no—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#533C2×C311yescharacteristic, complemented
49C2×#497C2×C311yescharacteristic, complemented
50C2×#489C2×C311yescharacteristic, complemented
51#2850C2×C311yescharacteristic
52#2001C2×C312yescomplemented
53#2000C2×C312yes—
54#1993C2×C311yescharacteristic, complemented
55#1960C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1878C2×C312yescomplemented
58#1749C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2312yes—
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C2×C3×Q16—42no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8D414yescomplemented
78C3×C4⋊Q8D414yescomplemented
79C3×#125—28no—
80C3×#124—22no—
81C3×#124—22no—
82C3×C22⋊Q8—28no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—42no—
86C3×#121—42no—
87C3×C4×Q8—28no—
88C3×C4×Q8—24no—
89C3×C4×Q8—24no—
90C3×#117—28no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—22no—
108C2×C3×C42—22no—
109C2×C3×C42—22no—
110C2×C3×C42D411yescharacteristic
111C2×C3×C42D411yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#103—24no—
115C3×#103—42no—
116C3×#103—42no—
117C3×#102—24no—
118C3×C22⋊C8—24no—
119C3×#97—22no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C22×Q16C22×C312yes—
126C22×SD16C22×C312yes—
127#533—24no—
128#533—24no—
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×C4⋊Q8C22×C311yescharacteristic
131C2×#125—21no—
132C2×#124C22×C311yescharacteristic
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8C22×C312yes—
135C2×C22⋊Q8C22×C311yescharacteristic
136C2×#121C22×C311yescharacteristic
137C2×C4×Q8—21no—
138C2×C4×Q8C22×C312yes—
139C2×#117—21no—
140C22×C42C22×C311yescharacteristic
141#497—24no—
142#489—24no—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148C2×#97C22×C311yescharacteristic
149C2×#97C22×C311yescharacteristic
150#391C22×C311yescharacteristic
151#390C22×C311yescharacteristic
152#388—21no—
153#388—21no—
154#386—21no—
155#386—21no—
156#382C22×C311yescharacteristic
157#340C22×C312yes—
158#340C22×C312yes—
159#336C22×C312yes—
160C24×C3—21no—
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8C2×D414yes—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—41no—
196C22×C3×C4—41no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C3×Q16—48no—
204C3×SD16—48no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8#4114yes—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4C2×D414yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—28no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42C2×D412yes—
234C3×C42C2×D412yes—
235C22×Q8—21no—
236C22×Q8C23×C311yescharacteristic
237C22×Q8C23×C312yes—
238C22×D4—21no—
239C22×D4C23×C311yescharacteristic
240C23×C4C3×D411yescharacteristic
241C23×C4C3×D411yescharacteristic
242C23×C4C23×C311yescharacteristic
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—42no—
249C22×C8C23×C312yes—
250C22×C8C23×C312yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8C3×D414yescomplemented
254C4⋊Q8C3×D414yescomplemented
255#125—28no—
256#124—22no—
257#124—22no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—42no—
262#121—42no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266#117—28no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C312yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42C3×D411yescharacteristic
287C2×C42C3×D411yescharacteristic
288C2×C42C23×C311yescharacteristic
289C2×C42C23×C311yescharacteristic
290#103—24no—
291#103—42no—
292#103—42no—
293#102—24no—
294C22⋊C8—24no—
295#97—22no—
296#97—22no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3#13712yes—
315C23×C3C22×D411yescharacteristic
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—48no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—84no—
326C2×C3×C4—28no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—44no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4#12012yes—
358C2×C3×C4#12012yes—
359C2×C3×C4#13712yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4H(5,𝔽2)12yes—
364C3×C8—24no—
365C3×C8—42no—
366C24—21no—
367C24C2×C3×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8C2×C3×D414yes—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C24×C311yescharacteristic
409Q16—48no—
410SD16—48no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8C3×#4114yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4C2×C3×D414yes—
427C22⋊C4—28no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C42—28no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42C2×C3×D412yes—
440C42C2×C3×D412yes—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3C2×#12011yescharacteristic
456C22×C3#53211yescharacteristic
457C22×C3#53511yescharacteristic
458C22×C3C2×#13711yescharacteristic
459C22×C3C2×#13711yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—41no—
466C3×C4—44no—
467C3×C4—41no—
468C3×C4—44no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4#54612yes—
473C3×C4#55012yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C3×#13712yes—
488C23C22×C3×D411yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496D4—42no—
497D4—42no—
498D4—84no—
499C2×C4—28no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4C3×#12012yes—
531C2×C4C3×#12012yes—
532C2×C4C3×#13712yes—
533C2×C4C22×C3×D411yescharacteristic
534C2×C4C22×C3×D411yescharacteristic
535C2×C4C2×C3×#4112yes—
536C2×C4C3×H(5,𝔽2)12yes—
537C8—24no—
538C8—42no—
539C2×C3—42no—
540C2×C3—161no—
541C2×C3#243111yescharacteristic
542C2×C3#300411yescharacteristic
543C2×C3#300511yescharacteristic
544C2×C3#317211yescharacteristic
545C2×C3#318111yescharacteristic
546C2×C3#321511yescharacteristic
547C2×C3#321611yescharacteristic
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22C2×C3×#12011yescharacteristic
563C22C3×#53211yescharacteristic
564C22C3×#53511yescharacteristic
565C22C2×C3×#13711yescharacteristic
566C22C2×C3×#13711yescharacteristic
567C22C3×#57511yescharacteristic
568C22C3×#57511yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—44no—
574C4—41no—
575C4—44no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4C3×#54612yes—
580C4C3×#55012yes—
581C3#2560311yescharacteristic, direct product factor, Sylow
582C2—42no—
583C2—161no—
584C2C3×#243111yescharacteristic
585C2C3×#300411yescharacteristic
586C2C3×#300511yescharacteristic
587C2C3×#317211yescharacteristic
588C2C3×#318111yescharacteristic
589C2C3×#321511yescharacteristic
590C2C3×#321611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#533C2semidirect
2. C2×C3×#497C2semidirect
3. C2×C3×#489C2semidirect
5. C3×#2001C2semidirect
7. C3×#1993C2semidirect
8. C3×#1960C2semidirect
9. C3×#1880C2semidirect
10. C3×#1878C2semidirect
11. C3×#1749C2semidirect
12. #25603C3direct (N × H)
48. C2×#533C2×C3semidirect
49. C2×#497C2×C3semidirect
50. C2×#489C2×C3semidirect
52. #2001C2×C3semidirect
54. #1993C2×C3semidirect
55. #1960C2×C3semidirect
56. #1880C2×C3semidirect
57. #1878C2×C3semidirect
58. #1749C2×C3semidirect
77. C3×C4⋊Q8D4semidirect
78. C3×C4⋊Q8D4semidirect
253. C4⋊Q8C3×D4semidirect
254. C4⋊Q8C3×D4semidirect
581. C3#25603direct (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×#4972
C2×C3×#4892
C3×#28502
C3×#20012
C3×#20002
C3×#19932
C3×#19602
C3×#18802
C3×#18782
C3×#17492
#256033