Finite Group Databasebeta

← C3×#25626 #10644233

Subgroups — C3×#25626 #10644233

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic, complemented
2C3×#3153C211yesmaximal, characteristic
3C3×#3140C211yesmaximal, characteristic, complemented
4C3×#2545C211yesmaximal, characteristic, complemented
5C3×#2005C212yesmaximal, complemented
6C3×#2003C212yesmaximal, complemented
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1882C212yesmaximal, complemented
11C3×C4×#101C211yesmaximal, characteristic, complemented
12#25626C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C3×#559—21no—
16C3×#557—21no—
17C3×#556—21no—
18C3×#556—22no—
19C3×#554—21no—
20C3×#552—22no—
21C3×#539—22no—
22C3×#536C2212yes—
23C3×#533C2211yescharacteristic
24C3×#532C2211yescharacteristic
25C2×C3×C4⋊Q8C2211yescharacteristic
26C3×#529—22no—
27C2×C3×#124C2211yescharacteristic
28C2×C3×#124C2211yescharacteristic
29C3×#520C2212yes—
30C2×C3×C4×Q8C2211yescharacteristic
31C3×#497—21no—
32C3×#496—21no—
33C3×#495—22no—
34C3×#489—21no—
35C3×#489—21no—
36C3×#488—22no—
37C2×C3×#104C2212yes—
38C2×C3×#104C2212yes—
39C3×#419C2212yes—
40C3×#418C2212yes—
41C2×C3×#101C2212yes—
42C3×C4×M16C2211yescharacteristic
43C3×C4×M16C2211yescharacteristic
44C3×#396—22no—
45C3×#391C2211yescharacteristic
46C3×#390C2211yescharacteristic
47C3×#388—22no—
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3507C2×C311yescharacteristic, complemented
53#3153C2×C311yescharacteristic
54#3140C2×C311yescharacteristic, complemented
55#2545C2×C311yescharacteristic, complemented
56#2005C2×C312yescomplemented
57#2003C2×C312yescomplemented
58#1986C2×C311yescharacteristic, complemented
59#1980C2×C311yescharacteristic, complemented
60#1883C2×C312yescomplemented
61#1882C2×C312yescomplemented
62C4×#101C2×C311yescharacteristic, complemented
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C3×#137—24no—
68C3×#137—42no—
69C3×AGL(1,ℤ/8)—24no—
70C3×AGL(1,ℤ/8)—42no—
71C2×C3×Q16—22no—
72C2×C3×SD16—22no—
73C2×C3×SD16—22no—
74C2×C3×D8—22no—
75C2×C3×M16C2312yes—
76C2×C3×M16C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—42no—
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D411yescharacteristic
81C3×C4⋊Q8D411yescharacteristic
82C3×#127D411yescharacteristic
83C3×#126—24no—
84C3×#126—24no—
85C3×#125—22no—
86C3×#125—24no—
87C3×#125D412yes—
88C3×#124—22no—
89C3×#124—22no—
90C3×#124—21no—
91C3×#124—21no—
92C3×#124—21no—
93C3×#124—21no—
94C3×#124—42no—
95C3×#124D412yes—
96C3×#123—24no—
97C3×#123—24no—
98C3×C22⋊Q8—24no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—22no—
103C3×#121—21no—
104C3×#121—21no—
105C3×C4×Q8—22no—
106C3×C4×Q8—22no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8—21no—
113C3×C4×D4—22no—
114C3×C4×D4—22no—
115C3×C4×D4—24no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×#117—24no—
119C3×#117—22no—
120C3×#117C2312yes—
121C2×C3×C4⋊C4—22no—
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4—22no—
124C2×C3×C4⋊C4D412yes—
125C2×C3×C4⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4—21no—
127C2×C3×C22⋊C4—22no—
128C2×C3×C22⋊C4—22no—
129C2×C3×C22⋊C4—21no—
130C2×C3×C42D412yes—
131C2×C3×C42C2311yescharacteristic
132C2×C3×C42C2311yescharacteristic
133C2×C3×C42C2311yescharacteristic
134C3×#104—42no—
135C3×#104—42no—
136C3×#103—22no—
137C3×#103—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#101—24no—
141C3×#97—21no—
142C3×#97—21no—
143C3×C4×C8—21no—
144C3×C4×C8—21no—
145C3×#95—21no—
146C3×#95—22no—
147C2×#137C22×C312yes—
148C2×AGL(1,ℤ/8)C22×C312yes—
149#559—21no—
150#557—21no—
151#556—21no—
152#556—22no—
153#554—21no—
154#552—22no—
155#539—22no—
156#536C22×C312yes—
157#533C22×C311yescharacteristic
158#532C22×C311yescharacteristic
159C2×C4⋊Q8C22×C311yescharacteristic
160#529—22no—
161C2×#124C22×C311yescharacteristic
162C2×#124C22×C311yescharacteristic
163#520C22×C312yes—
164C2×C4×Q8C22×C311yescharacteristic
165#497—21no—
166#496—21no—
167#495—22no—
168#489—21no—
169#489—21no—
170#488—22no—
171C2×#104C22×C312yes—
172C2×#104C22×C312yes—
173#419C22×C312yes—
174#418C22×C312yes—
175C2×#101C22×C312yes—
176C4×M16C22×C311yescharacteristic
177C4×M16C22×C311yescharacteristic
178#396—22no—
179#391C22×C311yescharacteristic
180#390C22×C311yescharacteristic
181#388—22no—
182C4×C4⋊C4C22×C311yescharacteristic
183#343C22×C312yes—
184#339C22×C312yes—
185#339C22×C312yes—
186C24×C3—21no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8C2×D412yes—
202C2×C3×D4—22no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—22no—
208C2×C3×D4—22no—
209C2×C3×D4C2×D412yes—
210C22×C3×C4—21no—
211C22×C3×C4—24no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C3×Q16—44no—
221C3×SD16—44no—
222C3×SD16—44no—
223C3×D8—44no—
224C3×M16—22no—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16#4114yes—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—44no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—44no—
254C3×C22⋊C4—44no—
255C3×C22⋊C4—41no—
256C3×C22⋊C4—41no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—41no—
259C3×C42—24no—
260C3×C42—22no—
261C3×C42—22no—
262C3×C42—22no—
263C3×C42—22no—
264C3×C42—22no—
265C3×C42C2×D411yescharacteristic
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D411yescharacteristic
268C3×C42C2×D411yescharacteristic
269C2×#41C23×C312yes—
270C22×Q8C23×C311yescharacteristic
271C22×Q8C23×C311yescharacteristic
272C22×D4C23×C311yescharacteristic
273#137—24no—
274#137—42no—
275AGL(1,ℤ/8)—24no—
276AGL(1,ℤ/8)—42no—
277C2×Q16—22no—
278C2×SD16—22no—
279C2×SD16—22no—
280C2×D8—22no—
281C2×M16C23×C312yes—
282C2×M16C23×C312yes—
283C4⋊Q8—22no—
284C4⋊Q8—42no—
285C4⋊Q8C3×D411yescharacteristic
286C4⋊Q8C3×D411yescharacteristic
287C4⋊Q8C3×D411yescharacteristic
288#127C3×D411yescharacteristic
289#126—24no—
290#126—24no—
291#125—22no—
292#125—24no—
293#125C3×D412yes—
294#124—22no—
295#124—22no—
296#124—21no—
297#124—21no—
298#124—21no—
299#124—21no—
300#124—42no—
301#124C3×D412yes—
302#123—24no—
303#123—24no—
304C22⋊Q8—24no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309#121—21no—
310#121—21no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—24no—
322C4×D4—21no—
323C4×D4—21no—
324#117—24no—
325#117—22no—
326#117C23×C312yes—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C3×D412yes—
331C2×C4⋊C4C23×C311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42C3×D412yes—
337C2×C42C23×C311yescharacteristic
338C2×C42C23×C311yescharacteristic
339C2×C42C23×C311yescharacteristic
340#104—42no—
341#104—42no—
342#103—22no—
343#103—22no—
344#102—22no—
345#102—22no—
346#101—24no—
347#97—21no—
348#97—21no—
349C4×C8—21no—
350C4×C8—21no—
351#95—21no—
352#95—22no—
353C23×C3—22no—
354C23×C3—42no—
355C23×C3—42no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8—42no—
367C3×Q8—81no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—81no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4#12011yescharacteristic
401C2×C3×C4#12011yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4112yes—
405C2×C3×C4H(5,𝔽2)12yes—
406C3×C8—24no—
407C3×C8—42no—
408C24—21no—
409#41—22no—
410#41—22no—
411#41—22no—
412#41—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8C2×C3×D412yes—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4C2×C3×D412yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D412yes—
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C24×C311yescharacteristic
442Q16—44no—
443SD16—44no—
444SD16—44no—
445D8—44no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16C3×#4114yes—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4C2×C3×D412yes—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42C2×C3×D411yescharacteristic
488C42C2×C3×D411yescharacteristic
489C42C2×C3×D411yescharacteristic
490C42C2×C3×D411yescharacteristic
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12011yescharacteristic
502C22×C3#53211yescharacteristic
503C22×C3#53511yescharacteristic
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4#54612yes—
517C3×C4#55012yes—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C22×C3×D411yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—81no—
533D4—22no—
534D4—22no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—81no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4C3×#12011yescharacteristic
564C2×C4C3×#12011yescharacteristic
565C2×C4C3×#12011yescharacteristic
566C2×C4C3×#12011yescharacteristic
567C2×C4C22×C3×D411yescharacteristic
568C2×C4C22×C3×D411yescharacteristic
569C2×C4C2×C3×#4112yes—
570C2×C4C3×H(5,𝔽2)12yes—
571C8—24no—
572C8—42no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—82no—
576C2×C3—161no—
577C2×C3#243111yescharacteristic
578C2×C3#301811yescharacteristic
579C2×C3#302011yescharacteristic
580C22—21no—
581C22—21no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—82no—
586C22—82no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22C2×C3×#12011yescharacteristic
591C22C3×#53211yescharacteristic
592C22C3×#53511yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4C3×#54612yes—
606C4C3×#55012yes—
607C3#2562611yescharacteristic, direct product factor, Sylow
608C2—21no—
609C2—21no—
610C2—82no—
611C2—161no—
612C2C3×#243111yescharacteristic
613C2C3×#301811yescharacteristic
614C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3153C2
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
1. C3×#3507C2semidirect
3. C3×#3140C2semidirect
4. C3×#2545C2semidirect
5. C3×#2005C2semidirect
6. C3×#2003C2semidirect
7. C3×#1986C2semidirect
8. C3×#1980C2semidirect
9. C3×#1883C2semidirect
10. C3×#1882C2semidirect
11. C3×C4×#101C2semidirect
12. #25626C3direct (N × H)
52. #3507C2×C3semidirect
54. #3140C2×C3semidirect
55. #2545C2×C3semidirect
56. #2005C2×C3semidirect
57. #2003C2×C3semidirect
58. #1986C2×C3semidirect
59. #1980C2×C3semidirect
60. #1883C2×C3semidirect
61. #1882C2×C3semidirect
62. C4×#101C2×C3semidirect
607. C3#25626direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#31532
C3×#31402
C3×#25452
C3×#20052
C3×#20032
C3×#19862
C3×#19802
C3×#18832
C3×#18822
C3×C4×#1012
#256263