Finite Group Databasebeta

← C3×#24179 #10642786

Subgroups — C3×#24179 #10642786

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#3037C211yesmaximal, characteristic
4C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2005C212yesmaximal, complemented
7C3×#1994C211yesmaximal, characteristic, complemented
8C3×#1962C211yesmaximal, characteristic
9C3×#1882C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#24179C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#559—21no—
15C3×#557—21no—
16C3×#556—21no—
17C3×#556—22no—
18C3×#554—21no—
19C3×#552—22no—
20C3×#539—22no—
21C3×#536C2212yes—
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#529—22no—
26C3×#520C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C3×#479—22no—
30C3×#477—21no—
31C3×#475—21no—
32C3×#464—22no—
33C3×#462—21no—
34C3×#460—21no—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C3×#423C2211yescharacteristic
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C3×#419C2212yes—
41C3×#417C2212yes—
42C2×C3×#101C2212yes—
43C3×#391C2211yescharacteristic
44C3×#391C2211yescharacteristic
45C3×#389—22no—
46C3×#384C2211yescharacteristic
47C3×C4×C4⋊C4—22no—
48C3×C43C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3507C2×C311yescharacteristic
53#3078C2×C311yescharacteristic, complemented
54#3037C2×C311yescharacteristic
55C4×C4⋊Q8C2×C311yescharacteristic, complemented
56#2038C2×C312yescomplemented
57#2005C2×C312yescomplemented
58#1994C2×C311yescharacteristic, complemented
59#1962C2×C311yescharacteristic
60#1882C2×C312yescomplemented
61#1859C2×C312yescomplemented
62#1801C2×C311yescharacteristic, complemented
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2311yescharacteristic
66C3×#137—24no—
67C3×#137—42no—
68C2×C3×Q16—22no—
69C2×C3×SD16—22no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8D411yescharacteristic
78C3×#127D411yescharacteristic
79C3×#126—24no—
80C3×#126—24no—
81C3×#125—22no—
82C3×#125—24no—
83C3×#125D412yes—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124D412yes—
87C3×#123—24no—
88C3×#123—24no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×#121—21no—
95C3×#121—21no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—24no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×#117—24no—
114C3×#117—22no—
115C3×#117C2312yes—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4—22no—
119C2×C3×C4⋊C4—21no—
120C2×C3×C4⋊C4D412yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C42—21no—
124C2×C3×C42—21no—
125C2×C3×C42—21no—
126C2×C3×C42D412yes—
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2311yescharacteristic
129C2×C3×C42C2311yescharacteristic
130C3×#107—22no—
131C3×#106—22no—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C4⋊C8—21no—
135C3×C4⋊C8—21no—
136C3×#104—24no—
137C3×#104—42no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#103—22no—
141C3×#102—22no—
142C3×#101—42no—
143C3×#95—22no—
144C3×#95—22no—
145C2×#137C22×C312yes—
146#559—21no—
147#557—21no—
148#556—21no—
149#556—22no—
150#554—21no—
151#552—22no—
152#539—22no—
153#536C22×C312yes—
154#533C22×C311yescharacteristic
155#532C22×C311yescharacteristic
156C2×C4⋊Q8C22×C311yescharacteristic
157#529—22no—
158#520C22×C312yes—
159C2×C4×Q8C22×C311yescharacteristic
160C2×C4×Q8C22×C311yescharacteristic
161#479—22no—
162#477—21no—
163#475—21no—
164#464—22no—
165#462—21no—
166#460—21no—
167#428C22×C312yes—
168#423C22×C311yescharacteristic
169#423C22×C311yescharacteristic
170C2×#104C22×C312yes—
171C2×#104C22×C312yes—
172#419C22×C312yes—
173#417C22×C312yes—
174C2×#101C22×C312yes—
175#391C22×C311yescharacteristic
176#391C22×C311yescharacteristic
177#389—22no—
178#384C22×C311yescharacteristic
179C4×C4⋊C4—22no—
180C43C22×C311yescharacteristic
181#343C22×C312yes—
182#339C22×C312yes—
183#339C22×C312yes—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8C2×D412yes—
202C2×C3×Q8#4112yes—
203C2×C3×D4—22no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—22no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×Q16—44no—
219C3×SD16—44no—
220C3×M16—22no—
221C3×M16—22no—
222C3×M16—22no—
223C3×M16—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—28no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
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×C42—24no—
254C3×C42—24no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42—22no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—41no—
263C3×C42—41no—
264C3×C42C2×D411yescharacteristic
265C3×C42C2×D411yescharacteristic
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D411yescharacteristic
268C3×C42#4112yes—
269C2×#41C23×C312yes—
270C22×Q8C23×C311yescharacteristic
271C22×Q8C23×C311yescharacteristic
272#137—24no—
273#137—42no—
274C2×Q16—22no—
275C2×SD16—22no—
276C2×M16C23×C312yes—
277C2×M16C23×C312yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8C3×D411yescharacteristic
282C4⋊Q8C3×D411yescharacteristic
283C4⋊Q8C3×D411yescharacteristic
284#127C3×D411yescharacteristic
285#126—24no—
286#126—24no—
287#125—22no—
288#125—24no—
289#125C3×D412yes—
290#124—22no—
291#124—22no—
292#124C3×D412yes—
293#123—24no—
294#123—24no—
295C22⋊Q8—24no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300#121—21no—
301#121—21no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—24no—
317C4×D4—21no—
318C4×D4—21no—
319#117—24no—
320#117—22no—
321#117C23×C312yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C3×D412yes—
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C4⋊C4C23×C311yescharacteristic
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C3×D412yes—
333C2×C42C23×C311yescharacteristic
334C2×C42C23×C311yescharacteristic
335C2×C42C23×C311yescharacteristic
336#107—22no—
337#106—22no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342#104—24no—
343#104—42no—
344#103—22no—
345#103—22no—
346#103—22no—
347#102—22no—
348#101—42no—
349#95—22no—
350#95—22no—
351C23×C3—22no—
352C23×C3C22×D411yescharacteristic
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—42no—
364C3×Q8—81no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—81no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
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—41no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—84no—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C3×C8—42no—
406C3×C8—42no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×C3×D412yes—
425C2×Q8C3×#4112yes—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×C3×D412yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4C2×C3×D412yes—
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D412yes—
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C24×C311yescharacteristic
441Q16—44no—
442SD16—44no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—22no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—44no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×C3×D412yes—
469C4⋊C4C2×C3×D412yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—41no—
486C42—41no—
487C42C2×C3×D411yescharacteristic
488C42C2×C3×D411yescharacteristic
489C42C2×C3×D411yescharacteristic
490C42C2×C3×D411yescharacteristic
491C42C3×#4112yes—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3C2×#12011yescharacteristic
498C22×C3#52511yescharacteristic
499C22×C3#53711yescharacteristic
500C3×C4—24no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4—42no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4#45612yes—
513C3×C4#54611yescharacteristic
514C3×C4#54711yescharacteristic
515C3×C4#54811yescharacteristic
516C3×C4#55011yescharacteristic
517C23—22no—
518C23C22×C3×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—81no—
531D4—22no—
532D4—22no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—81no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
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—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4C3×#12011yescharacteristic
562C2×C4C3×#12011yescharacteristic
563C2×C4C3×#12011yescharacteristic
564C2×C4C3×#12011yescharacteristic
565C2×C4C22×C3×D411yescharacteristic
566C2×C4C22×C3×D411yescharacteristic
567C2×C4C2×C3×#4111yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C2×C4C3×2−1+411yescharacteristic
571C8—42no—
572C8—42no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—82no—
576C2×C3#240711yescharacteristic
577C2×C3C2×#45611yescharacteristic
578C2×C3#301811yescharacteristic
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22C2×C3×#12011yescharacteristic
585C22C3×#52511yescharacteristic
586C22C3×#53711yescharacteristic
587C4—24no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4C3×#45612yes—
600C4C3×#54611yescharacteristic
601C4C3×#54711yescharacteristic
602C4C3×#54811yescharacteristic
603C4C3×#55011yescharacteristic
604C3#2417911yescharacteristic, direct product factor, Sylow
605C2—21no—
606C2—21no—
607C2—82no—
608C2C3×#240711yescharacteristic
609C2C2×C3×#45611yescharacteristic
610C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#532C22
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×#3078C2semidirect
4. C3×C4×C4⋊Q8C2semidirect
5. C3×#2038C2semidirect
6. C3×#2005C2semidirect
7. C3×#1994C2semidirect
9. C3×#1882C2semidirect
10. C3×#1859C2semidirect
11. C3×#1801C2semidirect
12. #24179C3direct (N × H)
53. #3078C2×C3semidirect
55. C4×C4⋊Q8C2×C3semidirect
56. #2038C2×C3semidirect
57. #2005C2×C3semidirect
58. #1994C2×C3semidirect
60. #1882C2×C3semidirect
61. #1859C2×C3semidirect
62. #1801C2×C3semidirect
604. C3#24179direct (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×#30782
C3×#30372
C3×C4×C4⋊Q82
C3×#20382
C3×#20052
C3×#19942
C3×#19622
C3×#18822
C3×#18592
C3×#18012
#241793