Finite Group Databasebeta

← #10690771

Subgroups — #10690771

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4726C211yesmaximal, characteristic, complemented
2#74048C211yesmaximal, characteristic
3C2×#4470C211yesmaximal, characteristic, complemented
4(C2×D3)⋊#95C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9117—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×ΓL(1,𝔽25)—21no—
13C2×#895C411yescharacteristic, complemented
14C2×#895C411yescharacteristic, complemented
15C2×#882—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C3×#380C2211yescharacteristic
18C2×C3×#95C2211yescharacteristic
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×(C2×D3)⋊C8—21no—
23#4987C411yescharacteristic, complemented
24#4987C411yescharacteristic, complemented
25#4958—21no—
26#4726—22no—
27#4726—22no—
28#4665—21no—
29#4665—21no—
30#4470—22no—
31#4451—22no—
32C2×#407—31no—
33#2847—31no—
34C2×#327—31no—
35C2×#327—31no—
36C22⋊#95—31no—
37#1294—31no—
38#1294D311yescharacteristic, complemented
39C24×D3D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C22×D12—21no—
46C22×D12—21no—
47C22×D12C2×C411yescharacteristic
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—21no—
52C22×C4×D3—42no—
53C22×C4×D3—42no—
54C22×C4×D3C2×C411yescharacteristic
55C22×C4×D3C2×C411yescharacteristic
56C22×C4×D3D411yescharacteristic
57C22×C4×D3D411yescharacteristic
58C22×C3×C8—21no—
59C2×C3×C4⋊C4C2×C411yescharacteristic
60C2×C3×C4⋊C4C2×C411yescharacteristic
61C2×C3×C22⋊C4—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—24no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×C4⋊Dic3C2×C411yescharacteristic
75C22×C3⋊C8—21no—
76C2×ΓL(1,𝔽25)—24no—
77C2×ΓL(1,𝔽25)—42no—
78C2×ΓL(1,𝔽25)—42no—
79#895—42no—
80#895—42no—
81#882—42no—
82#882—42no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yescomplemented
85C3×#95—21no—
86C3×#95—21no—
87C3×#95—21no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92C3⋊#95—42no—
93#830—21no—
94#830D412yescomplemented
95(C2×D3)⋊C8—24no—
96C24×C4—31no—
97C22×M16—61no—
98C2×#121—31no—
99C2×#121—31no—
100C2×#121—31no—
101C2×#120—61no—
102C22×C22⋊C4—61no—
103#407—62no—
104#407—62no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8—61no—
107C2×C22⋊C8C2×D311yescharacteristic
108#386—61no—
109#386—61no—
110#386—61no—
111#380—31no—
112#380—31no—
113#380C2×D311yescharacteristic
114C2×#95—31no—
115C2×#95—61no—
116C2×#95C2×D311yescharacteristic
117#327—62no—
118#327—62no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C23×D3—21no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—41no—
130C23×D3—41no—
131C23×D3C22⋊C411yescharacteristic
132C23×D3C22⋊C411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C22×C411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C2×#67—41no—
154C2×#67—41no—
155C2×#67—41no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3—41no—
166C22×Dic3—41no—
167C22×Dic3—41no—
168C22×Dic3C22⋊C411yescharacteristic
169C22×Dic3C22⋊C411yescharacteristic
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—41no—
173C2×D12—41no—
174C2×D12—42no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12—42no—
180C2×C4×D3—22no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—41no—
197C2×C4×D3—41no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207(C2×D3)⋊C4—42no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—42no—
210(C2×D3)⋊C4—48no—
211(C2×D3)⋊C4—41no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219ΓL(1,𝔽25)—48no—
220C25—31no—
221C22×D4—31no—
222C22×D4—61no—
223C22×D4—61no—
224C22×D4—61no—
225C23×C4—21no—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4—61no—
234C23×C4—61no—
235C23×C4—61no—
236C23×C4—122no—
237C23×C4—122no—
238C23×C4C22×D311yescharacteristic
239C2×M16—64no—
240C2×M16—122no—
241C2×M16—122no—
242C22×C8—21no—
243C22×C8—61no—
244#121—61no—
245#121—61no—
246#121—61no—
247#121—61no—
248#121—122no—
249#121—122no—
250#120—122no—
251#120—122no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4C4×D311yescharacteristic
254C2×C4⋊C4C4×D311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—122no—
263C2×C22⋊C4—122no—
264C22⋊C8—21no—
265C22⋊C8—32no—
266C22⋊C8—64no—
267C22⋊C8—61no—
268C22⋊C8D1212yescomplemented
269#95—21no—
270#95—21no—
271#95—21no—
272#95—61no—
273#95—61no—
274#95—61no—
275#95—61no—
276#95—122no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3C2×C22⋊C411yescharacteristic
282C23×C3#12111yescharacteristic
283C23×C3#12111yescharacteristic
284C23×C3AGL(1,ℤ/8)11yescharacteristic
285C23×C3AGL(1,ℤ/8)11yescharacteristic
286C23×C3AGL(1,ℤ/8)11yescharacteristic
287C23×C3#13711yescharacteristic
288C22×D3—24no—
289C22×D3—24no—
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—21no—
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3#9912yes—
324C22×D3#10412yes—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—81no—
343C2×C3×C4—81no—
344C2×C3×C4—81no—
345C2×C3×C4—81no—
346C2×C3×C4C4×D411yescharacteristic
347C2×C3×C4C4×D411yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12311yescharacteristic
350#67—81no—
351#67—81no—
352#67—81no—
353#67—81no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—81no—
368C2×Dic3—81no—
369C2×Dic3—81no—
370C2×Dic3—81no—
371C2×Dic3#9912yes—
372D12—81no—
373D12—81no—
374D12—84no—
375D12—84no—
376C4×D3—24no—
377C4×D3—24no—
378C4×D3—42no—
379C4×D3—42no—
380C4×D3—42no—
381C4×D3—42no—
382C4×D3—42no—
383C4×D3—42no—
384C3×C8—42no—
385C3⋊C8—42no—
386C24—31no—
387C24—31no—
388C24—62no—
389C24—62no—
390C24—61no—
391C24—61no—
392C24—61no—
393C24—61no—
394C24—61no—
395C24—61no—
396C24—61no—
397C24—121no—
398C24—121no—
399C24D3×D411yescharacteristic
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—122no—
405C2×D4—122no—
406C2×D4—122no—
407C2×D4—122no—
408C2×D4—122no—
409C2×D4—122no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—121no—
413C2×D4—121no—
414C2×D4—121no—
415C2×D4—121no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—121no—
447C22×C4—121no—
448C22×C4—121no—
449C22×C4—121no—
450C22×C4—124no—
451C22×C4—124no—
452C22×C4—124no—
453C22×C4—124no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4C2×C4×D311yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4#23511yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4#23911yescharacteristic
467M16—128no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—62no—
472C2×C8—121no—
473C2×C8—121no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—128no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—121no—
486C22⋊C4—121no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3#38611yescharacteristic
500C22×C3C2×#9911yescharacteristic
501C22×C3C2×#9911yescharacteristic
502C22×C3#41711yescharacteristic
503C22×C3#41811yescharacteristic
504C22×C3C2×#10411yescharacteristic
505C22×C3#42111yescharacteristic
506C22×C3#46811yescharacteristic
507C22×C3#46811yescharacteristic
508C22×C3#46911yescharacteristic
509C22×C3#46911yescharacteristic
510C2×D3—22no—
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—44no—
519C2×D3—44no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—82no—
540C3×C4—81no—
541C3×C4—81no—
542Dic3—22no—
543Dic3—41no—
544Dic3—82no—
545Dic3—81no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—32no—
551C23—32no—
552C23—64no—
553C23—64no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—122no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23D3×C22⋊C411yescharacteristic
588C23#88411yescharacteristic
589C23#89511yescharacteristic
590C23#91111yescharacteristic
591C23#91411yescharacteristic
592C23#91411yescharacteristic
593C23#91811yescharacteristic
594D4—241no—
595D4—241no—
596D4—244no—
597D4—244no—
598D4—241no—
599D4—241no—
600D4—241no—
601D4—241no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—32no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—241no—
646C2×C4—241no—
647C2×C4—241no—
648C2×C4—241no—
649C2×C4C4×D1211yescharacteristic
650C2×C4#88211yescharacteristic
651C2×C4#89311yescharacteristic
652C2×C4#89411yescharacteristic
653C8—42no—
654C8—122no—
655C2×C3—21no—
656C2×C3—21no—
657C2×C3—21no—
658C2×C3—21no—
659C2×C3#150411yescharacteristic
660C2×C3#150711yescharacteristic
661C2×C3#151011yescharacteristic
662C2×C3#151311yescharacteristic
663C2×C3#187111yescharacteristic
664C2×C3#189511yescharacteristic
665C2×C3#189811yescharacteristic
666D3—42no—
667D3—42no—
668D3—161no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—62no—
682C22—62no—
683C22—62no—
684C22—62no—
685C22—62no—
686C22—62no—
687C22—124no—
688C22—124no—
689C22—124no—
690C22—124no—
691C22—121no—
692C22—121no—
693C22—121no—
694C22—121no—
695C22—121no—
696C22—121no—
697C22—121no—
698C22—121no—
699C22—241no—
700C22—241no—
701C22—241no—
702C22—241no—
703C22—241no—
704C22—241no—
705C22—241no—
706C22#466511yescharacteristic
707C22D3×#9911yescharacteristic
708C22D3×#9911yescharacteristic
709C22#477011yescharacteristic
710C22#478211yescharacteristic
711C22#480311yescharacteristic
712C22#481211yescharacteristic
713C22D3×#10411yescharacteristic
714C22#482111yescharacteristic
715C22#486511yescharacteristic
716C22#488611yescharacteristic
717C4—21no—
718C4—21no—
719C4—21no—
720C4—21no—
721C4—62no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—121no—
726C4—242no—
727C4—241no—
728C3#911711yescharacteristic, complemented, Sylow
729C2—21no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2—122no—
734C2—122no—
735C2—481no—
736C2#6830311yescharacteristic
737C2#6830411yescharacteristic
738C2#6833411yescharacteristic
739C2#6833711yescharacteristic
740C2#6875511yescharacteristic
741C2#6888511yescharacteristic
742C2#6897011yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radical#74048C2
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9117
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
1. C2×#4726C2semidirect
3. C2×#4470C2semidirect
4. (C2×D3)⋊#95C2semidirect
5. C2×#4451C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
13. C2×#895C4semidirect
14. C2×#895C4semidirect
23. #4987C4semidirect
24. #4987C4semidirect
38. #1294D3semidirect
84. C3×C22⋊C8D4semidirect
94. #830D4semidirect
268. C22⋊C8D12semidirect
728. C3#9117semidirect

Maximal subgroups

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

subgroupindex
C2×#47262
#740482
C2×#44702
(C2×D3)⋊#952
C2×#44512
C3×#12942
#676242
#91173