Finite Group Databasebeta

← #10799082

Subgroups — #10799082

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#918C211yesmaximal, characteristic
2C2×#4881C211yesmaximal, characteristic, complemented
3C2×#4804C211yesmaximal, characteristic
4C2×D3×#103C211yesmaximal, characteristic, complemented
5C2×#4689C211yesmaximal, characteristic, complemented
6#76610C211yesmaximal, characteristic, complemented
7#76482C211yesmaximal, characteristic
8C3×#1969C211yesmaximal, characteristic, complemented
9#72067C211yesmaximal, characteristic, complemented
10#70629C211yesmaximal, characteristic, complemented
11#70258C211yesmaximal, characteristic, complemented
12#70257C211yesmaximal, characteristic, complemented
13#69774C211yesmaximal, characteristic, complemented
14#69679C211yesmaximal, characteristic, complemented
15#69678C211yesmaximal, characteristic, complemented
16#21234—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C22×D3×Q8C2211yescharacteristic
19C22×C3×Q16C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#918—24no—
23C2×#918—24no—
24C2×#918C418yescomplemented
25C22×#204C2211yescharacteristic
26C22×ΓL(1,𝔽25)C2211yescharacteristic
27C22×D3×C8C2211yescharacteristic
28C2×#892C2211yescharacteristic
29C2×D3×C4⋊C4C2211yescharacteristic
30C2×#872C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#391C2211yescharacteristic
37#5229C2211yescharacteristic
38#5229C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41C2×#821C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#4988C2211yescharacteristic
47#4973C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#808C2211yescharacteristic
51C2×#807C2211yescharacteristic
52#4939C2211yescharacteristic
53#4929C2211yescharacteristic
54C2×#803C2211yescharacteristic
55#4881—24no—
56#4804—24no—
57D3×#103—24no—
58#4689—24no—
59#4667—22no—
60#4665—22no—
61#4664—22no—
62#4646—22no—
63C22×#137—31no—
64C2×#428—31no—
65C2×#416—31no—
66C22×#103—31no—
67C2×#405—31no—
68#2311—31no—
69#2310—31no—
70#1973—31no—
71#1973—31no—
72#1971—31no—
73#1971—31no—
74#1969—31no—
75#1969D311yescharacteristic, complemented
76#1968—31no—
77#1968—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C2×#239—22no—
81C2×#239—22no—
82C2×#239—22no—
83C2×#239—22no—
84C2×#239—22no—
85C2×#239C2×C414yes—
86C2×D3×Q8—22no—
87C2×D3×Q8—22no—
88C2×D3×Q8—22no—
89C2×D3×Q8—22no—
90C2×D3×Q8—22no—
91C2×D3×Q8C2×C414yes—
92C22×D12—21no—
93C22×D12C2311yescharacteristic
94C22×C4×D3—21no—
95C22×C4×D3—21no—
96C22×C4×D3—21no—
97C22×C4×D3C2311yescharacteristic
98C22×Dic6—21no—
99C22×Dic6C2311yescharacteristic
100C2×C3×Q16—22no—
101C2×C3×Q16—22no—
102C2×C3×Q16C2×C414yescomplemented
103C22×C3×C8C2311yescharacteristic
104C22×C3×C8C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×#216—22no—
109C2×#216—22no—
110C2×#216C2×C414yescomplemented
111C2×#215—22no—
112C2×#215—22no—
113C2×#215C2×C414yescomplemented
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4D412yes—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4D412yes—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#918—232no—
127C2×#204—22no—
128C2×#204—22no—
129C2×#204C2×C414yescomplemented
130C2×ΓL(1,𝔽25)C2×C414yes—
131C2×ΓL(1,𝔽25)D414yes—
132C2×ΓL(1,𝔽25)D414yes—
133C2×D3×C8—22no—
134C2×D3×C8—22no—
135C2×D3×C8D414yes—
136#892—22no—
137#892—22no—
138D3×C4⋊C4—22no—
139D3×C4⋊C4—22no—
140#872—22no—
141#872—22no—
142C3×#107—22no—
143C3×#103—22no—
144C3×#103—22no—
145C3×C4×C8—22no—
146C3×#95—22no—
147C3×#95—22no—
148#835—22no—
149#835—22no—
150C3⋊#95—22no—
151C3⋊#95—22no—
152C3⋊#95—22no—
153C3⋊#95—22no—
154#821—22no—
155#817—22no—
156#816—22no—
157#815—22no—
158C8×Dic3—22no—
159#810—22no—
160#809—22no—
161#808—22no—
162#807—22no—
163#803—22no—
164C22×#41—31no—
165C23×Q8—31no—
166C2×#137—38no—
167C2×#137—64no—
168C2×#137—64no—
169C22×Q16—31no—
170C22×Q16C2×D311yescharacteristic
171C22×SD16—31no—
172C22×SD16—31no—
173C22×M16—31no—
174C23×C8—31no—
175C2×#117—31no—
176C2×#117—31no—
177C22×C4⋊C4—31no—
178#428—64no—
179C2×#107—31no—
180C2×#107C2×D311yescharacteristic
181C2×#106—31no—
182C2×#106—31no—
183#416—64no—
184C2×#103—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—64no—
189C2×#103C2×D311yescharacteristic
190C2×#103C2×D311yescharacteristic
191C2×#102—31no—
192C2×#102—31no—
193#405—64no—
194C2×#97—31no—
195C2×#97—31no—
196C2×C4×C8—31no—
197C2×C4×C8C2×D311yescharacteristic
198#391—31no—
199#391—31no—
200#391—31no—
201#391—31no—
202#391—62no—
203#391C2×D311yescharacteristic
204#391C2×D311yescharacteristic
205#390—31no—
206#390—31no—
207#386—62no—
208#381—62no—
209#380—62no—
210C23×D3—21no—
211C23×D3C2×D411yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8C22×C412yes—
223C2×C3×Q8C22×C412yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3C2×D412yes—
234C22×Dic3C2×D411yescharacteristic
235#239—28no—
236#239—28no—
237#239—44no—
238#239—44no—
239D3×Q8—28no—
240D3×Q8—28no—
241D3×Q8—44no—
242D3×Q8—44no—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—24no—
248C2×D12—21no—
249C2×D12—42no—
250C2×D12—42no—
251C2×D12C22×C412yes—
252C2×C4×D3—24no—
253C2×C4×D3—24no—
254C2×C4×D3—42no—
255C2×C4×D3—42no—
256C2×C4×D3—42no—
257C2×C4×D3—42no—
258C2×C4×D3—42no—
259C2×C4×D3—82no—
260C2×C4×D3C22×C412yes—
261C2×C4×D3C2×D412yes—
262C2×C4×D3C2×D412yes—
263C2×C4×D3C2×D412yes—
264C2×C4×D3#4112yes—
265C2×C4×D3#4112yes—
266C2×Dic6—21no—
267C2×Dic6—21no—
268C2×Dic6—21no—
269C2×Dic6—21no—
270C2×Dic6—21no—
271C2×Dic6—24no—
272C2×Dic6—42no—
273C2×Dic6—42no—
274C2×Dic6C22×C412yes—
275C3×Q16—28no—
276C2×C3×C8—21no—
277C2×C3×C8—21no—
278C2×C3×C8C22×C412yes—
279C2×C3×C8C2×D412yes—
280C2×C3×C8C2×D412yes—
281C2×C3×C8C2×D412yes—
282C3×C4⋊C4—21no—
283C3×C4⋊C4—21no—
284C3×C4⋊C4—21no—
285C3×C4⋊C4—21no—
286C3×C42—21no—
287C3×C42—21no—
288#216—28no—
289#215—28no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—42no—
292(C2×D3)⋊C4—82no—
293(C2×D3)⋊C4—82no—
294C4⋊Dic3—21no—
295C4⋊Dic3—21no—
296C4⋊Dic3—21no—
297C4⋊Dic3—21no—
298Dic3⋊C4—24no—
299Dic3⋊C4—42no—
300C4×Dic3—21no—
301C4×Dic3—21no—
302C4×Dic3—42no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8C22×C412yes—
306C2×C3⋊C8C2×D412yes—
307C2×C3⋊C8C2×D412yes—
308C2×C3⋊C8C2×D412yes—
309#204—28no—
310ΓL(1,𝔽25)C22⋊C4116yescomplemented
311D3×C8—28no—
312C2×#41—34no—
313C2×#41—62no—
314C2×#41—62no—
315C2×#41—62no—
316C2×#41—62no—
317C2×#41—62no—
318C22×Q8—34no—
319C22×Q8—31no—
320C22×Q8—61no—
321C22×Q8—62no—
322C22×Q8—62no—
323C22×Q8—62no—
324C22×Q8—62no—
325C22×Q8—62no—
326C22×Q8C22×D311yescharacteristic
327C22×Q8C22×D311yescharacteristic
328C22×D4—31no—
329C22×D4—61no—
330C23×C4—31no—
331C23×C4—61no—
332C23×C4—61no—
333C23×C4—61no—
334#137—632no—
335C2×Q16—22no—
336C2×Q16—22no—
337C2×Q16—34no—
338C2×Q16—62no—
339C2×Q16—62no—
340C2×Q16C4×D314yescomplemented
341C2×SD16—34no—
342C2×SD16—34no—
343C2×SD16—62no—
344C2×SD16—62no—
345C2×SD16—62no—
346C2×SD16—62no—
347C2×M16—34no—
348C2×M16—34no—
349C2×M16—34no—
350C22×C8—31no—
351C22×C8—31no—
352C22×C8—34no—
353C22×C8—62no—
354C22×C8—62no—
355C22×C8C22×D311yescharacteristic
356C22×C8C22×D311yescharacteristic
357#117—62no—
358#117—62no—
359#117—62no—
360#117—62no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—31no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—62no—
365C2×C4⋊C4—62no—
366C2×C4⋊C4—61no—
367C2×C4⋊C4C22×D311yescharacteristic
368C2×C4⋊C4C22×D311yescharacteristic
369C2×C22⋊C4—32no—
370C2×C22⋊C4—61no—
371C2×C22⋊C4—62no—
372C2×C22⋊C4—62no—
373C2×C42—31no—
374C2×C42—61no—
375C2×C42C22×D311yescharacteristic
376#107—22no—
377#107—62no—
378#106—62no—
379#106—62no—
380#103—22no—
381#103—22no—
382#103—62no—
383#103—62no—
384#103—62no—
385#103—62no—
386#102—62no—
387#102—62no—
388#97—62no—
389#97—62no—
390C4×C8—22no—
391C4×C8—62no—
392#95—22no—
393#95—22no—
394#95—62no—
395#95—62no—
396#95—62no—
397#95—62no—
398C23×C3C22×D411yescharacteristic
399C22×D3—22no—
400C22×D3—22no—
401C22×D3—22no—
402C22×D3—41no—
403C22×D3—41no—
404C22×D3—42no—
405C22×D3—42no—
406C22×D3C4×D412yes—
407C22×D3#12112yes—
408C22×D3#12712yes—
409C3×Q8—22no—
410C3×Q8—22no—
411C3×Q8—22no—
412C3×Q8—22no—
413C3×Q8—41no—
414C3×Q8—41no—
415C3×Q8—41no—
416C3×Q8—41no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—42no—
429C2×C3×C4—42no—
430C2×C3×C4C23×C411yescharacteristic
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—42no—
445C2×Dic3—42no—
446C2×Dic3C4×D412yes—
447C2×Dic3#12112yes—
448C2×Dic3#12712yes—
449D12—22no—
450D12—22no—
451D12—48no—
452D12—41no—
453D12—41no—
454C4×D3—48no—
455C4×D3—48no—
456C4×D3—84no—
457C4×D3C2×C22⋊C414yes—
458C4×D3C4×D414yes—
459C4×D3#12114yes—
460C4×D3#12414yes—
461Dic6—22no—
462Dic6—22no—
463Dic6—41no—
464Dic6—41no—
465Dic6—48no—
466C3×C8—22no—
467C3×C8C2×C22⋊C414yes—
468C3⋊C8—22no—
469C3⋊C8C2×C22⋊C414yes—
470C24—31no—
471C24—61no—
472#41—68no—
473#41—68no—
474#41—124no—
475#41—124no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—32no—
487C2×Q8—64no—
488C2×Q8—68no—
489C2×Q8—68no—
490C2×Q8—61no—
491C2×Q8—61no—
492C2×Q8—61no—
493C2×Q8—61no—
494C2×Q8—61no—
495C2×Q8—122no—
496C2×Q8—122no—
497C2×Q8—124no—
498C2×Q8—124no—
499C2×Q8C2×C4×D312yes—
500C2×Q8C2×C4×D312yes—
501C2×D4—32no—
502C2×D4—64no—
503C2×D4—61no—
504C2×D4—61no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—122no—
509C2×D4—122no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—31no—
515C22×C4—32no—
516C22×C4—32no—
517C22×C4—32no—
518C22×C4—32no—
519C22×C4—32no—
520C22×C4—32no—
521C22×C4—32no—
522C22×C4—64no—
523C22×C4—64no—
524C22×C4—61no—
525C22×C4—61no—
526C22×C4—61no—
527C22×C4—122no—
528C22×C4—122no—
529C22×C4—122no—
530C22×C4—122no—
531C22×C4—122no—
532C22×C4—242no—
533C22×C4D3×D412yes—
534C22×C4C23×D311yescharacteristic
535Q16—28no—
536Q16—68no—
537SD16—68no—
538SD16—68no—
539M16—316no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—32no—
543C2×C8—32no—
544C2×C8—32no—
545C2×C8—32no—
546C2×C8—68no—
547C2×C8—61no—
548C2×C8—61no—
549C2×C8C2×C4×D312yes—
550C2×C8D3×D412yes—
551C2×C8D3×D412yes—
552C2×C8D3×D412yes—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—64no—
558C4⋊C4—61no—
559C4⋊C4—61no—
560C4⋊C4—61no—
561C4⋊C4—61no—
562C4⋊C4—122no—
563C22⋊C4—64no—
564C22⋊C4—122no—
565C22⋊C4—242no—
566C22⋊C4—242no—
567C42—21no—
568C42—21no—
569C42—61no—
570C42—61no—
571C42—122no—
572C22×C3C2×C4×D411yescharacteristic
573C22×C3C2×#12111yescharacteristic
574C22×C3C2×#12711yescharacteristic
575C22×C3#57712yes—
576C22×C3#57812yes—
577C2×D3—22no—
578C2×D3—22no—
579C2×D3—24no—
580C2×D3—24no—
581C2×D3—44no—
582C2×D3—42no—
583C2×D3—81no—
584C2×D3—81no—
585C2×D3—82no—
586C2×D3#39014yes—
587C3×C4—42no—
588C3×C4—42no—
589C3×C4—42no—
590C3×C4—81no—
591C3×C4—81no—
592C3×C4C22×C22⋊C411yescharacteristic
593C3×C4C2×C4×D411yescharacteristic
594C3×C4C2×#12111yescharacteristic
595C3×C4C2×#12411yescharacteristic
596Dic3—42no—
597Dic3—42no—
598Dic3—81no—
599Dic3—81no—
600Dic3#39014yes—
601C23—32no—
602C23—32no—
603C23—32no—
604C23—62no—
605C23—62no—
606C23—62no—
607C23—121no—
608C23—122no—
609C23—122no—
610C23—121no—
611C23C2×D3×D411yescharacteristic
612Q8—22no—
613Q8—22no—
614Q8—22no—
615Q8—22no—
616Q8—41no—
617Q8—41no—
618Q8—41no—
619Q8—41no—
620Q8—62no—
621Q8—62no—
622Q8—128no—
623Q8—121no—
624Q8—121no—
625D4—62no—
626D4—62no—
627D4—128no—
628D4—121no—
629D4—121no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—32no—
636C2×C4—32no—
637C2×C4—32no—
638C2×C4—34no—
639C2×C4—34no—
640C2×C4—34no—
641C2×C4—34no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—128no—
655C2×C4—128no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4—122no—
661C2×C4—122no—
662C2×C4—244no—
663C2×C4C22×C4×D311yescharacteristic
664C2×C4C2×D3×D411yescharacteristic
665C2×C4C2×D3×D411yescharacteristic
666C2×C4C2×D3×D411yescharacteristic
667C2×C4D3×#4111yescharacteristic
668C2×C4D3×#4111yescharacteristic
669C8—22no—
670C8—34no—
671C8—62no—
672C8D3×C22⋊C414yes—
673C2×C3C2×#39011yescharacteristic
674C2×C3#294712yes—
675C2×C3#306312yes—
676C2×C3#315411yescharacteristic
677C2×C3#315611yescharacteristic
678D3—24no—
679D3—82no—
680C22—34no—
681C22—64no—
682C22—64no—
683C22—62no—
684C22—62no—
685C22—124no—
686C22—122no—
687C22—242no—
688C22—241no—
689C22—241no—
690C22C4×D3×D411yescharacteristic
691C22D3×#12111yescharacteristic
692C22D3×#12711yescharacteristic
693C22#576612yes—
694C22#577012yes—
695C4—34no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—81no—
700C4—81no—
701C4—122no—
702C4—122no—
703C4—241no—
704C4—241no—
705C4C2×D3×C22⋊C411yescharacteristic
706C4C4×D3×D411yescharacteristic
707C4D3×#12111yescharacteristic
708C4D3×#12411yescharacteristic
709C3#2123411yescharacteristic, complemented, Sylow
710C2—64no—
711C2—242no—
712C2D3×#39011yescharacteristic
713C2#7971912yes—
714C2#8076212yes—
715C2#8180111yescharacteristic
716C2#8182211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1969C2
socleC23×C3C22×D4
prime radicalC22×#239C22
cosocleC22×C3×C4C24
hypercenter#1969D3
hypocenterC3#21234
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1969

Complements

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

NH ≅ G/Ntype
2. C2×#4881C2semidirect
4. C2×D3×#103C2semidirect
5. C2×#4689C2semidirect
6. #76610C2semidirect
8. C3×#1969C2semidirect
9. #72067C2semidirect
10. #70629C2semidirect
11. #70258C2semidirect
12. #70257C2semidirect
13. #69774C2semidirect
14. #69679C2semidirect
15. #69678C2semidirect
24. C2×#918C4semidirect
75. #1969D3semidirect
102. C2×C3×Q16C2×C4semidirect
110. C2×#216C2×C4semidirect
113. C2×#215C2×C4semidirect
129. C2×#204C2×C4semidirect
310. ΓL(1,𝔽25)C22⋊C4semidirect
340. C2×Q16C4×D3semidirect
709. C3#21234semidirect

Maximal subgroups

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

subgroupindex
C22×#9182
C2×#48812
C2×#48042
C2×D3×#1032
C2×#46892
#766102
#764822
C3×#19692
#720672
#706292
#702582
#702572
#697742
#696792
#696782
#212343