Finite Group Databasebeta

← #10944301

Subgroups — #10944301

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic, complemented
2C2×C3×#469C211yesmaximal, characteristic, complemented
3C2×#5159C211yesmaximal, characteristic, complemented
4C2×#5157C211yesmaximal, characteristic, complemented
5#82709C212yesmaximal, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71681C212yesmaximal, complemented
8#70578C211yesmaximal, characteristic, complemented
9#70577C211yesmaximal, characteristic, complemented
10#70203C212yesmaximal, complemented
11#70202C212yesmaximal, complemented
12#32939—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#121C2212yes—
17C2×#938C2212yes—
18C2×#938C2212yes—
19C23×C3⋊C8C2211yescharacteristic
20C2×#907—24no—
21C2×#907—24no—
22C2×#907—24no—
23C22×ΓL(1,𝔽25)C2211yescharacteristic
24C22×ΓL(1,𝔽25)C2211yescharacteristic
25C22×D3×C8C2211yescharacteristic
26C22×D3×C8C2211yescharacteristic
27C3×#469—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#102C2212yes—
30#5223—22no—
31#5223—22no—
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5212C2212yes—
35#5159—24no—
36#5157—24no—
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39#4987C2212yes—
40#4987C2212yes—
41#4960—22no—
42#4947—22no—
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#469—31no—
48C2×#469D311yescharacteristic, complemented
49C2×#466—31no—
50#2615—32no—
51#1895—32no—
52#1893—32no—
53#1893—32no—
54#1809—31no—
55#1809—31no—
56#1807—31no—
57C22×C3×D4—22no—
58C22×C3×D4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67—22no—
62C22×#67D411yescharacteristic
63C22×#67D411yescharacteristic
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C22×D12D411yescharacteristic
71C22×C4×D3C2311yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6D411yescharacteristic
74C2×C3×D8—22no—
75C2×C3×D8—22no—
76C2×C3×D8—22no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16D414yescomplemented
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#121—24no—
83C3×#121—24no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89#938—24no—
90#938—24no—
91#938—24no—
92#938—24no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2312yes—
97C22×C3⋊C8—22no—
98C22×C3⋊C8—22no—
99C22×C3⋊C8—22no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#907—232no—
103C2×ΓL(1,𝔽25)—22no—
104C2×ΓL(1,𝔽25)—22no—
105C2×ΓL(1,𝔽25)—22no—
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)D414yescomplemented
109C2×D3×C8—22no—
110C2×D3×C8—22no—
111C2×D3×C8—22no—
112C2×D3×C8—22no—
113C2×D3×C8—22no—
114C2×D3×C8D414yescomplemented
115C3×#106—22no—
116C3×#102—24no—
117#832—24no—
118#832—24no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122#818—22no—
123#817—22no—
124C22×#41—31no—
125C22×D8C2×D311yescharacteristic
126C2×#131—64no—
127C2×#131—64no—
128C2×#131—64no—
129C22×M16—31no—
130C22×M16—31no—
131C22×M16C2×D311yescharacteristic
132C23×C8—31no—
133C23×C8—31no—
134C23×C8—31no—
135C2×#121—32no—
136C2×#121—32no—
137C2×#121C2×D312yes—
138#469—24no—
139#469—64no—
140#466—64no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×#102—32no—
145C2×#102—32no—
146C2×#102C2×D312yes—
147#393—62no—
148#392—62no—
149#392—62no—
150#392—62no—
151#380—32no—
152#380—32no—
153#380—32no—
154#340—32no—
155#340—32no—
156#340—32no—
157C24×C3—22no—
158C24×C3C2×D411yescharacteristic
159C23×D3C2×D411yescharacteristic
160C23×D3C2×D411yescharacteristic
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—44no—
168C2×C3×D4—82no—
169C2×C3×D4—82no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4#4112yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—22no—
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—22no—
185C2×#67—44no—
186C2×#67—44no—
187C2×#67—82no—
188C2×#67—82no—
189C2×#67—82no—
190C2×#67—82no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195#235—28no—
196#235—28no—
197#235—28no—
198#235—28no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3C2×D412yes—
211C2×C4×D3C2×D412yes—
212C2×C4×D3#4112yes—
213C2×C4×D3#4112yes—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C3×D8—44no—
218C3×M16—28no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C2×C3×C8C2×D412yes—
224C2×C3×C8C2×D412yes—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—44no—
228#217—44no—
229#217—44no—
230#217—82no—
231(C2×D3)⋊C4—82no—
232(C2×D3)⋊C4—82no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C2×C3⋊C8—28no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8C2×D412yes—
243C2×C3⋊C8C2×D412yes—
244ΓL(1,𝔽25)—28no—
245ΓL(1,𝔽25)—28no—
246D3×C8—28no—
247D3×C8—28no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C22×Q8—31no—
255C22×D4—22no—
256C22×D4—31no—
257C22×D4—31no—
258C22×D4—31no—
259C22×D4—62no—
260C22×D4—62no—
261C22×D4C22×D312yes—
262C23×C4—31no—
263C23×C4—31no—
264C23×C4C22×D311yescharacteristic
265C2×D8—22no—
266C2×D8—22no—
267C2×D8—22no—
268#131—632no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—34no—
272C2×M16—62no—
273C2×M16—62no—
274C2×M16—62no—
275C2×M16—62no—
276C2×M16—62no—
277C2×M16#6714yescomplemented
278C22×C8—31no—
279C22×C8—31no—
280C22×C8—34no—
281C22×C8—62no—
282C22×C8—62no—
283C22×C8—62no—
284C22×C8—62no—
285C22×C8—62no—
286C22×C8—62no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#121—24no—
292#121—24no—
293#121—64no—
294#121—64no—
295#121—64no—
296#121—64no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4C22×D312yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4—62no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4—62no—
306#107—62no—
307#106—22no—
308#106—62no—
309#102—24no—
310#102—64no—
311#102—64no—
312#95—62no—
313#95—62no—
314#95—62no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3#12112yes—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3#12112yes—
337C22×D3#12112yes—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—88no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4#12112yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359#67—28no—
360#67—28no—
361#67—88no—
362#67—88no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3#12112yes—
374C2×Dic3#12112yes—
375D12—28no—
376C4×D3—22no—
377C4×D3—22no—
378C4×D3—22no—
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—22no—
382C4×D3—22no—
383C4×D3—22no—
384Dic6—28no—
385C3×C8—22no—
386C3×C8#12014yes—
387C3⋊C8—22no—
388C3⋊C8—22no—
389C24—22no—
390C24—31no—
391C24—31no—
392C24D3×D411yescharacteristic
393#41—68no—
394#41—68no—
395#41—68no—
396#41—68no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—44no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—124no—
419C2×D4—124no—
420C2×D4—242no—
421C2×D4—242no—
422C2×D4—242no—
423C2×D4—242no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—32no—
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—62no—
444C22×C4—62no—
445C22×C4D3×D411yescharacteristic
446C22×C4#23712yes—
447C22×C4C2×#6712yes—
448C22×C4C23×D311yescharacteristic
449D8—44no—
450M16—28no—
451M16—68no—
452M16—68no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—32no—
456C2×C8—32no—
457C2×C8—68no—
458C2×C8—68no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—68no—
464C2×C8D3×D412yes—
465C2×C8D3×D412yes—
466C2×C8C2×#6712yes—
467C2×C8C2×#6712yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C22⋊C4—44no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C22⋊C4—242no—
478C22⋊C4—242no—
479C22⋊C4—242no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3C2×#12111yescharacteristic
493C22×C3C2×#12111yescharacteristic
494C22×C3C2×#12111yescharacteristic
495C22×C3#57612yes—
496C22×C3#57712yes—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—24no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C3×C4—22no—
508C3×C4—82no—
509C3×C4C2×#12011yescharacteristic
510C3×C4#52511yescharacteristic
511C3×C4#52511yescharacteristic
512C3×C4#52511yescharacteristic
513Dic3—22no—
514Dic3—22no—
515Dic3—82no—
516Dic3—82no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—32no—
522C23—32no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23#93812yes—
539C23C2×D3×D411yescharacteristic
540Q8—68no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—68no—
546D4—68no—
547D4—68no—
548D4—88no—
549D4—248no—
550D4—248no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4#93812yes—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C8—22no—
588C8—62no—
589C8—62no—
590C8#93714yes—
591C2×C3—42no—
592C2×C3—162no—
593C2×C3#284711yescharacteristic
594C2×C3#305811yescharacteristic
595C2×C3#306011yescharacteristic
596C2×C3#306212yes—
597C2×C3#306212yes—
598D3—42no—
599D3—42no—
600C22—22no—
601C22—24no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—62no—
609C22—64no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—121no—
618C22—121no—
619C22—122no—
620C22—122no—
621C22—122no—
622C22D3×#12111yescharacteristic
623C22D3×#12111yescharacteristic
624C22#577412yes—
625C22#577512yes—
626C22C2×#93811yescharacteristic
627C4—22no—
628C4—62no—
629C4—62no—
630C4—82no—
631C4—242no—
632C4—242no—
633C4#560611yescharacteristic
634C4#560611yescharacteristic
635C4C2×#93711yescharacteristic
636C4#582811yescharacteristic
637C3#3293911yescharacteristic, complemented, Sylow
638C2—42no—
639C2—122no—
640C2—122no—
641C2—162no—
642C2#8086511yescharacteristic
643C2#8086611yescharacteristic
644C2#8086712yes—
645C2#8272311yescharacteristic
646C2#8395112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#469C2
socleC23×C3C22×D4
hypercenterC2×#469D3
hypocenterC3#32939
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
2. C2×C3×#469C2semidirect
3. C2×#5159C2semidirect
4. C2×#5157C2semidirect
5. #82709C2semidirect
6. #71988C2semidirect
7. #71681C2semidirect
8. #70578C2semidirect
9. #70577C2semidirect
10. #70203C2semidirect
11. #70202C2semidirect
48. C2×#469D3semidirect
79. C2×C3×M16D4semidirect
108. C2×ΓL(1,𝔽25)D4semidirect
114. C2×D3×C8D4semidirect
277. C2×M16#67semidirect
637. C3#32939semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
C2×C3×#4692
C2×#51592
C2×#51572
#827092
#719882
#716812
#705782
#705772
#702032
#702022
#329393