Finite Group Databasebeta

← #10944307

Subgroups — #10944307

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×#5135C211yesmaximal, characteristic, complemented
3C2×#4883C211yesmaximal, characteristic, complemented
4C2×#4865C211yesmaximal, characteristic, complemented
5#74030C212yesmaximal, complemented
6C3×#1809C211yesmaximal, characteristic, complemented
7#70631C211yesmaximal, characteristic, complemented
8#70577C211yesmaximal, characteristic, complemented
9#69515C212yesmaximal, complemented
10#68885C212yesmaximal, complemented
11#68883C212yesmaximal, complemented
12#32939—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×#930C2212yes—
16C23×C3⋊C8C2211yescharacteristic
17C2×#907—24no—
18C2×#907—24no—
19C2×#907—24no—
20C22×D24C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C2×#895C2212yes—
26C2×#895C2212yes—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C3×#380C2212yes—
30#5135—24no—
31C2×#821C2212yes—
32C2×#817C2211yescharacteristic
33#4987C2212yes—
34#4987C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37#4960—22no—
38#4960—22no—
39#4883—24no—
40#4865—24no—
41#4673—22no—
42#4671—22no—
43C3×#340C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#469—31no—
48C2×#469—31no—
49C2×#466—31no—
50#2615—32no—
51#1895—32no—
52#1893—32no—
53#1893—32no—
54#1809—31no—
55#1809D311yescharacteristic, complemented
56#1807—31no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—22no—
59C22×#67D411yescharacteristic
60C22×#67D411yescharacteristic
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12D411yescharacteristic
70C22×D12C2312yes—
71C22×C4×D3C2311yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6D411yescharacteristic
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16D414yescomplemented
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82#930—24no—
83#930—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3C2312yes—
90C22×C3⋊C8—22no—
91C22×C3⋊C8—22no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95#907—232no—
96C2×D24—22no—
97C2×D24—22no—
98C2×D24—22no—
99C2×ΓL(1,𝔽25)—22no—
100C2×ΓL(1,𝔽25)—22no—
101C2×ΓL(1,𝔽25)—22no—
102C2×ΓL(1,𝔽25)—22no—
103C2×ΓL(1,𝔽25)—22no—
104C2×ΓL(1,𝔽25)D414yescomplemented
105C2×D3×C8—22no—
106C2×D3×C8—22no—
107C2×D3×C8—22no—
108C2×D3×C8—22no—
109C2×D3×C8—22no—
110C2×D3×C8D414yescomplemented
111#895—24no—
112#895—24no—
113#895—24no—
114#895—24no—
115C3×#107—22no—
116C3×#106—22no—
117C3×#95—22no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#821—24no—
121#817—22no—
122#809—24no—
123#809—24no—
124C22×#41—31no—
125C22×D8—31no—
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×#121—32no—
138#469—64no—
139#469—64no—
140#466—64no—
141C2×#107C2×D311yescharacteristic
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147#393—62no—
148#392—62no—
149#392—62no—
150#392—62no—
151#380—32no—
152#380—32no—
153#380C2×D312yes—
154#340—32no—
155#340—32no—
156#340C2×D312yes—
157C24×C3C2×D411yescharacteristic
158C23×D3—22no—
159C23×D3C2×D411yescharacteristic
160C23×D3C2×D411yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C2×#67—22no—
172C2×#67—22no—
173C2×#67—22no—
174C2×#67—22no—
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—44no—
178C2×#67—82no—
179C2×#67—82no—
180C22×Dic3—22no—
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183#235—28no—
184#235—28no—
185#235—28no—
186#235—28no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—44no—
197C2×D12—44no—
198C2×D12—82no—
199C2×D12—82no—
200C2×D12—82no—
201C2×D12—82no—
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×M16—28no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8C2×D412yes—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C2×C3×C8C2×D412yes—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C22⋊C4—82no—
229(C2×D3)⋊C4—44no—
230(C2×D3)⋊C4—44no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—82no—
233(C2×D3)⋊C4—82no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C2×C3⋊C8—28no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8C2×D412yes—
242C2×C3⋊C8C2×D412yes—
243D24—44no—
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—31no—
256C22×D4—31no—
257C22×D4—31no—
258C22×D4—32no—
259C22×D4—62no—
260C22×D4—62no—
261C22×D4—62no—
262C23×C4—31no—
263C23×C4—31no—
264C23×C4C22×D311yescharacteristic
265C2×D8—62no—
266C2×D8—62no—
267C2×D8—62no—
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×M16D1214yescomplemented
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—64no—
292#121—64no—
293#121—64no—
294#121—64no—
295#121—64no—
296#121—64no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4C22×D312yes—
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—22no—
307#106—22no—
308#106—62no—
309#102—64no—
310#102—64no—
311#102—64no—
312#95—22no—
313#95—62no—
314#95—62no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3#12112yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3#12112yes—
337C22×D3#12112yes—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12112yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357#67—28no—
358#67—28no—
359#67—88no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3#12112yes—
368C2×Dic3#12112yes—
369D12—28no—
370D12—42no—
371D12—42no—
372D12—42no—
373D12—42no—
374D12—88no—
375D12—88no—
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—31no—
390C24—31no—
391C24—62no—
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—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
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—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—242no—
419C2×D4—242no—
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—22no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—32no—
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×C4C2×D1212yes—
446C22×C4D3×D411yescharacteristic
447C22×C4#23912yes—
448C22×C4C23×D311yescharacteristic
449D8—124no—
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×C8C2×D1212yes—
465C2×C8C2×D1212yes—
466C2×C8D3×D412yes—
467C2×C8D3×D412yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C22⋊C4—82no—
475C22⋊C4—124no—
476C22⋊C4—124no—
477C22⋊C4—124no—
478C22⋊C4—242no—
479C22⋊C4—242no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3C2×#12111yescharacteristic
486C22×C3C2×#12111yescharacteristic
487C22×C3C2×#12111yescharacteristic
488C22×C3#57612yes—
489C22×C3#57712yes—
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—24no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—82no—
501C2×D3—82no—
502C2×D3—82no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3—82no—
507C3×C4—22no—
508C3×C4—82no—
509C3×C4—82no—
510C3×C4C2×#12011yescharacteristic
511C3×C4#52511yescharacteristic
512C3×C4#52511yescharacteristic
513C3×C4#52511yescharacteristic
514Dic3—22no—
515Dic3—22no—
516Dic3—82no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—32no—
522C23—32no—
523C23—61no—
524C23—61no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—61no—
530C23—61no—
531C23—122no—
532C23—122no—
533C23—122no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23—122no—
538C23#89512yes—
539C23C2×D3×D411yescharacteristic
540Q8—68no—
541D4—68no—
542D4—68no—
543D4—68no—
544D4—122no—
545D4—122no—
546D4—122no—
547D4—122no—
548D4—248no—
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—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4#89512yes—
581C2×C4C22×D1211yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×#23911yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—22no—
588C8—62no—
589C8—62no—
590C8#88214yes—
591C2×C3—42no—
592C2×C3#284711yescharacteristic
593C2×C3#305811yescharacteristic
594C2×C3#306011yescharacteristic
595C2×C3#306212yes—
596C2×C3#306212yes—
597D3—42no—
598D3—42no—
599D3—162no—
600C22—22no—
601C22—24no—
602C22—41no—
603C22—42no—
604C22—41no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—64no—
609C22—62no—
610C22—121no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—121no—
615C22—242no—
616C22—242no—
617C22—242no—
618C22—242no—
619C22—242no—
620C22—242no—
621C22—242no—
622C22C2×#89511yescharacteristic
623C22D3×#12111yescharacteristic
624C22D3×#12111yescharacteristic
625C22#577412yes—
626C22#577812yes—
627C4—22no—
628C4—62no—
629C4—62no—
630C4—82no—
631C4—82no—
632C4—242no—
633C4C2×#88211yescharacteristic
634C4#552411yescharacteristic
635C4#562711yescharacteristic
636C4#562711yescharacteristic
637C3#3293911yescharacteristic, complemented, Sylow
638C2—42no—
639C2—122no—
640C2—122no—
641C2—482no—
642C2#7404811yescharacteristic
643C2#7921312yes—
644C2#8080912yes—
645C2#8086511yescharacteristic
646C2#8090611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1809C2
socleC23×C3C22×D4
hypercenter#1809D3
hypocenterC3#32939
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
2. C2×#5135C2semidirect
3. C2×#4883C2semidirect
4. C2×#4865C2semidirect
5. #74030C2semidirect
6. C3×#1809C2semidirect
7. #70631C2semidirect
8. #70577C2semidirect
9. #69515C2semidirect
10. #68885C2semidirect
11. #68883C2semidirect
55. #1809D3semidirect
76. C2×C3×M16D4semidirect
104. C2×ΓL(1,𝔽25)D4semidirect
110. C2×D3×C8D4semidirect
277. C2×M16D12semidirect
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×#51352
C2×#48832
C2×#48652
#740302
C3×#18092
#706312
#705772
#695152
#688852
#688832
#329393