Finite Group Databasebeta

← #10712677

Subgroups — #10712677

Order 768.

1088 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2403C211yesmaximal, characteristic
2C22×#833C211yesmaximal, characteristic, complemented
3#72130C211yesmaximal, characteristic, complemented
4C2×#4536C212yesmaximal, complemented
5#67637C212yesmaximal, complemented
6#13082—31nomaximal, Sylow
7C23×C3×D4C411yescharacteristic, complemented
8C23×C3×D4C2211yescharacteristic
9C2×C3×#120—21no—
10C2×C3×#120C411yescharacteristic, complemented
11C2×C3×#120C2211yescharacteristic
12C22×#208—21no—
13C3×#392—21no—
14C3×#392—21no—
15C3×#386—21no—
16C3×#386C411yescharacteristic, complemented
17C3×#386C2211yescharacteristic
18C3×#379C411yescharacteristic, complemented
19#5247—21no—
20C2×#834—21no—
21C2×#834C2212yes—
22C2×#833—21no—
23C2×#833—24no—
24C2×#833C2212yes—
25#4536—24no—
26C22×#100—31no—
27#2403D311yescharacteristic
28C2×AΣL(1,𝔽16)—32no—
29#1781—31no—
30#1390—32no—
31C25×C3C2×C411yescharacteristic
32C25×C3C2×C411yescharacteristic
33C25×C3D411yescharacteristic
34C22×C3×D4—21no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—42no—
41C22×C3×D4—41no—
42C22×C3×D4—41no—
43C22×C3×D4—44no—
44C22×C3×D4—41no—
45C22×C3×D4—41no—
46C22×C3×D4C2×C411yescharacteristic
47C22×C3×D4D411yescharacteristic
48C22×C3×D4D411yescharacteristic
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C3×#120—22no—
53C3×#120—22no—
54C3×#120—44no—
55C3×#120—42no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4C2×C411yescharacteristic
63C2×C3×C22⋊C4C2×C411yescharacteristic
64C2×#217—22no—
65C2×#217—41no—
66C2×#217—41no—
67C2×#208—24no—
68C2×#208—22no—
69C2×#208—41no—
70C2×#208—42no—
71C22×C3⋊C8—41no—
72C3×#95—21no—
73C3×#95—21no—
74#834—22no—
75#834—22no—
76#834—42no—
77#833—24no—
78#833—22no—
79#833D414yescomplemented
80C23×D4Dic311yescharacteristic, complemented
81C23×D4C2×D311yescharacteristic
82C22×M16—61no—
83C2×#120—21no—
84C2×#120Dic311yescharacteristic, complemented
85C2×#120C2×D311yescharacteristic
86C2×#100—32no—
87C2×#100—64no—
88C2×#100—61no—
89C2×#99—32no—
90C2×#99—61no—
91#392—21no—
92#392—21no—
93#386—21no—
94#386Dic311yescharacteristic, complemented
95#386C2×D311yescharacteristic
96#379—61no—
97#379Dic311yescharacteristic, complemented
98AΣL(1,𝔽16)—64no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—21no—
102C24×C3—21no—
103C24×C3—21no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—41no—
108C24×C3—41no—
109C24×C3—42no—
110C24×C3—41no—
111C24×C3—42no—
112C24×C3—42no—
113C24×C3—41no—
114C24×C3—81no—
115C24×C3—81no—
116C24×C3—81no—
117C24×C3C22⋊C411yescharacteristic
118C24×C3C22⋊C411yescharacteristic
119C24×C3C22⋊C411yescharacteristic
120C24×C3C22×C411yescharacteristic
121C24×C3C2×D411yescharacteristic
122C2×C3×D4—24no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—44no—
136C2×C3×D4—44no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C2×C3×D4—44no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—81no—
154C2×C3×D4—81no—
155C2×C3×D4C2×D412yes—
156C2×C3×D4#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—42no—
166C22×C3×C4—41no—
167C22×C3×C4C22⋊C411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×Dic3—42no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—81no—
179#217—42no—
180#217—42no—
181#217—82no—
182#217—82no—
183#208—24no—
184#208—44no—
185#208—42no—
186C2×C3⋊C8—44no—
187C2×C3⋊C8—42no—
188C25C2×Dic311yescharacteristic
189C25C2×Dic311yescharacteristic
190C25#6711yescharacteristic
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—42no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4—44no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4C2×Dic311yescharacteristic
204C22×D4#6711yescharacteristic
205C22×D4#6711yescharacteristic
206C22×D4C22×D311yescharacteristic
207C23×C4C2×Dic311yescharacteristic
208C23×C4#6711yescharacteristic
209C2×M16—64no—
210C2×M16—62no—
211C2×M16—121no—
212C2×M16—122no—
213C22×C8—121no—
214#120—22no—
215#120—22no—
216#120—44no—
217#120—42no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—121no—
226C2×C22⋊C4—121no—
227C2×C22⋊C4C2×Dic311yescharacteristic
228C2×C22⋊C4C2×Dic311yescharacteristic
229#100—34no—
230#100—64no—
231#100—62no—
232#99—62no—
233#99—62no—
234#99—122no—
235#95—21no—
236#95—21no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—81no—
277C23×C3—81no—
278C23×C3—81no—
279C23×C3—81no—
280C23×C3—82no—
281C23×C3—84no—
282C23×C3—84no—
283C23×C3—84no—
284C23×C3—84no—
285C23×C3—84no—
286C23×C3—84no—
287C23×C3—82no—
288C23×C3#9911yescharacteristic
289C23×C3#9911yescharacteristic
290C23×C3C2×C22⋊C411yescharacteristic
291C23×C3C4×D412yes—
292C23×C3#12112yes—
293C3×D4—42no—
294C3×D4—42no—
295C3×D4—44no—
296C3×D4—44no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—44no—
300C3×D4—48no—
301C3×D4—44no—
302C3×D4—84no—
303C3×D4—84no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4#9912yes—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×Dic3—82no—
327C2×Dic3—82no—
328C2×Dic3—82no—
329C3⋊C8—44no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—41no—
345C24—81no—
346C24—81no—
347C24—81no—
348C24#21711yescharacteristic
349C24#21711yescharacteristic
350C24#21711yescharacteristic
351C24C22×Dic311yescharacteristic
352C24C2×#6711yescharacteristic
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4D3×D412yes—
387C2×D4#23712yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—122no—
399C22×C4#21711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401M16—64no—
402M16—124no—
403M16—122no—
404C2×C8—124no—
405C2×C8—122no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—81no—
415C22⋊C4—122no—
416C22⋊C4—122no—
417C22⋊C4—242no—
418C22⋊C4—242no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—22no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—42no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—81no—
448C22×C3—84no—
449C22×C3—84no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—84no—
456C22×C3—84no—
457C22×C3—84no—
458C22×C3—84no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3AΣL(1,𝔽16)12yescomplemented
462C22×C3#38611yescharacteristic
463C22×C3C2×#9911yescharacteristic
464C22×C3C2×#9911yescharacteristic
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—21no—
468C3×C4—42no—
469C3×C4—81no—
470C3×C4—81no—
471Dic3—162no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—82no—
523C23#83411yescharacteristic
524C23#83411yescharacteristic
525C23D4×Dic312yes—
526C23#93912yes—
527C23C2×#21711yescharacteristic
528D4—42no—
529D4—42no—
530D4—44no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—44no—
535D4—48no—
536D4—44no—
537D4—84no—
538D4—84no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—242no—
559C2×C4—242no—
560C2×C4—242no—
561C2×C4#83412yes—
562C2×C4#93511yescharacteristic
563C2×C4#93711yescharacteristic
564C8—124no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—22no—
568C2×C3—42no—
569C2×C3—41no—
570C2×C3—42no—
571C2×C3—81no—
572C2×C3—82no—
573C2×C3—81no—
574C2×C3#189811yescharacteristic
575C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
576C2×C3#212111yescharacteristic
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—81no—
606C22—84no—
607C22—84no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—81no—
618C22—81no—
619C22#453612yescomplemented
620C22C2×#83411yescharacteristic
621C22C2×#83411yescharacteristic
622C22#522011yescharacteristic
623C4—21no—
624C4—22no—
625C4—21no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—482no—
630C3#1308211yescharacteristic, complemented, Sylow
631C2—21no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—81no—
640C2C2×#453611yescharacteristic
641C2#7167011yescharacteristic
642C2#7184711yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
Fitting, prime radicalC3×#2403C2
socleC22×C3C2×#99
cosocleC22×C3×D4C23
hypercenter#2403D3
hypocenterC3#13082
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2403

Complements

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

NH ≅ G/Ntype
2. C22×#833C2semidirect
3. #72130C2semidirect
4. C2×#4536C2semidirect
5. #67637C2semidirect
7. C23×C3×D4C4semidirect
10. C2×C3×#120C4semidirect
16. C3×#386C4semidirect
18. C3×#379C4semidirect
79. #833D4semidirect
80. C23×D4Dic3semidirect
84. C2×#120Dic3semidirect
94. #386Dic3semidirect
97. #379Dic3semidirect
461. C22×C3AΣL(1,𝔽16)semidirect
619. C22#4536semidirect
630. C3#13082semidirect

Maximal subgroups

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

subgroupindex
C3×#24032
C22×#8332
#721302
C2×#45362
#676372
#130823