Finite Group Databasebeta

← #10804176

Subgroups — #10804176

Order 768.

1254 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5628C211yesmaximal, characteristic
2#83762C211yesmaximal, characteristic
3#78630C211yesmaximal, characteristic
4C3×#1901C211yesmaximal, characteristic, complemented
5#72003C211yesmaximal, characteristic, complemented
6#71698C211yesmaximal, characteristic, complemented
7#70115C212yesmaximal, complemented
8#70113C212yesmaximal, complemented
9#69986C211yesmaximal, characteristic, complemented
10#68993C212yesmaximal, complemented
11#68991C212yesmaximal, complemented
12#21756—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×#946C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C2×D3×M16C2212yes—
21#5628—22no—
22#5628—22no—
23#5628—22no—
24#5628—22no—
25C2×#897C2211yescharacteristic
26C2×#894C412yes—
27C2×#892C2211yescharacteristic
28C2×#888C412yes—
29C2×#885C412yes—
30C2×#883C412yes—
31C2×C3×#101C2212yes—
32C3×#407C2211yescharacteristic
33C2×#836C2212yes—
34#5207C2211yescharacteristic
35C2×#824C2212yes—
36C2×#824C2212yes—
37#5129—22no—
38#5126C2212yes—
39#5126C2212yes—
40#5126C2212yes—
41#5110—22no—
42#4727—22no—
43#4727—22no—
44C3×#343C2212yes—
45#4554C2212yes—
46#4554C2212yes—
47#4530C2212yes—
48C2×#536—31no—
49#2879—31no—
50#2879—31no—
51#1901—32no—
52#1901—31no—
53#1901—32no—
54#1901—31no—
55#1901—31no—
56#1901D311yescharacteristic, complemented
57#1822—32no—
58#1784—32no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67C2×C411yescharacteristic
62C22×#67C2×C411yescharacteristic
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—42no—
66C2×D3×Q8—42no—
67C2×#235—42no—
68C2×#235—42no—
69C2×#235D412yes—
70C2×#235D412yes—
71C2×#235D412yes—
72C2×#235D412yes—
73C22×D12C2×C411yescharacteristic
74C22×C4×D3C2×C412yes—
75C22×C4×D3C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2×C412yes—
78C22×Dic6C2×C411yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—21no—
81C2×C3×M16—42no—
82C2×C3×M16C2312yes—
83C2×C3×M16C2312yes—
84C22×C3×C8—21no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C2×C3×C4⋊C4C2×C412yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4C2×C412yes—
92C2×#217C2×C412yes—
93#946—22no—
94#946—22no—
95C2×(C2×D3)⋊C4C2×C412yes—
96C2×(C2×D3)⋊C4C2×C412yes—
97C2×(C2×D3)⋊C4C2×C412yes—
98C2×C4⋊Dic3C2×C412yes—
99C2×C4⋊Dic3C2311yescharacteristic
100#924—21no—
101#924—21no—
102#924—21no—
103#924—21no—
104C2×Dic3⋊C4C2×C412yes—
105C2×Dic3⋊C4C2×C412yes—
106C2×C4×Dic3C2×C412yes—
107C2×#208—21no—
108C2×#208—42no—
109C2×#208C2312yes—
110C2×#208C2312yes—
111C22×C3⋊C8—21no—
112D3×M16—42no—
113D3×M16—42no—
114C2×ΓL(1,𝔽25)—22no—
115C2×D3×C8—22no—
116#897—22no—
117#897—22no—
118#894—24no—
119#894—24no—
120#892—22no—
121#892—22no—
122#888—24no—
123#888—24no—
124#885—24no—
125#885—24no—
126#883—24no—
127#883—24no—
128C3×#101—42no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131#836—42no—
132#830—21no—
133#830—21no—
134#824—42no—
135#824—42no—
136(C2×D3)⋊C8—24no—
137(C2×D3)⋊C8—24no—
138(C2×D3)⋊C8—24no—
139C22×#41—31no—
140C23×Q8—31no—
141C22×M16—31no—
142C22×M16—32no—
143C22×M16C2×D311yescharacteristic
144#536—62no—
145#536—62no—
146#536—62no—
147#536—62no—
148C2×C4⋊Q8—32no—
149C2×#124—32no—
150C2×#123—32no—
151C2×#123—32no—
152C2×C22⋊Q8—31no—
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C2×C22⋊Q8C2×D311yescharacteristic
156C2×#117—31no—
157C2×#101—32no—
158C2×#101—32no—
159C2×#101—32no—
160C2×#101C2×D312yes—
161#408—62no—
162#408—62no—
163#408—62no—
164#408—62no—
165#407—31no—
166#407—32no—
167#407—32no—
168#407—32no—
169#407C2×D311yescharacteristic
170#343—32no—
171#343—32no—
172#343—32no—
173#343C2×D312yes—
174C24×C3C22×C411yescharacteristic
175C23×D3C22×C411yescharacteristic
176C23×D3C22×C411yescharacteristic
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4C22×C412yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—42no—
192C2×#67—42no—
193C2×#67C22⋊C412yes—
194C2×#67C22⋊C412yes—
195C2×#67C22⋊C412yes—
196C2×#67C22⋊C412yes—
197C22×Dic3C22×C412yes—
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200C22×Dic3C22×C412yes—
201D3×Q8—44no—
202D3×Q8—44no—
203D3×Q8—44no—
204D3×Q8—44no—
205#235—42no—
206#235—42no—
207#235—42no—
208#235—42no—
209#235—42no—
210#235—42no—
211#235—42no—
212#235—42no—
213C2×D12—42no—
214C2×D12C22⋊C412yes—
215C2×D12C22⋊C412yes—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—24no—
221C2×C4×D3—44no—
222C2×C4×D3C2×D412yes—
223C2×C4×D3C2×D412yes—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×D412yes—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—24no—
231C2×Dic6—42no—
232C2×Dic6—44no—
233C2×Dic6C22⋊C412yes—
234C2×Dic6C22⋊C412yes—
235C3×M16—22no—
236C3×M16—22no—
237C3×M16—22no—
238C3×M16—22no—
239C3×M16—42no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—42no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249#217—22no—
250#217—22no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—24no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—24no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—24no—
260C4×Dic3—24no—
261#208—22no—
262#208—22no—
263#208—22no—
264#208—22no—
265#208—42no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—42no—
269ΓL(1,𝔽25)—44no—
270D3×C8—44no—
271C2×#41—32no—
272C2×#41—32no—
273C2×#41—32no—
274C2×#41—32no—
275C2×#41—122no—
276C2×#41—122no—
277C22×Q8—31no—
278C22×Q8—32no—
279C22×Q8—31no—
280C22×Q8—62no—
281C22×Q8—62no—
282C22×Q8—122no—
283C22×Q8—122no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—32no—
291C23×C4C22×D311yescharacteristic
292C2×M16—21no—
293C2×M16—32no—
294C2×M16—32no—
295C2×M16—42no—
296C2×M16—62no—
297C2×M16—61no—
298C2×M16—122no—
299C2×M16—122no—
300C2×M16—122no—
301C2×M16C22×D312yes—
302C2×M16C22×D312yes—
303C22×C8—21no—
304C22×C8—62no—
305C22×C8—61no—
306C4⋊Q8—64no—
307C4⋊Q8—64no—
308#124—64no—
309#124—64no—
310#123—64no—
311#123—64no—
312#123—64no—
313#123—64no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C22⋊Q8—61no—
323C22⋊Q8—61no—
324C22⋊Q8—61no—
325C22⋊Q8—61no—
326#117—62no—
327#117—62no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4C4×D312yes—
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C22⋊C4—32no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4C4×D312yes—
339C2×C42—32no—
340#101—42no—
341#101—122no—
342#101—122no—
343#101—122no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—64no—
347C22⋊C8—64no—
348C22⋊C8—64no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C23×C3—22no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3C2×C22⋊C411yescharacteristic
355C23×C3C2×C22⋊C411yescharacteristic
356C23×C3C23×C411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3C2×C22⋊C412yes—
364C22×D3C2×C22⋊C412yes—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—42no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—42no—
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4H(5,𝔽2)12yes—
384C2×C3×C4H(5,𝔽2)12yes—
385#67—42no—
386#67—42no—
387#67—42no—
388#67—42no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—42no—
395C2×Dic3C2×C22⋊C412yes—
396C2×Dic3C2×C22⋊C412yes—
397D12—42no—
398D12—42no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—48no—
408Dic6—42no—
409Dic6—42no—
410Dic6—48no—
411Dic6—41no—
412Dic6—41no—
413Dic6—42no—
414C3×C8—42no—
415C3×C8—42no—
416C3⋊C8—42no—
417C3⋊C8—42no—
418C24—31no—
419C24—31no—
420C24C2×C4×D311yescharacteristic
421#41—122no—
422#41—122no—
423#41—122no—
424#41—122no—
425#41—122no—
426#41—122no—
427#41—122no—
428#41—122no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—32no—
434C2×Q8—32no—
435C2×Q8—64no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—62no—
439C2×Q8—62no—
440C2×Q8—124no—
441C2×Q8—122no—
442C2×Q8—124no—
443C2×Q8—124no—
444C2×Q8—124no—
445C2×Q8—124no—
446C2×D4—32no—
447C2×D4—32no—
448C2×D4—32no—
449C2×D4—32no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—32no—
465C22×C4—64no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—124no—
471C22×C4C2×C4×D312yes—
472C22×C4C2×C4×D311yescharacteristic
473C22×C4C2×C4×D312yes—
474C22×C4D3×D411yescharacteristic
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4D3×D411yescharacteristic
478C22×C4C23×D311yescharacteristic
479M16—22no—
480M16—22no—
481M16—22no—
482M16—22no—
483M16—42no—
484M16—62no—
485M16—62no—
486M16—62no—
487M16—62no—
488M16—124no—
489M16—122no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—42no—
493C2×C8—62no—
494C2×C8—62no—
495C2×C8—124no—
496C2×C8—122no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C22⋊C4—64no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C42—64no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—82no—
524C22×C3C22×C22⋊C411yescharacteristic
525C22×C3#51812yes—
526C2×D3—22no—
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—22no—
531C2×D3—22no—
532C2×D3—44no—
533C2×D3—82no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4#53411yescharacteristic
539C3×C4#53411yescharacteristic
540C3×C4#53412yes—
541Dic3—22no—
542Dic3—22no—
543Dic3—42no—
544Dic3—42no—
545C23—22no—
546C23—32no—
547C23—32no—
548C23—41no—
549C23—41no—
550C23—62no—
551C23—62no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23D3×C22⋊C411yescharacteristic
557C23D3×C22⋊C411yescharacteristic
558C23C22×C4×D311yescharacteristic
559Q8—41no—
560Q8—41no—
561Q8—42no—
562Q8—122no—
563Q8—122no—
564Q8—128no—
565Q8—121no—
566Q8—121no—
567Q8—122no—
568D4—122no—
569D4—122no—
570D4—122no—
571D4—122no—
572D4—122no—
573D4—122no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—42no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—64no—
600C2×C4—128no—
601C2×C4—122no—
602C2×C4D3×C22⋊C411yescharacteristic
603C2×C4D3×C22⋊C411yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4C2×D3×D411yescharacteristic
606C2×C4#100412yes—
607C2×C4#100912yes—
608C8—42no—
609C8—42no—
610C8—122no—
611C8—122no—
612C2×C3—22no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3#230811yescharacteristic
616C2×C3#288911yescharacteristic
617C2×C3#288911yescharacteristic
618D3—42no—
619D3—42no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—62no—
634C22—82no—
635C22—124no—
636C22—242no—
637C22C2×D3×C22⋊C411yescharacteristic
638C22#554512yes—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—62no—
644C4—62no—
645C4—122no—
646C4—122no—
647C4#558911yescharacteristic
648C4#560512yes—
649C4#565311yescharacteristic
650C3#2175611yescharacteristic, complemented, Sylow
651C2—22no—
652C2—41no—
653C2—41no—
654C2—122no—
655C2—122no—
656C2#7487411yescharacteristic
657C2#7870611yescharacteristic
658C2#7871511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1901C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1901D3
hypocenterC3#21756
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1901

Complements

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

NH ≅ G/Ntype
4. C3×#1901C2semidirect
5. #72003C2semidirect
6. #71698C2semidirect
7. #70115C2semidirect
8. #70113C2semidirect
9. #69986C2semidirect
10. #68993C2semidirect
11. #68991C2semidirect
56. #1901D3semidirect
650. C3#21756semidirect

Maximal subgroups

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

subgroupindex
C2×#56282
#837622
#786302
C3×#19012
#720032
#716982
#701152
#701132
#699862
#689932
#689912
#217563