Finite Group Databasebeta

← #10877180

Subgroups — #10877180

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2930C211yesmaximal, characteristic
2C2×#5121C211yesmaximal, characteristic, complemented
3#82915C211yesmaximal, characteristic
4#82914C211yesmaximal, characteristic
5#72016C211yesmaximal, characteristic, complemented
6#72015C211yesmaximal, characteristic, complemented
7#72000C211yesmaximal, characteristic, complemented
8#70533C211yesmaximal, characteristic, complemented
9#70532C211yesmaximal, characteristic, complemented
10#69995C211yesmaximal, characteristic, complemented
11#69994C211yesmaximal, characteristic, complemented
12#69974C211yesmaximal, characteristic, complemented
13#69973C211yesmaximal, characteristic, complemented
14#69894C211yesmaximal, characteristic, complemented
15#69893C211yesmaximal, characteristic, complemented
16#27676—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18C22×C3×M16C2211yescharacteristic
19C3×#525C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C3×#436—21no—
23C3×#436—21no—
24C3×#435—21no—
25C3×#435—21no—
26C3×#424C2211yescharacteristic
27C3×#408C2211yescharacteristic
28C3×#407C2211yescharacteristic
29C3×#405C2211yescharacteristic
30#5232C2211yescharacteristic
31#5232C2211yescharacteristic
32#5232C2211yescharacteristic
33#5232C2211yescharacteristic
34#5207C2211yescharacteristic
35#5207C2211yescharacteristic
36#5121—21no—
37#5121—21no—
38#5121—21no—
39#5121—21no—
40C2×#819C2211yescharacteristic
41C2×#819C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#817C2211yescharacteristic
45C2×#817C2211yescharacteristic
46#5045—21no—
47#5045—21no—
48#5044—21no—
49#5043—21no—
50#5035—21no—
51#5035—21no—
52#5034—21no—
53#5033—21no—
54#4995C2211yescharacteristic
55#4995C2211yescharacteristic
56#4995C2211yescharacteristic
57#4995C2211yescharacteristic
58#4554C2211yescharacteristic
59#4554C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4552C2211yescharacteristic
67#4552C2211yescharacteristic
68#3093—31no—
69#3092—31no—
70#2930D311yescharacteristic
71C2×#428—31no—
72#2090—31no—
73#2089—31no—
74#2079—31no—
75#2078—31no—
76#1952—31no—
77#1951—31no—
78#1822—31no—
79#1821—31no—
80#1816—31no—
81#1815—31no—
82#1811—31no—
83C2×C3×#41C2311yescharacteristic
84C23×C3×C4C2311yescharacteristic
85C23×Dic3—21no—
86C3×#131—22no—
87C3×#131—22no—
88C2×C3×M16—21no—
89C2×C3×M16—22no—
90C2×C3×M16—22no—
91C2×C3×M16D411yescharacteristic
92C2×C3×M16D411yescharacteristic
93C2×C3×M16D411yescharacteristic
94C2×C3×M16D411yescharacteristic
95C2×C3×M16C2311yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C22×C3×C8—21no—
98C22×C3×C8C2311yescharacteristic
99C22×C3×C8C2311yescharacteristic
100C3×#123—21no—
101C3×C22⋊Q8D411yescharacteristic
102C3×C22⋊Q8D411yescharacteristic
103C3×#121D411yescharacteristic
104C3×#121D411yescharacteristic
105C3×#120—21no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
108C3×#117C2311yescharacteristic
109C2×#217—21no—
110#926—21no—
111#926—21no—
112#926—21no—
113#926—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4×Dic3—21no—
124C2×#208C2311yescharacteristic
125C2×#208C2311yescharacteristic
126C2×#208C2311yescharacteristic
127C2×#208C2311yescharacteristic
128C3×C4⋊C8—21no—
129C3×C4⋊C8—21no—
130C3×C22⋊C8—21no—
131C3×C22⋊C8—21no—
132C3×C22⋊C8—21no—
133C3×C22⋊C8—21no—
134C3×#97—21no—
135C3×C4×C8—21no—
136#835—21no—
137#835—21no—
138#835—21no—
139#835—21no—
140#832—21no—
141#832—21no—
142#832—21no—
143#832—21no—
144#830—21no—
145#830—21no—
146#830—21no—
147#830—21no—
148#819—22no—
149#819—22no—
150#818—21no—
151#818—21no—
152#818—21no—
153#818—21no—
154#817—21no—
155#817—21no—
156#817—21no—
157#817—21no—
158#808—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166C2×#131C2×D311yescharacteristic
167C22×M16C2×D311yescharacteristic
168#525C2×D311yescharacteristic
169C2×#117—31no—
170C22×C4⋊C4—31no—
171#485—61no—
172#485—61no—
173#484—61no—
174#483—61no—
175#482—61no—
176#482—61no—
177#481—61no—
178#480—61no—
179#436—21no—
180#436—21no—
181#435—21no—
182#435—21no—
183C2×#108—31no—
184C2×#108—31no—
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—61no—
190#428—61no—
191#428—61no—
192#428—61no—
193C2×#107—31no—
194C2×#107—31no—
195C2×#106—31no—
196C2×#106—31no—
197#424C2×D311yescharacteristic
198#417—31no—
199#417—31no—
200#417—31no—
201#417—31no—
202#408C2×D311yescharacteristic
203#407—31no—
204#407—31no—
205#407C2×D311yescharacteristic
206#405C2×D311yescharacteristic
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341—31no—
216#341—31no—
217C24×C3#4111yescharacteristic
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C2×C3×Q8C2×D411yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4C2×D411yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4#4111yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3—21no—
241C22×Dic3—21no—
242C22×Dic3—41no—
243C22×Dic3—41no—
244C22×Dic3—82no—
245C3×M16—22no—
246C3×M16—22no—
247C3×M16—22no—
248C3×M16—22no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—42no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—42no—
259C2×C3×C8C2×D411yescharacteristic
260C2×C3×C8C2×D411yescharacteristic
261C2×C3×C8C2×D411yescharacteristic
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8#4112yes—
264C2×C3×C8#4112yes—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C4⋊C4C2×D411yescharacteristic
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C42—21no—
275#217—41no—
276#217—41no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—41no—
286C4⋊Dic3—41no—
287C4×Dic3—41no—
288C4×Dic3—41no—
289#208—21no—
290#208—21no—
291#208—21no—
292#208—21no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×#41C22×D311yescharacteristic
302C23×C4—61no—
303C23×C4C22×D311yescharacteristic
304#131—22no—
305#131—22no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—21no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—31no—
313C2×M16#6711yescharacteristic
314C2×M16#6711yescharacteristic
315C2×M16#6711yescharacteristic
316C2×M16#6711yescharacteristic
317C2×M16C22×D311yescharacteristic
318C2×M16C22×D311yescharacteristic
319C22×C8—21no—
320C22×C8C22×D311yescharacteristic
321C22×C8C22×D311yescharacteristic
322#123—21no—
323C22⋊Q8D1211yescharacteristic
324C22⋊Q8D1211yescharacteristic
325#121D1211yescharacteristic
326#121D1211yescharacteristic
327#120—21no—
328C4×D4—21no—
329C4×D4—21no—
330#117—61no—
331#117—61no—
332#117—61no—
333#117—61no—
334#117C22×D311yescharacteristic
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—61no—
346#108—62no—
347#108—62no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#107—61no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366#103—61no—
367#103—61no—
368#103—61no—
369#103—61no—
370#102—61no—
371#102—61no—
372#102—61no—
373#102—61no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—61no—
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382#97—21no—
383C4×C8—21no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—21no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3#12111yescharacteristic
390C23×C3#12111yescharacteristic
391C23×C3C2×#4111yescharacteristic
392C3×Q8—21no—
393C3×Q8—21no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—81no—
399C3×D4—81no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4#12111yescharacteristic
416C2×C3×C4#12111yescharacteristic
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422C2×C3×C42−1+411yescharacteristic
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—82no—
428C2×Dic3—82no—
429C2×Dic3—81no—
430C2×Dic3—81no—
431C2×Dic3—81no—
432C2×Dic3—81no—
433C3×C8—21no—
434C3×C8—21no—
435C3×C8—41no—
436C3×C8—41no—
437C3×C8#12312yes—
438C3×C8#12312yes—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8—41no—
442C3⋊C8—41no—
443C24#23511yescharacteristic
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448C2×Q8C2×D1211yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×D1211yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4—242no—
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4#23511yescharacteristic
468C22×C4C2×#6711yescharacteristic
469C22×C4C2×#6711yescharacteristic
470C22×C4C23×D311yescharacteristic
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—21no—
479M16—42no—
480M16—61no—
481M16—61no—
482M16—61no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487M16—61no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—42no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8#23712yes—
498C2×C8#23712yes—
499C2×C8C2×#6711yescharacteristic
500C2×C8C2×#6711yescharacteristic
501C2×C8C2×#6711yescharacteristic
502C2×C8C2×#6711yescharacteristic
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—121no—
514C4⋊C4—121no—
515C4⋊C4C2×D1211yescharacteristic
516C4⋊C4C2×D1211yescharacteristic
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—121no—
523C22⋊C4—121no—
524C42—21no—
525C42—121no—
526C42—121no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3C2×#12111yescharacteristic
541C22×C3#55211yescharacteristic
542C22×C3#55211yescharacteristic
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4C2×#12311yescharacteristic
549C3×C4#53611yescharacteristic
550C3×C4#54711yescharacteristic
551C3×C4#54711yescharacteristic
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555Dic3—81no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23#93011yescharacteristic
562C23#93011yescharacteristic
563C23C2×#23511yescharacteristic
564Q8—21no—
565Q8—21no—
566D4—21no—
567D4—21no—
568D4—41no—
569D4—41no—
570D4—81no—
571D4—81no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—242no—
592C2×C4—242no—
593C2×C4—241no—
594C2×C4—241no—
595C2×C4—241no—
596C2×C4—241no—
597C2×C4#93011yescharacteristic
598C2×C4#93011yescharacteristic
599C2×C4C22×D1211yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4C2×#23711yescharacteristic
602C2×C4#101011yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4C22×#6711yescharacteristic
605C8—21no—
606C8—21no—
607C8—41no—
608C8—41no—
609C8—121no—
610C8—121no—
611C8—121no—
612C8—121no—
613C8#93512yes—
614C8#93512yes—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—41no—
618C2×C3—41no—
619C2×C3—81no—
620C2×C3#248611yescharacteristic
621C2×C3#306811yescharacteristic
622C2×C3#306911yescharacteristic
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22#554111yescharacteristic
637C22#554111yescharacteristic
638C22C2×#93011yescharacteristic
639C4—21no—
640C4—21no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4—241no—
648C4#555511yescharacteristic
649C4#555511yescharacteristic
650C4C2×#93511yescharacteristic
651C4#582511yescharacteristic
652C3#2767611yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—41no—
656C2—41no—
657C2—81no—
658C2#8237311yescharacteristic
659C2#8381411yescharacteristic
660C2#8381611yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2930C2
socleC22×C3C2×#121
prime radicalC3×#525C22
hypercenter#2930D3
hypocenterC3#27676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2930

Complements

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

NH ≅ G/Ntype
2. C2×#5121C2semidirect
5. #72016C2semidirect
6. #72015C2semidirect
7. #72000C2semidirect
8. #70533C2semidirect
9. #70532C2semidirect
10. #69995C2semidirect
11. #69994C2semidirect
12. #69974C2semidirect
13. #69973C2semidirect
14. #69894C2semidirect
15. #69893C2semidirect
652. C3#27676semidirect

Maximal subgroups

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

subgroupindex
C3×#29302
C2×#51212
#829152
#829142
#720162
#720152
#720002
#705332
#705322
#699952
#699942
#699742
#699732
#698942
#698932
#276763