Finite Group Databasebeta

← #10855685

Subgroups — #10855685

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3046C211yesmaximal, characteristic
2#83899C211yesmaximal, characteristic
3#83397C211yesmaximal, characteristic, complemented
4#82562C211yesmaximal, characteristic, complemented
5#72103C211yesmaximal, characteristic, complemented
6#72100C211yesmaximal, characteristic, complemented
7#72087C211yesmaximal, characteristic
8#71863C211yesmaximal, characteristic, complemented
9#71665C211yesmaximal, characteristic, complemented
10#70534C211yesmaximal, characteristic, complemented
11#70533C211yesmaximal, characteristic, complemented
12#70301C211yesmaximal, characteristic, complemented
13#70298C211yesmaximal, characteristic, complemented
14#70263C211yesmaximal, characteristic, complemented
15#70252C211yesmaximal, characteristic, complemented
16#26193—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C2×C3×#135C2211yescharacteristic
19C3×#536C2211yescharacteristic
20C3×#520C2211yescharacteristic
21#5827C2211yescharacteristic
22#5827C2211yescharacteristic
23C2×#948C2211yescharacteristic
24C3×#464—21no—
25C3×#463—21no—
26C3×#462—21no—
27C3×#461—21no—
28C3×#424C2211yescharacteristic
29C3×#417C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×#837C2211yescharacteristic
32C2×#837C2211yescharacteristic
33#5234C2211yescharacteristic
34#5232C2211yescharacteristic
35#5232C2211yescharacteristic
36#5229C2211yescharacteristic
37C2×#836C2211yescharacteristic
38#5225C2211yescharacteristic
39C2×#835C2211yescharacteristic
40#5184—21no—
41Dic3×SD16—21no—
42Dic3×SD16—21no—
43#5152—21no—
44C2×#817C2211yescharacteristic
45#5101C2211yescharacteristic
46#5045—21no—
47#5045—21no—
48#5044—21no—
49#5043—21no—
50#4998—21no—
51#4998C2211yescharacteristic
52#4995—21no—
53#4995C2211yescharacteristic
54#4995C2211yescharacteristic
55#4977C2211yescharacteristic
56#4964—21no—
57#4964C2211yescharacteristic
58C2×#807—21no—
59C2×#807C2211yescharacteristic
60#4558C2211yescharacteristic
61#4555C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4552C2211yescharacteristic
65#4550C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3097—31no—
69#3046D311yescharacteristic
70#2952—31no—
71#2913—31no—
72#2097—31no—
73#2090—31no—
74#2078—31no—
75#2038—31no—
76#1930—31no—
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1876—31no—
81#1860—31no—
82#1860—31no—
83C2×C3×#41C2311yescharacteristic
84C2×C3×#41C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C3×#137—22no—
87C3×#137—22no—
88C3×#135—22no—
89C3×#135—22no—
90C2×C3×Q16—21no—
91C2×C3×Q16D411yescharacteristic
92C2×C3×SD16—21no—
93C2×C3×SD16D411yescharacteristic
94C2×C3×SD16D411yescharacteristic
95C2×C3×D8D411yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98C3×C4⋊Q8—21no—
99C3×#124—21no—
100C3×#123—41no—
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×D4—21no—
105C3×C4×D4—41no—
106C3×#117—21no—
107C3×#117C2311yescharacteristic
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4C2311yescharacteristic
110#948—22no—
111#948—22no—
112Q8×Dic3—21no—
113Q8×Dic3—21no—
114D4×Dic3—21no—
115D4×Dic3—21no—
116D4×Dic3—41no—
117D4×Dic3—41no—
118#926—21no—
119#926—21no—
120#926C2311yescharacteristic
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C2×#208—21no—
127C2×#208C2311yescharacteristic
128C2×#208C2311yescharacteristic
129C2×#208C2311yescharacteristic
130C22×C3⋊C8—21no—
131C22×C3⋊C8C2311yescharacteristic
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×#103—21no—
135C3×#103—21no—
136C3×#103—21no—
137C3×#102—21no—
138#837—22no—
139#837—22no—
140#836—22no—
141#835—21no—
142#835—21no—
143#835—21no—
144#835—21no—
145#835—21no—
146#835—21no—
147#832—21no—
148#832—21no—
149C3⋊#95—41no—
150C3⋊#95—41no—
151#830—21no—
152#830—21no—
153#817—21no—
154#817—21no—
155#815—21no—
156C8×Dic3—21no—
157#808—21no—
158#808—21no—
159#808—21no—
160#808D411yescharacteristic
161#808D411yescharacteristic
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166#807—21no—
167#807D411yescharacteristic
168#807D411yescharacteristic
169C2×#137C2×D311yescharacteristic
170C2×#135C2×D311yescharacteristic
171C2×#131—31no—
172#536C2×D311yescharacteristic
173#520—31no—
174#520—31no—
175#520C2×D311yescharacteristic
176#485—61no—
177#485—61no—
178#484—61no—
179#483—61no—
180#464—21no—
181#463—21no—
182#462—21no—
183#461—21no—
184#442—61no—
185#441—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188#428—31no—
189#428—31no—
190#428—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—61no—
197C2×#106—31no—
198#424C2×D311yescharacteristic
199C2×#104—31no—
200C2×#104—31no—
201#419—31no—
202#417—31no—
203#417—31no—
204#417C2×D311yescharacteristic
205C2×#103—31no—
206C2×#103C2×D311yescharacteristic
207C2×#101—31no—
208#408—31no—
209#405—31no—
210#391—31no—
211#382—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341—31no—
216#341—31no—
217#339—31no—
218#339—31no—
219#336—31no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C3×#41#4112yes—
227C3×#41#4112yes—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—21no—
234C2×C3×Q8C2×D411yescharacteristic
235C2×C3×Q8C2×D411yescharacteristic
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4C2×D411yescharacteristic
239C2×C3×D4C2×D411yescharacteristic
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—21no—
244C22×C3×C4—41no—
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—41no—
250C22×Dic3—41no—
251C3×Q16—22no—
252C3×Q16—42no—
253C3×SD16—22no—
254C3×SD16—22no—
255C3×SD16—42no—
256C3×D8—22no—
257C3×M16—21no—
258C3×M16—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8C2×D411yescharacteristic
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—41no—
267C3×C4⋊C4—41no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4C2×D411yescharacteristic
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C22⋊C4—21no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—41no—
274C3×C22⋊C4—41no—
275C3×C42—21no—
276C3×C42—41no—
277#217—21no—
278#217—41no—
279#217—41no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—41no—
285C4⋊Dic3—41no—
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—41no—
292C4×Dic3—41no—
293#208—24no—
294#208—21no—
295#208—21no—
296#208—21no—
297#208—21no—
298#208—21no—
299#208#4112yes—
300C2×C3⋊C8—24no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8C2×D411yescharacteristic
307C2×C3⋊C8C2×D411yescharacteristic
308C2×#41C22×D311yescharacteristic
309C2×#41C22×D311yescharacteristic
310C22×Q8C22×D311yescharacteristic
311#137—22no—
312#137—22no—
313#135—22no—
314#135—22no—
315C2×Q16—21no—
316C2×Q16#6711yescharacteristic
317C2×SD16—21no—
318C2×SD16#6711yescharacteristic
319C2×SD16#6711yescharacteristic
320C2×D8#6711yescharacteristic
321#131—62no—
322#131—62no—
323C2×M16—31no—
324C2×M16—31no—
325C2×M16—31no—
326C2×M16—61no—
327C2×M16C22×D311yescharacteristic
328C22×C8—31no—
329C22×C8—61no—
330C22×C8C22×D311yescharacteristic
331C4⋊Q8—21no—
332#124—21no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C4×Q8—21no—
337C4×Q8—61no—
338C4×Q8—61no—
339C4×D4—21no—
340C4×D4—41no—
341C4×D4—61no—
342C4×D4—61no—
343C4×D4—121no—
344C4×D4—121no—
345#117—21no—
346#117—31no—
347#117—61no—
348#117—61no—
349#117C22×D311yescharacteristic
350C2×C4⋊C4—21no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C42—31no—
357#107—31no—
358#107—31no—
359#107—61no—
360#107—61no—
361#107—61no—
362#107—61no—
363#107—61no—
364#106—31no—
365#106—31no—
366#106—61no—
367#106—61no—
368#106—61no—
369#106—61no—
370#106—61no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373#104—62no—
374#104—62no—
375#103—21no—
376#103—21no—
377#103—21no—
378#103—61no—
379#103—61no—
380#103—61no—
381#103—61no—
382#103—61no—
383#103—61no—
384#102—21no—
385#102—61no—
386#102—61no—
387#101—62no—
388C22⋊C8—61no—
389C22⋊C8—61no—
390#97—61no—
391C4×C8—61no—
392#95—121no—
393#95—121no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3H(5,𝔽2)11yescharacteristic
397C3×Q8—21no—
398C3×Q8—21no—
399C3×Q8—21no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×Q8—41no—
404C3×Q8#12312yes—
405C3×D4—22no—
406C3×D4—22no—
407C3×D4—21no—
408C3×D4—21no—
409C3×D4—21no—
410C3×D4—41no—
411C3×D4—41no—
412C3×D4#12312yes—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—81no—
426C2×C3×C4—161no—
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431C2×C3×C4C2×#4111yescharacteristic
432C2×C3×C42−1+411yescharacteristic
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—81no—
441C2×Dic3—81no—
442C2×Dic3—161no—
443C2×Dic3—161no—
444C3×C8—21no—
445C3×C8—21no—
446C3×C8—41no—
447C3⋊C8—21no—
448C3⋊C8—22no—
449C3⋊C8—21no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452#41—21no—
453#41—21no—
454#41—21no—
455#41—21no—
456#41—21no—
457#41—21no—
458#41#23712yes—
459#41#23712yes—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×#6711yescharacteristic
467C2×Q8C2×#6711yescharacteristic
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×#6711yescharacteristic
471C2×D4C2×#6711yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—41no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—121no—
481C22×C4—121no—
482C22×C4C23×D311yescharacteristic
483Q16—22no—
484Q16—42no—
485SD16—22no—
486SD16—22no—
487SD16—42no—
488D8—22no—
489M16—21no—
490M16—21no—
491M16—32no—
492M16—64no—
493M16—61no—
494M16—61no—
495M16—61no—
496M16—61no—
497M16—61no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—31no—
502C2×C8—31no—
503C2×C8—64no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8C2×#6711yescharacteristic
510C2×C8C2×#6711yescharacteristic
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—31no—
514C4⋊C4—31no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—121no—
523C4⋊C4—121no—
524C4⋊C4D3×D411yescharacteristic
525C4⋊C4D3×D411yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—61no—
531C22⋊C4—121no—
532C22⋊C4—121no—
533C42—21no—
534C42—41no—
535C42—61no—
536C42—61no—
537C42—61no—
538C42—121no—
539C42—121no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—22no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3#53911yescharacteristic
549C22×C3D4211yescharacteristic
550C22×C3#55311yescharacteristic
551C3×C4—22no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—41no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4C2×#12311yescharacteristic
558C3×C4#52511yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#54811yescharacteristic
561Dic3—41no—
562Dic3—41no—
563Dic3—81no—
564Dic3—81no—
565Dic3—81no—
566C23—21no—
567C23—21no—
568C23#100411yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—21no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8#93512yes—
577D4—22no—
578D4—22no—
579D4—21no—
580D4—21no—
581D4—21no—
582D4—41no—
583D4—41no—
584D4#93512yes—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—81no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—161no—
606C2×C4—241no—
607C2×C4—241no—
608C2×C4—481no—
609C2×C4—481no—
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4#101011yescharacteristic
615C2×C4C22×#6711yescharacteristic
616C8—21no—
617C8—21no—
618C8—41no—
619C8—61no—
620C8—61no—
621C8—62no—
622C8—121no—
623C8—121no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—42no—
627C2×C3—81no—
628C2×C3#244311yescharacteristic
629C2×C3#328911yescharacteristic
630C2×C3#329011yescharacteristic
631C22—21no—
632C22—21no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22#560111yescharacteristic
640C22#560211yescharacteristic
641C22D4×#6711yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—81no—
647C4—81no—
648C4—121no—
649C4—121no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4#559811yescharacteristic
654C4#560611yescharacteristic
655C4C2×#93511yescharacteristic
656C4#582411yescharacteristic
657C3#2619311yescharacteristic, complemented, Sylow
658C2—21no—
659C2—21no—
660C2—42no—
661C2—81no—
662C2#8265511yescharacteristic
663C2#8401211yescharacteristic
664C2#8401911yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3046C2
socleC22×C3D42
prime radicalC2×C3×#135C22
hypercenter#3046D3
hypocenterC3#26193
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3046

Complements

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

NH ≅ G/Ntype
3. #83397C2semidirect
4. #82562C2semidirect
5. #72103C2semidirect
6. #72100C2semidirect
8. #71863C2semidirect
9. #71665C2semidirect
10. #70534C2semidirect
11. #70533C2semidirect
12. #70301C2semidirect
13. #70298C2semidirect
14. #70263C2semidirect
15. #70252C2semidirect
657. C3#26193semidirect

Maximal subgroups

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

subgroupindex
C3×#30462
#838992
#833972
#825622
#721032
#721002
#720872
#718632
#716652
#705342
#705332
#703012
#702982
#702632
#702522
#261933