Finite Group Databasebeta

← #10824914

Subgroups — #10824914

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#137C211yesmaximal, characteristic
2#78867C211yesmaximal, characteristic
3C2×#4789C211yesmaximal, characteristic, complemented
4#78649C211yesmaximal, characteristic
5C3×#2052C211yesmaximal, characteristic
6#72109C211yesmaximal, characteristic
7#70717C211yesmaximal, characteristic, complemented
8#70692C211yesmaximal, characteristic, complemented
9#70302C211yesmaximal, characteristic
10#70265C211yesmaximal, characteristic, complemented
11#70120C211yesmaximal, characteristic, complemented
12Dic3×#101C211yesmaximal, characteristic, complemented
13#70004C211yesmaximal, characteristic
14#68981C211yesmaximal, characteristic, complemented
15#68973C211yesmaximal, characteristic, complemented
16#23720—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#951C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#417C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#101C2211yescharacteristic
27C3×#408C2211yescharacteristic
28#5234C2211yescharacteristic
29#5232C2211yescharacteristic
30C2×#836C2211yescharacteristic
31#5225C2211yescharacteristic
32#5184—21no—
33Dic3×Q16—21no—
34#5164—21no—
35Dic3×SD16—21no—
36C2×#824C2211yescharacteristic
37C2×#824C2211yescharacteristic
38#5121C2211yescharacteristic
39#5120C2211yescharacteristic
40#5119C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42Dic3×M16C2211yescharacteristic
43Dic3×M16C2211yescharacteristic
44C2×#819C2211yescharacteristic
45C2×#816C2211yescharacteristic
46#5102C2211yescharacteristic
47C2×#815C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4790—21no—
51#4789—21no—
52#4789—22no—
53#4789—22no—
54#4758—21no—
55#4757—21no—
56#4720—22no—
57#4719—22no—
58C3×#343C2211yescharacteristic
59#4558C2211yescharacteristic
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137—31no—
65C4×#137—31no—
66C2×#441—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2064—31no—
71#2064—31no—
72#2052—31no—
73#2052D311yescharacteristic
74#2047—31no—
75#2020—31no—
76#1778—31no—
77C4×#101—31no—
78C4×#101—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×#137—22no—
83C3×#137—22no—
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91#951—22no—
92#951—22no—
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95Q8×Dic3—21no—
96Q8×Dic3—21no—
97Q8×Dic3D411yescharacteristic
98Q8×Dic3D411yescharacteristic
99C2×#216—21no—
100C2×#216—22no—
101C2×#216—22no—
102C2×#216—22no—
103D4×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3D411yescharacteristic
106D4×Dic3D411yescharacteristic
107C2×#214—21no—
108#926—21no—
109#926C2311yescharacteristic
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#887—21no—
122#887—21no—
123#887—21no—
124#887—21no—
125C3×#108—22no—
126C3×#103—21no—
127C3×#103—22no—
128C3×#102—21no—
129C3×#101—22no—
130C3×C22⋊C8—22no—
131#836—22no—
132#835—21no—
133#835—21no—
134#835—21no—
135#832—21no—
136#830—22no—
137#824—22no—
138#824—22no—
139#819—22no—
140#818—21no—
141#817—21no—
142#816—22no—
143#816—21no—
144#816—21no—
145#815—21no—
146#815—21no—
147#815—21no—
148#815D412yes—
149#815D412yes—
150Dic3⋊C8—22no—
151Dic3⋊C8—22no—
152C8×Dic3—21no—
153C8×Dic3—21no—
154C8×Dic3—21no—
155#808—21no—
156#808—22no—
157#807—21no—
158C2×#137—31no—
159C2×#137C2×D311yescharacteristic
160C22×Q16—31no—
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×Q8—31no—
164#441—62no—
165#441—62no—
166#441—61no—
167#441—61no—
168#440—61no—
169#440—61no—
170C4×Q16—61no—
171C4×Q16—61no—
172C4×SD16—61no—
173C4×SD16—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108C2×D311yescharacteristic
178#428—31no—
179#428—31no—
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#419—31no—
185#417—31no—
186#417C2×D311yescharacteristic
187C2×#103—31no—
188C2×#103C2×D311yescharacteristic
189C2×#101—31no—
190C2×#101—31no—
191C2×#101—31no—
192C2×#101C2×D311yescharacteristic
193#408—31no—
194#408C2×D311yescharacteristic
195C4×M16—31no—
196C4×M16—31no—
197C4×M16—31no—
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343C2×D311yescharacteristic
203#341—31no—
204#341—31no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×Q8#4112yes—
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3#4111yescharacteristic
227C22×Dic3#4111yescharacteristic
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6#4112yes—
234C3×Q16—24no—
235C3×SD16—24no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16#4112yes—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D412yes—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250#217—21no—
251#217—21no—
252#217—21no—
253#216—28no—
254#216—24no—
255#214—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—41no—
261C4⋊Dic3—41no—
262C4⋊Dic3C2×D411yescharacteristic
263C4⋊Dic3C2×D411yescharacteristic
264Dic3⋊C4—41no—
265Dic3⋊C4—41no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—41no—
270C4×Dic3—41no—
271C4×Dic3—41no—
272C4×Dic3—41no—
273C4×Dic3C2×D411yescharacteristic
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208#4112yes—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C2×#41C22×D311yescharacteristic
290C22×Q8—31no—
291C22×Q8C22×D311yescharacteristic
292#137—22no—
293#137—22no—
294#137—62no—
295#137—62no—
296C2×Q16—21no—
297C2×Q16—62no—
298C2×Q16—62no—
299C2×Q16—61no—
300C2×Q16—62no—
301C2×SD16—21no—
302C2×SD16—61no—
303C2×M16—31no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16C22×D311yescharacteristic
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C4×Q8—31no—
312C4×Q8—31no—
313C4×Q8—61no—
314C4×Q8—61no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×Q8—61no—
320C4×Q8—61no—
321C4×D4—31no—
322C4×D4—31no—
323C4×D4—61no—
324C4×D4—61no—
325#117—31no—
326#117—61no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335#108—22no—
336#108—62no—
337#107—61no—
338#107—61no—
339#106—61no—
340#106—62no—
341#106—61no—
342C4⋊C8—62no—
343C4⋊C8—62no—
344#103—21no—
345#103—22no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—61no—
350#103—61no—
351#103—62no—
352#102—21no—
353#102—61no—
354#101—22no—
355#101—62no—
356#101—62no—
357#101—62no—
358C22⋊C8—22no—
359C22⋊C8—62no—
360#97—32no—
361#97—32no—
362#97—61no—
363#97—61no—
364#97—61no—
365C4×C8—61no—
366C4×C8—61no—
367C4×C8—61no—
368C23×C3—21no—
369C23×C3C2×#4111yescharacteristic
370C3×Q8—22no—
371C3×Q8—21no—
372C3×Q8—22no—
373C3×Q8—21no—
374C3×Q8—41no—
375C3×Q8—41no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—81no—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—81no—
403C2×Dic3—81no—
404C2×Dic3—81no—
405C2×Dic3#12111yescharacteristic
406C2×Dic3#12111yescharacteristic
407C2×Dic3#12111yescharacteristic
408C2×Dic3#12111yescharacteristic
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—22no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421#41—21no—
422#41—21no—
423#41—21no—
424#41—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—32no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8D3×D411yescharacteristic
437C2×Q8#23712yes—
438C2×D4—21no—
439C2×D4D3×D411yescharacteristic
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4C23×D311yescharacteristic
450Q16—24no—
451Q16—68no—
452Q16—64no—
453SD16—24no—
454SD16—64no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—32no—
461M16—61no—
462M16—61no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16#23712yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×D412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—31no—
482C4⋊C4—31no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—121no—
488C4⋊C4—121no—
489C4⋊C4—121no—
490C4⋊C4—121no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C22⋊C4—61no—
494C42—31no—
495C42—31no—
496C42—31no—
497C42—31no—
498C42—61no—
499C42—61no—
500C42—61no—
501C42—121no—
502C42—121no—
503C42—121no—
504C42—121no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12111yescharacteristic
511C22×C3#52911yescharacteristic
512C22×C3#52911yescharacteristic
513C3×C4—41no—
514C3×C4—42no—
515C3×C4—81no—
516C3×C4#52511yescharacteristic
517C3×C4#52511yescharacteristic
518C3×C4#52511yescharacteristic
519C3×C4#53211yescharacteristic
520Dic3—21no—
521Dic3—21no—
522Dic3—21no—
523Dic3—21no—
524Dic3—42no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528C23—21no—
529C23D3×#4111yescharacteristic
530Q8—21no—
531Q8—22no—
532Q8—21no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—62no—
537Q8—62no—
538Q8—121no—
539Q8—121no—
540D4—21no—
541D4—21no—
542D4—41no—
543D4—41no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—81no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—241no—
565C2×C4—241no—
566C2×C4—241no—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C8—22no—
574C8—22no—
575C8—41no—
576C8—41no—
577C8—62no—
578C8—62no—
579C8—121no—
580C8—121no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—81no—
584C2×C3#239211yescharacteristic
585C2×C3#306911yescharacteristic
586C2×C3#306911yescharacteristic
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22D3×#12111yescharacteristic
593C22#564911yescharacteristic
594C22#568011yescharacteristic
595C4—41no—
596C4—42no—
597C4—61no—
598C4—61no—
599C4—61no—
600C4—61no—
601C4—81no—
602C4—121no—
603C4—121no—
604C4—122no—
605C4—241no—
606C4#558611yescharacteristic
607C4#559611yescharacteristic
608C4#560611yescharacteristic
609C4#565611yescharacteristic
610C3#2372011yescharacteristic, complemented, Sylow
611C2—21no—
612C2—21no—
613C2—81no—
614C2#7689011yescharacteristic
615C2#8093111yescharacteristic
616C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2052C2
socleC22×C3C2×#121
prime radicalC2×C3×#41C23
hypercenter#2052D3
hypocenterC3#23720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2052

Complements

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

NH ≅ G/Ntype
3. C2×#4789C2semidirect
7. #70717C2semidirect
8. #70692C2semidirect
10. #70265C2semidirect
11. #70120C2semidirect
12. Dic3×#101C2semidirect
14. #68981C2semidirect
15. #68973C2semidirect
610. C3#23720semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1372
#788672
C2×#47892
#786492
C3×#20522
#721092
#707172
#706922
#703022
#702652
#701202
Dic3×#1012
#700042
#689812
#689732
#237203