Finite Group Databasebeta

← #10877435

Subgroups — #10877435

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74165C211yesmaximal, characteristic
2#74162C211yesmaximal, characteristic
3C3×#1835C211yesmaximal, characteristic, complemented
4C8×(C2×D3)⋊C4C211yesmaximal, characteristic
5#69767C211yesmaximal, characteristic, complemented
6#69560C211yesmaximal, characteristic, complemented
7#69542C211yesmaximal, characteristic, complemented
8#68910C211yesmaximal, characteristic, complemented
9#68882C211yesmaximal, characteristic
10#68879C211yesmaximal, characteristic
11#68866C211yesmaximal, characteristic
12#68850C211yesmaximal, characteristic, complemented
13#68813C211yesmaximal, characteristic
14#68810C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27698—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#4987C2211yescharacteristic
30#4975C2211yescharacteristic
31#4973C2211yescharacteristic
32#4972C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C4×(C2×D3)⋊C4C2211yescharacteristic
36#4929C2211yescharacteristic
37#4928C2211yescharacteristic
38#4676—21no—
39#4675—21no—
40#4675—21no—
41#4673—21no—
42#4655—21no—
43#4655—21no—
44#4652—21no—
45#4646—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2499—31no—
61#2439—31no—
62#2101—31no—
63#2099—31no—
64#2074—31no—
65#2024—31no—
66#1990—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1873—31no—
71#1835D311yescharacteristic, complemented
72#1825—31no—
73C8×C22⋊C4—31no—
74#1735—31no—
75C22×C4×D3C2311yescharacteristic
76C22×Dic6—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C2×D3×C8—22no—
106C2×D3×C8—22no—
107C2×D3×C8—41no—
108#897—22no—
109#897—22no—
110#892—22no—
111#892—22no—
112#892—22no—
113#892—41no—
114#888—22no—
115#888—41no—
116C3×C4×C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130(C2×D3)⋊C8—22no—
131#816—22no—
132#816—22no—
133C8×Dic3—41no—
134#810—41no—
135#810—41no—
136C23×C8—31no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—31no—
139C2×#117—31no—
140C2×#117—31no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8C2×D311yescharacteristic
148#397—61no—
149#397—61no—
150#396—61no—
151#393—61no—
152#391—31no—
153#391—31no—
154#391—61no—
155#389—31no—
156#389—31no—
157#387—61no—
158#387—61no—
159#387—61no—
160#384—31no—
161#382C2×D311yescharacteristic
162#382C2×D311yescharacteristic
163#380—31no—
164C4×C22⋊C4—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C23×D3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3#4112yes—
199C2×C4×D3#4112yes—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—42no—
207C2×Dic6—42no—
208C2×Dic6—81no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8#4112yes—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C42—21no—
224C3×C42—21no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—42no—
230(C2×D3)⋊C4—81no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—42no—
246C4×Dic3—42no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253D3×C8—44no—
254C22×Q8—31no—
255C22×Q8—61no—
256C23×C4—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—62no—
260C22×C8—62no—
261C22×C8—121no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4⋊Q8—62no—
265C4⋊Q8—121no—
266C22⋊Q8—62no—
267C22⋊Q8—62no—
268#117—62no—
269#117—62no—
270#117—62no—
271#117—121no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—31no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#103—62no—
296#103—62no—
297#103—121no—
298#103—121no—
299C22⋊C8—62no—
300C4×C8—22no—
301C4×C8—121no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—31no—
307#95—31no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315C23×C3C22×D411yescharacteristic
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3#12412yes—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C2×C3×C42−1+411yescharacteristic
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3#12412yes—
366C2×Dic3#13512yes—
367C2×Dic3#13512yes—
368C4×D3—22no—
369C4×D3—22no—
370C4×D3—22no—
371C4×D3—22no—
372Dic6—41no—
373Dic6—41no—
374Dic6—41no—
375Dic6—41no—
376Dic6—84no—
377C3×C8—22no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—31no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—122no—
389C2×Q8—122no—
390C2×Q8—241no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4C23×D311yescharacteristic
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—124no—
423C2×C8#23512yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—32no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—241no—
444C22⋊C4—62no—
445C22⋊C4—62no—
446C22⋊C4—122no—
447C22⋊C4—122no—
448C22⋊C4—122no—
449C22⋊C4—241no—
450C42—21no—
451C42—21no—
452C42—61no—
453C42—61no—
454C42—122no—
455C42—122no—
456C22×C3C2×#12411yescharacteristic
457C22×C3#54611yescharacteristic
458C22×C3#54711yescharacteristic
459C22×C3C2×#13511yescharacteristic
460C22×C3C2×#13511yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57811yescharacteristic
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4#52911yescharacteristic
477C3×C4#53911yescharacteristic
478C3×C4#55111yescharacteristic
479C3×C4#55211yescharacteristic
480Dic3—22no—
481Dic3—22no—
482Dic3—22no—
483Dic3—42no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487C23—32no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23C2×D3×D411yescharacteristic
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499Q8—244no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4C2×#23511yescharacteristic
544C2×C4#100711yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#100911yescharacteristic
549C8—22no—
550C8—41no—
551C8—121no—
552C8—121no—
553C2×C3#248811yescharacteristic
554C2×C3#313111yescharacteristic
555C2×C3#313711yescharacteristic
556C2×C3#329611yescharacteristic
557C2×C3#330011yescharacteristic
558C2×C3#334011yescharacteristic
559C2×C3#334211yescharacteristic
560D3—42no—
561C22—62no—
562C22—62no—
563C22—62no—
564C22—121no—
565C22—121no—
566C22—121no—
567C22—121no—
568C22#550711yescharacteristic
569C22D3×#12411yescharacteristic
570C22#567611yescharacteristic
571C22D3×#13511yescharacteristic
572C22D3×#13511yescharacteristic
573C22#577011yescharacteristic
574C22#577111yescharacteristic
575C4—41no—
576C4—41no—
577C4—62no—
578C4—62no—
579C4—62no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—122no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4#552711yescharacteristic
589C4#552811yescharacteristic
590C4#569211yescharacteristic
591C4#569311yescharacteristic
592C3#2769811yescharacteristic, complemented, Sylow
593C2—122no—
594C2#7423311yescharacteristic
595C2#7918911yescharacteristic
596C2#7919111yescharacteristic
597C2#8163711yescharacteristic
598C2#8163911yescharacteristic
599C2#8165211yescharacteristic
600C2#8165411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1835C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1835D3
hypocenterC3#27698
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
3. C3×#1835C2semidirect
5. #69767C2semidirect
6. #69560C2semidirect
7. #69542C2semidirect
8. #68910C2semidirect
12. #68850C2semidirect
14. #68810C2semidirect
15. #68803C2semidirect
71. #1835D3semidirect
592. C3#27698semidirect

Maximal subgroups

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

subgroupindex
#741652
#741622
C3×#18352
C8×(C2×D3)⋊C42
#697672
#695602
#695422
#689102
#688822
#688792
#688662
#688502
#688132
#688102
#688032
#276983