Finite Group Databasebeta

← #10854147

Subgroups — #10854147

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76066C211yesmaximal, characteristic, complemented
2#73639C211yesmaximal, characteristic
3C3×#1918C211yesmaximal, characteristic, complemented
4#70641C211yesmaximal, characteristic
5#70606C211yesmaximal, characteristic, complemented
6#69630C211yesmaximal, characteristic
7#69466C211yesmaximal, characteristic, complemented
8#68901C211yesmaximal, characteristic
9#68892C211yesmaximal, characteristic
10#68814C211yesmaximal, characteristic, complemented
11#68795C211yesmaximal, characteristic, complemented
12#68795C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic
14#68681C211yesmaximal, characteristic
15#68617C211yesmaximal, characteristic, complemented
16#26086—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#889—21no—
19C2×#872—21no—
20C2×C3×#107C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×#818C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4988C2211yescharacteristic
31#4981—21no—
32#4975C2211yescharacteristic
33#4975C2211yescharacteristic
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4940—21no—
44#4939C2211yescharacteristic
45#4935C2211yescharacteristic
46C4×Dic3⋊C4C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4669—21no—
49#4667—21no—
50#4657—21no—
51#4647—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2552—31no—
61#2354—31no—
62#2093—31no—
63#2093—31no—
64#2067—31no—
65#2067—31no—
66#1947—31no—
67#1947—31no—
68#1947—31no—
69#1939—31no—
70#1918—31no—
71#1918D311yescharacteristic, complemented
72#1845—31no—
73#1802—31no—
74C4×#102—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#893—22no—
112#893Q814yes—
113#889—44no—
114#872—44no—
115C3×#107—22no—
116C3×#95—21no—
117C3×#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95D411yescharacteristic
122C3⋊#95D411yescharacteristic
123#821—22no—
124#821—41no—
125#818—22no—
126#818—41no—
127Dic3⋊C8—22no—
128Dic3⋊C8—22no—
129C8×Dic3—22no—
130#809—22no—
131#809—41no—
132#807—41no—
133#807—41no—
134#807—41no—
135C4⋊(C3⋊C8)—22no—
136C2×#125—61no—
137C2×C4×D4—31no—
138C2×#117—61no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#107C2×D311yescharacteristic
145C2×C4⋊C8—31no—
146C2×C4⋊C8—31no—
147C2×C4⋊C8—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×C4×C8—31no—
153#398—61no—
154#398—61no—
155#390—31no—
156#390—31no—
157#389—31no—
158#389—61no—
159#389C2×D311yescharacteristic
160#388—61no—
161#388—61no—
162#384—31no—
163#384—31no—
164#384—31no—
165#384—31no—
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3—21no—
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×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12C2×Q812yes—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8#4112yes—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4C2×Q812yes—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—42no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—81no—
225(C2×D3)⋊C4—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—42no—
233C4⋊Dic3—81no—
234Dic3⋊C4—22no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—42no—
238Dic3⋊C4—42no—
239Dic3⋊C4—81no—
240Dic3⋊C4—81no—
241Dic3⋊C4—81no—
242Dic3⋊C4D814yes—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—42no—
246C4×Dic3—42no—
247C4×Dic3C2×Q812yes—
248C4×Dic3#4112yes—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8#4112yes—
255C22×D4—31no—
256C23×C4—61no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#125—124no—
262C4×D4—34no—
263C4×D4—62no—
264#117—124no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C42—21no—
290C2×C42—31no—
291C2×C42—31no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—22no—
296#107—62no—
297#107—121no—
298#107—121no—
299#107—121no—
300#107—121no—
301C4⋊C8—62no—
302C4⋊C8—62no—
303C4⋊C8—62no—
304#102—62no—
305#102—62no—
306#102—121no—
307#102—121no—
308C4×C8—62no—
309#95—21no—
310#95—21no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—22no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
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×D812yes—
342C2×C3×C4C22×Q811yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
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×Dic3C22⋊Q812yes—
365C2×Dic3C22⋊Q812yes—
366C2×Dic3C2×D812yes—
367C2×Dic3C2×D812yes—
368C2×Dic3#13712yes—
369D12—22no—
370D12—22no—
371D12—41no—
372D12—41no—
373C4×D3—48no—
374C3×C8—22no—
375C3×C8—41no—
376C3⋊C8—22no—
377C3⋊C8—41no—
378C24—61no—
379C2×D4—32no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—64no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—122no—
398C22×C4—122no—
399C22×C4D3×D411yescharacteristic
400C22×C4D3×D411yescharacteristic
401C22×C4C23×D311yescharacteristic
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—32no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8#23512yes—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—34no—
422C4⋊C4—42no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—122no—
435C4⋊C4—241no—
436C4⋊C4—241no—
437C4⋊C4—241no—
438C4⋊C4—241no—
439C4⋊C4D3×Q812yes—
440C22⋊C4—64no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C42—21no—
446C42—21no—
447C42—32no—
448C42—32no—
449C42—42no—
450C42—61no—
451C42—61no—
452C42—122no—
453C42—122no—
454C22×C3C2×C22⋊Q811yescharacteristic
455C22×C3#54711yescharacteristic
456C22×C3#55011yescharacteristic
457C22×C3C22×D811yescharacteristic
458C22×C3C2×#13711yescharacteristic
459C22×C3#57611yescharacteristic
460C22×C3#57811yescharacteristic
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4#53311yescharacteristic
475C3×C4#54411yescharacteristic
476C3×C4#55211yescharacteristic
477C3×C4#55311yescharacteristic
478Dic3—22no—
479Dic3—22no—
480Dic3—41no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484Dic3#47414yes—
485C23—62no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23C2×D3×D411yescharacteristic
493D4—62no—
494D4—62no—
495D4—121no—
496D4—121no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—41no—
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—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—128no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4D3×D812yes—
537C2×C4C2×#23511yescharacteristic
538C2×C4C2×D3×Q811yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C2×C4#100911yescharacteristic
542C2×C4#101011yescharacteristic
543C8—22no—
544C8—41no—
545C8—62no—
546C8—121no—
547C2×C3#244111yescharacteristic
548C2×C3C2×#47411yescharacteristic
549C2×C3#308311yescharacteristic
550C2×C3#333411yescharacteristic
551C2×C3#335711yescharacteristic
552C2×C3#339111yescharacteristic
553C2×C3#339811yescharacteristic
554D3—82no—
555C22—122no—
556C22—122no—
557C22—122no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22—241no—
562C22#555311yescharacteristic
563C22D3×C22⋊Q811yescharacteristic
564C22#565511yescharacteristic
565C22C2×D3×D811yescharacteristic
566C22#576911yescharacteristic
567C22D3×#13711yescharacteristic
568C22#577911yescharacteristic
569C4—22no—
570C4—34no—
571C4—42no—
572C4—41no—
573C4—62no—
574C4—62no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—121no—
579C4—241no—
580C4—241no—
581C4—241no—
582C4#553111yescharacteristic
583C4#557811yescharacteristic
584C4#568911yescharacteristic
585C4#570911yescharacteristic
586C3#2608611yescharacteristic, complemented, Sylow
587C2—242no—
588C2#7498411yescharacteristic
589C2#7957811yescharacteristic
590C2#7982111yescharacteristic
591C2D3×#47411yescharacteristic
592C2#8123411yescharacteristic
593C2#8129111yescharacteristic
594C2#8151311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1918C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1918D3
hypocenterC3#26086
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
1. #76066C2semidirect
3. C3×#1918C2semidirect
5. #70606C2semidirect
7. #69466C2semidirect
10. #68814C2semidirect
11. #68795C2semidirect
12. #68795C2semidirect
15. #68617C2semidirect
71. #1918D3semidirect
586. C3#26086semidirect

Maximal subgroups

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

subgroupindex
#760662
#736392
C3×#19182
#706412
#706062
#696302
#694662
#689012
#688922
#688142
#687952
#687802
#686812
#686172
#260863