Finite Group Databasebeta

← #10784395

Subgroups — #10784395

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79312C211yesmaximal, characteristic, complemented
2#79304C211yesmaximal, characteristic, complemented
3#79304C211yesmaximal, characteristic, complemented
4#79304C211yesmaximal, characteristic, complemented
5#79269C211yesmaximal, characteristic, complemented
6#79268C211yesmaximal, characteristic, complemented
7C3×#1454C211yesmaximal, characteristic, complemented
8#68096C211yesmaximal, characteristic, complemented
9#68094C211yesmaximal, characteristic, complemented
10#68093C211yesmaximal, characteristic, complemented
11#68092C211yesmaximal, characteristic, complemented
12#68091C211yesmaximal, characteristic, complemented
13#68090C211yesmaximal, characteristic, complemented
14#68059C211yesmaximal, characteristic, complemented
15#68058C211yesmaximal, characteristic, complemented
16#19742—31nomaximal, Sylow
17#5477C2211yescharacteristic
18#5477C2211yescharacteristic
19#5477C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#431C2211yescharacteristic
24#4846—22no—
25#4844—22no—
26#4844—22no—
27#4844—22no—
28#4842—22no—
29#4841—22no—
30#4738—22no—
31#4736—22no—
32#4735—22no—
33#4734—22no—
34#4734—22no—
35#4734—22no—
36#4695C2211yescharacteristic
37#4695C2211yescharacteristic
38#4695C2211yescharacteristic
39#4694C2211yescharacteristic
40#4694C2211yescharacteristic
41#4694C2211yescharacteristic
42#4694C2211yescharacteristic
43#4694C2211yescharacteristic
44#4694C2211yescharacteristic
45#4694C2211yescharacteristic
46#4693C2211yescharacteristic
47#4693C2211yescharacteristic
48#4693C2211yescharacteristic
49#4693C2211yescharacteristic
50#4693C2211yescharacteristic
51#4682C2211yescharacteristic
52#4682C2211yescharacteristic
53#4681C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4681C2211yescharacteristic
57#4681C2211yescharacteristic
58#4680C2211yescharacteristic
59#4680C2211yescharacteristic
60C3×#322C2211yescharacteristic
61C3×#322C2211yescharacteristic
62C3×#322C2211yescharacteristic
63C3×C82C2211yescharacteristic
64#3227—31no—
65#3226—31no—
66#3207—31no—
67#3207—31no—
68#3207—31no—
69#3206—31no—
70#1721—31no—
71#1719—31no—
72#1718—31no—
73#1717—31no—
74#1715—31no—
75#1714—31no—
76#1708—31no—
77#1706—31no—
78#1454D311yescharacteristic, complemented
79C2×#235—21no—
80C2×#235—21no—
81C2×#235—21no—
82C22×Dic6—21no—
83C2×C3×C42C2311yescharacteristic
84#930—21no—
85#930—21no—
86#930—21no—
87#930—21no—
88#930—21no—
89#930—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×D24—22no—
93C2×D24—22no—
94C2×D24—22no—
95C2×#204—22no—
96C2×#204—22no—
97C2×#204—22no—
98C2×#204—22no—
99C2×#204—22no—
100#875—21no—
101#875—21no—
102#875—21no—
103#874C2311yescharacteristic
104#874C2311yescharacteristic
105#874C2311yescharacteristic
106C4×D12—21no—
107C4×D12—21no—
108C4×D12—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12—21no—
112#869—21no—
113#869—21no—
114#869—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C3×C4⋊C8D412yes—
121C3×C4⋊C8D412yes—
122C3×C4⋊C8D412yes—
123C3×C22⋊C8D412yes—
124C3×C22⋊C8D412yes—
125C3×C22⋊C8D412yes—
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131C3×C4×C8C2311yescharacteristic
132#821—22no—
133#821—22no—
134#821—22no—
135#821—22no—
136#821—22no—
137#821—22no—
138#821—22no—
139#818—22no—
140#818—22no—
141#817—22no—
142#817—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148#532—31no—
149#532—31no—
150#532—31no—
151C2×C4⋊Q8—31no—
152#500—31no—
153#500—31no—
154#498—31no—
155#498—31no—
156#493—31no—
157#493—31no—
158#493—31no—
159#492—31no—
160#492—31no—
161#492—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#481—62no—
177#480—62no—
178#476—62no—
179#474—62no—
180#463—62no—
181#463—62no—
182#463—62no—
183#461—62no—
184#449—62no—
185#449—62no—
186#449—62no—
187#448—62no—
188#431C2×D311yescharacteristic
189#431C2×D311yescharacteristic
190#431C2×D311yescharacteristic
191#322C2×D311yescharacteristic
192#322C2×D311yescharacteristic
193#322C2×D311yescharacteristic
194C82C2×D311yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C2×#67—41no—
199C2×#67—41no—
200C2×#67—41no—
201C22×Dic3—41no—
202#235—82no—
203#235—82no—
204#235—82no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×C4×D3—41no—
215C2×C4×D3—41no—
216C2×C4×D3—41no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—41no—
223C2×Dic6—82no—
224C2×C3×C8C2×D412yes—
225C2×C3×C8C2×D412yes—
226C2×C3×C8C2×D412yes—
227C2×C3×C8C2×D412yes—
228C2×C3×C8C2×D412yes—
229C2×C3×C8C2×D412yes—
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2411yescharacteristic
234(C2×D3)⋊C4—41no—
235(C2×D3)⋊C4—41no—
236(C2×D3)⋊C4—41no—
237(C2×D3)⋊C4—41no—
238(C2×D3)⋊C4—41no—
239(C2×D3)⋊C4—41no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—41no—
251C4⋊Dic3—41no—
252D24—82no—
253D24—82no—
254D24—82no—
255#204—82no—
256#204—82no—
257#204—82no—
258#204—82no—
259#204—82no—
260C2×#41—61no—
261C2×#41—61no—
262C2×#41—61no—
263C22×Q8—61no—
264C2×SD16—62no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C2×SD16—62no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C4⋊Q8—31no—
273C4⋊Q8—31no—
274C4⋊Q8—31no—
275C4⋊Q8—31no—
276C4⋊Q8—31no—
277C4⋊Q8—61no—
278C4⋊Q8—61no—
279C4⋊Q8—61no—
280#127—31no—
281#127—31no—
282#127—31no—
283#124—61no—
284#124—61no—
285#124—61no—
286#121—61no—
287#121—61no—
288#121—61no—
289#121—61no—
290#121—61no—
291#121—61no—
292C4×D4—61no—
293C4×D4—61no—
294C4×D4—61no—
295C4×D4—61no—
296C4×D4—61no—
297C4×D4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C42C22×D311yescharacteristic
301#107—62no—
302#107—62no—
303#106—62no—
304#106—62no—
305C4⋊C8D1212yes—
306C4⋊C8D1212yes—
307C4⋊C8D1212yes—
308#103—62no—
309#103—62no—
310#103—62no—
311#103—62no—
312#103—62no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317#102—62no—
318#102—62no—
319#102—62no—
320C22⋊C8D1212yes—
321C22⋊C8D1212yes—
322C22⋊C8D1212yes—
323C4×C8C22×D311yescharacteristic
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C4×C8C22×D311yescharacteristic
327C4×C8C22×D311yescharacteristic
328C4×C8C22×D311yescharacteristic
329C23×C3#12711yescharacteristic
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4#12711yescharacteristic
337C2×C3×C4#12711yescharacteristic
338C2×C3×C4#12711yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342#67—82no—
343#67—82no—
344#67—82no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—82no—
351D12—42no—
352D12—42no—
353D12—42no—
354D12—42no—
355D12—42no—
356D12—42no—
357D12—42no—
358D12—42no—
359D12—42no—
360C4×D3—82no—
361C4×D3—82no—
362C4×D3—82no—
363Dic6—42no—
364Dic6—42no—
365Dic6—42no—
366Dic6—42no—
367Dic6—42no—
368Dic6—82no—
369C3×C8—22no—
370C3×C8—22no—
371C3×C8—22no—
372C3×C8—22no—
373C3×C8—22no—
374C3×C8—22no—
375#41—242no—
376#41—242no—
377#41—242no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—121no—
384C2×Q8—242no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—121no—
395C2×D4—121no—
396C2×D4—121no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4C2×D1211yescharacteristic
402C22×C4C2×D1211yescharacteristic
403C22×C4C2×D1211yescharacteristic
404SD16—242no—
405SD16—242no—
406SD16—242no—
407SD16—242no—
408SD16—242no—
409D8—242no—
410D8—242no—
411D8—242no—
412C2×C8C2×D1212yes—
413C2×C8C2×D1212yes—
414C2×C8C2×D1212yes—
415C2×C8C2×D1212yes—
416C2×C8C2×D1212yes—
417C2×C8C2×D1212yes—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—121no—
429C4⋊C4—121no—
430C22⋊C4—121no—
431C22⋊C4—121no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42C2×D1211yescharacteristic
437C42C2×D1211yescharacteristic
438C42C2×D1211yescharacteristic
439C42C23×D311yescharacteristic
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3C2×#12711yescharacteristic
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C2×D3—81no—
449C2×D3—81no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—81no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4#57612yes—
457C3×C4#57712yes—
458C3×C4#57712yes—
459Dic3—81no—
460Dic3—81no—
461Dic3—81no—
462Dic3—81no—
463Dic3—81no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23#87411yescharacteristic
468Q8—122no—
469Q8—122no—
470Q8—122no—
471Q8—122no—
472Q8—122no—
473Q8—242no—
474D4—122no—
475D4—122no—
476D4—122no—
477D4—122no—
478D4—122no—
479D4—122no—
480D4—122no—
481D4—122no—
482D4—122no—
483D4—242no—
484D4—242no—
485D4—242no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—242no—
495C2×C4—242no—
496C2×C4—242no—
497C2×C4—242no—
498C2×C4#87411yescharacteristic
499C2×C4#87411yescharacteristic
500C2×C4#87411yescharacteristic
501C2×C4C22×D1211yescharacteristic
502C2×C4C22×D1211yescharacteristic
503C2×C4C22×D1211yescharacteristic
504C8—22no—
505C8—22no—
506C8—22no—
507C8—22no—
508C8—22no—
509C8—22no—
510C2×C3—41no—
511C2×C3#315411yescharacteristic
512C2×C3#315411yescharacteristic
513C2×C3#315511yescharacteristic
514D3—161no—
515D3—161no—
516D3—161no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22—241no—
525C22—241no—
526C22—241no—
527C22—241no—
528C22—241no—
529C22C2×#87411yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—241no—
534C4—241no—
535C4—241no—
536C4—241no—
537C4—241no—
538C4#575112yes—
539C4#575112yes—
540C4#575212yes—
541C3#1974211yescharacteristic, complemented, Sylow
542C2—41no—
543C2—481no—
544C2—481no—
545C2—481no—
546C2#7856411yescharacteristic
547C2#7856411yescharacteristic
548C2#7856511yescharacteristic
subgroupG/H
centerC22C2×#874
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1454C2
socleC22×C3C2×#127
hypercenter#1454D3
hypocenterC3#19742
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1454

Complements

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

NH ≅ G/Ntype
1. #79312C2semidirect
2. #79304C2semidirect
3. #79304C2semidirect
4. #79304C2semidirect
5. #79269C2semidirect
6. #79268C2semidirect
7. C3×#1454C2semidirect
8. #68096C2semidirect
9. #68094C2semidirect
10. #68093C2semidirect
11. #68092C2semidirect
12. #68091C2semidirect
13. #68090C2semidirect
14. #68059C2semidirect
15. #68058C2semidirect
78. #1454D3semidirect
541. C3#19742semidirect

Maximal subgroups

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

subgroupindex
#793122
#793042
#792692
#792682
C3×#14542
#680962
#680942
#680932
#680922
#680912
#680902
#680592
#680582
#197423