Finite Group Databasebeta

← #10819946

Subgroups — #10819946

Order 768.

960 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
1#79168C211yesmaximal, characteristic, complemented
2#79167C211yesmaximal, characteristic, complemented
3#78572C211yesmaximal, characteristic, complemented
4#78561C211yesmaximal, characteristic, complemented
5C2×#4712C211yesmaximal, characteristic
6C3×#1773C211yesmaximal, characteristic, complemented
7#70709C211yesmaximal, characteristic, complemented
8#70708C211yesmaximal, characteristic, complemented
9#69903C211yesmaximal, characteristic, complemented
10#69902C211yesmaximal, characteristic, complemented
11#69829C211yesmaximal, characteristic, complemented
12#68766C211yesmaximal, characteristic, complemented
13#68760C211yesmaximal, characteristic, complemented
14#68760C211yesmaximal, characteristic, complemented
15#68759C211yesmaximal, characteristic, complemented
16#23202—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21#5521C2211yescharacteristic
22#5521C2211yescharacteristic
23#5521C2211yescharacteristic
24#5516C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×#874C2211yescharacteristic
27C2×C3×#108C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#405C2211yescharacteristic
30C3×#405C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×#824C2211yescharacteristic
33#5131C2211yescharacteristic
34#5131C2211yescharacteristic
35#5131C2211yescharacteristic
36C2×#823C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#823C2211yescharacteristic
39#5120C2211yescharacteristic
40C2×#819C2211yescharacteristic
41#5106C2211yescharacteristic
42#5101C2211yescharacteristic
43#5101C2211yescharacteristic
44#5004C2211yescharacteristic
45#5001C2211yescharacteristic
46#5001C2211yescharacteristic
47#5001C2211yescharacteristic
48#4889—22no—
49#4872—22no—
50#4869—22no—
51#4852—22no—
52#4712C418yescomplemented
53#4711—22no—
54#4710—22no—
55C4×D24—22no—
56C4×#204—22no—
57C3×#339C2211yescharacteristic
58C3×#339C2211yescharacteristic
59#4523C2211yescharacteristic
60#4523C2211yescharacteristic
61C2×#496—31no—
62#2949—31no—
63#2949—31no—
64#2948—31no—
65#2948—31no—
66#1978—31no—
67#1978—31no—
68#1977—31no—
69#1977—31no—
70#1888—31no—
71#1887—31no—
72#1887—31no—
73#1885—31no—
74#1773—31no—
75#1773D311yescharacteristic, complemented
76C22×D12—41no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C2×C3×M16C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#117C2311yescharacteristic
86C3×#117C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—41no—
89#926C2311yescharacteristic
90#926C2311yescharacteristic
91C2×#208C2311yescharacteristic
92C2×#208C2311yescharacteristic
93C2×D24—21no—
94C2×D24—21no—
95C2×D24—21no—
96C2×D24—21no—
97C2×D24—22no—
98C2×D24—22no—
99C2×D24C2×C414yescomplemented
100C2×D24C2×C414yescomplemented
101C2×#204—21no—
102C2×#204—21no—
103C2×#204—21no—
104C2×#204—21no—
105C2×#204—22no—
106C2×#204—22no—
107C2×#204C2×C414yescomplemented
108C2×#204C2×C414yescomplemented
109#893—22no—
110#893—22no—
111#893—22no—
112#887—22no—
113D3×C22⋊C4—41no—
114D3×C22⋊C4—41no—
115D3×C22⋊C4—41no—
116#877—41no—
117#875—41no—
118#875C2×C414yescomplemented
119#874—41no—
120#874C2×C414yescomplemented
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×Dic6—22no—
125C3×#108—22no—
126C3×#107D412yescomplemented
127C3×#106D412yescomplemented
128C3×#97—21no—
129C3×#97—21no—
130C3×#97C2×C414yescomplemented
131C3×C4×C8—21no—
132C3×C4×C8—21no—
133#824—41no—
134#823—41no—
135#823—41no—
136#823—41no—
137#821—22no—
138#821—22no—
139#821—22no—
140#819—22no—
141#818D412yescomplemented
142#817D412yescomplemented
143#816—22no—
144#815—21no—
145#815—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148#810—22no—
149#809—22no—
150#809—22no—
151#809—22no—
152C22×SD16—31no—
153C22×SD16—31no—
154C22×D8—31no—
155C22×D8—31no—
156C2×#127—31no—
157C2×#124—31no—
158#519—31no—
159#518—31no—
160#518—31no—
161#518—31no—
162#496—38no—
163#442—62no—
164#442—62no—
165#440—62no—
166#440—62no—
167C4×SD16—62no—
168C4×SD16—62no—
169C4×D8—62no—
170C4×D8—62no—
171C2×#108—31no—
172C2×#108C2×D311yescharacteristic
173#427—31no—
174#427C2×D311yescharacteristic
175#419—31no—
176#419—31no—
177#418—31no—
178#418—31no—
179#418—31no—
180#418—31no—
181#418—31no—
182#418—31no—
183C2×#101—31no—
184C2×#100—31no—
185C2×#100—31no—
186C2×#100—31no—
187#405—31no—
188#405—31no—
189#405C2×D311yescharacteristic
190#405C2×D311yescharacteristic
191C2×#97C2×D311yescharacteristic
192#339—31no—
193#339—31no—
194#339C2×D311yescharacteristic
195#339C2×D311yescharacteristic
196C23×D3—21no—
197C23×D3—21no—
198C23×D3—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
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×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—44no—
218C2×D12—84no—
219C2×D12C22×C412yes—
220C2×D12C22×C412yes—
221C2×D12C22×C412yes—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6C22×C412yes—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C22×C412yes—
238C2×C3×C8C22×C412yes—
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8#4112yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42C22×C412yes—
253#217—21no—
254#217—21no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3C2×D412yes—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265#208—21no—
266#208—21no—
267#208—21no—
268#208—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271D24—24no—
272D24—24no—
273D24—24no—
274D24—24no—
275#204—24no—
276#204—24no—
277#204—24no—
278#204—24no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—31no—
283C22×D4—121no—
284C2×SD16—34no—
285C2×SD16—34no—
286C2×SD16—61no—
287C2×SD16—61no—
288C2×SD16—62no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—62no—
292C2×D8—34no—
293C2×D8—34no—
294C2×D8—61no—
295C2×D8—61no—
296C2×D8—62no—
297C2×D8—61no—
298C2×D8—61no—
299C2×D8—62no—
300C2×M16—31no—
301C2×M16—31no—
302C2×M16C22×D311yescharacteristic
303C2×M16C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306#127—34no—
307#127—121no—
308#124—34no—
309#124—121no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318#117—31no—
319#117—31no—
320#117—121no—
321#117C22×D311yescharacteristic
322#117C22×D311yescharacteristic
323C2×C22⋊C4—121no—
324C2×C22⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C22⋊C4—121no—
327C2×C42C22×D311yescharacteristic
328#108—22no—
329#108—62no—
330#107—32no—
331#107D1212yescomplemented
332#106—32no—
333#106D1212yescomplemented
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#101—121no—
343#100—121no—
344#100—121no—
345#100—121no—
346#97—21no—
347#97—21no—
348#97—61no—
349#97—61no—
350#97C4×D314yescomplemented
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—61no—
354C4×C8—61no—
355C23×C3H(5,𝔽2)11yescharacteristic
356C22×D3—22no—
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—82no—
366C22×D3—82no—
367C22×D3—82no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—42no—
372C2×C3×C4C23×C411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—22no—
384D12—22no—
385D12—22no—
386D12—22no—
387D12—22no—
388D12—22no—
389D12—41no—
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394D12—41no—
395D12—88no—
396C4×D3—162no—
397C4×D3—162no—
398C4×D3—162no—
399Dic6—22no—
400Dic6—22no—
401Dic6—41no—
402Dic6—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8C4×D412yes—
406C3×C8C4×D412yes—
407C3×C8C4×D412yes—
408C3×C8C4×D412yes—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C24—61no—
412C24—61no—
413C24—61no—
414C2×Q8—32no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—32no—
421C2×D4—32no—
422C2×D4—32no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—124no—
439C2×D4—244no—
440C22×C4—21no—
441C22×C4—61no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4C23×D311yescharacteristic
446SD16—64no—
447SD16—64no—
448SD16—64no—
449SD16—64no—
450D8—64no—
451D8—64no—
452D8—64no—
453D8—64no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—61no—
459M16—61no—
460M16—61no—
461M16—61no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8C2×C4×D312yes—
467C2×C8C2×C4×D312yes—
468C2×C8C2×D1211yescharacteristic
469C2×C8C2×D1211yescharacteristic
470C2×C8#23512yes—
471C2×C8D3×D411yescharacteristic
472C2×C8D3×D411yescharacteristic
473C2×C8#23912yes—
474C4⋊C4—32no—
475C4⋊C4—32no—
476C4⋊C4—122no—
477C4⋊C4C2×D1212yes—
478C4⋊C4C2×D1212yes—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—61no—
482C22⋊C4—61no—
483C22⋊C4—124no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—61no—
491C42—61no—
492C42—122no—
493C42C2×C4×D312yes—
494C22×C3—22no—
495C22×C3#51811yescharacteristic
496C22×C3D4211yescharacteristic
497C22×C3#55911yescharacteristic
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—82no—
502C2×D3—82no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—161no—
508C2×D3—161no—
509C2×D3—161no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4C2×C4×D411yescharacteristic
514C3×C4C2×C4×D411yescharacteristic
515C3×C4#55011yescharacteristic
516C3×C4#55011yescharacteristic
517Dic3—42no—
518Dic3—42no—
519Dic3—42no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—242no—
530C23—242no—
531C23—242no—
532C23#100911yescharacteristic
533Q8—62no—
534Q8—62no—
535Q8—121no—
536Q8—121no—
537D4—62no—
538D4—62no—
539D4—62no—
540D4—62no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545D4—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—248no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—482no—
560C2×C4—482no—
561C2×C4—482no—
562C2×C4C22×C4×D311yescharacteristic
563C2×C4C22×D1211yescharacteristic
564C2×C4C2×#23511yescharacteristic
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×#23911yescharacteristic
567C2×C4#100911yescharacteristic
568C8—41no—
569C8—41no—
570C8—121no—
571C8—121no—
572C8C4×D1212yes—
573C8C4×D1212yes—
574C8#89312yes—
575C8#89312yes—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3#236611yescharacteristic
579C2×C3#328812yes—
580D3—82no—
581D3—82no—
582D3—82no—
583C22—22no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22—242no—
588C22—242no—
589C22—242no—
590C22—242no—
591C22—242no—
592C22—242no—
593C22—481no—
594C22—481no—
595C22—481no—
596C22#552111yescharacteristic
597C22D4×D1211yescharacteristic
598C22#558511yescharacteristic
599C4—42no—
600C4—42no—
601C4—42no—
602C4—122no—
603C4—122no—
604C4—122no—
605C4C2×C4×D1211yescharacteristic
606C4C2×#89311yescharacteristic
607C4#555311yescharacteristic
608C4#557711yescharacteristic
609C3#2320211yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—242no—
613C2—242no—
614C2—242no—
615C2#7365311yescharacteristic
616C2#7990012yes—
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3D42
prime radicalC2×#4712C2
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#23202
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1773

Complements

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

NH ≅ G/Ntype
1. #79168C2semidirect
2. #79167C2semidirect
3. #78572C2semidirect
4. #78561C2semidirect
6. C3×#1773C2semidirect
7. #70709C2semidirect
8. #70708C2semidirect
9. #69903C2semidirect
10. #69902C2semidirect
11. #69829C2semidirect
12. #68766C2semidirect
13. #68760C2semidirect
14. #68760C2semidirect
15. #68759C2semidirect
52. #4712C4semidirect
75. #1773D3semidirect
99. C2×D24C2×C4semidirect
100. C2×D24C2×C4semidirect
107. C2×#204C2×C4semidirect
108. C2×#204C2×C4semidirect
118. #875C2×C4semidirect
120. #874C2×C4semidirect
126. C3×#107D4semidirect
127. C3×#106D4semidirect
130. C3×#97C2×C4semidirect
141. #818D4semidirect
142. #817D4semidirect
331. #107D12semidirect
333. #106D12semidirect
350. #97C4×D3semidirect
609. C3#23202semidirect

Maximal subgroups

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

subgroupindex
#791682
#791672
#785722
#785612
C2×#47122
C3×#17732
#707092
#707082
#699032
#699022
#698292
#687662
#687602
#687592
#232023