Finite Group Databasebeta

← #10782871

Subgroups — #10782871

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81598C211yesmaximal, characteristic, complemented
2#81593C211yesmaximal, characteristic, complemented
3#81592C211yesmaximal, characteristic, complemented
4#81562C211yesmaximal, characteristic, complemented
5#81557C211yesmaximal, characteristic
6#81556C211yesmaximal, characteristic, complemented
7C3×#1711C211yesmaximal, characteristic, complemented
8#71214C211yesmaximal, characteristic, complemented
9#71213C211yesmaximal, characteristic, complemented
10#71212C211yesmaximal, characteristic, complemented
11#71182C211yesmaximal, characteristic, complemented
12#71181C211yesmaximal, characteristic, complemented
13#71180C211yesmaximal, characteristic, complemented
14#68061C211yesmaximal, characteristic, complemented
15D3×C82C211yesmaximal, characteristic, complemented
16#19600—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19#5725C2211yescharacteristic
20#5716C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#486C2211yescharacteristic
27#5093C2211yescharacteristic
28#5092C2211yescharacteristic
29#5088C2211yescharacteristic
30#5085C2211yescharacteristic
31#5080C2211yescharacteristic
32#5080C2211yescharacteristic
33#5079C2211yescharacteristic
34#5078C2211yescharacteristic
35#5078C2211yescharacteristic
36#5076C2211yescharacteristic
37#5073C2211yescharacteristic
38#5070C2211yescharacteristic
39#4813—22no—
40#4807—22no—
41#4804—22no—
42#4800—22no—
43#4790—22no—
44#4788—22no—
45#4778—22no—
46#4775—22no—
47#4772—22no—
48#4768—22no—
49#4758—22no—
50#4756—22no—
51#4694C2211yescharacteristic
52#4694C2211yescharacteristic
53#4694C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55C4×D3×C8C2211yescharacteristic
56C4×D3×C8C2211yescharacteristic
57#4681C2211yescharacteristic
58#4681C2211yescharacteristic
59#4681C2211yescharacteristic
60C3×C82C2211yescharacteristic
61C8×C3⋊C8C2211yescharacteristic
62C8×C3⋊C8C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3170—31no—
65#3169—31no—
66#3167—31no—
67#3167—31no—
68#3131—31no—
69#3131—31no—
70#1711—31no—
71#1711D311yescharacteristic, complemented
72#1710—31no—
73#1709—31no—
74#1708—31no—
75#1708—31no—
76#1707—31no—
77#1706—31no—
78C2×C82—31no—
79C2×#239—21no—
80C2×#237—21no—
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#127C2311yescharacteristic
85#947—21no—
86#946—21no—
87#946—21no—
88Q8×Dic3—21no—
89Q8×Dic3—21no—
90C2×#216—22no—
91C2×#215—22no—
92#938—21no—
93#938—21no—
94#936—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#214—22no—
98C2×#213—22no—
99C2×D3×C8D412yes—
100C2×D3×C8D412yes—
101C2×D3×C8D412yes—
102#897—21no—
103#897—21no—
104#895—21no—
105#895—21no—
106#893—21no—
107#893—21no—
108#892—21no—
109#892—21no—
110#890—21no—
111#890—21no—
112#887—21no—
113#887—21no—
114#874C2311yescharacteristic
115C42×D3C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128#835—22no—
129#832—22no—
130#821—22no—
131#821—22no—
132#821—22no—
133#816—22no—
134#816—22no—
135#816—22no—
136C8×Dic3D412yes—
137C8×Dic3D412yes—
138C8×Dic3D412yes—
139#810—22no—
140#809—22no—
141#808—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533—31no—
149#533—31no—
150#532—31no—
151#532—31no—
152#500—31no—
153#500—31no—
154#498—31no—
155#498—31no—
156#494—31no—
157#493—31no—
158#492—31no—
159#492—31no—
160#488—62no—
161#488—62no—
162#487—31no—
163#487—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487C2×D311yescharacteristic
168#487C2×D311yescharacteristic
169#487C2×D311yescharacteristic
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486C2×D311yescharacteristic
176#486C2×D311yescharacteristic
177#486C2×D311yescharacteristic
178#467—62no—
179#466—62no—
180#465—62no—
181#465—62no—
182C4×Q16—62no—
183C4×SD16—62no—
184C4×SD16—62no—
185C4×D8—62no—
186#416—62no—
187#416—62no—
188C2×C4×C8—31no—
189C2×C4×C8—31no—
190C2×C4×C8—31no—
191C82—31no—
192C82—31no—
193C82—31no—
194C82C2×D311yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×#67—41no—
202C22×Dic3—41no—
203#239—44no—
204#237—44no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—41no—
209C2×C4×D3—41no—
210C2×C4×D3C2×D411yescharacteristic
211C2×C4×D3C2×D411yescharacteristic
212C2×C4×D3C2×D411yescharacteristic
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×C3×C8C2×D412yes—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C42C2411yescharacteristic
226#217—41no—
227#217—41no—
228#216—44no—
229#215—44no—
230#214—44no—
231#213—44no—
232(C2×D3)⋊C4—41no—
233(C2×D3)⋊C4—41no—
234(C2×D3)⋊C4—41no—
235(C2×D3)⋊C4—41no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—41no—
243C4⋊Dic3—41no—
244Dic3⋊C4—41no—
245Dic3⋊C4—41no—
246C4×Dic3—41no—
247C4×Dic3—41no—
248C4×Dic3C2×D411yescharacteristic
249C4×Dic3C2×D411yescharacteristic
250C4×Dic3C2×D411yescharacteristic
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254D3×C8—24no—
255D3×C8—24no—
256D3×C8—24no—
257C2×#41—61no—
258C2×#41—61no—
259C2×Q16—62no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×D8—62no—
263C22×C8—32no—
264C22×C8—32no—
265C22×C8—32no—
266C4⋊Q8—31no—
267C4⋊Q8—31no—
268C4⋊Q8—31no—
269C4⋊Q8C22×D311yescharacteristic
270C4⋊Q8C22×D311yescharacteristic
271C4⋊Q8C22×D311yescharacteristic
272#127—31no—
273#127C22×D311yescharacteristic
274#125—61no—
275#125—61no—
276#124—61no—
277#124—61no—
278C22⋊Q8—61no—
279C22⋊Q8—61no—
280C22⋊Q8—61no—
281C22⋊Q8—61no—
282#121—61no—
283#121—61no—
284#121—61no—
285#121—61no—
286C4×Q8—61no—
287C4×Q8—61no—
288C4×Q8—61no—
289C4×Q8—61no—
290C4×D4—61no—
291C4×D4—61no—
292C4×D4—61no—
293C4×D4—61no—
294#117—61no—
295#117—61no—
296C2×C42—31no—
297#107—62no—
298#107—62no—
299#106—62no—
300#106—62no—
301#103—22no—
302#103—22no—
303#103—22no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317C4×C8—32no—
318C4×C8—32no—
319C4×C8—32no—
320C4×C8—31no—
321C4×C8—31no—
322C4×C8—31no—
323C4×C8C22×D311yescharacteristic
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C23×C3—41no—
327C22×D3—41no—
328C22×D3#12711yescharacteristic
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341#67—82no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—82no—
346C2×Dic3#12711yescharacteristic
347C2×Dic3#12711yescharacteristic
348C2×Dic3#12711yescharacteristic
349D12—42no—
350D12—42no—
351D12—42no—
352D12—82no—
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—82no—
356C4×D3#13514yes—
357Dic6—42no—
358Dic6—42no—
359Dic6—42no—
360C3×C8—22no—
361C3×C8—22no—
362C3×C8—22no—
363C3⋊C8—22no—
364C3⋊C8#13514yes—
365C3⋊C8#13514yes—
366#41—124no—
367#41—124no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—121no—
381C2×D4—121no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—121no—
386C22×C4—121no—
387Q16—124no—
388SD16—124no—
389SD16—124no—
390D8—124no—
391C2×C8—32no—
392C2×C8—32no—
393C2×C8—32no—
394C2×C8—64no—
395C2×C8—64no—
396C2×C8—64no—
397C2×C8D3×D412yes—
398C2×C8D3×D412yes—
399C2×C8D3×D412yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—121no—
413C4⋊C4—121no—
414C4⋊C4—121no—
415C4⋊C4—121no—
416C22⋊C4—121no—
417C22⋊C4—121no—
418C22⋊C4—121no—
419C22⋊C4—121no—
420C22⋊C4—121no—
421C22⋊C4—121no—
422C42—31no—
423C42—31no—
424C42—31no—
425C42—121no—
426C42—121no—
427C42C23×D311yescharacteristic
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3C2×#12711yescharacteristic
432C2×D3—21no—
433C2×D3—21no—
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3#49512yes—
438C3×C4—81no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4C2×#13512yes—
442C3×C4C2×#13512yes—
443C3×C4C2×#13512yes—
444Dic3—21no—
445Dic3—21no—
446Dic3—81no—
447Dic3—81no—
448Dic3—81no—
449Dic3#49512yes—
450C23—31no—
451C23—41no—
452C23—121no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—122no—
457Q8—122no—
458Q8—122no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—122no—
463D4—122no—
464D4—122no—
465D4—242no—
466D4—242no—
467C2×C4—31no—
468C2×C4—34no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—242no—
480C2×C4—242no—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C8—22no—
485C8—22no—
486C8—22no—
487C8—34no—
488C8—34no—
489C8—62no—
490C2×C3—161no—
491C2×C3C2×#49511yescharacteristic
492C2×C3#314711yescharacteristic
493C2×C3#314711yescharacteristic
494D3—22no—
495D3—161no—
496C22—32no—
497C22—61no—
498C22—61no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—241no—
503C22—241no—
504C22—241no—
505C22D3×#12711yescharacteristic
506C4—32no—
507C4—61no—
508C4—61no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—241no—
513C4—241no—
514C4—241no—
515C4D3×#13512yes—
516C4D3×#13512yes—
517C4D3×#13512yes—
518C3#1960011yescharacteristic, complemented, Sylow
519C2—62no—
520C2—161no—
521C2—481no—
522C2D3×#49511yescharacteristic
523C2#8182011yescharacteristic
524C2#8182011yescharacteristic
subgroupG/H
centerC22D3×#127
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1711C2
socleC22×C3C2×#127
prime radical#81557C2
hypercenter#1711D3
hypocenterC3#19600
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #81598C2semidirect
2. #81593C2semidirect
3. #81592C2semidirect
4. #81562C2semidirect
6. #81556C2semidirect
7. C3×#1711C2semidirect
8. #71214C2semidirect
9. #71213C2semidirect
10. #71212C2semidirect
11. #71182C2semidirect
12. #71181C2semidirect
13. #71180C2semidirect
14. #68061C2semidirect
15. D3×C82C2semidirect
71. #1711D3semidirect
518. C3#19600semidirect

Maximal subgroups

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

subgroupindex
#815982
#815932
#815922
#815622
#815572
#815562
C3×#17112
#712142
#712132
#712122
#711822
#711812
#711802
#680612
D3×C822
#196003