Finite Group Databasebeta

← #10784380

Subgroups — #10784380

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79313C211yesmaximal, characteristic, complemented
2#79306C211yesmaximal, characteristic, complemented
3#79304C211yesmaximal, characteristic
4#79270C211yesmaximal, characteristic, complemented
5#79269C211yesmaximal, characteristic, complemented
6#79268C211yesmaximal, characteristic, complemented
7C3×#1454C211yesmaximal, characteristic, complemented
8#68099C211yesmaximal, characteristic, complemented
9#68098C211yesmaximal, characteristic, complemented
10#68097C211yesmaximal, characteristic, complemented
11#68097C211yesmaximal, characteristic, complemented
12#68092C211yesmaximal, characteristic, complemented
13#68090C211yesmaximal, characteristic, complemented
14#68063C211yesmaximal, characteristic, complemented
15#68060C211yesmaximal, characteristic, complemented
16#19741—31nomaximal, Sylow
17#5477C2211yescharacteristic
18#5477C2211yescharacteristic
19#5470C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#431C2211yescharacteristic
24#4849—22no—
25#4847—22no—
26#4846—22no—
27#4845—22no—
28#4844—22no—
29#4842—22no—
30#4739—22no—
31#4737—22no—
32#4736—22no—
33#4735—22no—
34#4734—22no—
35#4733—22no—
36#4699C2211yescharacteristic
37#4699C2211yescharacteristic
38#4699C2211yescharacteristic
39#4695C2211yescharacteristic
40#4694C2211yescharacteristic
41#4694C2211yescharacteristic
42#4694C2211yescharacteristic
43#4694C2211yescharacteristic
44#4694C2211yescharacteristic
45#4694C2211yescharacteristic
46#4694C2211yescharacteristic
47#4694C2211yescharacteristic
48#4693C2211yescharacteristic
49#4693C2211yescharacteristic
50#4682C2211yescharacteristic
51#4682C2211yescharacteristic
52#4682C2211yescharacteristic
53#4681C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4681C2211yescharacteristic
57#4680C2211yescharacteristic
58#4680C2211yescharacteristic
59#4680C2211yescharacteristic
60C3×#322C2211yescharacteristic
61C3×#322C2211yescharacteristic
62C3×#322C2211yescharacteristic
63C3×C82C2211yescharacteristic
64#3228—31no—
65#3227—31no—
66#3226—31no—
67#3210—31no—
68#3209—31no—
69#3207—31no—
70#1723—31no—
71#1722—31no—
72#1720—31no—
73#1720—31no—
74#1717—31no—
75#1714—31no—
76#1709—31no—
77#1707—31no—
78#1454D311yescharacteristic, complemented
79C2×#235—21no—
80C2×#235—21no—
81C22×Dic6—21no—
82C2×C3×C42C2311yescharacteristic
83#930—21no—
84#930—21no—
85#930—21no—
86#930—21no—
87#926—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90#924—21no—
91#924—21no—
92C2×Dic12—22no—
93C2×Dic12—22no—
94C2×Dic12—22no—
95C2×D24—22no—
96C2×#204—22no—
97C2×#204—22no—
98#875—21no—
99#875—21no—
100#874C2311yescharacteristic
101#874C2311yescharacteristic
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106#870—21no—
107#869—21no—
108#869—21no—
109#869—21no—
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6—21no—
118C3×C4⋊C8D412yes—
119C3×C4⋊C8D412yes—
120C3×C4⋊C8D412yes—
121C3×C22⋊C8D412yes—
122C3×C22⋊C8D412yes—
123C3×C22⋊C8D412yes—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#821—22no—
131#821—22no—
132#821—22no—
133#821—22no—
134#821—22no—
135#821—22no—
136#821—22no—
137#821—22no—
138#818—22no—
139#818—22no—
140#818—22no—
141#817—22no—
142#817—22no—
143#817—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148#533—31no—
149#532—31no—
150#532—31no—
151C2×C4⋊Q8—31no—
152#500—31no—
153#500—31no—
154#500—31no—
155#498—31no—
156#498—31no—
157#498—31no—
158#494—31no—
159#494—31no—
160#494—31no—
161#493—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#486—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#482—62no—
177#481—62no—
178#480—62no—
179#478—62no—
180#476—62no—
181#474—62no—
182#464—62no—
183#463—62no—
184#462—62no—
185#452—62no—
186#451—62no—
187#449—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—
200C22×Dic3—41no—
201#235—82no—
202#235—82no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×C4×D3—41no—
210C2×C4×D3—41no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—41no—
218C2×Dic6—82no—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C2×C3×C8C2×D412yes—
224C2×C3×C8C2×D412yes—
225C3×C42C2×D411yescharacteristic
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2411yescharacteristic
229#217—41no—
230(C2×D3)⋊C4—41no—
231(C2×D3)⋊C4—41no—
232(C2×D3)⋊C4—41no—
233(C2×D3)⋊C4—41no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—41no—
247C4⋊Dic3—41no—
248Dic3⋊C4—41no—
249Dic3⋊C4—41no—
250C4×Dic3—41no—
251Dic12—82no—
252Dic12—82no—
253Dic12—82no—
254D24—82no—
255#204—82no—
256#204—82no—
257C2×#41—61no—
258C2×#41—61no—
259C22×Q8—61no—
260C2×Q16—62no—
261C2×Q16—62no—
262C2×Q16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C2×D8—62no—
266C4⋊Q8—31no—
267C4⋊Q8—31no—
268C4⋊Q8—31no—
269C4⋊Q8—31no—
270C4⋊Q8—31no—
271C4⋊Q8—31no—
272C4⋊Q8—61no—
273C4⋊Q8—61no—
274C4⋊Q8—61no—
275#127—31no—
276#127—31no—
277#125—61no—
278#124—61no—
279#124—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×D4—61no—
289C4×D4—61no—
290C4×D4—61no—
291C4×D4—61no—
292#117—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C42C22×D311yescharacteristic
296#107—62no—
297#107—62no—
298#107—62no—
299#106—62no—
300#106—62no—
301#106—62no—
302C4⋊C8D1212yes—
303C4⋊C8D1212yes—
304C4⋊C8D1212yes—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317C22⋊C8D1212yes—
318C22⋊C8D1212yes—
319C22⋊C8D1212yes—
320C4×C8C22×D311yescharacteristic
321C4×C8C22×D311yescharacteristic
322C4×C8C22×D311yescharacteristic
323C4×C8C22×D311yescharacteristic
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C23×C3#12711yescharacteristic
327C22×D3—41no—
328C22×D3—41no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4#12711yescharacteristic
333C2×C3×C4#12711yescharacteristic
334C2×C3×C4#12711yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338#67—82no—
339#67—82no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—82no—
347D12—42no—
348D12—42no—
349D12—42no—
350D12—42no—
351D12—42no—
352D12—42no—
353C4×D3—82no—
354C4×D3—82no—
355Dic6—42no—
356Dic6—42no—
357Dic6—42no—
358Dic6—42no—
359Dic6—42no—
360Dic6—42no—
361Dic6—82no—
362C3×C8—22no—
363C3×C8—22no—
364C3×C8—22no—
365C3×C8—22no—
366C3×C8—22no—
367C3×C8—22no—
368#41—242no—
369#41—242no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—121no—
377C2×Q8—242no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—121no—
385C2×D4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4C2×D1211yescharacteristic
390C22×C4C2×D1211yescharacteristic
391C22×C4C2×D1211yescharacteristic
392Q16—242no—
393Q16—242no—
394Q16—242no—
395SD16—242no—
396SD16—242no—
397D8—242no—
398C2×C8C2×D1212yes—
399C2×C8C2×D1212yes—
400C2×C8C2×D1212yes—
401C2×C8C2×D1212yes—
402C2×C8C2×D1212yes—
403C2×C8C2×D1212yes—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—121no—
417C4⋊C4—121no—
418C4⋊C4—121no—
419C4⋊C4—121no—
420C22⋊C4—121no—
421C22⋊C4—121no—
422C22⋊C4—121no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C42—121no—
426C42C2×D1211yescharacteristic
427C42C2×D1211yescharacteristic
428C42C2×D1211yescharacteristic
429C42C23×D311yescharacteristic
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3C2×#12711yescharacteristic
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4#57612yes—
444C3×C4#57712yes—
445C3×C4#57812yes—
446Dic3—81no—
447Dic3—81no—
448Dic3—81no—
449Dic3—81no—
450Dic3—81no—
451Dic3—81no—
452C23—121no—
453C23—121no—
454C23#87411yescharacteristic
455Q8—122no—
456Q8—122no—
457Q8—122no—
458Q8—122no—
459Q8—122no—
460Q8—122no—
461Q8—242no—
462D4—122no—
463D4—122no—
464D4—122no—
465D4—122no—
466D4—122no—
467D4—122no—
468D4—242no—
469D4—242no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—242no—
480C2×C4—242no—
481C2×C4—242no—
482C2×C4#87411yescharacteristic
483C2×C4#87411yescharacteristic
484C2×C4#87411yescharacteristic
485C2×C4C22×D1211yescharacteristic
486C2×C4C22×D1211yescharacteristic
487C2×C4C22×D1211yescharacteristic
488C8—22no—
489C8—22no—
490C8—22no—
491C8—22no—
492C8—22no—
493C8—22no—
494C2×C3—41no—
495C2×C3#315411yescharacteristic
496C2×C3#315511yescharacteristic
497C2×C3#315611yescharacteristic
498D3—161no—
499D3—161no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—241no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22C2×#87411yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—241no—
514C4—241no—
515C4—241no—
516C4—241no—
517C4—241no—
518C4—241no—
519C4#575112yes—
520C4#575212yes—
521C4#575312yes—
522C3#1974111yescharacteristic, complemented, Sylow
523C2—41no—
524C2—481no—
525C2—481no—
526C2#7856411yescharacteristic
527C2#7856511yescharacteristic
528C2#7857611yescharacteristic
subgroupG/H
centerC22C2×#874
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1454C2
socleC22×C3C2×#127
prime radical#79304C2
hypercenter#1454D3
hypocenterC3#19741
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. #79313C2semidirect
2. #79306C2semidirect
4. #79270C2semidirect
5. #79269C2semidirect
6. #79268C2semidirect
7. C3×#1454C2semidirect
8. #68099C2semidirect
9. #68098C2semidirect
10. #68097C2semidirect
11. #68097C2semidirect
12. #68092C2semidirect
13. #68090C2semidirect
14. #68063C2semidirect
15. #68060C2semidirect
78. #1454D3semidirect
522. C3#19741semidirect

Maximal subgroups

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

subgroupindex
#793132
#793062
#793042
#792702
#792692
#792682
C3×#14542
#680992
#680982
#680972
#680922
#680902
#680632
#680602
#197413