Finite Group Databasebeta

← #10786133

Subgroups — #10786133

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3354C211yesmaximal, characteristic, complemented
2#83647C211yesmaximal, characteristic, complemented
3#83511C211yesmaximal, characteristic, complemented
4#79616C211yesmaximal, characteristic, complemented
5#71928C211yesmaximal, characteristic, complemented
6#71911C211yesmaximal, characteristic, complemented
7#71344C211yesmaximal, characteristic, complemented
8#71341C211yesmaximal, characteristic, complemented
9#71145C211yesmaximal, characteristic, complemented
10#71133C211yesmaximal, characteristic, complemented
11#71109C211yesmaximal, characteristic, complemented
12#71103C211yesmaximal, characteristic, complemented
13#70878C211yesmaximal, characteristic, complemented
14#70874C211yesmaximal, characteristic, complemented
15#70862C211yesmaximal, characteristic, complemented
16#19862—31nomaximal, Sylow
17C3×#555C2211yescharacteristic
18C3×#547C2211yescharacteristic
19#5563C2211yescharacteristic
20#5563C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#482—21no—
23C3×#481—21no—
24C3×#476C2211yescharacteristic
25C3×#474C2211yescharacteristic
26C3×#469—21no—
27C3×C4×D8C2211yescharacteristic
28C3×#436C2211yescharacteristic
29C3×#428—21no—
30#5238—21no—
31#5233—21no—
32#5157—21no—
33#5093C2211yescharacteristic
34#5092C2211yescharacteristic
35#5084C2211yescharacteristic
36#5083C2211yescharacteristic
37#5079C2211yescharacteristic
38#5078C2211yescharacteristic
39#5048—21no—
40#5045—21no—
41#5038—21no—
42#5035—21no—
43C4×#213C2211yescharacteristic
44D4×C3⋊C8C2211yescharacteristic
45D4×C3⋊C8C2211yescharacteristic
46#5008C2211yescharacteristic
47#5008C2211yescharacteristic
48#5008C2211yescharacteristic
49#5008C2211yescharacteristic
50#5003—21no—
51#4998—21no—
52#4880—21no—
53#4784—21no—
54#4784—21no—
55C8×D12C2211yescharacteristic
56#4682C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59D4⋊(C3⋊C8)C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4483C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4479C2211yescharacteristic
66#4457C2211yescharacteristic
67#4455C2211yescharacteristic
68#3354D311yescharacteristic, complemented
69#3341—31no—
70#3341—31no—
71#3303—31no—
72#1723—31no—
73#1695—31no—
74#1683—31no—
75#1675—31no—
76#1671—31no—
77#1646—31no—
78#1635—31no—
79#1634—31no—
80#1634—31no—
81#1581—31no—
82#1580—31no—
83C2×C3×#41—21no—
84C2×C3×D8D411yescharacteristic
85C2×C3×M16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#124—41no—
89C3×#123—41no—
90C3×C22⋊Q8—21no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#121—21no—
94C3×C4×D4C2311yescharacteristic
95C3×C4×D4C2311yescharacteristic
96C3×C4×D4C2311yescharacteristic
97C3×#117—21no—
98C3×#117—21no—
99C2×C3×C4⋊C4—21no—
100C2×#216—41no—
101C2×#215—21no—
102#938—21no—
103#938—21no—
104C2×#213D411yescharacteristic
105#930—21no—
106#930—21no—
107#926—21no—
108#926—21no—
109C22×C3⋊C8—21no—
110C22×C3⋊C8—21no—
111C2×D3×C8—21no—
112#892—21no—
113#892—21no—
114#885—41no—
115#885—41no—
116C4×D12C2311yescharacteristic
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119C3×#107—21no—
120C3×#107D411yescharacteristic
121C3×#106—21no—
122C3×#106—21no—
123C3×#106—21no—
124C3×C4⋊C8C2311yescharacteristic
125C3×#102—21no—
126C3×#102—21no—
127C3×#102D411yescharacteristic
128C3×#102D411yescharacteristic
129C3×C22⋊C8—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#835—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832D411yescharacteristic
139#830—21no—
140#830—21no—
141(C2×D3)⋊C8—21no—
142#818—21no—
143#818—21no—
144#810—41no—
145#809—21no—
146#809D411yescharacteristic
147#808—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#808—21no—
152#807—21no—
153#807—21no—
154#807D411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#555—31no—
161#555—31no—
162#555C2×D311yescharacteristic
163#547C2×D311yescharacteristic
164#500—31no—
165#500—31no—
166#498C2×D311yescharacteristic
167#494—31no—
168#487—31no—
169#487—31no—
170#485—61no—
171#482—21no—
172#482—61no—
173#482—61no—
174#482—61no—
175#482—61no—
176#481—21no—
177#476—31no—
178#476—31no—
179#476—31no—
180#476—31no—
181#476—31no—
182#476C2×D311yescharacteristic
183#474C2×D311yescharacteristic
184#469—21no—
185#466—61no—
186#466—61no—
187#465—61no—
188#461—31no—
189#449—61no—
190C4×D8—31no—
191C4×D8C2×D311yescharacteristic
192#436C2×D311yescharacteristic
193C8×D4—31no—
194C8×D4—31no—
195C8×D4—31no—
196#428—21no—
197#427—61no—
198#427—61no—
199#416—61no—
200#335—31no—
201#334—31no—
202#333—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206D4⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C3×#41—42no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×D4—41no—
216C2×C3×D4—41no—
217C2×C3×D4C2×D411yescharacteristic
218C2×C3×D4C2×D411yescharacteristic
219C2×C3×D4C2×D411yescharacteristic
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—41no—
224C2×#67—41no—
225C2×#67—41no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C3×D8—22no—
231C3×M16—42no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C42—41no—
250C3×C42C2411yescharacteristic
251#217—41no—
252#217—41no—
253#216—82no—
254#215—81no—
255#213—41no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—41no—
258(C2×D3)⋊C4—41no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D411yescharacteristic
264C4×Dic3—41no—
265C4×Dic3—41no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—42no—
271C2×C3⋊C8—81no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274D3×C8—42no—
275C2×#41—21no—
276C2×Q16—121no—
277C2×SD16—61no—
278C2×D8—31no—
279C2×D8#6711yescharacteristic
280C2×M16—21no—
281C22×C8—61no—
282C22×C8—61no—
283C22×C8—61no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288#124—41no—
289#124—121no—
290#124—121no—
291#123—41no—
292C22⋊Q8—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296#121—61no—
297#121—61no—
298#121—61no—
299#121—61no—
300C4×D4—31no—
301C4×D4C22×D311yescharacteristic
302C4×D4C22×D311yescharacteristic
303C4×D4C22×D311yescharacteristic
304#117—21no—
305#117—21no—
306#117—61no—
307#117—61no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—21no—
311#107—21no—
312#107—31no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#107#6711yescharacteristic
318#106—21no—
319#106—21no—
320#106—21no—
321#106—61no—
322#106—61no—
323#106—61no—
324#106—61no—
325#106—61no—
326C4⋊C8—31no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8C22×D311yescharacteristic
330#103—61no—
331#103—61no—
332#103—61no—
333#103—121no—
334#102—21no—
335#102—21no—
336#102—31no—
337#102—31no—
338#102—61no—
339#102—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102#6711yescharacteristic
344#102#6711yescharacteristic
345C22⋊C8—21no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348C22⋊C8—61no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8C22×D311yescharacteristic
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C22×D3—21no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—81no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—82no—
366C3×D4—161no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—161no—
383#67—161no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387D12—21no—
388D12—21no—
389C4×D3—41no—
390C4×D3—41no—
391Dic6—81no—
392Dic6—81no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8#13512yes—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402#41—42no—
403#41—81no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×D4—31no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—121no—
412C2×D4—121no—
413C2×D4D3×D411yescharacteristic
414C2×D4C2×#6711yescharacteristic
415C2×D4C2×#6711yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—61no—
421Q16—242no—
422SD16—241no—
423D8—22no—
424D8—121no—
425M16—42no—
426C2×C8—21no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—122no—
434C2×C8—122no—
435C2×C8—241no—
436C2×C8C2×#6711yescharacteristic
437C2×C8C2×#6711yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—31no—
443C4⋊C4—41no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4D3×D411yescharacteristic
449C4⋊C4C2×#6711yescharacteristic
450C4⋊C4C2×#6711yescharacteristic
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—61no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C42—41no—
463C42—121no—
464C42—121no—
465C42C23×D311yescharacteristic
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3D4211yescharacteristic
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—41no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C2×#13511yescharacteristic
486C3×C4#57511yescharacteristic
487C3×C4#57711yescharacteristic
488C3×C4#57811yescharacteristic
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—61no—
496Q8—41no—
497Q8—41no—
498Q8—81no—
499Q8—241no—
500Q8—241no—
501D4—21no—
502D4—21no—
503D4—21no—
504D4—21no—
505D4—21no—
506D4—21no—
507D4—61no—
508D4—61no—
509D4—82no—
510D4—161no—
511D4—481no—
512D4—481no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—61no—
525C2×C4—82no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4C2×D3×D411yescharacteristic
531C2×C4#100411yescharacteristic
532C2×C4C22×#6711yescharacteristic
533C8—21no—
534C8—41no—
535C8—61no—
536C8—61no—
537C8—121no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8#95012yes—
542C2×C3—81no—
543C2×C3—81no—
544C2×C3—81no—
545C2×C3#326711yescharacteristic
546C2×C3#328311yescharacteristic
547C2×C3#328811yescharacteristic
548D3—81no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22D4×#6711yescharacteristic
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—121no—
569C4—241no—
570C4—241no—
571C4#577311yescharacteristic
572C4#577511yescharacteristic
573C4C2×#95011yescharacteristic
574C4#583811yescharacteristic
575C3#1986211yescharacteristic, complemented, Sylow
576C2—81no—
577C2—81no—
578C2—81no—
579C2—241no—
580C2#8352411yescharacteristic
581C2#8395411yescharacteristic
582C2#8401511yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3354C2
socleC22×C3D42
hypercenter#3354D3
hypocenterC3#19862
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3354

Complements

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

NH ≅ G/Ntype
1. C3×#3354C2semidirect
2. #83647C2semidirect
3. #83511C2semidirect
4. #79616C2semidirect
5. #71928C2semidirect
6. #71911C2semidirect
7. #71344C2semidirect
8. #71341C2semidirect
9. #71145C2semidirect
10. #71133C2semidirect
11. #71109C2semidirect
12. #71103C2semidirect
13. #70878C2semidirect
14. #70874C2semidirect
15. #70862C2semidirect
68. #3354D3semidirect
575. C3#19862semidirect

Maximal subgroups

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

subgroupindex
C3×#33542
#836472
#835112
#796162
#719282
#719112
#713442
#713412
#711452
#711332
#711092
#711032
#708782
#708742
#708622
#198623