Finite Group Databasebeta

← #10786175

Subgroups — #10786175

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81621C211yesmaximal, characteristic, complemented
2#81086C211yesmaximal, characteristic, complemented
3#80987C211yesmaximal, characteristic, complemented
4#79418C211yesmaximal, characteristic
5C3×#1683C211yesmaximal, characteristic, complemented
6#71227C211yesmaximal, characteristic, complemented
7#71145C211yesmaximal, characteristic, complemented
8#71109C211yesmaximal, characteristic, complemented
9#70839C211yesmaximal, characteristic, complemented
10#68589C211yesmaximal, characteristic, complemented
11#68559C211yesmaximal, characteristic, complemented
12#68497C211yesmaximal, characteristic, complemented
13#68492C211yesmaximal, characteristic, complemented
14#68450C211yesmaximal, characteristic, complemented
15#68435C211yesmaximal, characteristic, complemented
16#19864—31nomaximal, Sylow
17#5732C2211yescharacteristic
18#5730C2211yescharacteristic
19#5563C2211yescharacteristic
20#5563C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#474C2211yescharacteristic
24C3×C8×D4C2211yescharacteristic
25#5115—21no—
26#5106—21no—
27#5093C2211yescharacteristic
28#5092C2211yescharacteristic
29#5084C2211yescharacteristic
30#5083C2211yescharacteristic
31#5080C2211yescharacteristic
32#5078C2211yescharacteristic
33C4×#214C2211yescharacteristic
34#4888—21no—
35#4880—21no—
36#4868—21no—
37#4861—21no—
38#4850C2211yescharacteristic
39#4850C2211yescharacteristic
40#4842C2211yescharacteristic
41#4842C2211yescharacteristic
42#4838C2211yescharacteristic
43#4834C2211yescharacteristic
44#4819—21no—
45#4800—21no—
46#4796—21no—
47#4788—21no—
48#4784—21no—
49#4780—21no—
50#4778—21no—
51#4774—21no—
52#4737—21no—
53#4737—21no—
54C4×D24C2211yescharacteristic
55#4682C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57C3×#335C2211yescharacteristic
58C3×#333C2211yescharacteristic
59C3×#332C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61#4484C2211yescharacteristic
62#4483C2211yescharacteristic
63#4483C2211yescharacteristic
64#4481C2211yescharacteristic
65#4481C2211yescharacteristic
66#4479C2211yescharacteristic
67#4454C2211yescharacteristic
68#3360—31no—
69#3357—31no—
70#3341—31no—
71#3340—31no—
72#1723—31no—
73#1691—31no—
74#1683D311yescharacteristic, complemented
75#1676—31no—
76#1675—31no—
77#1646—31no—
78#1638—31no—
79#1634—31no—
80#1632—31no—
81#1581—31no—
82SD16⋊C8—31no—
83C22×C3×C8—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4×D4C2311yescharacteristic
87#947—21no—
88#946—41no—
89Q8×Dic3—21no—
90C2×#216—41no—
91C2×#215—21no—
92#938—21no—
93#938—21no—
94C2×#214D411yescharacteristic
95#930—21no—
96#930—21no—
97#926—21no—
98#926—21no—
99C2×D24D411yescharacteristic
100C2×ΓL(1,𝔽25)—21no—
101C2×D3×C8—21no—
102#898—41no—
103#897—21no—
104#897—21no—
105#893—21no—
106#892—21no—
107#892—21no—
108#892—21no—
109D3×C4⋊C4—21no—
110#890—21no—
111#885—41no—
112#885—41no—
113C4×D12C2311yescharacteristic
114C4×D12C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6C2311yescharacteristic
118C3×#107—21no—
119C3×#106—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#103—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×C22⋊C8—21no—
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#832D411yescharacteristic
131#821—21no—
132#821—21no—
133#821D411yescharacteristic
134#821D411yescharacteristic
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137#818—21no—
138#818—21no—
139#818—21no—
140#818—21no—
141#818D411yescharacteristic
142#817—21no—
143#817—21no—
144#816—21no—
145#816—21no—
146Dic3⋊C8—21no—
147C8×Dic3—21no—
148#810—41no—
149#810D411yescharacteristic
150#809—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808D411yescharacteristic
155#807—21no—
156#807—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#555—31no—
162#555—31no—
163#554—31no—
164#500—31no—
165#500—31no—
166#498—31no—
167#494—31no—
168#488—61no—
169#487—31no—
170#487C2×D311yescharacteristic
171#485—61no—
172#485—61no—
173#482—61no—
174#482—61no—
175#482—61no—
176#480—61no—
177#479—61no—
178#477—31no—
179#477—31no—
180#476—31no—
181#476—31no—
182#476—31no—
183#476C2×D311yescharacteristic
184#474C2×D311yescharacteristic
185#468—61no—
186#466—61no—
187#465—61no—
188#463—61no—
189#461—31no—
190C8×Q8—31no—
191C4×SD16—31no—
192C4×SD16—61no—
193C4×D8—31no—
194#436—31no—
195C8×D4—31no—
196C8×D4C2×D311yescharacteristic
197#428—61no—
198#427—61no—
199#427—61no—
200#335—31no—
201#335C2×D311yescharacteristic
202#333—31no—
203#333—31no—
204#333C2×D311yescharacteristic
205#332C2×D311yescharacteristic
206Q8⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—41no—
216C2×#67—41no—
217C2×D12C2×D411yescharacteristic
218C2×D12C2×D411yescharacteristic
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—41no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6C2×D411yescharacteristic
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—42no—
229C2×C3×C8C2×D411yescharacteristic
230C2×C3×C8C2×D411yescharacteristic
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4C2×D411yescharacteristic
236C3×C22⋊C4—21no—
237C3×C42C2411yescharacteristic
238#217—41no—
239#217—41no—
240#216—82no—
241#215—42no—
242#214—22no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—41no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—41no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3—41no—
256C4⋊Dic3C2×D411yescharacteristic
257C4⋊Dic3C2×D411yescharacteristic
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264C4×Dic3—41no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269D24—41no—
270ΓL(1,𝔽25)—42no—
271D3×C8—42no—
272C2×Q16—121no—
273C2×SD16—31no—
274C2×SD16—61no—
275C2×D8—31no—
276C2×M16—61no—
277C22×C8—21no—
278C22×C8—61no—
279C4⋊Q8—31no—
280C4⋊Q8—31no—
281C4⋊Q8C22×D311yescharacteristic
282C4⋊Q8C22×D311yescharacteristic
283#126—121no—
284#125—61no—
285#124—61no—
286#124—121no—
287#124—121no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290C22⋊Q8—121no—
291#121—61no—
292#121—61no—
293#121—61no—
294#121—61no—
295C4×Q8—31no—
296C4×Q8—61no—
297C4×D4—31no—
298C4×D4—31no—
299C4×D4—61no—
300C4×D4C22×D311yescharacteristic
301#117—61no—
302#117—61no—
303#117—61no—
304#117—61no—
305#117—61no—
306C2×C4⋊C4—61no—
307#107—21no—
308#107—31no—
309#107—61no—
310#107—61no—
311#107—61no—
312#107—61no—
313#107—61no—
314#107—61no—
315#106—21no—
316#106—31no—
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321#106—61no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—21no—
329#103—31no—
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—31no—
339#102—61no—
340#102—61no—
341#102—61no—
342C22⋊C8—21no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C4×C8—31no—
346C4×C8—31no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C23×C3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—81no—
355C3×D4—21no—
356C3×D4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365#67—161no—
366#67—161no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—21no—
374D12—21no—
375D12—21no—
376D12—21no—
377C4×D3—41no—
378C4×D3—41no—
379C4×D3—41no—
380C4×D3—41no—
381C4×D3—161no—
382Dic6—21no—
383Dic6—21no—
384Dic6—81no—
385Dic6—81no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—21no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8#13512yes—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—31no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×D4—31no—
401C2×D4—31no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4D3×D411yescharacteristic
405C22×C4—21no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—121no—
409Q16—242no—
410SD16—62no—
411SD16—122no—
412D8—121no—
413M16—122no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—42no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—122no—
423C2×C8D3×D411yescharacteristic
424C2×C8D3×D411yescharacteristic
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—31no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—121no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4D3×D411yescharacteristic
441C22⋊C4—21no—
442C22⋊C4—61no—
443C22⋊C4—61no—
444C22⋊C4—121no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C42—61no—
452C42—121no—
453C42—121no—
454C42—121no—
455C42C23×D311yescharacteristic
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3D4211yescharacteristic
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4C2×#13511yescharacteristic
471C3×C4#57511yescharacteristic
472C3×C4#57811yescharacteristic
473C3×C4#57811yescharacteristic
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—61no—
482C23—61no—
483Q8—41no—
484Q8—41no—
485Q8—61no—
486Q8—61no—
487Q8—81no—
488Q8—241no—
489Q8—241no—
490D4—21no—
491D4—21no—
492D4—61no—
493D4—61no—
494D4—61no—
495D4—61no—
496D4—481no—
497D4—481no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—481no—
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C2×C4#100911yescharacteristic
517C8—21no—
518C8—21no—
519C8—32no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—61no—
524C8—121no—
525C8—121no—
526C2×C3—81no—
527C2×C3#326911yescharacteristic
528C2×C3#328311yescharacteristic
529C2×C3#328811yescharacteristic
530D3—81no—
531D3—81no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—121no—
536C22—121no—
537C22—121no—
538C22—121no—
539C22—121no—
540C22—121no—
541C22#560911yescharacteristic
542C4—21no—
543C4—41no—
544C4—81no—
545C4—81no—
546C4—121no—
547C4—121no—
548C4—121no—
549C4—121no—
550C4—241no—
551C4—241no—
552C4#576611yescharacteristic
553C4D3×#13511yescharacteristic
554C4#577111yescharacteristic
555C4#577311yescharacteristic
556C3#1986411yescharacteristic, complemented, Sylow
557C2—81no—
558C2—241no—
559C2—241no—
560C2#8055411yescharacteristic
561C2#8077411yescharacteristic
562C2#8093211yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1683C2
socleC22×C3D42
prime radical#79418C2
hypercenter#1683D3
hypocenterC3#19864
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1683

Complements

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

NH ≅ G/Ntype
1. #81621C2semidirect
2. #81086C2semidirect
3. #80987C2semidirect
5. C3×#1683C2semidirect
6. #71227C2semidirect
7. #71145C2semidirect
8. #71109C2semidirect
9. #70839C2semidirect
10. #68589C2semidirect
11. #68559C2semidirect
12. #68497C2semidirect
13. #68492C2semidirect
14. #68450C2semidirect
15. #68435C2semidirect
74. #1683D3semidirect
556. C3#19864semidirect

Maximal subgroups

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

subgroupindex
#816212
#810862
#809872
#794182
C3×#16832
#712272
#711452
#711092
#708392
#685892
#685592
#684972
#684922
#684502
#684352
#198643