Finite Group Databasebeta

← #10883538

Subgroups — #10883538

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic, complemented
2#83928C211yesmaximal, characteristic, complemented
3#83910C211yesmaximal, characteristic, complemented
4#83734C211yesmaximal, characteristic, complemented
5#71977C211yesmaximal, characteristic, complemented
6#71715C212yesmaximal, complemented
7#71714C212yesmaximal, complemented
8#70665C211yesmaximal, characteristic, complemented
9#70657C211yesmaximal, characteristic, complemented
10#70294C212yesmaximal, complemented
11#70293C212yesmaximal, complemented
12#28180—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C3×#525C2212yes—
16#5822C2212yes—
17#5819C2212yes—
18#5791C2211yescharacteristic
19C2×#901C2211yescharacteristic
20C3×#467—21no—
21C3×#466—21no—
22C3×#465—22no—
23C3×#427C2211yescharacteristic
24C3×#416C2212yes—
25#5237C2212yes—
26#5236C2212yes—
27#5207C2211yescharacteristic
28#5207C2212yes—
29#5197—22no—
30#5196—22no—
31#5192—21no—
32#5177—22no—
33#5160—21no—
34#5111—22no—
35#5110C2211yescharacteristic
36C8×#67—22no—
37C2×#819C2211yescharacteristic
38C2×#819C2211yescharacteristic
39#5102C2211yescharacteristic
40#5101C2211yescharacteristic
41#5005C2212yes—
42#4552C2212yes—
43#4533C2212yes—
44#4533C2212yes—
45#4531C2212yes—
46#3067—31no—
47#3050D311yescharacteristic, complemented
48#2957—31no—
49#2929—31no—
50#2080—32no—
51#2010—32no—
52#1891—32no—
53#1881—32no—
54#1833—31no—
55#1814—31no—
56#1810—31no—
57C2×C3×#41C2312yes—
58C23×C3×C4C2311yescharacteristic
59C2×#235C2311yescharacteristic
60C3×#135—21no—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135—21no—
64C2×C3×Q16D411yescharacteristic, complemented
65C2×C3×SD16D412yescomplemented
66C2×C3×D8D411yescharacteristic, complemented
67C22×C3×C8—21no—
68C22×C3×C8—21no—
69C22×C3×C8—21no—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×#123—22no—
76C3×C22⋊Q8—22no—
77C3×#121—22no—
78C3×#120—22no—
79C3×C4×D4—42no—
80C3×#117C2312yes—
81#953—21no—
82#951—22no—
83#950—42no—
84#949—22no—
85#948—24no—
86#948—42no—
87#942—22no—
88#932—22no—
89#930—21no—
90#929—21no—
91C4×#67—22no—
92#926C2311yescharacteristic
93#924—21no—
94C2×#208—42no—
95C2×#208C2311yescharacteristic
96C2×#208C2312yes—
97C2×#208C2312yes—
98C2×#208C2311yescharacteristic
99#901—21no—
100#901—21no—
101#901—21no—
102#901—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×D3×C8—21no—
105C3×#107—21no—
106C3×#106—21no—
107C3×#103—22no—
108C3×#102—22no—
109#837—24no—
110#836—22no—
111#833—22no—
112#830—22no—
113#830—42no—
114(C2×D3)⋊C8—22no—
115#819—21no—
116#819—21no—
117#819—21no—
118#819—21no—
119#815D412yescomplemented
120Dic3⋊C8—22no—
121C8×Dic3D412yescomplemented
122#805—42no—
123#575—32no—
124C2×#135C2×D311yescharacteristic
125#568—32no—
126C2×#131—31no—
127C23×C8C2×D311yescharacteristic
128#525—31no—
129#525C2×D312yes—
130#473—61no—
131#472—61no—
132#471—62no—
133#467—21no—
134#466—21no—
135#465—22no—
136#444—62no—
137#443—62no—
138#435—62no—
139C8×D4—62no—
140C2×#108—31no—
141C2×#108—31no—
142#427C2×D311yescharacteristic
143#424—31no—
144#421—32no—
145#421—32no—
146#416C2×D312yes—
147#413—32no—
148#408—31no—
149#407—31no—
150#407—32no—
151#405—31no—
152#344—32no—
153#344—32no—
154#341—32no—
155H(3,ℤ/4)—32no—
156C24×C3#4111yescharacteristic
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—22no—
160C3×#41—22no—
161C2×C3×Q8C2×D412yes—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—22no—
175C2×#67—21no—
176#235—21no—
177#235—21no—
178#235—21no—
179#235—21no—
180C2×D12#4111yescharacteristic
181C2×C4×D3—21no—
182C2×Dic6#4111yescharacteristic
183C3×Q16—21no—
184C3×Q16—21no—
185C3×SD16—22no—
186C3×SD16—22no—
187C3×D8—21no—
188C3×D8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—41no—
193C2×C3×C8—41no—
194C2×C3×C8—41no—
195C2×C3×C8—41no—
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C2×C3×C8#4111yescharacteristic
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C42—22no—
208#217—22no—
209#217—21no—
210#216—42no—
211#215—42no—
212#214—42no—
213#213—42no—
214(C2×D3)⋊C4—22no—
215C4⋊Dic3—21no—
216Dic3⋊C4—22no—
217C4×Dic3C2×D412yes—
218#208—21no—
219#208—21no—
220#208—22no—
221#208—22no—
222#208—21no—
223#208—21no—
224#208—22no—
225#208—22no—
226#208—44no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—44no—
231C2×C3⋊C8C2×D412yes—
232ΓL(1,𝔽25)—41no—
233ΓL(1,𝔽25)—41no—
234D3×C8—41no—
235D3×C8—41no—
236C2×#41—31no—
237C2×#41C22×D312yes—
238C23×C4C22×D311yescharacteristic
239#137—62no—
240#137—62no—
241AGL(1,ℤ/8)—62no—
242AGL(1,ℤ/8)—62no—
243#135—21no—
244#135—21no—
245#135—21no—
246#135—21no—
247#135—122no—
248C2×Q16#6711yescharacteristic, complemented
249C2×SD16#6712yescomplemented
250C2×D8#6711yescharacteristic, complemented
251#131—64no—
252#131—61no—
253#131—61no—
254#131—61no—
255#131—61no—
256#131—122no—
257C2×M16—31no—
258C2×M16—31no—
259C2×M16—32no—
260C2×M16—32no—
261C2×M16—61no—
262C2×M16—122no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—61no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#123—22no—
273#123—61no—
274C22⋊Q8—22no—
275C22⋊Q8—61no—
276#121—22no—
277#121—61no—
278#120—22no—
279#120—61no—
280C4×D4—42no—
281C4×D4—62no—
282#117—31no—
283#117C22×D312yes—
284#108—61no—
285#108—61no—
286#108—61no—
287#108—61no—
288#107—21no—
289#106—21no—
290C4⋊C8—62no—
291#104—64no—
292#104—122no—
293#103—22no—
294#102—22no—
295#101—62no—
296#100—62no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299C22⋊C8—122no—
300#97—32no—
301C4×C8—32no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—42no—
307C23×C3C2×#4111yescharacteristic
308C22×D3—21no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—84no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333#67—44no—
334#67—41no—
335#67—41no—
336C2×Dic3—22no—
337C2×Dic3—21no—
338D12—21no—
339D12—21no—
340C4×D3—41no—
341C4×D3—41no—
342Dic6—21no—
343Dic6—21no—
344C3×C8—41no—
345C3×C8#12111yescharacteristic
346C3×C8#12111yescharacteristic
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3⋊C8—22no—
350C3⋊C8—41no—
351C3⋊C8—42no—
352C3⋊C8—42no—
353C24#23711yescharacteristic
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—61no—
359#41—61no—
360#41—61no—
361#41—61no—
362C2×Q8—31no—
363C2×Q8C2×#6712yes—
364C2×D4—22no—
365C2×D4—31no—
366C2×D4—42no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4C2×#6712yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—61no—
378C22×C4#23711yescharacteristic
379C22×C4C23×D311yescharacteristic
380Q16—21no—
381Q16—21no—
382Q16—122no—
383SD16—22no—
384SD16—22no—
385SD16—122no—
386SD16—122no—
387D8—21no—
388D8—21no—
389D8—122no—
390M16—61no—
391M16—61no—
392M16—62no—
393M16—61no—
394M16—62no—
395M16—62no—
396M16—61no—
397M16—62no—
398M16—124no—
399M16—121no—
400M16—121no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—32no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—61no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—124no—
413C2×C8—121no—
414C2×C8—121no—
415C2×C8D3×D411yescharacteristic
416C2×C8D3×D411yescharacteristic
417C2×C8#23711yescharacteristic
418C2×C8#23711yescharacteristic
419C2×C8C2×#6711yescharacteristic
420C2×C8C2×#6711yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C22⋊C4—22no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—62no—
429C22⋊C4—62no—
430C22⋊C4—61no—
431C42—22no—
432C42—32no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—82no—
443C22×C3#52511yescharacteristic
444C22×C3#54611yescharacteristic
445C22×C3#55311yescharacteristic
446C2×D3—42no—
447C2×D3—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—82no—
451C3×C4C2×#12111yescharacteristic
452C3×C4#53511yescharacteristic
453C3×C4#54611yescharacteristic
454C3×C4#55311yescharacteristic
455Dic3—42no—
456Dic3—41no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—42no—
462C23—61no—
463C23C2×#23711yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—61no—
467Q8—61no—
468D4—22no—
469D4—22no—
470D4—42no—
471D4—42no—
472D4—61no—
473D4—61no—
474D4—84no—
475D4—124no—
476D4—121no—
477D4—121no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23711yescharacteristic
495C2×C4#100411yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C2×C4C22×#6711yescharacteristic
499C8—41no—
500C8—62no—
501C8—121no—
502C8—122no—
503C8—122no—
504C8#93911yescharacteristic
505C8#93911yescharacteristic
506C8#93911yescharacteristic
507C8#93911yescharacteristic
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—42no—
512C2×C3—82no—
513C2×C3#253511yescharacteristic
514D3—81no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—122no—
526C22—121no—
527C22#560611yescharacteristic
528C22#561811yescharacteristic
529C22#580611yescharacteristic
530C4—41no—
531C4—42no—
532C4—82no—
533C4—122no—
534C4—121no—
535C4#560711yescharacteristic
536C4#561411yescharacteristic
537C4C2×#93911yescharacteristic
538C4#582911yescharacteristic
539C3#2818011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—42no—
544C2—82no—
545C2—241no—
546C2#8266611yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3050C2
socleC2×C3#2535
hypercenter#3050D3
hypocenterC3#28180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C3×#3050C2semidirect
2. #83928C2semidirect
3. #83910C2semidirect
4. #83734C2semidirect
5. #71977C2semidirect
6. #71715C2semidirect
7. #71714C2semidirect
8. #70665C2semidirect
9. #70657C2semidirect
10. #70294C2semidirect
11. #70293C2semidirect
47. #3050D3semidirect
64. C2×C3×Q16D4semidirect
65. C2×C3×SD16D4semidirect
66. C2×C3×D8D4semidirect
119. #815D4semidirect
121. C8×Dic3D4semidirect
248. C2×Q16#67semidirect
249. C2×SD16#67semidirect
250. C2×D8#67semidirect
539. C3#28180semidirect

Maximal subgroups

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

subgroupindex
C3×#30502
#839282
#839102
#837342
#719772
#717152
#717142
#706652
#706572
#702942
#702932
#281803