Finite Group Databasebeta

← #10883555

Subgroups — #10883555

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83771C211yesmaximal, characteristic, complemented
2#83756C211yesmaximal, characteristic, complemented
3#79256C211yesmaximal, characteristic, complemented
4#79249C211yesmaximal, characteristic, complemented
5C3×#1811C211yesmaximal, characteristic, complemented
6#70719C211yesmaximal, characteristic, complemented
7#70698C211yesmaximal, characteristic, complemented
8#69993C212yesmaximal, complemented
9#69992C212yesmaximal, complemented
10#68947C212yesmaximal, complemented
11#68935C212yesmaximal, complemented
12#28181—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14#5791C2211yescharacteristic
15#5791C2212yes—
16#5748C2212yes—
17#5745C2212yes—
18C2×#904C2211yescharacteristic
19C2×#901C2211yescharacteristic
20C2×C3×#108C2211yescharacteristic
21C2×C3×#108C2211yescharacteristic
22C3×#407C2212yes—
23#5207C2211yescharacteristic
24#5139C2212yes—
25#5139C2212yes—
26#5137—21no—
27#5135—21no—
28#5134—22no—
29#5132C2212yes—
30#5128—22no—
31#5127—22no—
32#5113C2212yes—
33#5110C2211yescharacteristic
34#5106C2211yescharacteristic
35#5102C2211yescharacteristic
36#5101C2211yescharacteristic
37#4898—22no—
38#4897—21no—
39#4896—21no—
40#4895—22no—
41#4894—22no—
42C3×#341C2212yes—
43#4559C2212yes—
44#4557C2212yes—
45#4556C2212yes—
46#3067—31no—
47#3054—31no—
48#2957—31no—
49#2930—31no—
50#2080—32no—
51#2010—32no—
52#1891—32no—
53#1881—32no—
54#1833—31no—
55#1814—31no—
56#1811D311yescharacteristic, complemented
57C23×C3×C4C2311yescharacteristic
58C2×#235C2312yes—
59C2×#235C2311yescharacteristic
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16C2312yes—
67C2×C3×M16C2312yes—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70#953—21no—
71#953—22no—
72#930—21no—
73#930—22no—
74#929—21no—
75#929—22no—
76C4×#67—22no—
77C4×#67—42no—
78#926C2312yes—
79#926C2311yescharacteristic
80#924—21no—
81#924—22no—
82C2×#208C2311yescharacteristic
83C2×#208C2311yescharacteristic
84#909—22no—
85#909—22no—
86#908—22no—
87#908—22no—
88#907—24no—
89D3×M16—42no—
90C2×Dic12D411yescharacteristic, complemented
91#904—21no—
92#904—21no—
93#904—21no—
94#904—21no—
95#904—42no—
96C2×D24D411yescharacteristic, complemented
97C2×#204D412yescomplemented
98#901—21no—
99#901—21no—
100#901—21no—
101#901—21no—
102#901—42no—
103C2×ΓL(1,𝔽25)—21no—
104C2×D3×C8—21no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#108—21no—
108C3×#108—21no—
109C3×C22⋊C8—42no—
110#830—22no—
111#825—24no—
112#825—42no—
113#824—22no—
114#823—22no—
115#821—22no—
116(C2×D3)⋊C8—22no—
117#818—21no—
118#817—21no—
119#816—22no—
120#815D412yescomplemented
121Dic3⋊C8—22no—
122C8×Dic3D412yescomplemented
123#575—32no—
124C2×#135—31no—
125#568—32no—
126C2×#131—31no—
127C22×M16C2×D311yescharacteristic
128#525—31no—
129#525—32no—
130#473—61no—
131#472—61no—
132#471—62no—
133#470—61no—
134#469—61no—
135#468—62no—
136#444—62no—
137#443—62no—
138#436—62no—
139#435—62no—
140C2×#108C2×D311yescharacteristic
141C2×#108C2×D311yescharacteristic
142#427—31no—
143#424—31no—
144#421—32no—
145#421—32no—
146#416—32no—
147#413—32no—
148#408—31no—
149#407—31no—
150#407C2×D312yes—
151#405—31no—
152#344—32no—
153#344—32no—
154#341C2×D312yes—
155H(3,ℤ/4)—32no—
156C24×C3#4111yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4#4111yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—21no—
166C2×#67—22no—
167C2×#67—22no—
168C2×#67—42no—
169#235—22no—
170#235—21no—
171#235—21no—
172#235—22no—
173#235—22no—
174#235—21no—
175#235—22no—
176#235—21no—
177C2×D12C2×D412yes—
178C2×D12#4111yescharacteristic
179C2×C4×D3—21no—
180C2×C4×D3—22no—
181C2×Dic6C2×D412yes—
182C2×Dic6#4111yescharacteristic
183C3×M16—22no—
184C3×M16—22no—
185C3×M16—22no—
186C3×M16—22no—
187C3×M16—41no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16—41no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C2×C3×C8#4111yescharacteristic
202#217—22no—
203#217—21no—
204#217—22no—
205#217—42no—
206(C2×D3)⋊C4—22no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210Dic3⋊C4—22no—
211Dic3⋊C4—42no—
212C4×Dic3—22no—
213C4×Dic3C2×D412yes—
214#208—21no—
215#208—21no—
216#208—21no—
217#208—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8C2×D412yes—
220Dic12—21no—
221Dic12—21no—
222Dic12—42no—
223D24—21no—
224D24—21no—
225D24—42no—
226#204—22no—
227#204—22no—
228#204—42no—
229#204—42no—
230ΓL(1,𝔽25)—44no—
231ΓL(1,𝔽25)—41no—
232ΓL(1,𝔽25)—41no—
233D3×C8—44no—
234D3×C8—41no—
235D3×C8—41no—
236C2×#41—31no—
237C2×#41—32no—
238C23×C4C22×D311yescharacteristic
239#137—62no—
240#137—62no—
241AGL(1,ℤ/8)—62no—
242AGL(1,ℤ/8)—62no—
243#135—61no—
244#135—61no—
245#135—61no—
246#135—61no—
247#135—122no—
248C2×Q16—31no—
249C2×SD16—32no—
250C2×D8—31no—
251#131—64no—
252#131—61no—
253#131—61no—
254#131—61no—
255#131—61no—
256#131—122no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—31no—
264C2×M16—31no—
265C2×M16—61no—
266C2×M16—122no—
267C2×M16C22×D312yes—
268C2×M16C22×D312yes—
269C22×C8—61no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#123—61no—
273#123—62no—
274C22⋊Q8—61no—
275C22⋊Q8—62no—
276#121—61no—
277#121—62no—
278#120—61no—
279#120—62no—
280C4×D4—62no—
281C4×D4—122no—
282#117—31no—
283#117—32no—
284#108—21no—
285#108—21no—
286#108—21no—
287#108—21no—
288#107—61no—
289#106—61no—
290C4⋊C8—62no—
291#104—64no—
292#104—122no—
293#103—62no—
294#102—62no—
295#101—62no—
296#100—62no—
297C22⋊C8—42no—
298C22⋊C8—62no—
299C22⋊C8—62no—
300#97—32no—
301C4×C8—32no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—81no—
306C23×C3C2×#4111yescharacteristic
307C22×D3—21no—
308C22×D3—22no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322#67—42no—
323#67—41no—
324#67—41no—
325#67—42no—
326#67—82no—
327#67—84no—
328C2×Dic3—22no—
329C2×Dic3—21no—
330C2×Dic3—22no—
331C2×Dic3—42no—
332D12—22no—
333D12—21no—
334D12—22no—
335D12—21no—
336C4×D3—41no—
337C4×D3—42no—
338C4×D3—41no—
339C4×D3—42no—
340Dic6—22no—
341Dic6—21no—
342Dic6—22no—
343Dic6—21no—
344C3×C8—41no—
345C3×C8—42no—
346C3×C8—42no—
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3×C8#12111yescharacteristic
350C3×C8#12111yescharacteristic
351C3⋊C8—22no—
352C3⋊C8—41no—
353C24#23911yescharacteristic
354#41—61no—
355#41—62no—
356#41—62no—
357#41—61no—
358#41—62no—
359#41—61no—
360#41—62no—
361#41—61no—
362C2×Q8—32no—
363C2×Q8—31no—
364C2×D4—32no—
365C2×D4—31no—
366C2×D4—61no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—122no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4#23911yescharacteristic
379C22×C4C23×D311yescharacteristic
380Q16—61no—
381Q16—61no—
382Q16—122no—
383SD16—62no—
384SD16—62no—
385SD16—122no—
386SD16—122no—
387D8—61no—
388D8—61no—
389D8—122no—
390M16—22no—
391M16—22no—
392M16—22no—
393M16—22no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—61no—
399M16—61no—
400M16—61no—
401M16—61no—
402M16—124no—
403M16—121no—
404M16—121no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—32no—
411C2×C8—61no—
412C2×C8—124no—
413C2×C8—121no—
414C2×C8—121no—
415C2×C8#23511yescharacteristic
416C2×C8#23511yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8D3×D411yescharacteristic
420C2×C8D3×D411yescharacteristic
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—122no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—61no—
428C22⋊C4—62no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C42—32no—
432C42—62no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—42no—
438C22×C3—41no—
439C22×C3#52511yescharacteristic
440C22×C3#54611yescharacteristic
441C22×C3#55311yescharacteristic
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—82no—
446C3×C4—41no—
447C3×C4C2×#12111yescharacteristic
448C3×C4#53511yescharacteristic
449C3×C4#54611yescharacteristic
450C3×C4#55311yescharacteristic
451Dic3—42no—
452Dic3—42no—
453Dic3—41no—
454Dic3—82no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—61no—
459C23—62no—
460C23—81no—
461C23C2×#23911yescharacteristic
462Q8—62no—
463Q8—61no—
464Q8—61no—
465Q8—62no—
466D4—62no—
467D4—61no—
468D4—61no—
469D4—62no—
470D4—121no—
471D4—122no—
472D4—122no—
473D4—121no—
474D4—242no—
475D4—244no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—122no—
491C2×C4C2×#23511yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4#100411yescharacteristic
495C2×C4D3×#4111yescharacteristic
496C2×C4#100911yescharacteristic
497C8—41no—
498C8—42no—
499C8—42no—
500C8—62no—
501C8—121no—
502C8#88411yescharacteristic
503C8#88411yescharacteristic
504C8#88411yescharacteristic
505C8#88411yescharacteristic
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—81no—
510C2×C3#253511yescharacteristic
511D3—81no—
512D3—82no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—41no—
519C22—122no—
520C22—121no—
521C22—122no—
522C22—242no—
523C22#551411yescharacteristic
524C22#562711yescharacteristic
525C22#563011yescharacteristic
526C4—41no—
527C4—122no—
528C4—121no—
529C4—122no—
530C4—242no—
531C4C2×#88411yescharacteristic
532C4#553011yescharacteristic
533C4#560711yescharacteristic
534C4#561211yescharacteristic
535C3#2818111yescharacteristic, complemented, Sylow
536C2—21no—
537C2—21no—
538C2—21no—
539C2—81no—
540C2—241no—
541C2—242no—
542C2#7393311yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1811C2
socleC2×C3#2535
hypercenter#1811D3
hypocenterC3#28181
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1811

Complements

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

NH ≅ G/Ntype
1. #83771C2semidirect
2. #83756C2semidirect
3. #79256C2semidirect
4. #79249C2semidirect
5. C3×#1811C2semidirect
6. #70719C2semidirect
7. #70698C2semidirect
8. #69993C2semidirect
9. #69992C2semidirect
10. #68947C2semidirect
11. #68935C2semidirect
56. #1811D3semidirect
90. C2×Dic12D4semidirect
96. C2×D24D4semidirect
97. C2×#204D4semidirect
120. #815D4semidirect
122. C8×Dic3D4semidirect
535. C3#28181semidirect

Maximal subgroups

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

subgroupindex
#837712
#837562
#792562
#792492
C3×#18112
#707192
#706982
#699932
#699922
#689472
#689352
#281813