Finite Group Databasebeta

← #10883556

Subgroups — #10883556

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#83805C211yesmaximal, characteristic, complemented
2#83795C211yesmaximal, characteristic, complemented
3#79249C211yesmaximal, characteristic, complemented
4#79181C211yesmaximal, characteristic, complemented
5C3×#1814C211yesmaximal, characteristic, complemented
6#70712C211yesmaximal, characteristic, complemented
7#70657C211yesmaximal, characteristic, complemented
8#70416C212yesmaximal, complemented
9#70406C212yesmaximal, complemented
10#69043C212yesmaximal, complemented
11#69036C212yesmaximal, complemented
12#28181—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5828C2211yescharacteristic
15#5822C2212yes—
16#5745C2212yes—
17C2×#904C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19#5524C2212yes—
20C2×C3×#108C2211yescharacteristic
21C3×#427C2211yescharacteristic
22C3×#421C2212yes—
23#5234C2211yescharacteristic
24#5143—21no—
25#5142—21no—
26#5141—22no—
27#5139C2212yes—
28#5132C2212yes—
29#5128—22no—
30#5126C2211yescharacteristic
31#5126C2212yes—
32#5119C2211yescharacteristic
33#5111—22no—
34C2×#819C2211yescharacteristic
35#5101C2211yescharacteristic
36#5002C2212yes—
37#4895—22no—
38#4894—22no—
39#4886—22no—
40#4867—21no—
41#4865—21no—
42C3×#344C2212yes—
43#4558C2212yes—
44#4533C2212yes—
45#4531C2212yes—
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#1814D311yescharacteristic, complemented
56#1811—31no—
57C2×C3×#41C2311yescharacteristic
58C2×#235C2312yes—
59C22×C4×D3C2311yescharacteristic
60C3×#131—21no—
61C3×#131—21no—
62C3×#131—21no—
63C3×#131—21no—
64C2×C3×M16—21no—
65C2×C3×M16C2311yescharacteristic
66C2×C3×M16C2312yes—
67C22×C3×C8—21no—
68C22×C3×C8C2311yescharacteristic
69C3×#117C2312yes—
70#951—22no—
71#950—42no—
72#949—22no—
73#946—21no—
74#942—22no—
75#938—21no—
76#937—21no—
77#935—21no—
78#932—22no—
79C4×#67—22no—
80#926C2311yescharacteristic
81C2×#208C2312yes—
82C2×#208C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84#907—24no—
85D3×M16—42no—
86C2×Dic12D411yescharacteristic, complemented
87#904—21no—
88#904—21no—
89#904—21no—
90#904—21no—
91C2×D24D411yescharacteristic, complemented
92C2×#204D412yescomplemented
93#901—42no—
94C2×ΓL(1,𝔽25)—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×ΓL(1,𝔽25)—21no—
100#897—22no—
101#895—22no—
102#886—22no—
103#882—22no—
104C4×D12—42no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#107—21no—
108C3×#106—21no—
109C3×#104—42no—
110#830—22no—
111#825—24no—
112#824—22no—
113#823—22no—
114(C2×D3)⋊C8—22no—
115(C2×D3)⋊C8—42no—
116#819—21no—
117#819—21no—
118#815D412yescomplemented
119Dic3⋊C8—22no—
120C8×Dic3D412yescomplemented
121#810—22no—
122#809—22no—
123#575—32no—
124C2×#135—31no—
125#568—32no—
126C2×#131C2×D311yescharacteristic
127C22×M16—31no—
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×#108—31no—
141C2×#108C2×D311yescharacteristic
142#427C2×D311yescharacteristic
143#424—31no—
144#421—32no—
145#421C2×D312yes—
146#416—32no—
147#413—32no—
148#408—31no—
149#407—31no—
150#407—32no—
151#405—31no—
152#344—32no—
153#344C2×D312yes—
154#341—32no—
155H(3,ℤ/4)—32no—
156C23×D3#4111yescharacteristic
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—21no—
161C2×C3×Q8#4111yescharacteristic
162C2×C3×D4—21no—
163C2×C3×D4#4111yescharacteristic
164C22×C3×C4—21no—
165C22×C3×C4C2411yescharacteristic
166C2×#67—22no—
167C2×#67—22no—
168C22×Dic3#4111yescharacteristic
169#235—22no—
170#235—22no—
171#235—22no—
172#235—22no—
173C2×D12—42no—
174C2×D12C2×D412yes—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—21no—
181C2×C4×D3—22no—
182C2×Dic6C2×D412yes—
183C3×M16—21no—
184C3×M16—22no—
185C3×M16—21no—
186C3×M16—22no—
187C3×M16—41no—
188C3×M16—41no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—41no—
192C2×C3×C8—41no—
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8#4111yescharacteristic
198C2×C3×C8#4111yescharacteristic
199C3×C4⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C42—22no—
202#217—22no—
203#217—21no—
204#216—42no—
205#215—42no—
206#214—42no—
207#213—42no—
208(C2×D3)⋊C4—22no—
209(C2×D3)⋊C4—42no—
210(C2×D3)⋊C4—42no—
211C4⋊Dic3—21no—
212C4⋊Dic3—42no—
213Dic3⋊C4—22no—
214C4×Dic3C2×D412yes—
215#208—21no—
216#208—21no—
217#208—22no—
218#208—22no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8C2×D412yes—
224Dic12—21no—
225Dic12—21no—
226D24—21no—
227D24—21no—
228#204—22no—
229#204—22no—
230ΓL(1,𝔽25)—44no—
231ΓL(1,𝔽25)—41no—
232ΓL(1,𝔽25)—41no—
233ΓL(1,𝔽25)—41no—
234ΓL(1,𝔽25)—41no—
235D3×C8—44no—
236C2×#41—32no—
237C2×#41C22×D311yescharacteristic
238C23×C4—31no—
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—21no—
252#131—21no—
253#131—21no—
254#131—21no—
255#131—64no—
256#131—122no—
257C2×M16—21no—
258C2×M16—31no—
259C2×M16—32no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16—61no—
264C2×M16—61no—
265C2×M16—61no—
266C2×M16—122no—
267C2×M16C22×D311yescharacteristic
268C2×M16C22×D312yes—
269C22×C8—21no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272#123—62no—
273#123—61no—
274C22⋊Q8—61no—
275C22⋊Q8—62no—
276#121—61no—
277#121—62no—
278#120—62no—
279#120—61no—
280C4×D4—62no—
281C4×D4—122no—
282#117—31no—
283#117C22×D312yes—
284#108—21no—
285#108—21no—
286#108—61no—
287#108—61no—
288#107—21no—
289#106—21no—
290C4⋊C8—62no—
291#104—42no—
292#104—64no—
293#103—62no—
294#102—62no—
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×C3C2×#4111yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—22no—
308C22×D3—81no—
309C3×Q8—21no—
310C3×Q8—21no—
311C3×D4—21no—
312C3×D4—21no—
313C3×D4—41no—
314C3×D4—41no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×C3×C4H(5,𝔽2)11yescharacteristic
325#67—42no—
326#67—42no—
327#67—82no—
328C2×Dic3—22no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—21no—
333D12—22no—
334D12—22no—
335D12—84no—
336C4×D3—41no—
337C4×D3—42no—
338C4×D3—41no—
339C4×D3—41no—
340C4×D3—41no—
341C4×D3—42no—
342Dic6—22no—
343Dic6—22no—
344C3×C8—41no—
345C3×C8—42no—
346C3×C8#12111yescharacteristic
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3×C8#12111yescharacteristic
350C3⋊C8—22no—
351C3⋊C8—41no—
352C3⋊C8—42no—
353C24—31no—
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—62no—
359#41—62no—
360#41—62no—
361#41—62no—
362C2×Q8—32no—
363C2×Q8#23911yescharacteristic
364C2×D4—21no—
365C2×D4—32no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4#23911yescharacteristic
370C22×C4—21no—
371C22×C4—31no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4C23×D311yescharacteristic
380Q16—61no—
381Q16—61no—
382Q16—122no—
383SD16—62no—
384SD16—62no—
385SD16—122no—
386SD16—122no—
387D8—61no—
388D8—61no—
389D8—122no—
390M16—21no—
391M16—21no—
392M16—22no—
393M16—22no—
394M16—41no—
395M16—41no—
396M16—61no—
397M16—61no—
398M16—62no—
399M16—62no—
400M16—124no—
401M16—121no—
402M16—121no—
403M16—121no—
404M16—121no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—32no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—62no—
414C2×C8—124no—
415C2×C8#23511yescharacteristic
416C2×C8#23511yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8D3×D411yescharacteristic
420C2×C8D3×D411yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—122no—
425C22⋊C4—22no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—61no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C42—22no—
432C42—32no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—41no—
436C22×C3#52511yescharacteristic
437C22×C3#54611yescharacteristic
438C22×C3#55311yescharacteristic
439C2×D3—42no—
440C2×D3—41no—
441C2×D3—41no—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—82no—
446C3×C4—41no—
447C3×C4—82no—
448C3×C4C2×#12111yescharacteristic
449C3×C4#53511yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#55311yescharacteristic
452Dic3—42no—
453Dic3—42no—
454Dic3—41no—
455C23—21no—
456C23—61no—
457C23—61no—
458C23—61no—
459C23—62no—
460C23—241no—
461C23D3×#4111yescharacteristic
462Q8—21no—
463Q8—21no—
464Q8—62no—
465Q8—62no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—62no—
471D4—62no—
472D4—122no—
473D4—122no—
474D4—244no—
475D4—242no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4C2×#23511yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4#100411yescharacteristic
495C2×C4C2×#23911yescharacteristic
496C2×C4#100911yescharacteristic
497C8—41no—
498C8—42no—
499C8—62no—
500C8—121no—
501C8—122no—
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—42no—
515C22—41no—
516C22—122no—
517C22—121no—
518C22—121no—
519C22—122no—
520C22—121no—
521C22—122no—
522C22—242no—
523C22#553011yescharacteristic
524C22#561211yescharacteristic
525C22#562711yescharacteristic
526C4—41no—
527C4—82no—
528C4—122no—
529C4—121no—
530C4—122no—
531C4C2×#88411yescharacteristic
532C4#551411yescharacteristic
533C4#560711yescharacteristic
534C4#563011yescharacteristic
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×#1814C2
socleC2×C3#2535
hypercenter#1814D3
hypocenterC3#28181
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
1. #83805C2semidirect
2. #83795C2semidirect
3. #79249C2semidirect
4. #79181C2semidirect
5. C3×#1814C2semidirect
6. #70712C2semidirect
7. #70657C2semidirect
8. #70416C2semidirect
9. #70406C2semidirect
10. #69043C2semidirect
11. #69036C2semidirect
55. #1814D3semidirect
86. C2×Dic12D4semidirect
91. C2×D24D4semidirect
92. C2×#204D4semidirect
118. #815D4semidirect
120. 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
#838052
#837952
#792492
#791812
C3×#18142
#707122
#706572
#704162
#704062
#690432
#690362
#281813