Finite Group Databasebeta

← #10772463

Subgroups — #10772463

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86205C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3#83215C211yesmaximal, characteristic, complemented
4#83209C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#71238C214yesmaximal, complemented
7#71231C214yesmaximal, complemented
8#69735C211yesmaximal, characteristic, complemented
9#69734C211yesmaximal, characteristic, complemented
10#18636—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5817—22no—
14C2×#942—21no—
15#5734C2212yes—
16#5733—22no—
17#5732—21no—
18#5730C2212yes—
19#5728—21no—
20#5726—21no—
21#5725—21no—
22#5514—22no—
23#5510—21no—
24#5504—21no—
25#5477C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#487—22no—
28C3×#487C2214yes—
29C2×C3×#103—22no—
30C2×C3×C4×C8C2211yescharacteristic
31#5225—21no—
32#5090—21no—
33#5090C2212yes—
34#5088—21no—
35#5087C2214yes—
36#5083—21no—
37#5083C2212yes—
38#5078—21no—
39#4967—22no—
40#4925C2211yescharacteristic
41#4924C2211yescharacteristic
42#4546C2214yes—
43#4459C2212yes—
44#4459C2214yes—
45#4456C2212yes—
46#3511—31no—
47#3226—31no—
48C2×#487D311yescharacteristic, complemented
49#3039—31no—
50#1646—34no—
51#1644—34no—
52#1548—31no—
53#1492—31no—
54#1476—31no—
55C22×C3×Q8—21no—
56C22×C3×Q8D412yes—
57C2×#239—22no—
58C2×#235—21no—
59C2×#235D411yescharacteristic
60C22×C3×C8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8D412yes—
67C3×C4⋊Q8C2312yes—
68C3×C4⋊Q8C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74#947—22no—
75#947—24no—
76#946—22no—
77#946—24no—
78Q8×Dic3—22no—
79C2×#216—22no—
80#942—44no—
81C2×#215—22no—
82#930—24no—
83#930—21no—
84#930D412yes—
85C4×#67—24no—
86C4×#67—21no—
87C4×#67—21no—
88#926D411yescharacteristic
89C2×#208—21no—
90#898—24no—
91#897—22no—
92#897—22no—
93#895—22no—
94#895—22no—
95#895—22no—
96#894—24no—
97#893—22no—
98#893—22no—
99#892—22no—
100#892—22no—
101D3×C4⋊C4—22no—
102#890—22no—
103#887—22no—
104#875—21no—
105#874D411yescharacteristic
106C4×D12—21no—
107C4×D12D411yescharacteristic
108C4×D12C2311yescharacteristic
109C4×D12C2312yes—
110#872—21no—
111C42×D3—21no—
112#869D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×#103—28no—
116C3×C4×C8—21no—
117C3×C4×C8C2312yes—
118#835—22no—
119#835—22no—
120#830—21no—
121#809—24no—
122#807—24no—
123C4⋊(C3⋊C8)—21no—
124C4⋊(C3⋊C8)C2312yes—
125C4⋊(C3⋊C8)C2312yes—
126C4×C3⋊C8—21no—
127C2×#137—61no—
128#556—61no—
129#555—32no—
130#554—32no—
131#553—62no—
132#548—62no—
133#547—61no—
134#542—62no—
135#537—61no—
136#533—61no—
137#532—31no—
138#532—61no—
139C2×C4⋊Q8C2×D311yescharacteristic
140C2×C4⋊Q8C2×D311yescharacteristic
141#520—61no—
142C4×#41—31no—
143#500—61no—
144#487—22no—
145#487C2×D314yes—
146#486—61no—
147#480—62no—
148#474—34no—
149#462—32no—
150#462—61no—
151#461—32no—
152#461—61no—
153#431—31no—
154#423—31no—
155#419—61no—
156C2×#103—22no—
157C2×C4×C8C2×D311yescharacteristic
158#333—34no—
159Q8⋊C8—32no—
160D4⋊C8—34no—
161D4⋊C8—32no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—44no—
165C2×C3×Q8—41no—
166C2×C3×Q8—42no—
167C2×C3×Q8—42no—
168C2×C3×Q8C2×D414yes—
169C22×C3×C4—22no—
170C22×C3×C4—41no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C2×#67—21no—
175C2×#67—22no—
176#239—84no—
177#235—22no—
178#235—22no—
179#235—42no—
180#235—42no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—24no—
184C2×D12C2×D412yes—
185C2×D12C2×D411yescharacteristic
186C2×C4×D3—21no—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—22no—
190C2×C4×D3—24no—
191C2×Dic6—21no—
192C2×Dic6C2×D411yescharacteristic
193C2×C3×C8—22no—
194C2×C3×C8—42no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2411yescharacteristic
207#217—21no—
208#216—44no—
209#215—44no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—21no—
214(C2×D3)⋊C4—22no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—22no—
217C4⋊Dic3—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3C2×D412yes—
220C4⋊Dic3C2×D411yescharacteristic
221C4⋊Dic3C2×D411yescharacteristic
222Dic3⋊C4—24no—
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225C4×Dic3—24no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229#208—42no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—31no—
233C2×#41—62no—
234C2×#41—61no—
235C22×Q8—21no—
236C22×Q8#6712yes—
237#137—124no—
238C2×Q16—62no—
239C2×SD16—62no—
240C2×M16—61no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—31no—
248C4⋊Q8#6712yes—
249C4⋊Q8C22×D312yes—
250C4⋊Q8C22×D312yes—
251#127—31no—
252#126—64no—
253#125—62no—
254#124—62no—
255#124—64no—
256#124—61no—
257#123—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—64no—
262#121—32no—
263#121—62no—
264#121—62no—
265#121—62no—
266#121—64no—
267#121—61no—
268C4×Q8—31no—
269C4×Q8—31no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×D4—31no—
273C4×D4—31no—
274C4×D4—32no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—64no—
278C4×D4—61no—
279C4×D4—61no—
280C4×D4—61no—
281#117—31no—
282#117—62no—
283#117—62no—
284#117—61no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—62no—
290C2×C42—61no—
291C2×C42C22×D311yescharacteristic
292#107—64no—
293C4⋊C8—32no—
294C4⋊C8—32no—
295C4⋊C8—61no—
296#103—28no—
297#103—62no—
298#103—62no—
299#102—64no—
300C22⋊C8—61no—
301C4×C8—21no—
302C4×C8—61no—
303C4×C8C22×D312yes—
304C23×C3#12011yescharacteristic
305C22×D3—21no—
306C22×D3—22no—
307C3×Q8—44no—
308C3×Q8—44no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—82no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—82no—
317C2×C3×C4—82no—
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4#13512yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326#67—42no—
327#67—44no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332D12—21no—
333D12—21no—
334D12—24no—
335D12—42no—
336D12—42no—
337D12—84no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—84no—
344Dic6—21no—
345Dic6—21no—
346Dic6—42no—
347C3×C8—42no—
348C3⋊C8—42no—
349C3⋊C8—42no—
350#41—62no—
351#41—62no—
352#41—122no—
353#41—122no—
354#41—244no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—31no—
358C2×Q8—44no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—61no—
363C2×Q8C2×#6714yes—
364C2×D4—32no—
365C2×D4—31no—
366C2×D4—64no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—64no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4D3×D411yescharacteristic
379C22×C4D3×D411yescharacteristic
380C22×C4C2×#6711yescharacteristic
381Q16—124no—
382SD16—124no—
383M16—122no—
384C2×C8—22no—
385C2×C8—42no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—31no—
395C4⋊C4—31no—
396C4⋊C4—32no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—64no—
400C4⋊C4—64no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—62no—
410C22⋊C4—61no—
411C22⋊C4—62no—
412C42—64no—
413C42—61no—
414C42—61no—
415C42—61no—
416C42D3×D411yescharacteristic
417C42D3×D411yescharacteristic
418C42C2×#6711yescharacteristic
419C42C23×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3#44912yes—
423C22×C3C2×#12011yescharacteristic
424C2×D3—41no—
425C2×D3—42no—
426C2×D3—41no—
427C2×D3—42no—
428C2×D3—41no—
429C2×D3—42no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—44no—
433C3×C4—82no—
434C3×C4#44912yes—
435C3×C4#45512yes—
436C3×C4C2×#13511yescharacteristic
437C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C3×C4#57812yes—
439Dic3—41no—
440Dic3—42no—
441Dic3—41no—
442Dic3—41no—
443C23—61no—
444C23—62no—
445C23#93711yescharacteristic
446Q8—44no—
447Q8—44no—
448Q8—42no—
449Q8—42no—
450Q8—61no—
451Q8—61no—
452Q8—82no—
453Q8—122no—
454D4—61no—
455D4—61no—
456D4—64no—
457D4—122no—
458D4—124no—
459D4—122no—
460D4—122no—
461D4—244no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—244no—
478C2×C4#91412yes—
479C2×C4#91612yes—
480C2×C4#93711yescharacteristic
481C2×C4#93711yescharacteristic
482C2×C4#93711yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C22×#6711yescharacteristic
486C8—42no—
487C8—122no—
488C8—122no—
489C2×C3—22no—
490C2×C3C2×#44911yescharacteristic
491C2×C3#301111yescharacteristic
492C2×C3C2×#45511yescharacteristic
493D3—81no—
494D3—82no—
495C22—21no—
496C22—21no—
497C22—121no—
498C22—122no—
499C22—121no—
500C22—122no—
501C22—121no—
502C22—122no—
503C22#517212yes—
504C22C2×#93711yescharacteristic
505C4—21no—
506C4—21no—
507C4—44no—
508C4—82no—
509C4—121no—
510C4—121no—
511C4—122no—
512C4—121no—
513C4#506112yes—
514C4#517212yes—
515C4C2×#91411yescharacteristic
516C4C2×#91611yescharacteristic
517C4#576612yes—
518C3#1863611yescharacteristic, complemented, Sylow
519C2—22no—
520C2—241no—
521C2—242no—
522C2C2×#506111yescharacteristic
523C2C2×#517211yescharacteristic
524C2#8388411yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC2×C3×#487C2
socleC22×C3C2×#120
prime radical#86205C2
hypercenterC2×#487D3
hypocenterC3#18636
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
2. C2×C3×#487C2semidirect
3. #83215C2semidirect
4. #83209C2semidirect
5. #71774C2semidirect
6. #71238C2semidirect
7. #71231C2semidirect
8. #69735C2semidirect
9. #69734C2semidirect
48. C2×#487D3semidirect
518. C3#18636semidirect

Maximal subgroups

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

subgroupindex
#862052
C2×C3×#4872
#832152
#832092
#717742
#712382
#712312
#697352
#697342
#186363