Finite Group Databasebeta

← #10849397

Subgroups — #10849397

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86205C211yesmaximal, characteristic
2C3×#2868C211yesmaximal, characteristic, complemented
3#83212C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5#71790C211yesmaximal, characteristic, complemented
6#69593C214yesmaximal, complemented
7#69592C214yesmaximal, complemented
8#69395C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#25666—31nomaximal, Sylow
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14#5817—22no—
15#5734—21no—
16#5733—22no—
17#5732—21no—
18#5730—21no—
19#5728—21no—
20#5726—21no—
21#5725—21no—
22#5514—22no—
23#5510C2212yes—
24#5504C2212yes—
25#5477C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#384—22no—
28C3×#384C2214yes—
29C3×C43C2211yescharacteristic
30#5094—21no—
31#5093—21no—
32#5091—21no—
33#5089—22no—
34#5088—21no—
35#5087—21no—
36#5084—21no—
37#4966C2212yes—
38#4966C2214yes—
39#4964C2212yes—
40C2×#805C2212yes—
41C2×#805C2214yes—
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4530C2214yes—
45#4523C2212yes—
46#3511—31no—
47#3141—31no—
48#3079—31no—
49#2868D311yescharacteristic, complemented
50#2039—34no—
51#1997—31no—
52#1962—31no—
53#1859—34no—
54C4×#104—31no—
55C22×C3×Q8—21no—
56C22×C3×Q8D412yes—
57C2×#239—22no—
58C2×#235C2311yescharacteristic
59C2×#235C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—48no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—28no—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78#947—22no—
79#947—24no—
80#946—22no—
81#946—24no—
82Q8×Dic3—22no—
83#930—21no—
84#930—21no—
85#930—24no—
86C4×#67—21no—
87C4×#67—21no—
88C4×#67—24no—
89#926C2311yescharacteristic
90C2×#208C2312yes—
91C2×#208C2312yes—
92#898—24no—
93#897—22no—
94#897—22no—
95#895—22no—
96#895—22no—
97#895—22no—
98#894—24no—
99#893—22no—
100#893—22no—
101#892—22no—
102#892—22no—
103D3×C4⋊C4—22no—
104#890—22no—
105#887—22no—
106#875D412yes—
107#874D411yescharacteristic
108C4×D12—21no—
109C4×D12—21no—
110C4×D12D411yescharacteristic
111C4×D12D411yescharacteristic
112#872—21no—
113C42×D3—21no—
114#869D411yescharacteristic
115C4×Dic6D411yescharacteristic
116C4×Dic6D411yescharacteristic
117#810—24no—
118#810—22no—
119#809—22no—
120#809—24no—
121#808—22no—
122#807—22no—
123#805—24no—
124#805—28no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127#803—21no—
128C4×C3⋊C8—21no—
129#556—61no—
130#555—61no—
131#554—61no—
132#553—62no—
133#548—62no—
134#547—61no—
135#542—62no—
136#537—32no—
137#533—61no—
138#532—31no—
139#532—61no—
140C2×C4⋊Q8—22no—
141C2×C4⋊Q8C2×D311yescharacteristic
142C2×C4⋊Q8C2×D311yescharacteristic
143#520—32no—
144C4×#41—31no—
145#489—62no—
146#487—61no—
147#486—61no—
148#477—61no—
149#476—61no—
150#475—61no—
151#474—61no—
152#428—32no—
153#423—31no—
154C2×#104—34no—
155C2×#104—32no—
156#417—32no—
157#417—34no—
158C4×M16—31no—
159#384—22no—
160#384C2×D314yes—
161C43C2×D311yescharacteristic
162#343—34no—
163#339—32no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—44no—
167C2×C3×Q8—44no—
168C2×C3×Q8—41no—
169C2×C3×Q8—41no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—21no—
177C2×#67—22no—
178#239—84no—
179#235—22no—
180#235—24no—
181#235—22no—
182#235—24no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—24no—
186C2×D12C2×D411yescharacteristic
187C2×D12C2×D412yes—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—22no—
192C2×C4×D3—24no—
193C2×Dic6C2×D411yescharacteristic
194C2×Dic6C2×D412yes—
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—416no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C42—24no—
205C3×C42—24no—
206C3×C42—22no—
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42C2×D411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42#4114yescomplemented
212#217—21no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220C4⋊Dic3—24no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D411yescharacteristic
224C4⋊Dic3C2×D411yescharacteristic
225Dic3⋊C4—24no—
226Dic3⋊C4—21no—
227Dic3⋊C4—21no—
228C4×Dic3—24no—
229C4×Dic3—21no—
230C4×Dic3—21no—
231C4×Dic3—21no—
232#208—24no—
233#208—22no—
234#208—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×#41—31no—
238C2×#41—32no—
239C2×#41—62no—
240C22×Q8—21no—
241C22×Q8#6712yes—
242C2×M16—32no—
243C2×M16—32no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—31no—
253C4⋊Q8—48no—
254#127—31no—
255#126—64no—
256#125—62no—
257#124—32no—
258#124—62no—
259#124—64no—
260#123—64no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C22⋊Q8—64no—
265#121—62no—
266#121—62no—
267#121—62no—
268#121—61no—
269#121—61no—
270#121—64no—
271C4×Q8—31no—
272C4×Q8—31no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—31no—
276C4×D4—31no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—64no—
282C4×D4—61no—
283C4×D4—61no—
284#117—31no—
285#117—62no—
286#117—62no—
287#117—61no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4#6712yes—
292C2×C4⋊C4C22×D312yes—
293C2×C4⋊C4C22×D312yes—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—61no—
297C2×C42C22×D312yes—
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#106—62no—
301C4⋊C8—61no—
302C4⋊C8—61no—
303#104—68no—
304#104—64no—
305#103—62no—
306#103—64no—
307#102—62no—
308#102—64no—
309#97—61no—
310C4×C8—61no—
311C23×C3C22×D411yescharacteristic
312C22×D3—21no—
313C22×D3—22no—
314C3×Q8—44no—
315C3×Q8—82no—
316C3×Q8—82no—
317C3×Q8—82no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—42no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C42−1+412yes—
335#67—42no—
336#67—44no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—22no—
340C2×Dic3—21no—
341D12—21no—
342D12—21no—
343D12—24no—
344D12—42no—
345D12—42no—
346D12—84no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—84no—
353Dic6—21no—
354Dic6—21no—
355Dic6—24no—
356C3⋊C8—42no—
357C3⋊C8—42no—
358#41—64no—
359#41—62no—
360#41—64no—
361#41—62no—
362#41—244no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—31no—
366C2×Q8—32no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×D4—31no—
372C2×D4—32no—
373C2×D4—64no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—61no—
377C2×D4—61no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—64no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—62no—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6714yes—
388C22×C4C23×D311yescharacteristic
389M16—64no—
390M16—62no—
391M16—62no—
392C2×C8—62no—
393C2×C8—62no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—31no—
401C4⋊C4—31no—
402C4⋊C4—416no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—64no—
406C4⋊C4—64no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—61no—
415C22⋊C4—61no—
416C22⋊C4—61no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—64no—
423C42—61no—
424C42—61no—
425C42—61no—
426C42D3×D411yescharacteristic
427C42D3×D411yescharacteristic
428C42D3×D411yescharacteristic
429C42D3×D411yescharacteristic
430C42#23914yescomplemented
431C22×C3—21no—
432C22×C3—21no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54812yes—
435C2×D3—42no—
436C2×D3—41no—
437C2×D3—41no—
438C2×D3—42no—
439C2×D3—41no—
440C2×D3—42no—
441C3×C4—28no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—44no—
445C3×C4—82no—
446C3×C4#45312yes—
447C3×C4#45612yes—
448C3×C4#53211yescharacteristic
449C3×C4#53611yescharacteristic
450C3×C4#54712yes—
451Dic3—41no—
452Dic3—42no—
453Dic3—41no—
454Dic3—41no—
455C23—61no—
456C23—62no—
457C23C22×#6711yescharacteristic
458Q8—44no—
459Q8—61no—
460Q8—61no—
461Q8—64no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465D4—61no—
466D4—61no—
467D4—64no—
468D4—124no—
469D4—122no—
470D4—122no—
471D4—122no—
472D4—244no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—244no—
492C2×C4#93711yescharacteristic
493C2×C4#93711yescharacteristic
494C2×C4#93711yescharacteristic
495C2×C4#93711yescharacteristic
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23912yes—
499C2×C4#100712yes—
500C8—122no—
501C8—122no—
502C2×C3—22no—
503C2×C3#243211yescharacteristic
504C2×C3C2×#45311yescharacteristic
505C2×C3C2×#45611yescharacteristic
506D3—81no—
507D3—82no—
508C22—21no—
509C22—21no—
510C22—121no—
511C22—122no—
512C22—122no—
513C22—121no—
514C22—121no—
515C22—122no—
516C22C2×#93711yescharacteristic
517C22#581712yes—
518C4—28no—
519C4—21no—
520C4—21no—
521C4—44no—
522C4—82no—
523C4—121no—
524C4—122no—
525C4—121no—
526C4—121no—
527C4#507712yes—
528C4#509512yes—
529C4#572411yescharacteristic
530C4#572511yescharacteristic
531C4#572612yes—
532C3#2566611yescharacteristic, complemented, Sylow
533C2—22no—
534C2—241no—
535C2—242no—
536C2C2×#507711yescharacteristic
537C2C2×#509511yescharacteristic
538C2#8325511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2868C2
socleC22×C3C2×#120
prime radical#86205C2
hypercenter#2868D3
hypocenterC3#25666
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
2. C3×#2868C2semidirect
3. #83212C2semidirect
4. #83199C2semidirect
5. #71790C2semidirect
6. #69593C2semidirect
7. #69592C2semidirect
8. #69395C2semidirect
9. C4×#805C2semidirect
49. #2868D3semidirect
211. C3×C42#41semidirect
430. C42#239semidirect
532. C3#25666semidirect

Maximal subgroups

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

subgroupindex
#862052
C3×#28682
#832122
#831992
#717902
#695932
#695922
#693952
C4×#8052
#256663