Finite Group Databasebeta

← #10848856

Subgroups — #10848856

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5733C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic, complemented
4#74193C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69830C211yesmaximal, characteristic, complemented
7#69551C211yesmaximal, characteristic, complemented
8#68908C212yesmaximal, complemented
9#68896C212yesmaximal, complemented
10#68888C212yesmaximal, complemented
11#68887C212yesmaximal
12#25613—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—21no—
15C2×#946—21no—
16C22×Dic12C2212yes—
17C22×#204C2212yes—
18#5733—24no—
19#5733—24no—
20C2×#898—21no—
21C2×#897C2211yescharacteristic
22C2×#897C2212yes—
23C2×#895C2211yescharacteristic
24C2×#894—21no—
25C2×#893C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#875C2211yescharacteristic
28C2×#872C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C3×#382C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33#5093—24no—
34#4973C2212yes—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×C4×C3⋊C8C2211yescharacteristic
40#4715—24no—
41#4674—21no—
42#4674—21no—
43#4661—21no—
44#4661—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47#4550C2212yes—
48C2×#542—31no—
49C2×#497—31no—
50C2×#487—31no—
51#2621—31no—
52#2001—32no—
53#2000—32no—
54#1993D311yescharacteristic, complemented
55#1958—31no—
56#1878—32no—
57#1877—32no—
58#1750—31no—
59C22×C3×Q8—21no—
60C22×D12—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C4×D3C2311yescharacteristic
64C22×Dic6C2312yes—
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74#947—28no—
75#946—28no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D411yescharacteristic
83C2×(C2×D3)⋊C4D411yescharacteristic
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4D411yescharacteristic
90C2×Dic3⋊C4D411yescharacteristic
91C2×C4×Dic3—21no—
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C2×Dic12—24no—
95C2×Dic12—24no—
96C2×Dic12—42no—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—42no—
100#898—28no—
101#897—22no—
102#897—22no—
103#897—42no—
104#897—42no—
105#895—22no—
106#895—22no—
107#894—28no—
108#893—22no—
109#893—22no—
110#887—22no—
111#887—22no—
112#875—22no—
113#875D414yes—
114#872—22no—
115#872—22no—
116#869—22no—
117#869—22no—
118C3×#97—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123(C2×D3)⋊C8—24no—
124#810—24no—
125#810—24no—
126#810—42no—
127#809—42no—
128C4×C3⋊C8—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×#126—61no—
136C2×#124—31no—
137C2×#124—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—32no—
141C2×C22⋊Q8—61no—
142C2×#121—31no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146#497—64no—
147#487—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97C2×D311yescharacteristic
154C2×C4×C8—31no—
155#398—61no—
156#398—61no—
157#397—61no—
158#397—61no—
159#391—32no—
160#382C2×D311yescharacteristic
161#340C2×D312yes—
162#340C2×D312yes—
163#336—32no—
164C23×D3—21no—
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—82no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12—24no—
183C2×D12—82no—
184C2×D12C2×D412yes—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—24no—
190C2×C4×D3—82no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—21no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6C2×D412yes—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8#4114yes—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—24no—
222C4⋊Dic3—44no—
223C4⋊Dic3—42no—
224C4⋊Dic3—42no—
225Dic3⋊C4—24no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—24no—
229C4×Dic3—21no—
230C4×Dic3—21no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234Dic12—48no—
235#204—48no—
236C22×Q8—21no—
237C22×Q8—31no—
238C22×Q8—32no—
239C22×D4—31no—
240C22×D4—61no—
241C23×C4—31no—
242C23×C4—61no—
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8C22×D312yes—
251C4⋊Q8—22no—
252C4⋊Q8—62no—
253C4⋊Q8—62no—
254C4⋊Q8D1214yes—
255#126—68no—
256#124—34no—
257#124—68no—
258#124—62no—
259#123—68no—
260C22⋊Q8—62no—
261C22⋊Q8—68no—
262C22⋊Q8—62no—
263C22⋊Q8—122no—
264C22⋊Q8—122no—
265#121—62no—
266#121—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×D4—62no—
270C4×D4—62no—
271#117—62no—
272#117—62no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4C22×D312yes—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—62no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#103—64no—
296#103—64no—
297#103—122no—
298#102—122no—
299C22⋊C8—64no—
300#97—22no—
301C4×C8—62no—
302#95—21no—
303#95—21no—
304#95—62no—
305#95—62no—
306C23×C3C22×D411yescharacteristic
307C22×D3—21no—
308C22×D3—22no—
309C22×D3—21no—
310C22×D3—21no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—42no—
317C3×Q8—84no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12012yes—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C4H(5,𝔽2)12yes—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346D12—24no—
347D12—42no—
348D12—84no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—84no—
352Dic6—24no—
353Dic6—42no—
354Dic6—42no—
355Dic6—42no—
356Dic6—42no—
357Dic6—42no—
358C3×C8—24no—
359C3⋊C8—42no—
360C24—31no—
361C24—61no—
362C2×Q8—24no—
363C2×Q8—32no—
364C2×Q8—61no—
365C2×Q8—62no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8—82no—
374C2×D4—32no—
375C2×D4—64no—
376C2×D4—61no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—242no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—64no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—242no—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4C23×D311yescharacteristic
397Q16—128no—
398SD16—128no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8#23914yes—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—124no—
415C4⋊C4—122no—
416C4⋊C4—122no—
417C4⋊C4C2×D1212yes—
418C4⋊C4C2×D1212yes—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—62no—
423C22⋊C4—62no—
424C22⋊C4—64no—
425C22⋊C4—64no—
426C22⋊C4—124no—
427C42—64no—
428C42—61no—
429C42—61no—
430C42C2×D1212yes—
431C42D3×D412yes—
432C22×C3C2×#12011yescharacteristic
433C22×C3#53211yescharacteristic
434C22×C3#53511yescharacteristic
435C22×C3#57712yes—
436C22×C3#57812yes—
437C2×D3—42no—
438C2×D3—41no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—82no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—82no—
452C3×C4#54612yes—
453C3×C4#55012yes—
454Dic3—42no—
455Dic3—41no—
456Dic3—41no—
457Dic3—41no—
458Dic3—82no—
459C23—61no—
460C23—61no—
461C23—62no—
462C23—61no—
463C23—62no—
464C23—62no—
465C23—121no—
466C23—121no—
467C23—122no—
468C23—122no—
469C23C2×D3×D411yescharacteristic
470Q8—64no—
471Q8—84no—
472Q8—122no—
473Q8—122no—
474Q8—122no—
475Q8—122no—
476Q8—122no—
477D4—64no—
478D4—122no—
479D4—244no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—244no—
505C2×C4#88212yes—
506C2×C4#88212yes—
507C2×C4C22×D1211yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×#23912yes—
510C2×C4#100912yes—
511C8—24no—
512C8—122no—
513C2×C3#243111yescharacteristic
514C2×C3#300911yescharacteristic
515C2×C3#301111yescharacteristic
516C2×C3#317812yes—
517C2×C3#322512yes—
518D3—81no—
519D3—82no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—121no—
524C22—121no—
525C22—122no—
526C22—121no—
527C22—122no—
528C22—242no—
529C22—242no—
530C22C2×#88211yescharacteristic
531C22#567411yescharacteristic
532C22#572511yescharacteristic
533C22#577512yes—
534C22#577912yes—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—42no—
539C4—82no—
540C4—121no—
541C4—121no—
542C4—122no—
543C4—121no—
544C4—242no—
545C4#557712yes—
546C4#563012yes—
547C3#2561311yescharacteristic, complemented, Sylow
548C2—241no—
549C2—242no—
550C2#7662611yescharacteristic
551C2#7892911yescharacteristic
552C2#7896111yescharacteristic
553C2#8168912yes—
554C2#8196012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1993D3
hypocenterC3#25613
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#5093C2semidirect
3. C2×#4715C2semidirect
4. #74193C2semidirect
5. C3×#1993C2semidirect
6. #69830C2semidirect
7. #69551C2semidirect
8. #68908C2semidirect
9. #68896C2semidirect
10. #68888C2semidirect
54. #1993D3semidirect
547. C3#25613semidirect

Maximal subgroups

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

subgroupindex
C2×#57332
C2×#50932
C2×#47152
#741932
C3×#19932
#698302
#695512
#689082
#688962
#688882
#688872
#256133