Finite Group Databasebeta

← #10848853

Subgroups — #10848853

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×#5534C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic, complemented
4C2×#4681C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic, complemented
6#69964C212yesmaximal, complemented
7#69962C212yesmaximal
8#69828C211yesmaximal, characteristic, complemented
9#69773C211yesmaximal, characteristic, complemented
10#68685C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#25613—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×#924C2211yescharacteristic
16C2×#924C2212yes—
17C22×Dic12C2212yes—
18C22×#204C2212yes—
19#5534—24no—
20#5534—24no—
21C2×#898—21no—
22C2×#896—21no—
23C2×#885—21no—
24C2×#883—21no—
25C2×#875C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#869C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C22⋊C8C2212yes—
31C2×C3×#97C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C2×#821C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#4981—21no—
38#4981—21no—
39#4953—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42#4929C2212yes—
43#4715—24no—
44#4681—24no—
45C3×#336C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#542—31no—
49C2×#497—31no—
50C2×#487—31no—
51#2621—31no—
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1958—31no—
56#1878—32no—
57#1877—32no—
58#1750D311yescharacteristic, complemented
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6—21no—
64C22×Dic6C2312yes—
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×#117—22no—
69C3×#117—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C22⋊C4D411yescharacteristic
73C2×C3×C22⋊C4D411yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77C2×#217—22no—
78#930—22no—
79#930—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90#924—22no—
91#924—22no—
92#924—42no—
93#924—42no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—22no—
97C2×C4×Dic3—21no—
98C2×Dic12—24no—
99C2×Dic12—24no—
100C2×Dic12—42no—
101C2×#204—24no—
102C2×#204—24no—
103C2×#204—42no—
104#898—28no—
105#896—28no—
106#885—28no—
107#883—28no—
108#875—22no—
109#875D414yes—
110C4×D12—22no—
111C4×D12—22no—
112#869—22no—
113#869—22no—
114#869—22no—
115#869D414yes—
116C4×Dic6—22no—
117C4×Dic6—22no—
118C3×C22⋊C8—24no—
119C3×#97—22no—
120C3×C4×C8—22no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125#821—42no—
126#816—24no—
127#816—24no—
128#816—42no—
129C22×Q16—32no—
130C22×SD16—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×C4⋊Q8—31no—
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×#117C2×D311yescharacteristic
146#497—64no—
147#487—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8C2×D312yes—
153C2×#97C2×D311yescharacteristic
154C2×C4×C8C2×D311yescharacteristic
155#398—61no—
156#398—61no—
157#397—61no—
158#397—61no—
159#391—32no—
160#382—31no—
161#340—32no—
162#340—32no—
163#336C2×D312yes—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—82no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—82no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—24no—
199C2×Dic6—82no—
200C2×Dic6C2×D412yes—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8#4114yes—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215#217—24no—
216#217—44no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221C4⋊Dic3—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—42no—
227C4⋊Dic3—42no—
228C4⋊Dic3C2×D412yes—
229C4⋊Dic3C2×D412yes—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—44no—
233C4×Dic3—24no—
234Dic12—48no—
235#204—48no—
236C22×Q8—31no—
237C22×Q8—32no—
238C22×Q8—61no—
239C22×D4—31no—
240C22×D4—61no—
241C23×C4—61no—
242C23×C4C22×D311yescharacteristic
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8C22×D312yes—
250C22×C8C22×D312yes—
251C4⋊Q8—34no—
252C4⋊Q8—62no—
253C4⋊Q8—62no—
254C4⋊Q8—62no—
255#126—68no—
256#124—34no—
257#124—68no—
258#124—62no—
259#123—68no—
260C22⋊Q8—68no—
261C22⋊Q8—62no—
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—22no—
272#117—22no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4D1211yescharacteristic
283C2×C4⋊C4D1211yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4D1211yescharacteristic
291C2×C22⋊C4D1211yescharacteristic
292C2×C42—61no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#103—64no—
296#103—64no—
297#103—122no—
298#102—122no—
299C22⋊C8—24no—
300#97—22no—
301C4×C8—22no—
302#95—61no—
303#95—61no—
304#95—62no—
305#95—62no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3C22×D411yescharacteristic
312C22×D3—22no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—42no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4#12012yes—
329C2×C3×C4#12012yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C4H(5,𝔽2)12yes—
334#67—84no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348D12—24no—
349D12—42no—
350C4×D3—84no—
351Dic6—24no—
352Dic6—42no—
353Dic6—42no—
354Dic6—42no—
355Dic6—42no—
356Dic6—42no—
357Dic6—84no—
358C3×C8—24no—
359C3×C8—42no—
360C24—61no—
361C24C2×D1211yescharacteristic
362C2×Q8—32no—
363C2×Q8—64no—
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—242no—
374C2×D4—32no—
375C2×D4—64no—
376C2×D4—62no—
377C2×D4—61no—
378C2×D4—62no—
379C2×D4—242no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—64no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—242no—
391C22×C4C2×D1211yescharacteristic
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4C2×D1211yescharacteristic
395C22×C4C2×D1211yescharacteristic
396C22×C4C23×D311yescharacteristic
397Q16—128no—
398SD16—128no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8#23514yes—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—64no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—124no—
417C4⋊C4—122no—
418C4⋊C4—122no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—64no—
426C22⋊C4—124no—
427C42—21no—
428C42—21no—
429C42—64no—
430C42D3×D412yes—
431C42D3×D412yes—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#53211yescharacteristic
439C22×C3#53511yescharacteristic
440C22×C3#57712yes—
441C22×C3#57812yes—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—82no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4#54612yes—
453C3×C4#55012yes—
454Dic3—42no—
455Dic3—42no—
456Dic3—42no—
457Dic3—82no—
458Dic3—82no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—21no—
464C23—62no—
465C23—121no—
466C23—122no—
467C23—122no—
468C23—121no—
469C23C22×D1211yescharacteristic
470Q8—64no—
471Q8—122no—
472Q8—122no—
473Q8—122no—
474Q8—122no—
475Q8—122no—
476Q8—244no—
477D4—64no—
478D4—122no—
479D4—244no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—122no—
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×C4C2×#23512yes—
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×D3×D411yescharacteristic
510C2×C4#100912yes—
511C8—24no—
512C8—42no—
513C2×C3—81no—
514C2×C3#243111yescharacteristic
515C2×C3#300911yescharacteristic
516C2×C3#301111yescharacteristic
517C2×C3#317812yes—
518C2×C3#322512yes—
519D3—82no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—242no—
529C22—242no—
530C22#547711yescharacteristic
531C22C2×#88211yescharacteristic
532C22#552511yescharacteristic
533C22#575112yes—
534C22#575312yes—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—122no—
541C4—122no—
542C4—122no—
543C4—242no—
544C4—242no—
545C4#555012yes—
546C4#555312yes—
547C3#2561311yescharacteristic, complemented, Sylow
548C2—81no—
549C2—242no—
550C2#7358611yescharacteristic
551C2#7857012yes—
552C2#7867111yescharacteristic
553C2#7867911yescharacteristic
554C2#7939312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1750D3
hypocenterC3#25613
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. #82262C2semidirect
3. C2×#4715C2semidirect
4. C2×#4681C2semidirect
5. C3×#1750C2semidirect
6. #69964C2semidirect
8. #69828C2semidirect
9. #69773C2semidirect
10. #68685C2semidirect
11. #68672C2semidirect
58. #1750D3semidirect
547. C3#25613semidirect

Maximal subgroups

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

subgroupindex
C2×#55342
#822622
C2×#47152
C2×#46812
C3×#17502
#699642
#699622
#698282
#697732
#686852
#686722
#256133