Finite Group Databasebeta

← #10847368

Subgroups — #10847368

Order 768.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#4778C212yesmaximal, complemented
3#76287C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71768C211yesmaximal, characteristic, complemented
6#70202C212yesmaximal, complemented
7#70171C212yesmaximal, complemented
8#70169C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68689C211yesmaximal, characteristic, complemented
11D3×#336C211yesmaximal, characteristic, complemented
12#25495—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×#938—21no—
16C2×#938C2212yes—
17C2×#936—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20C22×#214C2212yes—
21C22×D3×C8C2212yes—
22#5716—28no—
23C2×#897C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C3×#389C2211yescharacteristic
29C2×#832C2212yes—
30C2×#816C2212yes—
31#4989—21no—
32#4987C2211yescharacteristic
33#4980—21no—
34#4973C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#808C2212yes—
37#4929C2211yescharacteristic
38#4778—28no—
39#4664—21no—
40#4655—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#465—32no—
48#2855—31no—
49#2058—31no—
50#2041D311yescharacteristic, complemented
51#2024—31no—
52#2014—31no—
53#1893—32no—
54#1877—32no—
55#1864—32no—
56C2×#336—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—44no—
64C2×#237—44no—
65C22×C4×D3D411yescharacteristic
66C22×C4×D3D411yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—21no—
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×#127—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78#938—28no—
79#938—24no—
80#938D418yes—
81#936—28no—
82D4×Dic3—28no—
83D4×Dic3—24no—
84D4×Dic3—24no—
85C2×#214—24no—
86C2×#214—24no—
87C2×#214—42no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C2×D3×C8—24no—
100C2×D3×C8—24no—
101C2×D3×C8—42no—
102#897—24no—
103C42×D3—24no—
104#869—24no—
105C3×#102—24no—
106C3×#102—42no—
107C3×#95—21no—
108#832—24no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115#816—24no—
116#808—24no—
117C22×#41—61no—
118C22×SD16—32no—
119C23×C8—32no—
120#532—68no—
121C2×C4⋊Q8—31no—
122C2×#127C2×D311yescharacteristic
123C2×#124—61no—
124C2×C22⋊Q8—31no—
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×D4—32no—
128C2×C4×D4—61no—
129C22×C42—31no—
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×#102C2×D312yes—
136#391—31no—
137#391—31no—
138#389—31no—
139#389C2×D311yescharacteristic
140#387—61no—
141#387—61no—
142#385—61no—
143#381—61no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336—32no—
148#336—31no—
149#336C2×D311yescharacteristic
150C24×C3—22no—
151C23×D3C2×D411yescharacteristic
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4C2×D414yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—28no—
170C2×#67—44no—
171C2×#67—44no—
172C22×Dic3—28no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—44no—
177C22×Dic3—44no—
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181#237—432no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—41no—
190C2×C4×D3SD1614yescomplemented
191C2×C4×D3C2×D414yes—
192C2×Dic6—24no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—42no—
198C2×Dic6—42no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—42no—
206C3×C42—22no—
207#217—28no—
208#217—28no—
209#214—48no—
210(C2×D3)⋊C4—42no—
211(C2×D3)⋊C4—42no—
212C4⋊Dic3—24no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—42no—
216C4⋊Dic3C2×D414yes—
217Dic3⋊C4—42no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8#4114yescomplemented
224D3×C8—48no—
225C2×#41—68no—
226C2×#41—124no—
227C2×#41—124no—
228C22×Q8—31no—
229C22×Q8—61no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—62no—
233C22×D4C22×D312yes—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—62no—
238C2×SD16—64no—
239C2×SD16—64no—
240C2×SD16—122no—
241C22×C8—32no—
242C22×C8—64no—
243C22×C8—64no—
244C22×C8—122no—
245C22×C8C22×D312yes—
246C4⋊Q8—64no—
247#127—24no—
248#124—68no—
249C22⋊Q8—64no—
250#121—38no—
251#121—64no—
252#121—68no—
253C4×D4—64no—
254C4×D4—68no—
255C4×D4—64no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C42—31no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—64no—
272C2×C42C22×D311yescharacteristic
273#106—64no—
274#103—64no—
275#102—24no—
276#102—42no—
277#102—64no—
278#95—21no—
279#95—61no—
280#95—61no—
281#95—61no—
282#95—61no—
283#95—61no—
284#95—61no—
285C23×C3—24no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3C22×D411yescharacteristic
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—21no—
297C22×D3#12012yes—
298C22×D3C2×SD1612yes—
299C22×D3AGL(1,ℤ/8)12yes—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—48no—
303C3×D4—48no—
304C3×D4—42no—
305C3×D4—42no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4C2×SD1612yes—
317C2×C3×C4#13512yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C4C2×#4112yes—
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322#67—416no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—24no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—416no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3#12012yes—
341C2×Dic3C2×SD1612yes—
342C4×D3—28no—
343C4×D3—24no—
344C4×D3—24no—
345C4×D3—44no—
346Dic6—48no—
347Dic6—42no—
348Dic6—42no—
349C3×C8—42no—
350C3⋊C8—24no—
351C24—22no—
352C24—31no—
353#41—1232no—
354C2×Q8—64no—
355C2×Q8—62no—
356C2×Q8—61no—
357C2×Q8—62no—
358C2×Q8—61no—
359C2×Q8—122no—
360C2×Q8—122no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—68no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4D3×D414yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—34no—
381C22×C4—34no—
382C22×C4—31no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—68no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—121no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C23×D311yescharacteristic
400SD16—128no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—34no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—128no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—34no—
412C4⋊C4—42no—
413C4⋊C4—64no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—122no—
417C4⋊C4—122no—
418C22⋊C4—68no—
419C22⋊C4—68no—
420C22⋊C4—122no—
421C22⋊C4—122no—
422C42—22no—
423C42—62no—
424C42—62no—
425C42—62no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#53211yescharacteristic
435C22×C3#53711yescharacteristic
436C22×C3C22×SD1611yescharacteristic
437C22×C3C2×#13511yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3#57511yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3#45014yes—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—81no—
450C3×C4—81no—
451C3×C4#54612yes—
452C3×C4#54712yes—
453Dic3—22no—
454Dic3—22no—
455Dic3—44no—
456Dic3—41no—
457Dic3—81no—
458Dic3—81no—
459C23—24no—
460C23—32no—
461C23—32no—
462C23—32no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23C2×D3×D411yescharacteristic
474Q8—128no—
475Q8—122no—
476Q8—122no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—42no—
482D4—42no—
483D4—1216no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—32no—
489C2×C4—32no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—68no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—64no—
507C2×C4—124no—
508C2×C4—1216no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4#91212yes—
518C2×C4D3×SD1612yes—
519C2×C4C2×D3×D412yes—
520C2×C4#100411yescharacteristic
521C2×C4D3×#4112yes—
522C2×C4#101011yescharacteristic
523C8—42no—
524C8—64no—
525C2×C3—84no—
526C2×C3#243011yescharacteristic
527C2×C3C2×#45011yescharacteristic
528C2×C3#300211yescharacteristic
529C2×C3#316711yescharacteristic
530C2×C3#318011yescharacteristic
531C2×C3#324611yescharacteristic
532C2×C3#324711yescharacteristic
533D3—24no—
534C22—34no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22D3×#12011yescharacteristic
549C22#560811yescharacteristic
550C22#566811yescharacteristic
551C22C2×#91211yescharacteristic
552C22C2×D3×SD1611yescharacteristic
553C22D3×AGL(1,ℤ/8)11yescharacteristic
554C22#577311yescharacteristic
555C4—22no—
556C4—22no—
557C4—62no—
558C4—62no—
559C4—81no—
560C4—81no—
561C4—124no—
562C4—121no—
563C4—241no—
564C4—241no—
565C4#561012yes—
566C4#565512yes—
567C3#2549511yescharacteristic, complemented, Sylow
568C2—64no—
569C2—84no—
570C2#7711611yescharacteristic
571C2#8000111yescharacteristic
572C2D3×#45011yescharacteristic
573C2#8034311yescharacteristic
574C2#8036611yescharacteristic
575C2#8155611yescharacteristic
576C2#8170711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2041D3
hypocenterC3#25495
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#4778C2semidirect
3. #76287C2semidirect
4. C3×#2041C2semidirect
5. #71768C2semidirect
6. #70202C2semidirect
7. #70171C2semidirect
8. #70169C2semidirect
9. #69605C2semidirect
10. #68689C2semidirect
11. D3×#336C2semidirect
50. #2041D3semidirect
190. C2×C4×D3SD16semidirect
223. C2×C3⋊C8#41semidirect
567. C3#25495semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#47782
#762872
C3×#20412
#717682
#702022
#701712
#701692
#696052
#686892
D3×#3362
#254953