Finite Group Databasebeta

← #10691632

Subgroups — #10691632

Order 768.

966 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
1D4×#217C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic
3C3×#1786C211yesmaximal, characteristic, complemented
4#69944C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6#67678C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#217—21no—
11C2×D4×Dic3—21no—
12C22×#213C412yescomplemented
13C2×#930C2211yescharacteristic
14C2×C3×#102—22no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5247—21no—
18#5221—21no—
19#5220—22no—
20#5217—21no—
21#5217—21no—
22#5217—22no—
23C2×#832C412yescomplemented
24#5212C2211yescharacteristic
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#806—21no—
28C2×#806—21no—
29#4470—22no—
30C2×#447—31no—
31D4×C22⋊C4—31no—
32#1964—31no—
33#1786D311yescharacteristic, complemented
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C25×C3—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—44no—
42C22×C3×D4C2×C412yes—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C22×C3×D4D412yes—
46C23×C3×C4C2311yescharacteristic
47C22×#67—21no—
48C23×Dic3—22no—
49C22×D12C2×C411yescharacteristic, complemented
50C22×C3×C8—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—21no—
56C2×#217—22no—
57C2×#217—42no—
58C2×#217—42no—
59C2×#217D411yescharacteristic
60C2×#217D411yescharacteristic
61D4×Dic3—44no—
62C2×#213—22no—
63C2×#213—22no—
64C2×#213—44no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3C2×C411yescharacteristic, complemented
72C2×C4×Dic3—21no—
73C22×C3⋊C8C2×C412yescomplemented
74C3×#102—44no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95—21no—
78#832—22no—
79#832—22no—
80C3⋊#95—21no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#806—42no—
85#806—42no—
86C23×D4C2×D311yescharacteristic
87C22×D8—32no—
88C2×#121—31no—
89C2×C4×D4—61no—
90C22×C22⋊C4—61no—
91#447—61no—
92#447—61no—
93#447—62no—
94C2×#102—22no—
95C2×#102—32no—
96C2×#99—61no—
97C2×#99—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#390—61no—
101#386—62no—
102#380—31no—
103#380C2×D311yescharacteristic
104#379—61no—
105C4×C22⋊C4—31no—
106#327—62no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—44no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3—82no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4C22⋊C412yes—
133C2×C3×D4C22⋊C412yes—
134C22×C3×C4—42no—
135C22×C3×C4—41no—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3—44no—
153C22×Dic3—44no—
154C22×Dic3—82no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—42no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C4⋊C4—42no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163#217—21no—
164#217—21no—
165#217—21no—
166#217—44no—
167#217—44no—
168#217—41no—
169#217—41no—
170#217—82no—
171#217D812yescomplemented
172#213—44no—
173#213—44no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—42no—
179C4×Dic3—42no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—42no—
182C25—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—31no—
187C22×D4—44no—
188C22×D4—61no—
189C22×D4C4×D312yes—
190C22×D4D1211yescharacteristic
191C22×D4D1211yescharacteristic
192C22×D4#6712yes—
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—62no—
196C2×D8—62no—
197C2×D8—124no—
198C22×C8—21no—
199C22×C8—32no—
200#121—61no—
201#121—61no—
202#121—61no—
203#121—61no—
204C4×D4—124no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—31no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—122no—
217C2×C22⋊C4—122no—
218C2×C42—61no—
219#102—44no—
220#102—62no—
221#102—62no—
222#99—122no—
223#99—122no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—21no—
229#95—32no—
230#95—61no—
231C23×C3—24no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—44no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—82no—
250C23×C3—82no—
251C23×C3—84no—
252C23×C3—84no—
253C23×C3—84no—
254C23×C3—84no—
255C23×C3C4×D411yescharacteristic
256C23×C3#12011yescharacteristic
257C23×C3#12111yescharacteristic
258C23×C3C2×D811yescharacteristic
259C23×C3#13511yescharacteristic
260C23×C3AGL(1,ℤ/8)11yescharacteristic
261C23×C3AGL(1,ℤ/8)11yescharacteristic
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—81no—
266C22×D3—81no—
267C3×D4—24no—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—44no—
272C3×D4—44no—
273C3×D4—42no—
274C3×D4#9914yescomplemented
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4C2×C22⋊C411yescharacteristic
285C2×C3×C4C4×D411yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4#12311yescharacteristic
288#67—81no—
289#67—81no—
290#67—81no—
291#67—81no—
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—88no—
305D12—42no—
306D12—81no—
307C3×C8—81no—
308C3⋊C8—42no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—44no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—61no—
319C24—82no—
320C24D3×D411yescharacteristic
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—122no—
337C2×D4—121no—
338C2×D4—121no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4(C2×D3)⋊C412yes—
344C2×D4(C2×D3)⋊C412yes—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—124no—
352C22×C4—124no—
353C22×C4—242no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4#23711yescharacteristic
358C22×C4#23711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360D8—124no—
361D8—124no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—62no—
365C2×C8—122no—
366C4⋊C4—42no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—122no—
370C22⋊C4—32no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—242no—
383C42—122no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—24no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—84no—
403C22×C3—84no—
404C22×C3—84no—
405C22×C3—84no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3#38611yescharacteristic
409C22×C3C4×D811yescharacteristic
410C22×C3#44211yescharacteristic
411C22×C3#44711yescharacteristic
412C22×C3#44911yescharacteristic
413C22×C3#45911yescharacteristic
414C22×C3#46311yescharacteristic
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—161no—
419C2×D3—161no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—161no—
424C3×C4C2×#9912yes—
425Dic3—42no—
426Dic3—81no—
427Dic3—82no—
428Dic3—81no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—241no—
457C23—241no—
458C23#67⋊C411yescharacteristic
459C23#88211yescharacteristic
460C23D3×D811yescharacteristic
461C23#91111yescharacteristic
462C23#91411yescharacteristic
463C23#91611yescharacteristic
464C23#93911yescharacteristic
465D4—24no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—44no—
470D4—44no—
471D4—42no—
472D4—122no—
473D4—241no—
474D4—241no—
475D4—241no—
476D4—241no—
477D4—241no—
478D4#80614yescomplemented
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—62no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—248no—
501C2×C4#67⋊C411yescharacteristic
502C2×C4#88611yescharacteristic
503C2×C4C2×(C2×D3)⋊C411yescharacteristic
504C2×C4#93911yescharacteristic
505C8—81no—
506C8—122no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—44no—
514C2×C3—82no—
515C2×C3#187911yescharacteristic
516C2×C3#189811yescharacteristic
517C2×C3#190411yescharacteristic
518D3—161no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—24no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—82no—
542C22—82no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22—481no—
547C22—481no—
548C22#475611yescharacteristic
549C22#477311yescharacteristic
550C22#478111yescharacteristic
551C22#478511yescharacteristic
552C22#495811yescharacteristic
553C22#515011yescharacteristic
554C22#516511yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—122no—
559C4—161no—
560C4—242no—
561C4—241no—
562C4—241no—
563C4C2×#80612yes—
564C3#925911yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—44no—
572C2—82no—
573C2—481no—
574C2#6949611yescharacteristic
575C2#7017211yescharacteristic
576C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1786C2
socleC22×C3#386
prime radicalC2×#5217C2
cosocleC23×C3×C4C23
hypercenter#1786D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. D4×#217C2semidirect
3. C3×#1786C2semidirect
4. #69944C2semidirect
5. #69528C2semidirect
6. #67678C2semidirect
7. #67625C2semidirect
12. C22×#213C4semidirect
23. C2×#832C4semidirect
33. #1786D3semidirect
49. C22×D12C2×C4semidirect
71. C2×C4⋊Dic3C2×C4semidirect
73. C22×C3⋊C8C2×C4semidirect
171. #217D8semidirect
274. C3×D4#99semidirect
478. D4#806semidirect
564. C3#9259semidirect

Maximal subgroups

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

subgroupindex
D4×#2172
C2×#52172
C3×#17862
#699442
#695282
#676782
#676252
#92593