Finite Group Databasebeta

← #10691637

Subgroups — #10691637

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
1C2×#5036C211yesmaximal, characteristic
2C22⋊C4×D12C211yesmaximal, characteristic, complemented
3C3×#1964C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic, complemented
5#69521C211yesmaximal, characteristic, complemented
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×#213C412yescomplemented
12C2×D3×C22⋊C4—21no—
13C2×C4×D12—21no—
14C2×C3×#99—21no—
15C2×C3×#99—21no—
16C3×#380C2211yescharacteristic
17C3×C4×C22⋊C4C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5036—21no—
22#5036—22no—
23#5036—21no—
24C2×#809—22no—
25C2×#809C412yescomplemented
26#4955—21no—
27#4939—21no—
28#4665—22no—
29#4537—22no—
30C2×#447—31no—
31D4×C22⋊C4—31no—
32#1964D311yescharacteristic, complemented
33#1786—31no—
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C24×D3—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C411yescharacteristic, complemented
40C23×C3×C4C2311yescharacteristic
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—44no—
45C22×D12C2×C412yes—
46C22×D12D411yescharacteristic
47C22×D12D412yes—
48C22×D12D411yescharacteristic
49C22×C4×D3—22no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C22⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#213—22no—
63C2×#213—22no—
64C2×#213—44no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8C2×C412yescomplemented
71D3×C22⋊C4—42no—
72D3×C22⋊C4—42no—
73C4×D12—44no—
74C3×#99—42no—
75C3×#99—42no—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95D412yes—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#809—22no—
84#809—22no—
85#809—44no—
86C23×D4—31no—
87C22×D8—32no—
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—61no—
90C22×C22⋊C4—61no—
91#447—61no—
92#447—62no—
93#447—61no—
94C2×#102—32no—
95C2×#102—62no—
96C2×#99—21no—
97C2×#99—21no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100#390—61no—
101#386—62no—
102#380—31no—
103#380C2×D311yescharacteristic
104#379—61no—
105C4×C22⋊C4C2×D311yescharacteristic
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C23×D3—21no—
111C23×D3—22no—
112C23×D3—22no—
113C23×D3—44no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3—41no—
118C23×D3—82no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—41no—
133C22×C3×C4C22×C411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×Dic3—22no—
140C2×D12—21no—
141C2×D12—24no—
142C2×D12—24no—
143C2×D12—24no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—21no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12—42no—
150C2×D12—44no—
151C2×D12—44no—
152C2×D12—44no—
153C2×D12C22⋊C412yes—
154C2×D12C22⋊C412yes—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—82no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—42no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4D812yescomplemented
169C3×C42—42no—
170#217—41no—
171#217—41no—
172#213—44no—
173#213—44no—
174(C2×D3)⋊C4—44no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177C4⋊Dic3—42no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—42no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8—41no—
182C25—61no—
183C22×D4—21no—
184C22×D4—32no—
185C22×D4—31no—
186C22×D4—32no—
187C22×D4—31no—
188C22×D4—61no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—124no—
192C22×D4C4×D311yescharacteristic, complemented
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—62no—
196C2×D8—62no—
197C2×D8—124no—
198C22×C8—32no—
199C22×C8—61no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C4×D4—124no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C4×D311yescharacteristic, complemented
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—122no—
215C2×C22⋊C4—122no—
216C2×C22⋊C4#6711yescharacteristic
217C2×C22⋊C4#6711yescharacteristic
218C2×C42—21no—
219#102—62no—
220#102—62no—
221#102—124no—
222#99—42no—
223#99—42no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—21no—
229#95—21no—
230#95—32no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—81no—
237C23×C3—81no—
238C23×C3C4×D411yescharacteristic
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3C2×D811yescharacteristic
242C23×C3#13511yescharacteristic
243C23×C3AGL(1,ℤ/8)11yescharacteristic
244C23×C3AGL(1,ℤ/8)11yescharacteristic
245C22×D3—24no—
246C22×D3—22no—
247C22×D3—22no—
248C22×D3—44no—
249C22×D3—44no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—82no—
262C22×D3—82no—
263C22×D3—84no—
264C22×D3—84no—
265C22×D3—84no—
266C22×D3—84no—
267C3×D4—42no—
268C3×D4—81no—
269C3×D4—81no—
270C3×D4—81no—
271C3×D4—81no—
272C3×D4—81no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—81no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C4×D411yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12311yescharacteristic
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298D12—24no—
299D12—22no—
300D12—22no—
301D12—22no—
302D12—44no—
303D12—44no—
304D12—42no—
305D12#9914yescomplemented
306C4×D3—88no—
307C3⋊C8—42no—
308C3⋊C8—81no—
309C24—21no—
310C24—61no—
311C24—61no—
312C24—62no—
313C24—62no—
314C24—124no—
315C24—121no—
316C24—121no—
317C24—121no—
318C24—121no—
319C24—242no—
320C24C2×#6711yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—32no—
324C2×D4—32no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—64no—
333C2×D4—64no—
334C2×D4—64no—
335C2×D4—62no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—62no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4—122no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—124no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—62no—
351C22×C4—124no—
352C22×C4—124no—
353C22×C4—242no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4#23511yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23711yescharacteristic
360D8—124no—
361D8—124no—
362C2×C8—62no—
363C2×C8—122no—
364C2×C8—122no—
365C2×C8—121no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—122no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—242no—
382C22⋊C4#21312yescomplemented
383C42—42no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—161no—
398C22×C3—161no—
399C22×C3#38611yescharacteristic
400C22×C3C4×D811yescharacteristic
401C22×C3#44211yescharacteristic
402C22×C3#44711yescharacteristic
403C22×C3#44911yescharacteristic
404C22×C3#45911yescharacteristic
405C22×C3#46311yescharacteristic
406C2×D3—24no—
407C2×D3—44no—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—84no—
415C2×D3—84no—
416C2×D3—84no—
417C2×D3—84no—
418C2×D3—82no—
419C2×D3—82no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—42no—
424C3×C4—81no—
425C3×C4—81no—
426C3×C4—161no—
427C3×C4C2×#9912yes—
428Dic3—82no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—64no—
435C23—62no—
436C23—62no—
437C23—81no—
438C23—81no—
439C23—124no—
440C23—124no—
441C23—122no—
442C23—122no—
443C23—122no—
444C23—121no—
445C23—121no—
446C23—122no—
447C23—122no—
448C23—121no—
449C23—121no—
450C23—122no—
451C23—122no—
452C23—242no—
453C23—242no—
454C23—244no—
455C23—244no—
456C23—244no—
457C23—244no—
458C23C4×#6711yescharacteristic
459C23C2×#21311yescharacteristic
460C23#93211yescharacteristic
461C23#93711yescharacteristic
462C23#93811yescharacteristic
463C23#94911yescharacteristic
464C23#95011yescharacteristic
465D4—34no—
466D4—42no—
467D4—64no—
468D4—62no—
469D4—62no—
470D4—62no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—124no—
477D4—124no—
478D4—122no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—248no—
501C2×C4D3×C22⋊C411yescharacteristic
502C2×C4#67⋊C411yescharacteristic
503C2×C4#88311yescharacteristic
504C2×C4#88411yescharacteristic
505C8—122no—
506C8—241no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—161no—
514C2×C3#187911yescharacteristic
515C2×C3#189811yescharacteristic
516C2×C3#190411yescharacteristic
517D3—44no—
518D3—82no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—64no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—124no—
534C22—124no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22—122no—
540C22—161no—
541C22—161no—
542C22—244no—
543C22—244no—
544C22—244no—
545C22—244no—
546C22—242no—
547C22—242no—
548C22#495611yescharacteristic
549C22C4×#21311yescharacteristic
550C22#501411yescharacteristic
551C22#503611yescharacteristic
552C22#503711yescharacteristic
553C22#505711yescharacteristic
554C22#507211yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—161no—
562C4—242no—
563C4D3×#9912yes—
564C3#925911yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—124no—
572C2—161no—
573C2—242no—
574C2#6965711yescharacteristic
575C2#7003611yescharacteristic
576C2#7004911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#386
prime radicalC2×#5036C2
cosocleC23×C3×C4C23
hypercenter#1964D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×D12C2semidirect
3. C3×#1964C2semidirect
4. #71675C2semidirect
5. #69521C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
11. C22×#213C4semidirect
25. C2×#809C4semidirect
32. #1964D3semidirect
39. C22×C3×D4C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
70. C22×C3⋊C8C2×C4semidirect
168. C3×C22⋊C4D8semidirect
192. C22×D4C4×D3semidirect
207. C2×C4⋊C4C4×D3semidirect
305. D12#99semidirect
382. C22⋊C4#213semidirect
564. C3#9259semidirect

Maximal subgroups

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

subgroupindex
C2×#50362
C22⋊C4×D122
C3×#19642
#716752
#695212
#676252
#676172
#92593