Finite Group Databasebeta

← #10691636

Subgroups — #10691636

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×C3×#447C211yesmaximal, characteristic
2D4×#217C211yesmaximal, characteristic, complemented
3#72128C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#67677C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×D8C412yescomplemented
11C2×C3×#121C2211yescharacteristic
12C22×#217—21no—
13C2×D4×Dic3—21no—
14C3×#447—21no—
15C3×#447—22no—
16C3×#447—21no—
17C2×C3×#102C412yescomplemented
18C2×C3×C22⋊C8C2211yescharacteristic
19#5247—21no—
20#5221—21no—
21#5220—22no—
22C2×#834—21no—
23C2×#834—21no—
24C2×#832—22no—
25#5212C2211yescharacteristic
26#5212C2211yescharacteristic
27C4×#217C2211yescharacteristic
28C2×#830C2211yescharacteristic
29#4537—22no—
30C2×#447D311yescharacteristic
31D4×C22⋊C4—31no—
32#1964—31no—
33#1786—31no—
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—21no—
42C22×C3×D4—44no—
43C22×C3×D4C2×C412yes—
44C22×C3×D4C2×C411yescharacteristic, complemented
45C22×C3×D4D411yescharacteristic
46C22×C3×D4D412yes—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—22no—
50C2×C3×D8—22no—
51C2×C3×D8—22no—
52C2×C3×D8—44no—
53C22×C3×C8C2×C412yescomplemented
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
59C2×C3×C22⋊C4—21no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217—22no—
65C2×#217—42no—
66C2×#217—42no—
67C2×#217D411yescharacteristic
68C2×#217D411yescharacteristic
69D4×Dic3—44no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4×Dic3—21no—
73C22×C3⋊C8—21no—
74C3×#102—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78#834—42no—
79#834—42no—
80#832—44no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86C23×D4C2×D311yescharacteristic
87C22×D8Dic312yescomplemented
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—61no—
90C22×C22⋊C4—61no—
91#447—21no—
92#447—22no—
93#447—21no—
94C2×#102—62no—
95C2×#102Dic312yescomplemented
96C2×#99—61no—
97C2×#99—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#390—61no—
101#386—62no—
102#380—31no—
103#380—31no—
104#379—61no—
105C4×C22⋊C4—31no—
106#327—62no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—21no—
112C24×C3—44no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3—41no—
117C24×C3—82no—
118C24×C3C2×D411yescharacteristic
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—21no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4C22⋊C412yes—
142C2×C3×D4C22⋊C412yes—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—44no—
155C22×Dic3—44no—
156C22×Dic3—41no—
157C22×Dic3—82no—
158C3×D8—44no—
159C3×D8—44no—
160C2×C3×C8—22no—
161C2×C3×C8—42no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166#217—21no—
167#217—21no—
168#217—21no—
169#217—44no—
170#217—44no—
171#217—41no—
172#217—41no—
173#217—41no—
174#217—41no—
175#217—82no—
176#217D812yescomplemented
177C4⋊Dic3—42no—
178C4⋊Dic3—42no—
179C4×Dic3—42no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8—41no—
182C25—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—44no—
188C22×D4C2×Dic312yes—
189C22×D4C2×Dic311yescharacteristic, complemented
190C22×D4#6711yescharacteristic
191C22×D4#6712yes—
192C22×D4#6711yescharacteristic
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—22no—
196C2×D8—22no—
197C2×D8—44no—
198C22×C8—61no—
199C22×C8C2×Dic312yescomplemented
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C4×D4—124no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C2×Dic311yescharacteristic, complemented
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—22no—
220#102—22no—
221#102—124no—
222#99—122no—
223#99—122no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—32no—
229#95—61no—
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—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—82no—
253C23×C3—82no—
254C23×C3—81no—
255C23×C3—81no—
256C23×C3—84no—
257C23×C3—84no—
258C23×C3—84no—
259C23×C3—84no—
260C23×C3C4×D411yescharacteristic
261C23×C3#12011yescharacteristic
262C23×C3#12111yescharacteristic
263C23×C3C2×D811yescharacteristic
264C23×C3#13511yescharacteristic
265C23×C3AGL(1,ℤ/8)11yescharacteristic
266C23×C3AGL(1,ℤ/8)11yescharacteristic
267C3×D4—24no—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—44no—
272C3×D4—44no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—81no—
276C3×D4—81no—
277C3×D4—81no—
278C3×D4—81no—
279C3×D4—81no—
280C3×D4#9914yescomplemented
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4C2×C22⋊C411yescharacteristic
291C2×C3×C4C4×D411yescharacteristic
292C2×C3×C4#12111yescharacteristic
293C2×C3×C4#12311yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—88no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C3×C8—42no—
308C3⋊C8—81no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—44no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—82no—
320C24D3×D411yescharacteristic
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4#21712yes—
344C2×D4#21712yes—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4—121no—
353C22×C4—242no—
354C22×C4D3×D411yescharacteristic
355C22×C4#23711yescharacteristic
356C22×C4#23711yescharacteristic
357C22×C4C22×Dic311yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360D8—44no—
361D8—44no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—122no—
365C2×C8—121no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—122no—
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—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—161no—
412C22×C3—161no—
413C22×C3#38611yescharacteristic
414C22×C3C4×D811yescharacteristic
415C22×C3#44211yescharacteristic
416C22×C3#44711yescharacteristic
417C22×C3#44911yescharacteristic
418C22×C3#45911yescharacteristic
419C22×C3#46311yescharacteristic
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—81no—
424C3×C4C2×#9912yes—
425Dic3—42no—
426Dic3—81no—
427Dic3—82no—
428Dic3—161no—
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—42no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—82no—
451C23—82no—
452C23—81no—
453C23—81no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23D3×D811yescharacteristic
459C23#91111yescharacteristic
460C23#91111yescharacteristic
461C23#91211yescharacteristic
462C23D4×Dic311yescharacteristic
463C23#93711yescharacteristic
464C23#93911yescharacteristic
465D4—24no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—44no—
470D4—44no—
471D4—42no—
472D4—42no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478D4#83414yescomplemented
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—62no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—248no—
498C2×C4—241no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4D4×Dic311yescharacteristic
502C2×C4#93511yescharacteristic
503C2×C4#93911yescharacteristic
504C2×C4C2×#21711yescharacteristic
505C8—42no—
506C8—241no—
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—161no—
516C2×C3#187911yescharacteristic
517C2×C3#189811yescharacteristic
518C2×C3#190411yescharacteristic
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—81no—
544C22—81no—
545C22—81no—
546C22—161no—
547C22—161no—
548C22Dic3×D811yescharacteristic
549C22#515011yescharacteristic
550C22#515211yescharacteristic
551C22#515311yescharacteristic
552C22#515611yescharacteristic
553C22#515811yescharacteristic
554C22#522011yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—81no—
559C4—122no—
560C4—242no—
561C4—241no—
562C4—481no—
563C4C2×#83412yes—
564C3#925911yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—44no—
572C2—82no—
573C2—161no—
574C2#7167011yescharacteristic
575C2#7203711yescharacteristic
576C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#447C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenterC2×#447D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
2. D4×#217C2semidirect
3. #72128C2semidirect
4. #71686C2semidirect
5. #71675C2semidirect
6. #67677C2semidirect
7. #67676C2semidirect
10. C22×C3×D8C4semidirect
17. C2×C3×#102C4semidirect
44. C22×C3×D4C2×C4semidirect
53. C22×C3×C8C2×C4semidirect
58. C2×C3×C4⋊C4C2×C4semidirect
87. C22×D8Dic3semidirect
95. C2×#102Dic3semidirect
176. #217D8semidirect
189. C22×D4C2×Dic3semidirect
199. C22×C8C2×Dic3semidirect
207. C2×C4⋊C4C2×Dic3semidirect
280. C3×D4#99semidirect
478. D4#834semidirect
564. C3#9259semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4472
D4×#2172
#721282
#716862
#716752
#676772
#676762
#92593