Finite Group Databasebeta

← #10691634

Subgroups — #10691634

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
1C3×D4×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic
3#72128C211yesmaximal, characteristic, complemented
4#69509C211yesmaximal, characteristic, complemented
5#69456C211yesmaximal, characteristic, complemented
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×C4×D4—21no—
11C22×C3×C22⋊C4—21no—
12C22×#213C412yescomplemented
13C2×#930C2211yescharacteristic
14C3×#390—21no—
15C3×#386—22no—
16C3×#380C2211yescharacteristic
17C3×#379—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5217—22no—
20#5217—21no—
21#5217—21no—
22C2×#832—22no—
23C2×#832C412yescomplemented
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#806—21no—
28C2×#806—21no—
29#4451—22no—
30C2×#447—31no—
31D4×C22⋊C4D311yescharacteristic, complemented
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—44no—
42C22×C3×D4C2×C412yes—
43C22×C3×D4D412yes—
44C22×C3×D4D411yescharacteristic
45C22×C3×D4D411yescharacteristic
46C23×C3×C4—22no—
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C22×D12C2×C411yescharacteristic, complemented
50C3×C4×D4—44no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—42no—
57C2×C3×C22⋊C4—42no—
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C22⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×#217—21no—
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
72C22×C3⋊C8—21no—
73C22×C3⋊C8C2×C412yescomplemented
74C3×#95—21no—
75C3×#95D412yes—
76#832—22no—
77#832—22no—
78#832—44no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#806—42no—
85#806—42no—
86C23×D4C2×D311yescharacteristic
87C22×D8—32no—
88C2×#121—31no—
89C2×C4×D4—21no—
90C22×C22⋊C4—21no—
91#447—62no—
92#447—61no—
93#447—61no—
94C2×#102—32no—
95C2×#102—62no—
96C2×#99—61no—
97C2×#99—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100#390—21no—
101#386—22no—
102#380—31no—
103#380C2×D311yescharacteristic
104#379—21no—
105C4×C22⋊C4C2×D311yescharacteristic
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—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—42no—
138C22×C3×C4—44no—
139C22×C3×C4—44no—
140C22×C3×C4—82no—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C22×Dic3—21no—
154C22×Dic3—41no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—42no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—44no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—82no—
167C3×C22⋊C4D812yescomplemented
168C3×C42—42no—
169#217—41no—
170#217—41no—
171#213—44no—
172#213—44no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—42no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—42no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8—41no—
182C25—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—31no—
187C22×D4—44no—
188C22×D4—61no—
189C22×D4C4×D312yes—
190C22×D4D1212yes—
191C22×D4#6711yescharacteristic
192C22×D4#6711yescharacteristic
193C23×C4—22no—
194C23×C4C22×D311yescharacteristic
195C2×D8—62no—
196C2×D8—62no—
197C2×D8—124no—
198C22×C8—32no—
199C22×C8—61no—
200#121—61no—
201#121—61no—
202#121—61no—
203#121—61no—
204C4×D4—44no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4#6711yescharacteristic
217C2×C22⋊C4#6711yescharacteristic
218C2×C42—21no—
219#102—62no—
220#102—62no—
221#102—124no—
222#99—122no—
223#99—122no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—21no—
229#95—61no—
230#95#6712yes—
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—22no—
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—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—88no—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C4×D411yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12311yescharacteristic
295#67—81no—
296#67—81no—
297#67—81no—
298#67—81no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—81no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305D12—42no—
306D12—81no—
307C3⋊C8—42no—
308C3⋊C8—81no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—44no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—61no—
319C24—82no—
320C24C2×#6711yescharacteristic
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—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—61no—
352C22×C4—82no—
353C22×C4—121no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4#23511yescharacteristic
357C22×C4#23511yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360D8—124no—
361D8—124no—
362C2×C8—62no—
363C2×C8—122no—
364C2×C8—122no—
365C2×C8—121no—
366C4⋊C4—42no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—122no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—82no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
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—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—42no—
424C3×C4—81no—
425C3×C4—82no—
426C3×C4C2×#9912yes—
427Dic3—81no—
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—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—241no—
457C23—241no—
458C23C4×#6711yescharacteristic
459C23#93011yescharacteristic
460C23C2×#21311yescharacteristic
461C23#93211yescharacteristic
462C23#94911yescharacteristic
463C23#95011yescharacteristic
464C23#95311yescharacteristic
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—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—88no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—241no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4C2×(C2×D3)⋊C411yescharacteristic
502C2×C4C4×#6711yescharacteristic
503C2×C4#92911yescharacteristic
504C2×C4#93011yescharacteristic
505C8—122no—
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#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—
548C22C4×#21311yescharacteristic
549C22#501411yescharacteristic
550C22#501511yescharacteristic
551C22#501611yescharacteristic
552C22#521311yescharacteristic
553C22#521711yescharacteristic
554C22#523811yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—42no—
559C4—82no—
560C4—81no—
561C4—241no—
562C4—481no—
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#7154311yescharacteristic
575C2#7161311yescharacteristic
576C2#7165111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×D4×C22⋊C4C2
socleC22×C3#386
prime radicalC2×#5217C2
cosocleC23×C3×C4C23
hypercenterD4×C22⋊C4D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C22⋊C4C2semidirect
3. #72128C2semidirect
4. #69509C2semidirect
5. #69456C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
12. C22×#213C4semidirect
23. C2×#832C4semidirect
31. D4×C22⋊C4D3semidirect
49. C22×D12C2×C4semidirect
71. C2×C4⋊Dic3C2×C4semidirect
73. C22×C3⋊C8C2×C4semidirect
167. C3×C22⋊C4D8semidirect
274. C3×D4#99semidirect
382. C22⋊C4#213semidirect
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
C3×D4×C22⋊C42
C2×#52172
#721282
#695092
#694562
#676252
#676172
#92593