Finite Group Databasebeta

← #10691633

Subgroups — #10691633

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×#4732C211yesmaximal, characteristic
2C22⋊C4×D12C211yesmaximal, characteristic, complemented
3#69959C211yesmaximal, characteristic, complemented
4#69944C211yesmaximal, characteristic, complemented
5#69456C211yesmaximal, characteristic, complemented
6C3×#1282C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×D24C412yescomplemented
12C2×D3×C22⋊C4—21no—
13C2×C4×D12—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5212C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#821—22no—
20C2×#821C412yescomplemented
21#4955—21no—
22C2×#806—21no—
23C2×#806—21no—
24#4939—21no—
25#4732—21no—
26#4732—22no—
27#4732—21no—
28#4665—22no—
29#4470—22no—
30C2×#447—31no—
31D4×C22⋊C4—31no—
32#1964—31no—
33#1786—31no—
34#1519—31no—
35#1289—31no—
36#1282D311yescharacteristic, complemented
37C24×D3—21no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—44no—
44C22×D12C2×C412yes—
45C22×D12C2×C411yescharacteristic, complemented
46C22×D12D411yescharacteristic
47C22×D12D412yes—
48C22×D12D411yescharacteristic
49C22×C4×D3—22no—
50C22×C3×C8—21no—
51C22×C3×C8C2×C412yescomplemented
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3C2×C411yescharacteristic, complemented
68C2×D24—22no—
69C2×D24—22no—
70C2×D24—44no—
71D3×C22⋊C4—42no—
72D3×C22⋊C4—42no—
73C4×D12—44no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#821—22no—
82#821—22no—
83#821—44no—
84#806—42no—
85#806—42no—
86C23×D4—31no—
87C22×D8—32no—
88C2×#121—31no—
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—61no—
97C2×#99—61no—
98C2×C22⋊C8C2×D311yescharacteristic
99C2×C22⋊C8C2×D311yescharacteristic
100#390—61no—
101#386—62no—
102#380—31no—
103#380—31no—
104#379—61no—
105C4×C22⋊C4C2×D311yescharacteristic
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C23×D3—21no—
110C23×D3—22no—
111C23×D3—22no—
112C23×D3—21no—
113C23×D3—44no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3—41no—
118C23×D3—82no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—41no—
137C2×D12—21no—
138C2×D12—24no—
139C2×D12—24no—
140C2×D12—24no—
141C2×D12—21no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—42no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12—44no—
150C2×D12—44no—
151C2×D12—44no—
152C2×D12—42no—
153C2×D12C22⋊C412yes—
154C2×D12C22⋊C412yes—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—82no—
158C2×C3×C8—22no—
159C2×C3×C8—42no—
160C2×C3×C8—42no—
161C2×C3×C8—41no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—21no—
165C3×C22⋊C4D812yescomplemented
166C3×C42—42no—
167#217—41no—
168#217—41no—
169#217—41no—
170#217—41no—
171(C2×D3)⋊C4—44no—
172(C2×D3)⋊C4—44no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—82no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—42no—
179C4⋊Dic3—42no—
180D24—44no—
181D24—44no—
182C25—61no—
183C22×D4—32no—
184C22×D4—31no—
185C22×D4—31no—
186C22×D4—32no—
187C22×D4—31no—
188C22×D4—61no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—61no—
192C22×D4—124no—
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—62no—
196C2×D8—62no—
197C2×D8—124no—
198C22×C8—21no—
199C22×C8C4×D312yescomplemented
200#121—61no—
201#121—61no—
202#121—61no—
203#121—61no—
204C4×D4—124no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—122no—
215C2×C22⋊C4—122no—
216C2×C22⋊C4D1211yescharacteristic
217C2×C22⋊C4D1211yescharacteristic
218C2×C42—21no—
219#102—62no—
220#102—62no—
221#102—124no—
222#99—122no—
223#99—122no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228#95—21no—
229#95—32no—
230#95—61no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3C4×D411yescharacteristic
234C23×C3#12011yescharacteristic
235C23×C3#12111yescharacteristic
236C23×C3C2×D811yescharacteristic
237C23×C3#13511yescharacteristic
238C23×C3AGL(1,ℤ/8)11yescharacteristic
239C23×C3AGL(1,ℤ/8)11yescharacteristic
240C22×D3—24no—
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—44no—
244C22×D3—44no—
245C22×D3—42no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—42no—
249C22×D3—42no—
250C22×D3—42no—
251C22×D3—42no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—82no—
260C22×D3—82no—
261C22×D3—84no—
262C22×D3—84no—
263C22×D3—84no—
264C22×D3—84no—
265C22×D3—81no—
266C22×D3—81no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4C2×C22⋊C411yescharacteristic
280C2×C3×C4C4×D411yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12311yescharacteristic
283#67—81no—
284#67—81no—
285#67—81no—
286#67—81no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296D12—24no—
297D12—22no—
298D12—22no—
299D12—22no—
300D12—44no—
301D12—44no—
302D12—42no—
303D12—42no—
304D12—81no—
305D12#9914yescomplemented
306C4×D3—88no—
307C3×C8—42no—
308C3×C8—81no—
309C24—61no—
310C24—61no—
311C24—62no—
312C24—62no—
313C24—61no—
314C24—124no—
315C24—121no—
316C24—121no—
317C24—121no—
318C24—121no—
319C24—242no—
320C24C2×D1211yescharacteristic
321C2×D4—32no—
322C2×D4—32no—
323C2×D4—64no—
324C2×D4—64no—
325C2×D4—64no—
326C2×D4—61no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—61no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—122no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—124no—
337C2×D4—124no—
338C2×D4—124no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C2×D4—121no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4—121no—
353C22×C4—242no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4#23511yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4#23911yescharacteristic
360D8—124no—
361D8—124no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—122no—
369C4⋊C4—122no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—124no—
374C22⋊C4—124no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—242no—
382C22⋊C4D2412yescomplemented
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#38611yescharacteristic
395C22×C3C4×D811yescharacteristic
396C22×C3#44211yescharacteristic
397C22×C3#44711yescharacteristic
398C22×C3#44911yescharacteristic
399C22×C3#45911yescharacteristic
400C22×C3#46311yescharacteristic
401C2×D3—24no—
402C2×D3—44no—
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—42no—
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—84no—
410C2×D3—84no—
411C2×D3—84no—
412C2×D3—84no—
413C2×D3—82no—
414C2×D3—82no—
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×C4C2×#9912yes—
426Dic3—82no—
427Dic3—81no—
428Dic3—161no—
429C23—41no—
430C23—41no—
431C23—64no—
432C23—62no—
433C23—62no—
434C23—124no—
435C23—124no—
436C23—122no—
437C23—122no—
438C23—122no—
439C23—121no—
440C23—121no—
441C23—122no—
442C23—122no—
443C23—121no—
444C23—121no—
445C23—122no—
446C23—122no—
447C23—121no—
448C23—121no—
449C23—121no—
450C23—242no—
451C23—242no—
452C23—241no—
453C23—241no—
454C23—244no—
455C23—244no—
456C23—244no—
457C23—244no—
458C23C4×D1211yescharacteristic
459C23#88211yescharacteristic
460C23#89511yescharacteristic
461C23C2×D2411yescharacteristic
462C23#90411yescharacteristic
463C23#90811yescharacteristic
464C23#90811yescharacteristic
465D4—34no—
466D4—64no—
467D4—62no—
468D4—62no—
469D4—62no—
470D4—124no—
471D4—124no—
472D4—122no—
473D4—122no—
474D4—241no—
475D4—241no—
476D4—241no—
477D4—241no—
478D4—241no—
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—122no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—248no—
498C2×C4—241no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4D3×C22⋊C411yescharacteristic
502C2×C4#88411yescharacteristic
503C2×C4#89311yescharacteristic
504C2×C4#89411yescharacteristic
505C8—42no—
506C8—81no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3#187911yescharacteristic
514C2×C3#189811yescharacteristic
515C2×C3#190411yescharacteristic
516D3—44no—
517D3—82no—
518D3—161no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—64no—
530C22—124no—
531C22—124no—
532C22—122no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—244no—
538C22—244no—
539C22—244no—
540C22—244no—
541C22—242no—
542C22—242no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22—481no—
547C22—481no—
548C22#466511yescharacteristic
549C22C4×D2411yescharacteristic
550C22#471111yescharacteristic
551C22#473211yescharacteristic
552C22#473411yescharacteristic
553C22#484311yescharacteristic
554C22#484411yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—42no—
559C4—81no—
560C4—242no—
561C4—241no—
562C4—481no—
563C4D3×#9912yes—
564C3#925911yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—124no—
572C2—242no—
573C2—481no—
574C2#6867311yescharacteristic
575C2#6897011yescharacteristic
576C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC2×#4732C2
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×D12C2semidirect
3. #69959C2semidirect
4. #69944C2semidirect
5. #69456C2semidirect
6. C3×#1282C2semidirect
7. #67676C2semidirect
11. C22×D24C4semidirect
20. C2×#821C4semidirect
36. #1282D3semidirect
45. C22×D12C2×C4semidirect
51. C22×C3×C8C2×C4semidirect
67. C2×C4⋊Dic3C2×C4semidirect
165. C3×C22⋊C4D8semidirect
199. C22×C8C4×D3semidirect
305. D12#99semidirect
382. C22⋊C4D24semidirect
564. C3#9259semidirect

Maximal subgroups

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

subgroupindex
C2×#47322
C22⋊C4×D122
#699592
#699442
#694562
C3×#12822
#676762
#92593