Finite Group Databasebeta

← #10814907

Subgroups — #10814907

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4852C211yesmaximal, characteristic
2#73641C211yesmaximal, characteristic, complemented
3#73639C211yesmaximal, characteristic
4C3×C4×#106C211yesmaximal, characteristic, complemented
5#70574C211yesmaximal, characteristic
6#70554C211yesmaximal, characteristic, complemented
7Dic3×#106C211yesmaximal, characteristic, complemented
8C8×Dic3⋊C4C211yesmaximal, characteristic, complemented
9#69774C211yesmaximal, characteristic
10#69464C211yesmaximal, characteristic, complemented
11#69463C211yesmaximal, characteristic
12#68904C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic
14#68793C211yesmaximal, characteristic, complemented
15#68792C211yesmaximal, characteristic
16#22805—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#872—21no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#817C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4988C2211yescharacteristic
33#4975C2211yescharacteristic
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4940—21no—
45#4939C2211yescharacteristic
46#4933—21no—
47C4×Dic3⋊C4—21no—
48C4×Dic3⋊C4C2211yescharacteristic
49#4929C2211yescharacteristic
50#4852C418yescomplemented
51#4669—21no—
52#4667—21no—
53#4652—21no—
54#4650—21no—
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2355—31no—
63#2354—31no—
64#1968—31no—
65#1968—31no—
66#1927—31no—
67#1927—31no—
68#1926—31no—
69#1926—31no—
70#1844—31no—
71#1803—31no—
72#1802—31no—
73C4×#106—31no—
74C4×#106D311yescharacteristic, complemented
75C8×C4⋊C4—31no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×#204—22no—
110C2×#204—22no—
111C2×#204C2×C414yes—
112#893—22no—
113#893C2×C414yes—
114#887—22no—
115#887C2×C414yes—
116#872—44no—
117C3×#106—22no—
118C3×#106C2×C414yes—
119C3×C4×C8—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#821—41no—
131#817—22no—
132#816—41no—
133Dic3⋊C8—22no—
134C8×Dic3—22no—
135C8×Dic3C2×C414yes—
136#810—41no—
137#810C2×C414yes—
138#809—41no—
139#809C2×C414yes—
140#808—41no—
141#808—41no—
142C22×SD16—31no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—61no—
146C4×SD16—38no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×#106C2×D311yescharacteristic
152C2×C4⋊C8—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×C4×C8—31no—
160C2×C4×C8—31no—
161C2×C4×C8C2×D311yescharacteristic
162#391—31no—
163#391—31no—
164#390—31no—
165#390—31no—
166#388—61no—
167#388—61no—
168#387—61no—
169#387—61no—
170#384—31no—
171#384—31no—
172#382—61no—
173#380—61no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—61no—
178C4×C4⋊C4C2×D311yescharacteristic
179C4×C4⋊C4C2×D311yescharacteristic
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#336—31no—
185#336—31no—
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C22×C412yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C22×C412yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8C22×C412yes—
220C2×C3×C8#4112yes—
221C2×C3×C8#4112yes—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C22×C412yes—
229C3×C4⋊C4C22×C412yes—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—42no—
233C3×C42—81no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243Dic3⋊C4—24no—
244Dic3⋊C4—22no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249Dic3⋊C4SD1614yescomplemented
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—24no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C4×Dic3C22×C412yes—
256C4×Dic3#4112yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C22×C412yes—
263#204—28no—
264C22×Q8—31no—
265C22×D4—31no—
266C23×C4—61no—
267C2×SD16—34no—
268C2×SD16—62no—
269C2×SD16—62no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8C22×D311yescharacteristic
273C22×C8C22×D311yescharacteristic
274C4×Q8—34no—
275C4×Q8—62no—
276C4×D4—34no—
277C4×D4—62no—
278#117—124no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#106—22no—
306#106—62no—
307#106—121no—
308#106—121no—
309#106C4×D314yes—
310C4⋊C8—62no—
311#103—34no—
312#103—121no—
313#103—121no—
314#102—34no—
315#102—121no—
316#102—121no—
317C4×C8—22no—
318C4×C8—34no—
319C4×C8—62no—
320#95—21no—
321#95—21no—
322#95—31no—
323#95—31no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4C2×SD1612yes—
356C2×C3×C4C23×C411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3C4×D412yes—
379C2×Dic3C4×D412yes—
380C2×Dic3C2×SD1612yes—
381C2×Dic3C2×SD1612yes—
382C2×Dic3#13512yes—
383D12—22no—
384D12—22no—
385D12—41no—
386D12—41no—
387C4×D3—48no—
388Dic6—22no—
389Dic6—22no—
390Dic6—41no—
391Dic6—41no—
392C3×C8—41no—
393C3×C8#11714yescomplemented
394C3⋊C8—22no—
395C3⋊C8—41no—
396C24—61no—
397C2×Q8—32no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×D4—32no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—68no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8C2×C4×D312yes—
437C2×C8#23512yes—
438C2×C8#23512yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—34no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4C2×C4×D312yes—
457C4⋊C4C2×C4×D312yes—
458C22⋊C4—64no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—42no—
468C42—64no—
469C42—61no—
470C42—61no—
471C42—81no—
472C42—122no—
473C42—241no—
474C22×C3C2×C4×D411yescharacteristic
475C22×C3#54711yescharacteristic
476C22×C3#55011yescharacteristic
477C22×C3C22×SD1611yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57711yescharacteristic
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4C2×#11711yescharacteristic
495C3×C4#52011yescharacteristic
496C3×C4#55511yescharacteristic
497C3×C4#55611yescharacteristic
498Dic3—22no—
499Dic3—22no—
500Dic3—41no—
501Dic3—42no—
502Dic3—81no—
503Dic3—81no—
504Dic3C4×SD1614yes—
505C23—62no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23C2×D3×D411yescharacteristic
513Q8—62no—
514Q8—62no—
515Q8—121no—
516Q8—121no—
517D4—62no—
518D4—62no—
519D4—121no—
520D4—121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—128no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4D3×SD1612yes—
561C2×C4C22×C4×D311yescharacteristic
562C2×C4C2×#23511yescharacteristic
563C2×C4C2×#23511yescharacteristic
564C2×C4#100711yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4#100911yescharacteristic
567C8—41no—
568C8—62no—
569C8—121no—
570C8#87214yescomplemented
571C2×C3#235411yescharacteristic
572C2×C3C2×C4×SD1611yescharacteristic
573C2×C3#295211yescharacteristic
574C2×C3#333511yescharacteristic
575C2×C3#334011yescharacteristic
576C2×C3#339211yescharacteristic
577C2×C3#339411yescharacteristic
578D3—82no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—241no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22#550711yescharacteristic
587C22C4×D3×D411yescharacteristic
588C22#555311yescharacteristic
589C22C2×D3×SD1611yescharacteristic
590C22#576211yescharacteristic
591C22D3×#13511yescharacteristic
592C22#577011yescharacteristic
593C4—22no—
594C4—34no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—62no—
599C4—62no—
600C4—81no—
601C4—81no—
602C4—121no—
603C4—122no—
604C4—241no—
605C4—241no—
606C4C2×#87211yescharacteristic
607C4#553211yescharacteristic
608C4#556811yescharacteristic
609C4#557411yescharacteristic
610C3#2280511yescharacteristic, complemented, Sylow
611C2—242no—
612C2#7363911yescharacteristic
613C2#7912411yescharacteristic
614C2#7918911yescharacteristic
615C2#7963711yescharacteristic
616C2#7963911yescharacteristic
617C2C4×D3×SD1611yescharacteristic
618C2#7965011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenterC4×#106D3
hypocenterC3#22805
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #73641C2semidirect
4. C3×C4×#106C2semidirect
6. #70554C2semidirect
7. Dic3×#106C2semidirect
8. C8×Dic3⋊C4C2semidirect
10. #69464C2semidirect
12. #68904C2semidirect
14. #68793C2semidirect
50. #4852C4semidirect
74. C4×#106D3semidirect
249. Dic3⋊C4SD16semidirect
393. C3×C8#117semidirect
570. C8#872semidirect
610. C3#22805semidirect

Maximal subgroups

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

subgroupindex
C2×#48522
#736412
#736392
C3×C4×#1062
#705742
#705542
Dic3×#1062
C8×Dic3⋊C42
#697742
#694642
#694632
#689042
#688922
#687932
#687922
#228053