Finite Group Databasebeta

← #10876058

Subgroups — #10876058

Order 768.

724 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2351C211yesmaximal, characteristic, complemented
2#73692C211yesmaximal, characteristic
3#71633C211yesmaximal, characteristic
4#71633C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#71628C211yesmaximal, characteristic, complemented
8#71626C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic
10#69617C211yesmaximal, characteristic
11#69611C211yesmaximal, characteristic, complemented
12#69603C211yesmaximal, characteristic, complemented
13#69557C211yesmaximal, characteristic
14#69542C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27578—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#898—21no—
22C3×#389—21no—
23C3×#389—21no—
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#382—21no—
27C3×#382C2211yescharacteristic
28C3×C4×C4⋊C4—21no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#4994—21no—
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41#4935C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4674—21no—
47#4670—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2351D311yescharacteristic, complemented
62#2101—31no—
63#2099—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1803—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—41no—
82C3×C4×Q8—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#215—22no—
100C2×#215—22no—
101C2×#215—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#898—44no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126#835—22no—
127#835—22no—
128#835—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#809—22no—
132#809—41no—
133#809—41no—
134#807—22no—
135C4⋊(C3⋊C8)—41no—
136#803—41no—
137C22×SD16—31no—
138C2×#126—61no—
139C2×C4×Q8C2×D311yescharacteristic
140C2×C4×Q8C2×D311yescharacteristic
141C2×C4×Q8C2×D311yescharacteristic
142C2×#107—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—21no—
157#389—21no—
158#389—31no—
159#389C2×D311yescharacteristic
160#389C2×D311yescharacteristic
161#388—61no—
162#384—31no—
163#382—21no—
164#382C2×D311yescharacteristic
165C4×C4⋊C4—21no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C23×D3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C2×C3×Q8#4112yes—
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×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4#4112yes—
220C3×C42—24no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—42no—
228C3×C42—42no—
229#215—44no—
230(C2×D3)⋊C4—42no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—81no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3—81no—
244Dic3⋊C4—42no—
245C4×Dic3—42no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C22×Q8C22×D311yescharacteristic
259C22×D4—31no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267#126—124no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—41no—
273C4×Q8—41no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4#6711yescharacteristic
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#103—62no—
306#103—62no—
307#103—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—61no—
317#95—61no—
318#95#6711yescharacteristic
319#95#6711yescharacteristic
320C23×C3C22×D411yescharacteristic
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8#12614yescomplemented
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4AGL(1,ℤ/8)12yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C42−1+411yescharacteristic
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—41no—
381C3⋊C8—41no—
382C3⋊C8—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C24—61no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8#23512yes—
390C2×Q8#23712yes—
391C2×Q8#23912yes—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C23×D311yescharacteristic
412SD16—124no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—241no—
446C4⋊C4#23512yes—
447C22⋊C4—122no—
448C22⋊C4—122no—
449C22⋊C4—122no—
450C22⋊C4—122no—
451C22⋊C4—241no—
452C22⋊C4—241no—
453C42—24no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—42no—
461C42—42no—
462C42—122no—
463C22×C3#52511yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3#54811yescharacteristic
466C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
467C22×C3#57511yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C3×C4—24no—
478C3×C4—22no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4C2×#12611yescharacteristic
488C3×C4#53911yescharacteristic
489C3×C4#55211yescharacteristic
490C3×C4#55311yescharacteristic
491Dic3—81no—
492Dic3—81no—
493Dic3—81no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23C22×#6711yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8#89814yescomplemented
506D4—121no—
507D4—121no—
508D4—121no—
509D4—121no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4#94912yes—
549C2×C4C2×#23511yescharacteristic
550C2×C4C2×#23511yescharacteristic
551C2×C4C2×#23711yescharacteristic
552C2×C4#100411yescharacteristic
553C2×C4C2×#23911yescharacteristic
554C2×C4#100711yescharacteristic
555C8—121no—
556C8—121no—
557C8—121no—
558C8—121no—
559C2×C3#248111yescharacteristic
560C2×C3#311511yescharacteristic
561C2×C3#312111yescharacteristic
562C2×C3#330811yescharacteristic
563C2×C3#332611yescharacteristic
564C2×C3#336811yescharacteristic
565C2×C3#337711yescharacteristic
566D3—161no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22—241no—
574C22#579111yescharacteristic
575C22#580611yescharacteristic
576C22#581711yescharacteristic
577C22C2×#94911yescharacteristic
578C22#582211yescharacteristic
579C22#583711yescharacteristic
580C22#583811yescharacteristic
581C4—22no—
582C4—24no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—241no—
592C4—241no—
593C4—241no—
594C4C2×#89811yescharacteristic
595C4#551311yescharacteristic
596C4#551411yescharacteristic
597C4#551511yescharacteristic
598C3#2757811yescharacteristic, complemented, Sylow
599C2—481no—
600C2#8217511yescharacteristic
601C2#8253111yescharacteristic
602C2#8255211yescharacteristic
603C2#8353611yescharacteristic
604C2#8354011yescharacteristic
605C2#8369011yescharacteristic
606C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2351C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2351D3
hypocenterC3#27578
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C3×#2351C2semidirect
6. #71631C2semidirect
7. #71628C2semidirect
8. #71626C2semidirect
11. #69611C2semidirect
12. #69603C2semidirect
14. #69542C2semidirect
15. #69483C2semidirect
61. #2351D3semidirect
331. C3×Q8#126semidirect
505. Q8#898semidirect
598. C3#27578semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
#736922
#716332
#716312
#716282
#716262
#696302
#696172
#696112
#696032
#695572
#695422
#694832
#275783