Finite Group Databasebeta

← #10875893

Subgroups — #10875893

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2#73692C211yesmaximal, characteristic
3#71633C211yesmaximal, characteristic
4#71633C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#71629C211yesmaximal, characteristic, complemented
8#71626C211yesmaximal, characteristic, complemented
9#69629C211yesmaximal, characteristic
10#69617C211yesmaximal, characteristic
11#69612C211yesmaximal, characteristic, complemented
12#69601C211yesmaximal, characteristic, complemented
13#69585C211yesmaximal, characteristic
14#69571C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27567—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×#384—21no—
27C3×#384C2211yescharacteristic
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×#808C2211yescharacteristic
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—
61Q8×C4⋊C4D311yescharacteristic, complemented
62#2100—31no—
63#2099—31no—
64#2071—31no—
65#2065—31no—
66#1936—31no—
67#1919—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—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⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#215—22no—
102C2×#215—22no—
103C2×#215—22no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#898—44no—
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#808—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×#106—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—21no—
163#384—31no—
164#384C2×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—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—81no—
219C3×C4⋊C4—81no—
220C3×C4⋊C4—81no—
221C3×C4⋊C4#4112yes—
222C3×C42—24no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—42no—
230C3×C42—42no—
231#215—44no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246Dic3⋊C4—42no—
247C4×Dic3—42no—
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—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—31no—
262C2×SD16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269#126—124no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—41no—
275C4×Q8—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307#106—62no—
308C4⋊C8—121no—
309#103—62no—
310#103—62no—
311#103—62no—
312#102—62no—
313#102—121no—
314#102—121no—
315#97—121no—
316#95—21no—
317#95—21no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3C22×D411yescharacteristic
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8#12614yescomplemented
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
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—21no—
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×C4—41no—
362C2×C3×C4AGL(1,ℤ/8)12yes—
363C2×C3×C4#13712yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
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×Dic3—41no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—41no—
383C3⋊C8—41no—
384C3⋊C8—41no—
385C3⋊C8—41no—
386C3⋊C8—41no—
387C24—61no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8#23512yes—
392C2×Q8#23712yes—
393C2×Q8#23912yes—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C23×D311yescharacteristic
414SD16—124no—
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—
425C2×C8—61no—
426C2×C8—61no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—122no—
448C4⋊C4—122no—
449C4⋊C4—241no—
450C4⋊C4#23512yes—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C22⋊C4—122no—
454C22⋊C4—122no—
455C22⋊C4—241no—
456C22⋊C4—241no—
457C42—24no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—42no—
465C42—42no—
466C42—122no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54811yescharacteristic
470C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
471C22×C3C2×#13711yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—22no—
482C3×C4—24no—
483C3×C4—22no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4C2×#12611yescharacteristic
492C3×C4#53911yescharacteristic
493C3×C4#55211yescharacteristic
494C3×C4#55311yescharacteristic
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23C22×#6711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8#89814yescomplemented
510D4—121no—
511D4—121no—
512D4—121no—
513D4—121no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
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—21no—
532C2×C4—21no—
533C2×C4—21no—
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—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4#94212yes—
552C2×C4#94912yes—
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23511yescharacteristic
555C2×C4C2×#23711yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4C2×#23911yescharacteristic
558C2×C4#100711yescharacteristic
559C8—121no—
560C8—121no—
561C8—121no—
562C8—121no—
563C2×C3#248111yescharacteristic
564C2×C3#311211yescharacteristic
565C2×C3#312511yescharacteristic
566C2×C3#332011yescharacteristic
567C2×C3#333211yescharacteristic
568C2×C3#336811yescharacteristic
569C2×C3#336911yescharacteristic
570D3—161no—
571C22—241no—
572C22—241no—
573C22—241no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22—241no—
578C22#579111yescharacteristic
579C22#580611yescharacteristic
580C22C2×#94211yescharacteristic
581C22#581711yescharacteristic
582C22C2×#94911yescharacteristic
583C22#583611yescharacteristic
584C22#583711yescharacteristic
585C4—22no—
586C4—22no—
587C4—24no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4C2×#89811yescharacteristic
599C4#551311yescharacteristic
600C4#551411yescharacteristic
601C4#551511yescharacteristic
602C3#2756711yescharacteristic, complemented, Sylow
603C2—481no—
604C2#8217511yescharacteristic
605C2#8244411yescharacteristic
606C2#8257611yescharacteristic
607C2#8355311yescharacteristic
608C2#8357411yescharacteristic
609C2#8369011yescharacteristic
610C2#8370411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×Q8×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterQ8×C4⋊C4D3
hypocenterC3#27567
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C4⋊C4C2semidirect
6. #71631C2semidirect
7. #71629C2semidirect
8. #71626C2semidirect
11. #69612C2semidirect
12. #69601C2semidirect
14. #69571C2semidirect
15. #69483C2semidirect
61. Q8×C4⋊C4D3semidirect
333. C3×Q8#126semidirect
509. Q8#898semidirect
602. C3#27567semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C4⋊C42
#736922
#716332
#716312
#716292
#716262
#696292
#696172
#696122
#696012
#695852
#695712
#694832
#275673