Finite Group Databasebeta

← #10795153

Subgroups — #10795153

Order 768.

1294 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C3×#2920C211yesmaximal, characteristic, complemented
3#82049C211yesmaximal, characteristic, complemented
4#70528C212yesmaximal, complemented
5#70527C212yesmaximal, complemented
6#69989C212yesmaximal
7#69974C212yesmaximal, complemented
8C4×#824C212yesmaximal, complemented
9C4×#823C212yesmaximal, complemented
10#20728—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×C4×#67C412yes—
14C2×C4×#67C412yes—
15C2×#926C2211yescharacteristic
16C4×#235—21no—
17C4×#235—21no—
18C4×#235—21no—
19C4×#235—21no—
20C4×#235—21no—
21C4×#235—21no—
22C4×#235—21no—
23C4×#235—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#872C412yes—
27C2×C42×D3C412yes—
28C2×C4×Dic6C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#432—21no—
31C3×#432—21no—
32C3×#431—21no—
33C3×#431—21no—
34C3×#423C2212yes—
35C3×#407C2212yes—
36C3×C4×M16C2212yes—
37#5207C2212yes—
38C2×#824C2214yes—
39C2×#823C2214yes—
40#4926—21no—
41#4926—21no—
42#4925—21no—
43#4925—21no—
44#4924C2212yes—
45C4×#208C2212yes—
46#4554C2214yes—
47#4554C2214yes—
48C2×C4×#41—31no—
49#2920—31no—
50#2920D311yescharacteristic, complemented
51#1822—32no—
52#1801—32no—
53#1800—32no—
54#1785—32no—
55C4×#101—32no—
56C4×#100—32no—
57C23×C3×C4C2×C411yescharacteristic
58C23×C3×C4C2×C411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C412yes—
61C2×#235—21no—
62C2×#235—21no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—24no—
66C2×#235—24no—
67C22×D12C2311yescharacteristic
68C22×C4×D3C2×C412yes—
69C22×C4×D3C2×C412yes—
70C22×C4×D3C2×C412yes—
71C22×Dic6C2311yescharacteristic
72C2×C3×M16C2314yes—
73C2×C3×C42C2×C411yescharacteristic
74C2×C3×C42C2×C411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217C2×C412yes—
86C4×#67—22no—
87C4×#67—22no—
88C4×#67—22no—
89C4×#67—22no—
90C4×#67—22no—
91C4×#67—22no—
92C4×#67—22no—
93C4×#67—22no—
94C2×(C2×D3)⋊C4C2×C412yes—
95C2×(C2×D3)⋊C4C2×C412yes—
96#926—21no—
97#926—21no—
98#926—21no—
99#926—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4C2×C412yes—
103C2×Dic3⋊C4C2×C412yes—
104C2×C4×Dic3C2×C412yes—
105C2×C4×Dic3C2×C412yes—
106C2×C4×Dic3C2×C412yes—
107C2×#208C2314yes—
108C4×D12—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12—21no—
112C4×D12—21no—
113C4×D12—21no—
114C4×D12—21no—
115C4×D12—21no—
116#872—22no—
117#872—22no—
118#872—22no—
119#872—22no—
120C42×D3—22no—
121C42×D3—22no—
122C42×D3—22no—
123C42×D3—22no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C4×Dic6—21no—
128C4×Dic6—21no—
129C4×Dic6—21no—
130C4×Dic6—21no—
131C4×Dic6—21no—
132C3×C4⋊C8—22no—
133C3×C4⋊C8—22no—
134C3×C22⋊C8—22no—
135C3×C22⋊C8—22no—
136C3×#97—22no—
137C3×C4×C8—22no—
138#830—22no—
139#830—22no—
140#824—28no—
141#823—28no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144#803—22no—
145C4×C3⋊C8—22no—
146C22×#41—31no—
147C4×#41—61no—
148C4×#41—61no—
149C4×#41—61no—
150C4×#41—61no—
151C4×#41—61no—
152C4×#41—61no—
153C4×#41—61no—
154C4×#41—61no—
155C2×C4×Q8—31no—
156C2×C4×Q8—31no—
157C2×C4×D4—32no—
158C2×C4×D4—32no—
159C2×C4×D4—31no—
160C2×C4×D4—31no—
161C2×#117—31no—
162C2×#117—32no—
163C22×C42—32no—
164C22×C42C2×D311yescharacteristic
165#432—21no—
166#432—21no—
167#432—61no—
168#432—61no—
169#431—21no—
170#431—21no—
171#431—61no—
172#431—61no—
173#423—32no—
174#423C2×D312yes—
175C2×#101—34no—
176C2×#100—34no—
177#407—32no—
178#407C2×D312yes—
179C4×M16—32no—
180C4×M16C2×D312yes—
181#343—34no—
182#343—34no—
183C24×C3C22×C411yescharacteristic
184C23×D3C22×C412yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4C22⋊C411yescharacteristic
190C22×C3×C4C22⋊C411yescharacteristic
191C22×C3×C4C22⋊C411yescharacteristic
192C22×C3×C4C22⋊C411yescharacteristic
193C22×C3×C4C22⋊C411yescharacteristic
194C22×C3×C4C22⋊C411yescharacteristic
195C22×C3×C4C22⋊C411yescharacteristic
196C22×C3×C4C22⋊C411yescharacteristic
197C22×C3×C4C22×C411yescharacteristic
198C22×C3×C4C22×C411yescharacteristic
199C22×C3×C4C22×C411yescharacteristic
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—24no—
210C2×#67—24no—
211C22×Dic3C22×C412yes—
212C22×Dic3C22×C412yes—
213C22×Dic3C22×C412yes—
214#235—28no—
215#235—28no—
216#235—28no—
217#235—28no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—22no—
221C2×D12#4114yes—
222C2×C4×D3—24no—
223C2×C4×D3—22no—
224C2×C4×D3—22no—
225C2×C4×D3—24no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—44no—
231C2×C4×D3—44no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—22no—
235C2×Dic6#4114yes—
236C3×M16—24no—
237C3×M16—24no—
238C2×C3×C8—24no—
239C3×C42C22⋊C411yescharacteristic
240C3×C42C22⋊C411yescharacteristic
241C3×C42C22⋊C411yescharacteristic
242C3×C42C22⋊C411yescharacteristic
243C3×C42C22⋊C411yescharacteristic
244C3×C42C22⋊C411yescharacteristic
245C3×C42C22⋊C411yescharacteristic
246C3×C42C22⋊C411yescharacteristic
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255#217—22no—
256#217—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—22no—
274C4×Dic3—22no—
275#208—24no—
276#208—24no—
277C2×C3⋊C8—24no—
278C2×#41—64no—
279C2×#41—64no—
280C2×#41—61no—
281C2×#41—61no—
282C2×#41—61no—
283C2×#41—61no—
284C22×Q8—31no—
285C22×D4—32no—
286C22×D4—31no—
287C23×C4—32no—
288C23×C4—32no—
289C23×C4—32no—
290C23×C4C4×D311yescharacteristic
291C23×C4C4×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C2×M16—34no—
294C2×M16C22×D314yes—
295C4×Q8—61no—
296C4×Q8—61no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4—62no—
311C4×D4—61no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4—61no—
319#117—61no—
320#117—61no—
321#117—61no—
322#117—61no—
323#117—62no—
324#117—62no—
325#117—62no—
326#117—62no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C22⋊C4—32no—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C42—32no—
335C2×C42—32no—
336C2×C42—32no—
337C2×C42—62no—
338C2×C42—62no—
339C2×C42—62no—
340C2×C42—62no—
341C2×C42C4×D311yescharacteristic
342C2×C42C4×D311yescharacteristic
343C2×C42D1211yescharacteristic
344C2×C42D1211yescharacteristic
345C2×C42D1211yescharacteristic
346C2×C42D1211yescharacteristic
347C2×C42#6711yescharacteristic
348C2×C42#6711yescharacteristic
349C2×C42#6711yescharacteristic
350C2×C42#6711yescharacteristic
351C2×C42C22×D311yescharacteristic
352C2×C42C22×D311yescharacteristic
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8—62no—
356C4⋊C8—62no—
357#101—68no—
358#100—68no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—62no—
362C22⋊C8—62no—
363#97—22no—
364#97—62no—
365C4×C8—22no—
366C4×C8—62no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3C2×C22⋊C411yescharacteristic
372C23×C3C2×C22⋊C411yescharacteristic
373C23×C3C23×C411yescharacteristic
374C22×D3—22no—
375C22×D3—22no—
376C22×D3—42no—
377C22×D3—42no—
378C22×D3#11714yes—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C2×C22⋊C411yescharacteristic
398C2×C3×C4C2×C22⋊C411yescharacteristic
399C2×C3×C4C2×C22⋊C411yescharacteristic
400C2×C3×C4C2×C22⋊C411yescharacteristic
401C2×C3×C4C2×C22⋊C411yescharacteristic
402C2×C3×C4C2×C22⋊C411yescharacteristic
403C2×C3×C4C2×C22⋊C411yescharacteristic
404C2×C3×C4C2×C22⋊C411yescharacteristic
405C2×C3×C4C2×C22⋊C411yescharacteristic
406C2×C3×C4C2×C22⋊C411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4112yes—
411#67—28no—
412#67—28no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—42no—
418C2×Dic3—42no—
419C2×Dic3#11714yes—
420D12—24no—
421D12—24no—
422C4×D3—28no—
423C4×D3—28no—
424C4×D3—44no—
425C4×D3—44no—
426C4×D3—44no—
427C4×D3—44no—
428Dic6—24no—
429Dic6—24no—
430C3×C8—44no—
431C3⋊C8—44no—
432C24—32no—
433C24C2×C4×D311yescharacteristic
434#41—68no—
435#41—68no—
436#41—68no—
437#41—68no—
438C2×Q8—34no—
439C2×Q8—61no—
440C2×Q8—62no—
441C2×Q8—61no—
442C2×D4—34no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—62no—
446C2×D4—62no—
447C2×D4—61no—
448C2×D4—62no—
449C2×D4—61no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—64no—
458C22×C4—64no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—124no—
466C22×C4—124no—
467C22×C4(C2×D3)⋊C411yescharacteristic
468C22×C4(C2×D3)⋊C411yescharacteristic
469C22×C4(C2×D3)⋊C411yescharacteristic
470C22×C4(C2×D3)⋊C411yescharacteristic
471C22×C4(C2×D3)⋊C411yescharacteristic
472C22×C4(C2×D3)⋊C411yescharacteristic
473C22×C4(C2×D3)⋊C411yescharacteristic
474C22×C4(C2×D3)⋊C411yescharacteristic
475C22×C4C2×C4×D311yescharacteristic
476C22×C4C2×C4×D311yescharacteristic
477C22×C4C2×C4×D311yescharacteristic
478C22×C4C2×C4×D311yescharacteristic
479C22×C4C2×C4×D311yescharacteristic
480C22×C4C2×D1211yescharacteristic
481C22×C4C2×D1211yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C23×D311yescharacteristic
485M16—24no—
486M16—24no—
487M16—64no—
488M16—64no—
489C2×C8—24no—
490C2×C8—64no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—62no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C42—62no—
506C42—62no—
507C42—62no—
508C42—62no—
509C42—62no—
510C42—62no—
511C42(C2×D3)⋊C411yescharacteristic
512C42(C2×D3)⋊C411yescharacteristic
513C42(C2×D3)⋊C411yescharacteristic
514C42(C2×D3)⋊C411yescharacteristic
515C42(C2×D3)⋊C411yescharacteristic
516C42(C2×D3)⋊C411yescharacteristic
517C42(C2×D3)⋊C411yescharacteristic
518C42(C2×D3)⋊C411yescharacteristic
519C42C2×D1211yescharacteristic
520C42C2×D1211yescharacteristic
521C42C2×D1211yescharacteristic
522C42C2×D1211yescharacteristic
523C42C2×#6711yescharacteristic
524C42C2×#6711yescharacteristic
525C42C2×#6711yescharacteristic
526C42C2×#6711yescharacteristic
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3C22×C22⋊C411yescharacteristic
538C22×C3C2×#11711yescharacteristic
539C22×C3C2×#11711yescharacteristic
540C2×D3—24no—
541C2×D3—44no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—44no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4#41312yes—
554C3×C4#41312yes—
555C3×C4#52512yes—
556C3×C4#52512yes—
557Dic3—24no—
558Dic3—42no—
559Dic3—42no—
560C23—21no—
561C23—21no—
562C23—34no—
563C23—41no—
564C23—41no—
565C23—62no—
566C23—62no—
567C23—122no—
568C23—122no—
569C23C2×(C2×D3)⋊C411yescharacteristic
570C23C2×(C2×D3)⋊C411yescharacteristic
571C23C22×C4×D311yescharacteristic
572Q8—64no—
573Q8—64no—
574D4—68no—
575D4—68no—
576D4—64no—
577D4—64no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—34no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—68no—
598C2×C4—68no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—124no—
604C2×C4—124no—
605C2×C4—124no—
606C2×C4—124no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4C2×(C2×D3)⋊C411yescharacteristic
610C2×C4C2×(C2×D3)⋊C411yescharacteristic
611C2×C4C2×(C2×D3)⋊C411yescharacteristic
612C2×C4C2×(C2×D3)⋊C411yescharacteristic
613C2×C4C2×(C2×D3)⋊C411yescharacteristic
614C2×C4C2×(C2×D3)⋊C411yescharacteristic
615C2×C4C2×(C2×D3)⋊C411yescharacteristic
616C2×C4C2×(C2×D3)⋊C411yescharacteristic
617C2×C4C2×(C2×D3)⋊C411yescharacteristic
618C2×C4C2×(C2×D3)⋊C411yescharacteristic
619C2×C4C22×D1211yescharacteristic
620C2×C4D3×#4112yes—
621C2×C4D3×#4112yes—
622C2×C4C22×#6711yescharacteristic
623C8—44no—
624C8—124no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—41no—
628C2×C3—41no—
629C2×C3#228011yescharacteristic
630C2×C3C2×#41311yescharacteristic
631C2×C3C2×#41311yescharacteristic
632D3—44no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—42no—
642C22—42no—
643C22—64no—
644C22—124no—
645C22—124no—
646C22—122no—
647C22—122no—
648C22D3×#11711yescharacteristic
649C22D3×#11711yescharacteristic
650C22C22×(C2×D3)⋊C411yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—64no—
660C4—122no—
661C4—122no—
662C4#513212yes—
663C4#513212yes—
664C4#552412yes—
665C4#582812yes—
666C3#2072811yescharacteristic, complemented, Sylow
667C2—21no—
668C2—21no—
669C2—41no—
670C2—41no—
671C2—124no—
672C2#8226711yescharacteristic
673C2C2×#513211yescharacteristic
674C2C2×#513211yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2920C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#2920D3
hypocenterC3#20728
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2920

Complements

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

NH ≅ G/Ntype
2. C3×#2920C2semidirect
3. #82049C2semidirect
4. #70528C2semidirect
5. #70527C2semidirect
7. #69974C2semidirect
8. C4×#824C2semidirect
9. C4×#823C2semidirect
50. #2920D3semidirect
666. C3#20728semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C3×#29202
#820492
#705282
#705272
#699892
#699742
C4×#8242
C4×#8232
#207283