Finite Group Databasebeta

← #10800535

Subgroups — #10800535

Order 768.

1222 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C3×#2283C211yesmaximal, characteristic, complemented
3C4×#926C211yesmaximal, characteristic, complemented
4#69535C214yesmaximal, complemented
5#69529C212yesmaximal, complemented
6#69528C212yesmaximal
7C4×#806C214yesmaximal, complemented
8#21421—31nomaximal, Sylow
9C22×C3×C42C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#929C2212yes—
12C2×C4×#67C2212yes—
13C2×#926C2212yes—
14C2×#924C2211yescharacteristic
15#5479—21no—
16#5479—21no—
17#5479—21no—
18#5479—21no—
19#5479—21no—
20#5479—21no—
21#5479—21no—
22#5479—21no—
23C2×#876C412yescomplemented
24C2×#875C411yescharacteristic, complemented
25C2×C4×D12C411yescharacteristic, complemented
26C2×#872C412yescomplemented
27C2×#870C411yescharacteristic, complemented
28C2×C4×Dic6C411yescharacteristic, complemented
29C3×#383—21no—
30C3×#382—21no—
31C3×#381C2212yes—
32C3×#380C2212yes—
33C3×C4×C4⋊C4—21no—
34C3×C4×C22⋊C4C2212yes—
35C3×#376—21no—
36C4×#217C2212yes—
37C4×#217C2212yes—
38C2×#806C2218yes—
39C4×C4⋊Dic3—21no—
40C4×C4⋊Dic3—21no—
41#4932—21no—
42C42×Dic3—21no—
43#4527C2218yes—
44C2×#529—31no—
45#2283D311yescharacteristic, complemented
46C4×#117—31no—
47#1996—32no—
48#1964—32no—
49#1899—34no—
50C4×#99—34no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×#67C2312yes—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3C2×C412yes—
57C22×Dic6C2×C411yescharacteristic
58C2×C3×C4⋊C4—22no—
59C2×C3×C22⋊C4C2314yes—
60C2×C3×C42—22no—
61C2×C3×C42C2×C411yescharacteristic
62C2×C3×C42C2×C411yescharacteristic
63C2×C3×C42C2×C411yescharacteristic
64C2×C3×C42C2×C411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×#217C2312yes—
74C2×#217C2314yes—
75#930—21no—
76#930—21no—
77#930—21no—
78#930—21no—
79#929—22no—
80#929—22no—
81#929—22no—
82#929—22no—
83C4×#67—22no—
84C4×#67—22no—
85C4×#67—22no—
86C4×#67—22no—
87C2×(C2×D3)⋊C4C2×C412yes—
88C2×(C2×D3)⋊C4C2×C412yes—
89C2×(C2×D3)⋊C4C2×C412yes—
90C2×(C2×D3)⋊C4C2×C412yes—
91#926—28no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95#924—21no—
96#924—21no—
97#924—21no—
98#924—21no—
99C2×Dic3⋊C4C2×C412yes—
100C2×Dic3⋊C4C2×C412yes—
101C2×Dic3⋊C4C2×C412yes—
102C2×Dic3⋊C4C2×C412yes—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3C2×C412yes—
107#876—22no—
108#876—22no—
109#876—22no—
110#876—22no—
111#875—21no—
112#875—21no—
113#875—21no—
114#875—21no—
115C4×D12—21no—
116C4×D12—21no—
117C4×D12—21no—
118C4×D12—21no—
119#872—22no—
120#872—22no—
121#872—22no—
122#872—22no—
123#870—21no—
124#870—21no—
125#870—21no—
126#870—21no—
127C4×Dic6—21no—
128C4×Dic6—21no—
129C4×Dic6—21no—
130C4×Dic6—21no—
131C3×#95—22no—
132C3×#95—22no—
133C3×#95—22no—
134C3×#95—22no—
135C3⋊#95—22no—
136C3⋊#95—22no—
137#806—216no—
138#529—61no—
139#529—61no—
140#529—61no—
141#529—61no—
142#529—61no—
143#529—61no—
144#529—61no—
145#529—61no—
146C2×#126—32no—
147C2×#125—31no—
148C2×#124—31no—
149C2×#123—32no—
150C2×C22⋊Q8—31no—
151C2×#121—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—32no—
154C2×C4×D4—31no—
155C2×#117—32no—
156C2×#117—32no—
157C22×C42C2×D311yescharacteristic
158C2×#99—38no—
159#383—21no—
160#382—21no—
161#381C2×D312yes—
162#380C2×D312yes—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—61no—
165C4×C4⋊C4—61no—
166C4×C22⋊C4—32no—
167C4×C22⋊C4—32no—
168C4×C22⋊C4C2×D312yes—
169#376—21no—
170#376—61no—
171C43—61no—
172#342—38no—
173C24×C3C2411yescharacteristic
174C23×D3C22×C412yes—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4C22×C411yescharacteristic
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—28no—
200C22×Dic3—24no—
201C22×Dic3C22×C412yes—
202C22×Dic3C22×C412yes—
203C22×Dic3C22×C412yes—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—42no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—44no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—42no—
213C3×C4⋊C4—44no—
214C3×C22⋊C4—28no—
215C3×C42—44no—
216C3×C42C22⋊C411yescharacteristic
217C3×C42C22⋊C411yescharacteristic
218C3×C42C22⋊C411yescharacteristic
219C3×C42C22⋊C411yescharacteristic
220C3×C42C22⋊C411yescharacteristic
221C3×C42C22⋊C411yescharacteristic
222C3×C42C22⋊C411yescharacteristic
223C3×C42C22⋊C411yescharacteristic
224C3×C42C22⋊C411yescharacteristic
225C3×C42C22⋊C411yescharacteristic
226C3×C42C22⋊C411yescharacteristic
227C3×C42C22⋊C411yescharacteristic
228C3×C42C22⋊C411yescharacteristic
229C3×C42C22⋊C411yescharacteristic
230C3×C42C22⋊C411yescharacteristic
231C3×C42C22⋊C411yescharacteristic
232#217—22no—
233#217—22no—
234#217—24no—
235#217#4118yes—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244C4⋊Dic3—28no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257C4×Dic3—28no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—44no—
261C4×Dic3—44no—
262C22×Q8—31no—
263C22×D4—32no—
264C22×D4—31no—
265C23×C4—32no—
266C23×C4C22×D311yescharacteristic
267C23×C4C22×D311yescharacteristic
268C23×C4C22×D311yescharacteristic
269#126—62no—
270#126—62no—
271#126—62no—
272#126—62no—
273#125—61no—
274#125—61no—
275#125—61no—
276#125—61no—
277#124—61no—
278#124—61no—
279#124—61no—
280#124—61no—
281#123—62no—
282#123—62no—
283#123—62no—
284#123—62no—
285C22⋊Q8—61no—
286C22⋊Q8—61no—
287C22⋊Q8—61no—
288C22⋊Q8—61no—
289#121—61no—
290#121—61no—
291#121—61no—
292#121—61no—
293C4×Q8—61no—
294C4×Q8—61no—
295C4×Q8—61no—
296C4×Q8—61no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304C4×D4—61no—
305#117—68no—
306#117—62no—
307#117—62no—
308#117—62no—
309#117—62no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—62no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4C22×D314yes—
325C2×C42—22no—
326C2×C42—32no—
327C2×C42—62no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42C4×D311yescharacteristic
331C2×C42C4×D311yescharacteristic
332C2×C42C4×D311yescharacteristic
333C2×C42C4×D311yescharacteristic
334C2×C42D1211yescharacteristic
335C2×C42D1211yescharacteristic
336C2×C42D1211yescharacteristic
337C2×C42D1211yescharacteristic
338C2×C42#6711yescharacteristic
339C2×C42#6711yescharacteristic
340C2×C42#6711yescharacteristic
341C2×C42#6711yescharacteristic
342#99—616no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—62no—
348#95—62no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C23×C411yescharacteristic
352C23×C3C22×D411yescharacteristic
353C23×C3C22×D411yescharacteristic
354C23×C3C2×#4114yes—
355C22×D3—22no—
356C22×D3—24no—
357C22×D3—42no—
358C22×D3—42no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—44no—
378C2×C3×C4—44no—
379C2×C3×C4—44no—
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C2×C22⋊C411yescharacteristic
387C2×C3×C4C2×C22⋊C411yescharacteristic
388C2×C3×C4C2×C22⋊C411yescharacteristic
389C2×C3×C4C2×C22⋊C411yescharacteristic
390C2×C3×C4C2×C22⋊C411yescharacteristic
391C2×C3×C4C2×C22⋊C411yescharacteristic
392#67—44no—
393#67—44no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—28no—
399C2×Dic3—44no—
400C2×Dic3—44no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403D12—42no—
404D12—42no—
405C4×D3—44no—
406C4×D3—44no—
407Dic6—42no—
408Dic6—42no—
409C24—32no—
410C24C23×D311yescharacteristic
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—122no—
414C2×D4—62no—
415C2×D4—68no—
416C2×D4—62no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—122no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—124no—
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C2×C4×D311yescharacteristic
433C22×C4C2×C4×D311yescharacteristic
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×C4×D311yescharacteristic
436C22×C4C2×C4×D311yescharacteristic
437C22×C4C2×D1211yescharacteristic
438C22×C4C2×D1211yescharacteristic
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×D1211yescharacteristic
442C22×C4C2×D1211yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C4⋊C4—44no—
450C4⋊C4—68no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C22⋊C4—28no—
464C22⋊C4—38no—
465C22⋊C4—62no—
466C22⋊C4—64no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—62no—
472C22⋊C4—62no—
473C22⋊C4—62no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C42—44no—
477C42—68no—
478C42—62no—
479C42—62no—
480C42—124no—
481C42—124no—
482C42(C2×D3)⋊C411yescharacteristic
483C42(C2×D3)⋊C411yescharacteristic
484C42(C2×D3)⋊C411yescharacteristic
485C42(C2×D3)⋊C411yescharacteristic
486C42(C2×D3)⋊C411yescharacteristic
487C42(C2×D3)⋊C411yescharacteristic
488C42(C2×D3)⋊C411yescharacteristic
489C42(C2×D3)⋊C411yescharacteristic
490C42(C2×D3)⋊C411yescharacteristic
491C42(C2×D3)⋊C411yescharacteristic
492C42(C2×D3)⋊C411yescharacteristic
493C42(C2×D3)⋊C411yescharacteristic
494C42(C2×D3)⋊C411yescharacteristic
495C42(C2×D3)⋊C411yescharacteristic
496C42(C2×D3)⋊C411yescharacteristic
497C42(C2×D3)⋊C411yescharacteristic
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—44no—
505C22×C3C22×C22⋊C411yescharacteristic
506C22×C3C4×#4112yes—
507C22×C3#52512yes—
508C22×C3#53212yes—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—84no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—84no—
524C3×C4#41012yes—
525C3×C4#41012yes—
526Dic3—48no—
527Dic3—42no—
528Dic3—42no—
529Dic3—42no—
530C23—41no—
531C23—41no—
532C23—62no—
533C23—64no—
534C23—122no—
535C23—122no—
536C23C22×C4×D311yescharacteristic
537C23C22×D1211yescharacteristic
538C23D3×#4114yes—
539C23C22×#6711yescharacteristic
540Q8—122no—
541Q8—122no—
542D4—122no—
543D4—124no—
544D4—124no—
545D4—122no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—64no—
568C2×C4—68no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—124no—
573C2×C4—124no—
574C2×C4—124no—
575C2×C4—124no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4C2×(C2×D3)⋊C411yescharacteristic
579C2×C4C2×(C2×D3)⋊C411yescharacteristic
580C2×C4C2×(C2×D3)⋊C411yescharacteristic
581C2×C4C2×(C2×D3)⋊C411yescharacteristic
582C2×C4C2×(C2×D3)⋊C411yescharacteristic
583C2×C4C2×(C2×D3)⋊C411yescharacteristic
584C2×C4C2×(C2×D3)⋊C411yescharacteristic
585C2×C4C2×(C2×D3)⋊C411yescharacteristic
586C2×C4C2×(C2×D3)⋊C411yescharacteristic
587C2×C4C2×(C2×D3)⋊C411yescharacteristic
588C2×C4C2×(C2×D3)⋊C411yescharacteristic
589C2×C4C2×(C2×D3)⋊C411yescharacteristic
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3#229711yescharacteristic
594C2×C3C2×#41011yescharacteristic
595C2×C3C2×#41011yescharacteristic
596D3—82no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—44no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—244no—
608C22#552212yes—
609C22#552412yes—
610C22C22×(C2×D3)⋊C411yescharacteristic
611C22#583212yes—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—84no—
623C4—128no—
624C4—122no—
625C4—122no—
626C4—122no—
627C4#500612yes—
628C4#500612yes—
629C3#2142111yescharacteristic, complemented, Sylow
630C2—22no—
631C2—22no—
632C2—22no—
633C2—242no—
634C2#8222811yescharacteristic
635C2C2×#500611yescharacteristic
636C2C2×#500611yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2283C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#930C22
cosocleC24×C3C24
hypercenter#2283D3
hypocenterC3#21421
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. C3×#2283C2semidirect
3. C4×#926C2semidirect
4. #69535C2semidirect
5. #69529C2semidirect
7. C4×#806C2semidirect
23. C2×#876C4semidirect
24. C2×#875C4semidirect
25. C2×C4×D12C4semidirect
26. C2×#872C4semidirect
27. C2×#870C4semidirect
28. C2×C4×Dic6C4semidirect
45. #2283D3semidirect
629. C3#21421semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#22832
C4×#9262
#695352
#695292
#695282
C4×#8062
#214213