Finite Group Databasebeta

← #10804996

Subgroups — #10804996

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C3×#2283C211yesmaximal, characteristic, complemented
3#82062C211yesmaximal, characteristic, complemented
4#69535C214yesmaximal, complemented
5#69530C214yesmaximal, complemented
6#69457C212yesmaximal, complemented
7#69456C212yesmaximal
8#21875—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—
36#5212C2212yes—
37#5212C2212yes—
38C2×#806C2218yes—
39#4936—21no—
40#4935—21no—
41C4×C4⋊Dic3—21no—
42C4×C4⋊Dic3—21no—
43#4527C2218yes—
44C2×#529—31no—
45#2285—31no—
46#2283D311yescharacteristic, complemented
47#1996—32no—
48#1964—32no—
49#1899—34no—
50#1799—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×#217C2314yes—
74C2×#217C2312yes—
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—44no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2×C411yescharacteristic
96C2×C4⋊Dic3C2×C411yescharacteristic
97#924—21no—
98#924—21no—
99#924—21no—
100#924—21no—
101C2×Dic3⋊C4C2×C412yes—
102C2×Dic3⋊C4C2×C412yes—
103C2×Dic3⋊C4C2×C412yes—
104C2×Dic3⋊C4C2×C412yes—
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—48no—
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#384—61no—
160#383—21no—
161#382—21no—
162#382—61no—
163#381C2×D312yes—
164#380—32no—
165#380—32no—
166#380C2×D312yes—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—61no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4C2×D312yes—
171#376—21no—
172#342—38no—
173C24×C3C2411yescharacteristic
174C23×D3C22×C412yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
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
197C22×C3×C4C2×D411yescharacteristic
198C2×#67—22no—
199C2×#67—22no—
200C2×#67—24no—
201C2×#67—24no—
202C22×Dic3—24no—
203C22×Dic3C22×C412yes—
204C22×Dic3C22×C412yes—
205C22×Dic3C22×C412yes—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—42no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—44no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—42no—
215C3×C4⋊C4—44no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C42—44no—
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
232C3×C42C22⋊C411yescharacteristic
233C3×C42C22⋊C411yescharacteristic
234C3×C42C22⋊C411yescharacteristic
235#217—22no—
236#217—22no—
237#217—28no—
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—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—44no—
251C4⋊Dic3—44no—
252C4⋊Dic3—44no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261C4×Dic3—22no—
262C4×Dic3—22no—
263C4×Dic3—44no—
264C22×Q8—31no—
265C22×D4—32no—
266C22×D4—31no—
267C23×C4—32no—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D311yescharacteristic
271#126—62no—
272#126—62no—
273#126—62no—
274#126—62no—
275#125—61no—
276#125—61no—
277#125—61no—
278#125—61no—
279#124—61no—
280#124—61no—
281#124—61no—
282#124—61no—
283#123—62no—
284#123—62no—
285#123—62no—
286#123—62no—
287C22⋊Q8—61no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290C22⋊Q8—61no—
291#121—61no—
292#121—61no—
293#121—61no—
294#121—61no—
295C4×Q8—61no—
296C4×Q8—61no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307#117—62no—
308#117—62no—
309#117—62no—
310#117—62no—
311#117—124no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—62no—
321C2×C4⋊C4—62no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—34no—
328C2×C22⋊C4C22×D314yes—
329C2×C42—22no—
330C2×C42—32no—
331C2×C42—62no—
332C2×C42C4×D311yescharacteristic
333C2×C42C4×D311yescharacteristic
334C2×C42C4×D311yescharacteristic
335C2×C42C4×D311yescharacteristic
336C2×C42D1211yescharacteristic
337C2×C42D1211yescharacteristic
338C2×C42D1211yescharacteristic
339C2×C42D1211yescharacteristic
340C2×C42#6711yescharacteristic
341C2×C42#6711yescharacteristic
342C2×C42#6711yescharacteristic
343C2×C42#6711yescharacteristic
344#99—128no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—62no—
350#95—62no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C23×C411yescharacteristic
354C23×C3C22×D411yescharacteristic
355C23×C3C22×D411yescharacteristic
356C23×C3H(5,𝔽2)12yes—
357C23×C32−1+412yes—
358C22×D3—22no—
359C22×D3—24no—
360C22×D3—42no—
361C22×D3—42no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—44no—
383C2×C3×C4—44no—
384C2×C3×C4—44no—
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
392C2×C3×C4C2×C22⋊C411yescharacteristic
393C2×C3×C4C2×C22⋊C411yescharacteristic
394C2×C3×C4C2×C22⋊C411yescharacteristic
395C2×C3×C4C2×C22⋊C411yescharacteristic
396C2×C3×C4C2×C22⋊C411yescharacteristic
397#67—42no—
398#67—42no—
399#67—44no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—44no—
407C2×Dic3—44no—
408C2×Dic3—44no—
409D12—42no—
410D12—42no—
411C4×D3—44no—
412C4×D3—44no—
413Dic6—42no—
414Dic6—42no—
415C24—32no—
416C24C23×D311yescharacteristic
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—122no—
420C2×D4—62no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—62no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—122no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—64no—
438C22×C4—124no—
439C22×C4C2×C4×D311yescharacteristic
440C22×C4C2×C4×D311yescharacteristic
441C22×C4C2×C4×D311yescharacteristic
442C22×C4C2×C4×D311yescharacteristic
443C22×C4C2×C4×D311yescharacteristic
444C22×C4C2×C4×D311yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C4⋊C4—44no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—124no—
471C4⋊C4—124no—
472C4⋊C4—124no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—68no—
486C42—44no—
487C42—62no—
488C42—62no—
489C42—124no—
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
498C42(C2×D3)⋊C411yescharacteristic
499C42(C2×D3)⋊C411yescharacteristic
500C42(C2×D3)⋊C411yescharacteristic
501C42(C2×D3)⋊C411yescharacteristic
502C42(C2×D3)⋊C411yescharacteristic
503C42(C2×D3)⋊C411yescharacteristic
504C42(C2×D3)⋊C411yescharacteristic
505C42(C2×D3)⋊C411yescharacteristic
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3C22×C22⋊C411yescharacteristic
515C22×C3#52012yes—
516C22×C3#53511yescharacteristic
517C22×C3#53611yescharacteristic
518C22×C3#53712yes—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—84no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—84no—
534C3×C4#41012yes—
535C3×C4#41012yes—
536Dic3—42no—
537Dic3—42no—
538Dic3—42no—
539Dic3—84no—
540C23—41no—
541C23—41no—
542C23—62no—
543C23—64no—
544C23—122no—
545C23—122no—
546C23C22×C4×D311yescharacteristic
547C23C22×D1211yescharacteristic
548C23#100912yes—
549C23#101012yes—
550C23C22×#6711yescharacteristic
551Q8—122no—
552Q8—122no—
553D4—122no—
554D4—122no—
555D4—122no—
556D4—124no—
557D4—122no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—64no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4—124no—
592C2×C4C2×(C2×D3)⋊C411yescharacteristic
593C2×C4C2×(C2×D3)⋊C411yescharacteristic
594C2×C4C2×(C2×D3)⋊C411yescharacteristic
595C2×C4C2×(C2×D3)⋊C411yescharacteristic
596C2×C4C2×(C2×D3)⋊C411yescharacteristic
597C2×C4C2×(C2×D3)⋊C411yescharacteristic
598C2×C4C2×(C2×D3)⋊C411yescharacteristic
599C2×C4C2×(C2×D3)⋊C411yescharacteristic
600C2×C4C2×(C2×D3)⋊C411yescharacteristic
601C2×C4C2×(C2×D3)⋊C411yescharacteristic
602C2×C4C2×(C2×D3)⋊C411yescharacteristic
603C2×C4C2×(C2×D3)⋊C411yescharacteristic
604C2×C3—21no—
605C2×C3—22no—
606C2×C3—22no—
607C2×C3—21no—
608C2×C3#231111yescharacteristic
609C2×C3C2×#41011yescharacteristic
610C2×C3C2×#41011yescharacteristic
611D3—82no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—42no—
619C22—42no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—244no—
624C22#552312yes—
625C22#552511yescharacteristic
626C22#552611yescharacteristic
627C22C22×(C2×D3)⋊C411yescharacteristic
628C22#583312yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—84no—
640C4—122no—
641C4—122no—
642C4—122no—
643C4—244no—
644C4#500612yes—
645C4#500612yes—
646C3#2187511yescharacteristic, complemented, Sylow
647C2—22no—
648C2—21no—
649C2—22no—
650C2—21no—
651C2—242no—
652C2C2×#500611yescharacteristic
653C2C2×#500611yescharacteristic
654C2#8227011yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2283C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#930C22
cosocleC24×C3C24
hypercenter#2283D3
hypocenterC3#21875
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. C3×#2283C2semidirect
3. #82062C2semidirect
4. #69535C2semidirect
5. #69530C2semidirect
6. #69457C2semidirect
23. C2×#876C4semidirect
24. C2×#875C4semidirect
25. C2×C4×D12C4semidirect
26. C2×#872C4semidirect
27. C2×#870C4semidirect
28. C2×C4×Dic6C4semidirect
46. #2283D3semidirect
646. C3#21875semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#22832
#820622
#695352
#695302
#694572
#694562
#218753