Finite Group Databasebeta

← #10804416

Subgroups — #10804416

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C2×#4719C211yesmaximal, characteristic, complemented
3C2×#4718C211yesmaximal, characteristic, complemented
4C3×#1779C211yesmaximal, characteristic, complemented
5#72126C211yesmaximal, characteristic, complemented
6#69955C211yesmaximal, characteristic, complemented
7#69955C211yesmaximal, characteristic, complemented
8#69917C214yesmaximal, complemented
9#69916C214yesmaximal, complemented
10#21788—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67C411yescharacteristic
13C23×C4×D3C2211yescharacteristic
14C22×#217C411yescharacteristic
15C2×C4×#67—24no—
16C2×C4×#67—24no—
17C2×C4×#67C412yes—
18C2×C4×#67C412yes—
19C2×C4×#67C2212yes—
20C2×C4×#67C2212yes—
21C22×(C2×D3)⋊C4C411yescharacteristic
22C22×Dic3⋊C4C411yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2214yescomplemented
27C2×#830C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#830C2214yescomplemented
30C2×(C2×D3)⋊C8C2214yescomplemented
31C2×(C2×D3)⋊C8C2214yescomplemented
32C2×#815C2212yescomplemented
33C2×Dic3⋊C8C2212yescomplemented
34C2×Dic3⋊C8C2212yescomplemented
35C2×C8×Dic3C2212yescomplemented
36#4719—24no—
37#4718—24no—
38C22×C4×D4—31no—
39C2×#432—31no—
40C2×#431—31no—
41#1955—34no—
42#1954—34no—
43#1779—31no—
44#1779—31no—
45#1779—31no—
46#1779D311yescharacteristic, complemented
47C25×C3C2×C411yescharacteristic
48C24×D3C2×C411yescharacteristic
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—44no—
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4C2×C412yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67—28no—
57C22×#67—24no—
58C22×#67C2×C412yes—
59C22×#67C2×C412yes—
60C23×Dic3C2×C411yescharacteristic
61C23×Dic3C2×C411yescharacteristic
62C23×Dic3C2×C411yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—44no—
66C22×C4×D3C2×C412yes—
67C22×C4×D3C2312yes—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C2×#217—24no—
71C2×#217C2×C412yes—
72C2×#217C2×C412yes—
73C4×#67—28no—
74C4×#67—28no—
75C4×#67—24no—
76C4×#67—24no—
77C4×#67—24no—
78C4×#67—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4C2×C412yes—
81C2×(C2×D3)⋊C4C2×C412yes—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4C2×C412yes—
84C2×Dic3⋊C4C2×C412yes—
85C2×C4×Dic3C2×C411yescharacteristic
86C2×C4×Dic3C2×C411yescharacteristic
87C2×C4×Dic3D414yes—
88C2×C4×Dic3D414yes—
89C2×C4×Dic3C2311yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92C22×C3⋊C8C2312yes—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—44no—
96#830—22no—
97#830—22no—
98#830—44no—
99(C2×D3)⋊C8—44no—
100(C2×D3)⋊C8—44no—
101#815—24no—
102Dic3⋊C8—24no—
103Dic3⋊C8—24no—
104C8×Dic3—24no—
105C23×D4—31no—
106C24×C4—31no—
107C24×C4C2×D311yescharacteristic
108C2×C4×D4—32no—
109C2×C4×D4—32no—
110C2×C4×D4—32no—
111C2×C4×D4—32no—
112C2×C4×D4—64no—
113C2×C4×D4—64no—
114C22×C4⋊C4—31no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—31no—
117C22×C42—31no—
118#432—64no—
119#431—64no—
120C2×C4⋊C8—32no—
121C2×C4⋊C8—32no—
122C2×C22⋊C8—31no—
123C2×C22⋊C8—31no—
124C2×C22⋊C8—34no—
125C2×C22⋊C8—34no—
126C2×C22⋊C8—34no—
127C2×C22⋊C8C2×D311yescharacteristic
128C2×C22⋊C8C2×D311yescharacteristic
129C2×C22⋊C8C2×D314yescomplemented
130C2×#97—32no—
131C2×C4×C8—32no—
132C24×C3—22no—
133C24×C3—24no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—42no—
137C24×C3C22×C412yes—
138C24×C3C22×C411yescharacteristic
139C23×D3—22no—
140C23×D3—24no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—42no—
144C23×D3C22×C412yes—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—48no—
156C22×C3×C4—48no—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2411yescharacteristic
162C2×#67—232no—
163C2×#67—24no—
164C2×#67—24no—
165C2×#67—24no—
166C2×#67—24no—
167C2×#67—24no—
168C2×#67M1618yes—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—42no—
174C22×Dic3C22⋊C414yes—
175C22×Dic3C22⋊C414yes—
176C22×Dic3C22×C412yes—
177C22×Dic3C22×C411yescharacteristic
178C22×Dic3C22×C411yescharacteristic
179C22×Dic3C22×C411yescharacteristic
180C22×Dic3C22×C411yescharacteristic
181C2×C4×D3—24no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—24no—
185C2×C4×D3—24no—
186C2×C4×D3—42no—
187C2×C4×D3—42no—
188C2×C4×D3—48no—
189C2×C4×D3—48no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—24no—
193C2×C3×C8—24no—
194#217—24no—
195#217—24no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198Dic3⋊C4—24no—
199Dic3⋊C4—24no—
200C4×Dic3C22⋊C414yes—
201C4×Dic3C22⋊C414yes—
202C4×Dic3C2×D412yes—
203C4×Dic3C2×D412yes—
204C4×Dic3C2×D412yes—
205C4×Dic3C2×D412yes—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—24no—
210C25—31no—
211C25C4×D311yescharacteristic
212C22×D4—32no—
213C22×D4—32no—
214C22×D4—64no—
215C22×D4—68no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—32no—
223C23×C4—44no—
224C23×C4—62no—
225C23×C4—62no—
226C23×C4—124no—
227C23×C4C4×D311yescharacteristic
228C23×C4C4×D312yes—
229C23×C4C22×D312yes—
230C23×C4C22×D311yescharacteristic
231C22×C8—32no—
232C22×C8—32no—
233C22×C8C22×D312yes—
234C22×C8C22×D312yes—
235C4×D4—68no—
236C4×D4—68no—
237C4×D4—64no—
238C4×D4—64no—
239C4×D4—64no—
240C4×D4—64no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—64no—
244C2×C22⋊C4—32no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4—32no—
247C2×C22⋊C4—32no—
248C2×C22⋊C4—64no—
249C2×C22⋊C4—64no—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42—34no—
253C2×C42—34no—
254C2×C42—31no—
255C2×C42—31no—
256C4⋊C8—64no—
257C4⋊C8—64no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—44no—
261C22⋊C8—62no—
262C22⋊C8—62no—
263C22⋊C8—124no—
264C22⋊C8—124no—
265C22⋊C8—124no—
266#97—64no—
267C4×C8—64no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—28no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—48no—
278C23×C3—48no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3C2×C22⋊C412yes—
282C23×C3C2×M1614yes—
283C23×C3#13112yes—
284C23×C3#13112yes—
285C23×C3C23×C411yescharacteristic
286C22×D3—28no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—48no—
294C22×D3—48no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3C2×M1614yes—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—48no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4C2×C22⋊C412yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4H(5,𝔽2)14yes—
312#67—216no—
313#67—216no—
314C2×Dic3—22no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—28no—
320C2×Dic3—28no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—44no—
325C2×Dic3C2×C22⋊C412yes—
326C2×Dic3C2×C22⋊C412yes—
327C2×Dic3C2×C22⋊C412yes—
328C2×Dic3C2×C22⋊C412yes—
329C2×Dic3C2×M1614yes—
330C4×D3—48no—
331C4×D3—44no—
332C4×D3—44no—
333C3×C8—42no—
334C3×C8—42no—
335C3⋊C8—42no—
336C3⋊C8—42no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—32no—
342C24—42no—
343C24—64no—
344C24—62no—
345C24—62no—
346C24—62no—
347C24—122no—
348C24C2×C4×D312yes—
349C24C2×C4×D311yescharacteristic
350C2×D4—38no—
351C2×D4—64no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—632no—
356C2×D4—64no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—31no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—32no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—64no—
384C22×C4—64no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—128no—
388C22×C4—128no—
389C22×C4—122no—
390C22×C4C2×C4×D312yes—
391C22×C4C2×C4×D311yescharacteristic
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C22×C4C23×D311yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—64no—
402C2×C8—64no—
403C4⋊C4—64no—
404C4⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C42—34no—
410C42—34no—
411C42—32no—
412C42—32no—
413C42—32no—
414C42—32no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—416no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3#40814yes—
428C22×C3C22×C22⋊C411yescharacteristic
429C22×C3#51812yes—
430C22×C3C22×M1612yes—
431C22×C3C2×#13111yescharacteristic
432C22×C3C2×#13111yescharacteristic
433C2×D3—24no—
434C2×D3—24no—
435C2×D3—24no—
436C2×D3—416no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—42no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4#53512yes—
444C3×C4#53512yes—
445Dic3—22no—
446Dic3—22no—
447Dic3—22no—
448Dic3—42no—
449Dic3#40714yes—
450C23—22no—
451C23—22no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—34no—
460C23—48no—
461C23—48no—
462C23—44no—
463C23—44no—
464C23—68no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23—62no—
471C23—128no—
472C23—128no—
473C23—124no—
474C23—124no—
475C23D3×C22⋊C412yes—
476C23#90112yes—
477C23D3×M1614yes—
478C23#90712yes—
479C23C22×C4×D311yescharacteristic
480D4—616no—
481D4—616no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—32no—
489C2×C4—34no—
490C2×C4—32no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—68no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—62no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—68no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—128no—
508C2×C4—124no—
509C2×C4—124no—
510C2×C4—124no—
511C2×C4D3×C22⋊C412yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4#100414yes—
515C8—42no—
516C8—42no—
517C8—122no—
518C8—122no—
519C2×C3—24no—
520C2×C3—44no—
521C2×C3#230911yescharacteristic
522C2×C3C2×#40711yescharacteristic
523C2×C3C2×#40811yescharacteristic
524C2×C3#297412yes—
525C2×C3#297412yes—
526D3—44no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—22no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—416no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—64no—
540C22—64no—
541C22—64no—
542C22—1216no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22#472714yes—
548C22C2×D3×C22⋊C411yescharacteristic
549C22#548612yes—
550C22C2×#90111yescharacteristic
551C22C2×D3×M1612yes—
552C22C2×#90711yescharacteristic
553C4—22no—
554C4—34no—
555C4—42no—
556C4—62no—
557C4—62no—
558C4—62no—
559C4—122no—
560C4#559212yes—
561C4#559212yes—
562C3#2178811yescharacteristic, complemented, Sylow
563C2—24no—
564C2—44no—
565C2—124no—
566C2#7381911yescharacteristic
567C2C2×#472711yescharacteristic
568C2#7860312yes—
569C2D3×#40711yescharacteristic
570C2#7863312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1779C2
prime radicalC23×#67C4
cosocleC22×C3×C4C24
hypercenter#1779D3
hypocenterC3#21788
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#4719C2semidirect
3. C2×#4718C2semidirect
4. C3×#1779C2semidirect
5. #72126C2semidirect
6. #69955C2semidirect
7. #69955C2semidirect
8. #69917C2semidirect
9. #69916C2semidirect
26. C2×C3×C22⋊C8C22semidirect
29. C2×#830C22semidirect
30. C2×(C2×D3)⋊C8C22semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×#815C22semidirect
33. C2×Dic3⋊C8C22semidirect
34. C2×Dic3⋊C8C22semidirect
35. C2×C8×Dic3C22semidirect
46. #1779D3semidirect
129. C2×C22⋊C8C2×D3semidirect
562. C3#21788semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#672
C2×#47192
C2×#47182
C3×#17792
#721262
#699552
#699172
#699162
#217883