Finite Group Databasebeta

← #10763289

Subgroups — #10763289

Order 768.

952 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#533C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic
3C2×#4705C211yesmaximal, characteristic
4C3×#1531C211yesmaximal, characteristic, complemented
5#71822C211yesmaximal, characteristic, complemented
6#70813C211yesmaximal, characteristic, complemented
7#69801C211yesmaximal, characteristic, complemented
8#68369C214yesmaximal, complemented
9#68363C214yesmaximal, complemented
10#17672—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12#5728C411yescharacteristic
13#5728C411yescharacteristic
14D3×C4⋊Q8C411yescharacteristic
15D3×C4⋊Q8C411yescharacteristic
16#5688C2212yes—
17D3×#125C2212yes—
18D3×C22⋊Q8—21no—
19D3×C22⋊Q8—21no—
20#5622—21no—
21#5622—21no—
22#5578—21no—
23#5578—21no—
24C4×D3×Q8—21no—
25C4×D3×Q8—21no—
26D3×#117C412yes—
27#5517C412yes—
28C2×C42×D3C2211yescharacteristic
29#5470C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×(C2×D3)⋊C8—21no—
33C2×Dic3⋊C8—21no—
34#4926C2211yescharacteristic
35C2×#803C2211yescharacteristic
36#4837—21no—
37#4833—21no—
38#4726—21no—
39#4718—21no—
40#4707C2212yes—
41#4705—21no—
42#4705—21no—
43#4705C2212yes—
44#4705C2212yes—
45#4705C2212yes—
46C3×#330C2214yescomplemented
47#4543C2214yescomplemented
48#4465C2214yescomplemented
49#4465C2214yescomplemented
50C2×#533—31no—
51#2920—31no—
52C2×#432—31no—
53#1537—34no—
54#1531—31no—
55#1531—31no—
56#1531—31no—
57#1531D311yescharacteristic, complemented
58#1528—34no—
59C2×D3×Q8—21no—
60C2×D3×Q8—21no—
61C22×C4×D3C2×C411yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C3×C8—21no—
65C3×C4⋊Q8C2×C411yescharacteristic
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×#125C2312yes—
68C3×C22⋊Q8—21no—
69C3×C22⋊Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×#117C2×C412yes—
73C2×C3×C42C2311yescharacteristic
74#946—21no—
75#946—21no—
76Q8×Dic3—21no—
77Q8×Dic3—21no—
78#944—21no—
79#944—21no—
80#926C2×C412yes—
81#924—21no—
82#924—21no—
83C2×C4×Dic3C2×C411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C22×C3⋊C8—21no—
87C2×ΓL(1,𝔽25)—22no—
88#897—21no—
89#897—21no—
90#896—22no—
91#896—22no—
92#892C2×C412yes—
93#892C2×C412yes—
94D3×C4⋊C4—21no—
95D3×C4⋊C4—21no—
96D3×C4⋊C4—22no—
97D3×C4⋊C4—22no—
98D3×C4⋊C4C2×C412yes—
99D3×C4⋊C4C2×C412yes—
100#890C2×C412yes—
101#890C2×C412yes—
102#889—22no—
103#889—22no—
104#888—21no—
105#888—21no—
106#888C2×C412yes—
107#888C2×C412yes—
108#887—21no—
109#887—21no—
110#887—22no—
111#887—22no—
112D3×C22⋊C4—22no—
113#878—22no—
114#878—22no—
115#877—22no—
116#872—22no—
117C42×D3—22no—
118C42×D3C2×C411yescharacteristic
119C42×D3C2×C411yescharacteristic
120C42×D3D411yescharacteristic
121C42×D3D411yescharacteristic
122C42×D3D411yescharacteristic
123C42×D3D411yescharacteristic
124C42×D3C2311yescharacteristic
125C42×D3C2311yescharacteristic
126#870C2312yes—
127#869C2×C411yescharacteristic
128#869C2×C411yescharacteristic
129C4×Dic6—21no—
130C4×Dic6—21no—
131C4×Dic6—22no—
132C4×Dic6—22no—
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×#97—21no—
136C3×#97C2312yes—
137C3×#97C2312yes—
138#830—21no—
139(C2×D3)⋊C8—22no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C4⋊(C3⋊C8)—21no—
147#803—21no—
148#803C2312yes—
149#803C2312yes—
150#533—31no—
151#533—31no—
152#533—32no—
153#533—32no—
154#533—31no—
155#533—61no—
156#533—61no—
157#533—61no—
158#533—61no—
159#533C2×D311yescharacteristic
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×#125—32no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×C4×Q8—61no—
166C2×C4×Q8—61no—
167C2×#117—32no—
168C22×C42—31no—
169#432—31no—
170#432—32no—
171#432—32no—
172#432—32no—
173#432—61no—
174#432—61no—
175#432C2×D311yescharacteristic
176#431—61no—
177#431—61no—
178#423—61no—
179C2×C4⋊C8—61no—
180#407—61no—
181C2×C22⋊C8—61no—
182C4×M16—32no—
183C2×#97—31no—
184C2×#97C2×D311yescharacteristic
185#330—34no—
186#330—34no—
187#330—34no—
188#330C2×D314yescomplemented
189C23×D3C22⋊C411yescharacteristic
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×Dic3C22⋊C411yescharacteristic
196C22×Dic3C22⋊C411yescharacteristic
197C22×Dic3C22⋊C411yescharacteristic
198D3×Q8—44no—
199D3×Q8—44no—
200C2×C4×D3—24no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3C22⋊C411yescharacteristic
206C2×C4×D3C22⋊C411yescharacteristic
207C2×C4×D3C22⋊C411yescharacteristic
208C2×C4×D3C22⋊C411yescharacteristic
209C2×C4×D3C22⋊C411yescharacteristic
210C2×C4×D3C22⋊C411yescharacteristic
211C2×C4×D3C22×C411yescharacteristic
212C2×C4×D3C22×C411yescharacteristic
213C2×C4×D3C2×D411yescharacteristic
214C2×C4×D3C2×D411yescharacteristic
215C2×C4×D3C2×D411yescharacteristic
216C2×C4×D3C2×D411yescharacteristic
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—42no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4C22×C412yes—
229C3×C4⋊C4C22×C412yes—
230C3×C22⋊C4—22no—
231C3×C42—22no—
232C3×C42C22×C411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2411yescharacteristic
236#217—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—22no—
242C4⋊Dic3—22no—
243C4⋊Dic3C22×C412yes—
244C4⋊Dic3C22×C412yes—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3C22⋊C411yescharacteristic
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—42no—
271ΓL(1,𝔽25)—44no—
272C22×Q8—61no—
273C22×Q8—61no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C2×M16—62no—
278C22×C8—21no—
279C22×C8—61no—
280C4⋊Q8—32no—
281C4⋊Q8—32no—
282C4⋊Q8—31no—
283C4⋊Q8—31no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8C4×D311yescharacteristic
289C4⋊Q8C4×D311yescharacteristic
290#125—32no—
291#125—32no—
292#125—32no—
293#125—62no—
294#125—62no—
295#125C22×D312yes—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—62no—
301C22⋊Q8—62no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306C22⋊Q8—61no—
307C22⋊Q8—61no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—61no—
313C4×Q8—61no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×Q8—61no—
320#117—32no—
321#117—32no—
322#117—32no—
323#117—62no—
324#117—62no—
325#117C4×D312yes—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—62no—
329C2×C4⋊C4—62no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C22⋊C4—62no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—62no—
345C2×C42C22×D311yescharacteristic
346C4⋊C8—21no—
347C4⋊C8—62no—
348C4⋊C8—62no—
349C4⋊C8—62no—
350C4⋊C8—61no—
351C22⋊C8—21no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355C22⋊C8—61no—
356#97—21no—
357#97—32no—
358#97—32no—
359#97—61no—
360#97C22×D312yes—
361#97C22×D312yes—
362C4×C8—62no—
363C23×C3C2×C22⋊C411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—41no—
367C22×D3C2×C22⋊C411yescharacteristic
368C22×D3C2×C22⋊C411yescharacteristic
369C3×Q8—42no—
370C3×Q8—42no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—41no—
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C23×C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3#10114yes—
387C2×Dic3C2×C22⋊C411yescharacteristic
388C2×Dic3C2×C22⋊C411yescharacteristic
389C2×Dic3C2×C22⋊C411yescharacteristic
390C2×Dic3C2×C22⋊C411yescharacteristic
391C2×Dic3C2×C22⋊C411yescharacteristic
392C2×Dic3C2×C22⋊C411yescharacteristic
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—44no—
406C4×D3—44no—
407Dic6—44no—
408Dic6—44no—
409Dic6—42no—
410Dic6—42no—
411C3×C8—42no—
412C3×C8—42no—
413C3⋊C8—42no—
414C3⋊C8—42no—
415C24—31no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—124no—
423C2×Q8—124no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4C2×C4×D311yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4D3×D411yescharacteristic
447M16—124no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—42no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8—122no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—32no—
459C4⋊C4—32no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4C2×C4×D312yes—
473C4⋊C4C2×C4×D312yes—
474C22⋊C4—22no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C42—22no—
479C42—31no—
480C42—31no—
481C42—31no—
482C42—31no—
483C42—31no—
484C42—31no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—62no—
492C42—62no—
493C42—62no—
494C42C2×C4×D311yescharacteristic
495C42D3×D411yescharacteristic
496C42D3×D411yescharacteristic
497C42C23×D311yescharacteristic
498C22×C3—21no—
499C22×C3—21no—
500C22×C3C2×#10112yes—
501C22×C3C22×C22⋊C411yescharacteristic
502C2×D3—21no—
503C2×D3—21no—
504C2×D3—21no—
505C2×D3—21no—
506C2×D3—21no—
507C2×D3—44no—
508C2×D3#42112yes—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4#42112yes—
517Dic3—21no—
518Dic3—21no—
519Dic3—21no—
520Dic3—21no—
521Dic3—42no—
522Dic3—42no—
523Dic3C2×#10112yes—
524Dic3C2×#10112yes—
525C23—31no—
526C23—31no—
527C23—62no—
528C23—62no—
529C23—121no—
530C23D3×C22⋊C411yescharacteristic
531Q8—42no—
532Q8—42no—
533Q8—122no—
534Q8—124no—
535Q8—124no—
536Q8—122no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—31no—
542C2×C4—31no—
543C2×C4—34no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—41no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—124no—
564C2×C4—124no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4D3×C22⋊C411yescharacteristic
569C2×C4D3×C22⋊C411yescharacteristic
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C8—42no—
574C8—42no—
575C8—122no—
576C8—122no—
577C2×C3—22no—
578C2×C3C22×#10111yescharacteristic
579C2×C3C2×#42111yescharacteristic
580C2×C3C2×#42111yescharacteristic
581D3—22no—
582D3—41no—
583C22—21no—
584C22—21no—
585C22—32no—
586C22—61no—
587C22—61no—
588C22—61no—
589C22—61no—
590C22—61no—
591C22—124no—
592C22D3×#10112yes—
593C22C2×D3×C22⋊C411yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—32no—
600C4—32no—
601C4—42no—
602C4—42no—
603C4—61no—
604C4—61no—
605C4—61no—
606C4—61no—
607C4—122no—
608C4—122no—
609C4#482112yes—
610C3#1767211yescharacteristic, complemented, Sylow
611C2—22no—
612C2—62no—
613C2—121no—
614C2C2×D3×#10111yescharacteristic
615C2C2×#482111yescharacteristic
616C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1531C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1531D3
hypocenterC3#17672
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
4. C3×#1531C2semidirect
5. #71822C2semidirect
6. #70813C2semidirect
7. #69801C2semidirect
8. #68369C2semidirect
9. #68363C2semidirect
46. C3×#330C22semidirect
47. #4543C22semidirect
48. #4465C22semidirect
49. #4465C22semidirect
57. #1531D3semidirect
188. #330C2×D3semidirect
610. C3#17672semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
#793312
C2×#47052
C3×#15312
#718222
#708132
#698012
#683692
#683632
#176723