Finite Group Databasebeta

← #10763300

Subgroups — #10763300

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86219C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3C2×#4705C211yesmaximal, characteristic, complemented
4C3×#1530C211yesmaximal, characteristic, complemented
5#71798C211yesmaximal, characteristic, complemented
6#70813C211yesmaximal, characteristic, complemented
7#69801C211yesmaximal, characteristic, complemented
8#68348C214yesmaximal, complemented
9#68344C214yesmaximal, complemented
10#17674—31nomaximal, Sylow
11C3×#532C2211yescharacteristic
12#5828C412yes—
13Dic3×#41C412yes—
14#5728C411yescharacteristic
15#5728C411yescharacteristic
16#5716C411yescharacteristic
17#5716C411yescharacteristic
18#5680C2212yes—
19#5680C2212yes—
20#5606—21no—
21#5606—21no—
22#5600—21no—
23#5600—21no—
24#5557—21no—
25#5557—21no—
26C4×#237—21no—
27C4×#237—21no—
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×#329C2214yescomplemented
47#4540C2214yescomplemented
48#4465C2214yescomplemented
49#4465C2214yescomplemented
50#3444—31no—
51#2920—31no—
52C2×#432—31no—
53#1535—34no—
54#1531—31no—
55#1531—31no—
56#1530—31no—
57#1530D311yescharacteristic, complemented
58#1529—34no—
59C2×C3×#41C2×C412yes—
60C2×#237—21no—
61C2×#237—21no—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C3×C8—21no—
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×#127C2×C411yescharacteristic
68C3×#124C2312yes—
69C3×#121—21no—
70C3×#121—21no—
71C3×C4×D4—21no—
72C3×C4×D4—21no—
73C2×C3×C42C2311yescharacteristic
74#946C2×C412yes—
75#946C2×C412yes—
76Q8×Dic3C2×C412yes—
77Q8×Dic3C2×C412yes—
78#938—21no—
79#938—21no—
80#938C2×C412yes—
81#938C2×C412yes—
82#937—22no—
83#936—21no—
84#936—21no—
85#935—22no—
86D4×Dic3—22no—
87D4×Dic3—22no—
88D4×Dic3—21no—
89D4×Dic3—21no—
90D4×Dic3C2×C412yes—
91D4×Dic3C2×C412yes—
92C4×#67—22no—
93C4×#67—22no—
94#926—22no—
95#926C2×C412yes—
96#924—21no—
97#924—21no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C22×C3⋊C8—21no—
103C2×ΓL(1,𝔽25)—22no—
104#897—21no—
105#897—21no—
106#892—21no—
107#892—21no—
108#890—21no—
109#890—21no—
110#887—21no—
111#887—21no—
112#883—22no—
113#883—22no—
114#67⋊C4—22no—
115#67⋊C4—22no—
116#879—22no—
117#879—22no—
118#877—22no—
119#877—22no—
120C42×D3C2×C411yescharacteristic
121C42×D3C2×C411yescharacteristic
122C42×D3D411yescharacteristic
123C42×D3D411yescharacteristic
124C42×D3D411yescharacteristic
125C42×D3D411yescharacteristic
126C42×D3C2311yescharacteristic
127C42×D3C2311yescharacteristic
128#870C2312yes—
129#869C2×C411yescharacteristic
130#869C2×C411yescharacteristic
131C4×Dic6—21no—
132C4×Dic6—21no—
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—31no—
153#532—31no—
154#532—31no—
155#532C2×D311yescharacteristic
156#529—32no—
157#529—32no—
158#529—61no—
159#529—61no—
160#529—61no—
161#529—61no—
162#525—32no—
163#525—61no—
164#525—61no—
165C4×#41—32no—
166C4×#41—61no—
167C4×#41—61no—
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#329—34no—
188#329C2×D314yescomplemented
189C23×D3C22⋊C411yescharacteristic
190C3×#41—44no—
191C2×C3×Q8C22×C412yes—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C22×C412yes—
196C22×C3×C4—22no—
197C22×C3×C4C22×C411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C2×#67—22no—
201C2×#67—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3C22⋊C411yescharacteristic
205C22×Dic3C22⋊C411yescharacteristic
206C22×Dic3C22⋊C411yescharacteristic
207#237—44no—
208#237—44no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3C22⋊C411yescharacteristic
213C2×C4×D3C22⋊C411yescharacteristic
214C2×C4×D3C22⋊C411yescharacteristic
215C2×C4×D3C22⋊C411yescharacteristic
216C2×C4×D3C22⋊C411yescharacteristic
217C2×C4×D3C22⋊C411yescharacteristic
218C2×C4×D3C22×C411yescharacteristic
219C2×C4×D3C22×C411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—42no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C42C22×C411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2411yescharacteristic
237#217—22no—
238#217—22no—
239#217—22no—
240#217—22no—
241#217—22no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—22no—
248C4⋊Dic3C22×C412yes—
249C4⋊Dic3C22×C412yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3—22no—
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22⋊C411yescharacteristic
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C22×C411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3C2×D411yescharacteristic
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—42no—
272ΓL(1,𝔽25)—44no—
273C2×#41—61no—
274C2×#41—61no—
275C2×#41C4×D312yes—
276C23×C4—31no—
277C23×C4—31no—
278C23×C4—31no—
279C2×M16—62no—
280C22×C8—21no—
281C22×C8—61no—
282C4⋊Q8—31no—
283C4⋊Q8—31no—
284C4⋊Q8C4×D311yescharacteristic
285#127C4×D311yescharacteristic
286#126—62no—
287#126—62no—
288#125—32no—
289#125—61no—
290#125—61no—
291#124—61no—
292#124—61no—
293#124C22×D312yes—
294#123—62no—
295#123—62no—
296#123—62no—
297C22⋊Q8—32no—
298C22⋊Q8—32no—
299C22⋊Q8—61no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302C22⋊Q8—61no—
303#121—21no—
304#121—21no—
305#121—32no—
306#121—32no—
307#121—61no—
308#121—61no—
309#120—62no—
310C4×Q8—32no—
311C4×Q8—32no—
312C4×Q8—61no—
313C4×Q8—61no—
314C4×Q8—61no—
315C4×Q8—61no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—32no—
319C4×D4—32no—
320C4×D4—61no—
321C4×D4—61no—
322C4×D4—62no—
323C4×D4—62no—
324C4×D4—62no—
325C4×D4—62no—
326C4×D4—62no—
327C4×D4—62no—
328#117—32no—
329#117—62no—
330#117—62no—
331#117—62no—
332#117—61no—
333#117—61no—
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—31no—
345C2×C42—62no—
346C2×C42C22×D311yescharacteristic
347C4⋊C8—21no—
348C4⋊C8—62no—
349C4⋊C8—62no—
350C4⋊C8—62no—
351C4⋊C8—61no—
352C22⋊C8—21no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355C22⋊C8—62no—
356C22⋊C8—61no—
357#97—21no—
358#97—32no—
359#97—32no—
360#97—61no—
361#97C22×D312yes—
362#97C22×D312yes—
363C4×C8—62no—
364C23×C3—22no—
365C23×C3C2×C22⋊C411yescharacteristic
366C22×D3—22no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3C2×C22⋊C411yescharacteristic
370C22×D3C2×C22⋊C411yescharacteristic
371C3×Q8—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—44no—
375C3×D4—42no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—44no—
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C2×C22⋊C411yescharacteristic
384C2×C3×C4C23×C411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387#67—44no—
388#67—44no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—44no—
395C2×Dic3—44no—
396C2×Dic3C2×C22⋊C411yescharacteristic
397C2×Dic3C2×C22⋊C411yescharacteristic
398C2×Dic3C2×C22⋊C411yescharacteristic
399C2×Dic3C2×C22⋊C411yescharacteristic
400C2×Dic3C2×C22⋊C411yescharacteristic
401C2×Dic3C2×C22⋊C411yescharacteristic
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414Dic6—42no—
415Dic6—42no—
416C3×C8—42no—
417C3×C8—42no—
418C3⋊C8—42no—
419C3⋊C8—42no—
420C24—31no—
421#41—44no—
422#41—124no—
423#41—124no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8C2×C4×D312yes—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4C2×C4×D312yes—
433C22×C4—22no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4C2×C4×D311yescharacteristic
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457M16—124no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—42no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—122no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—62no—
475C4⋊C4—62no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—31no—
494C42—31no—
495C42—31no—
496C42—31no—
497C42—62no—
498C42—62no—
499C42—62no—
500C42C2×C4×D311yescharacteristic
501C42D3×D411yescharacteristic
502C42D3×D411yescharacteristic
503C42C23×D311yescharacteristic
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3#41312yes—
510C22×C3C22×C22⋊C411yescharacteristic
511C2×D3—21no—
512C2×D3—21no—
513C2×D3—21no—
514C2×D3—21no—
515C2×D3—21no—
516C2×D3—21no—
517C2×D3—44no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—42no—
524C3×C4#42112yes—
525Dic3—21no—
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—21no—
530Dic3—42no—
531Dic3—42no—
532Dic3#42112yes—
533C23—22no—
534C23—31no—
535C23—31no—
536C23—62no—
537C23—121no—
538C23—121no—
539C23D3×C22⋊C411yescharacteristic
540Q8—42no—
541Q8—122no—
542Q8—122no—
543D4—42no—
544D4—42no—
545D4—44no—
546D4—42no—
547D4—124no—
548D4—124no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—124no—
577C2×C4—121no—
578C2×C4—124no—
579C2×C4D3×C22⋊C411yescharacteristic
580C2×C4D3×C22⋊C411yescharacteristic
581C2×C4D3×C22⋊C411yescharacteristic
582C2×C4C22×C4×D311yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C8—42no—
586C8—42no—
587C8—122no—
588C8—122no—
589C2×C3—22no—
590C2×C3—82no—
591C2×C3C2×#41311yescharacteristic
592C2×C3C2×#42111yescharacteristic
593C2×C3C2×#42111yescharacteristic
594D3—41no—
595D3—41no—
596C22—21no—
597C22—21no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—61no—
602C22—61no—
603C22—61no—
604C22—61no—
605C22—61no—
606C22—61no—
607C22—124no—
608C22#474512yes—
609C22C2×D3×C22⋊C411yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—32no—
616C4—42no—
617C4—61no—
618C4—61no—
619C4—61no—
620C4—61no—
621C4—61no—
622C4—122no—
623C4—122no—
624C4#482112yes—
625C3#1767411yescharacteristic, complemented, Sylow
626C2—22no—
627C2—82no—
628C2—121no—
629C2—121no—
630C2C2×#474511yescharacteristic
631C2C2×#482111yescharacteristic
632C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1530C2
socleC22×C3C22×C22⋊C4
prime radical#86219C2
cosocleC3×C42C24
hypercenter#1530D3
hypocenterC3#17674
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C2×#4705C2semidirect
4. C3×#1530C2semidirect
5. #71798C2semidirect
6. #70813C2semidirect
7. #69801C2semidirect
8. #68348C2semidirect
9. #68344C2semidirect
46. C3×#329C22semidirect
47. #4540C22semidirect
48. #4465C22semidirect
49. #4465C22semidirect
57. #1530D3semidirect
188. #329C2×D3semidirect
625. C3#17674semidirect

Maximal subgroups

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

subgroupindex
#862192
#793312
C2×#47052
C3×#15302
#717982
#708132
#698012
#683482
#683442
#176743