Finite Group Databasebeta

← #10801899

Subgroups — #10801899

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×#4816C211yesmaximal, characteristic
4C2×#4707C211yesmaximal, characteristic
5C4×#897C211yesmaximal, characteristic
6C3×#1763C211yesmaximal, characteristic, complemented
7#70271C211yesmaximal, characteristic
8#70258C211yesmaximal, characteristic, complemented
9#70253C211yesmaximal, characteristic, complemented
10#69813C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68884C211yesmaximal, characteristic
13#68870C211yesmaximal, characteristic
14#68858C211yesmaximal, characteristic, complemented
15#68850C211yesmaximal, characteristic, complemented
16#21539—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C2×#897C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#5030—24no—
36#4989—21no—
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984C2211yescharacteristic
40#4978—21no—
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×(C2×D3)⋊C4—21no—
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51C4×Dic3⋊C4—21no—
52C2×#803C2211yescharacteristic
53#4816C418yescomplemented
54#4707—24no—
55#4668—21no—
56#4653—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#470—31no—
64C2×#441—31no—
65C2×C4×M16—31no—
66C4×C22⋊Q8—31no—
67C4×C22⋊Q8—31no—
68#1969—31no—
69#1941—31no—
70#1933—31no—
71#1933—31no—
72#1894—31no—
73#1894—31no—
74#1809—31no—
75#1776—31no—
76#1763—31no—
77#1763D311yescharacteristic, complemented
78C22×C3×Q8C2311yescharacteristic
79C22×C4×D3D411yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#946—22no—
95#946C2×C414yes—
96C2×#216—22no—
97C2×#216—22no—
98C2×#216C2×C414yescomplemented
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×ΓL(1,𝔽25)—22no—
119C2×ΓL(1,𝔽25)—22no—
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)C2×C414yes—
124#897—22no—
125#897C2×C414yes—
126C42×D3—22no—
127C42×D3—22no—
128C4×Dic6—22no—
129C4×Dic6—22no—
130C3×#103—22no—
131C3×#103C2×C414yescomplemented
132C3×#97—22no—
133C3×#95—21no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#816—22no—
145#816C2×C414yescomplemented
146C8×Dic3—22no—
147C8×Dic3—22no—
148#810—22no—
149#808—22no—
150#808C2×C414yescomplemented
151#807—22no—
152#803D414yes—
153C22×Q16—31no—
154C22×M16—31no—
155C22×M16—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8C2×D311yescharacteristic
160C22×C42—31no—
161#470—38no—
162#441—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C4×M16—64no—
173C2×#97—31no—
174C2×#97C2×D311yescharacteristic
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391—31no—
178#391—31no—
179#389—31no—
180#389—31no—
181#387—61no—
182#387—61no—
183#385—61no—
184#385—61no—
185#380—31no—
186#380—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—61no—
189C4×C4⋊C4—61no—
190C4×C4⋊C4C2×D311yescharacteristic
191C4×C22⋊C4—61no—
192C4×C22⋊C4—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C23×D3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—41no—
229C2×C4×D3—41no—
230C2×C4×D3C22×C412yes—
231C2×C4×D3#4112yes—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C22×C412yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C22×C412yes—
243C2×C3×C8#4112yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4C22×C412yes—
251C3×C42—42no—
252C3×C42—42no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255#216—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260C4⋊Dic3—21no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3C22×C412yes—
268Dic3⋊C4—24no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C22×C412yes—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8C2×D412yes—
282C2×C3⋊C8#4112yes—
283ΓL(1,𝔽25)—28no—
284ΓL(1,𝔽25)—28no—
285C22×Q8—31no—
286C22×Q8C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C2×Q16—34no—
291C2×Q16—62no—
292C2×Q16—62no—
293C2×M16—34no—
294C2×M16—62no—
295C2×M16—62no—
296C2×M16—62no—
297C2×M16—62no—
298C2×M16—62no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22⋊Q8—34no—
304C22⋊Q8—34no—
305C22⋊Q8—62no—
306C22⋊Q8—62no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—62no—
310C4×Q8—62no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—62no—
333C2×C42—62no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42C22×D311yescharacteristic
337#107—62no—
338#106—34no—
339#106—62no—
340#103—22no—
341#103—34no—
342#103—62no—
343#103—62no—
344#103—62no—
345#103C4×D314yescomplemented
346#97—22no—
347#97—34no—
348C4×C8—62no—
349C4×C8—62no—
350#95—21no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3C4×D412yes—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#12112yes—
389C2×C3×C4#12112yes—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3C4×D412yes—
415C2×Dic3#13712yes—
416C2×Dic3#13712yes—
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—44no—
422C4×D3—44no—
423Dic6—22no—
424Dic6—22no—
425Dic6—41no—
426Dic6—41no—
427C3×C8—22no—
428C3×C8—22no—
429C3⋊C8—22no—
430C3⋊C8C4×D414yes—
431C24—31no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—32no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8C2×C4×D312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—121no—
466C22×C4—121no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470Q16—68no—
471M16—68no—
472M16—68no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—32no—
478C2×C8—32no—
479C2×C8—32no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8C2×C4×D312yes—
484C2×C8#23512yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—42no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4C2×C4×D312yes—
503C22⋊C4—64no—
504C22⋊C4—64no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C42—42no—
508C42—42no—
509C42—62no—
510C42—62no—
511C42—61no—
512C42—61no—
513C42—122no—
514C42—122no—
515C42#23512yes—
516C42D3×D412yes—
517C22×C3C2×C4×D411yescharacteristic
518C22×C3C2×#12111yescharacteristic
519C22×C3#53211yescharacteristic
520C22×C3C2×#13711yescharacteristic
521C22×C3C2×#13711yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57511yescharacteristic
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4C2×C4×D411yescharacteristic
538C3×C4C4×#4111yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#52911yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—22no—
544Dic3—42no—
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—32no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C2×D3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—121no—
563Q8—121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—124no—
596C2×C4—124no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4#88412yes—
608C2×C4#88412yes—
609C2×C4C22×C4×D311yescharacteristic
610C2×C4C2×#23511yescharacteristic
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—22no—
616C8—22no—
617C8—34no—
618C8—62no—
619C2×C3C4×#12111yescharacteristic
620C2×C3C4×#13711yescharacteristic
621C2×C3C4×#13711yescharacteristic
622C2×C3#305611yescharacteristic
623C2×C3#305611yescharacteristic
624C2×C3#318311yescharacteristic
625C2×C3#318511yescharacteristic
626D3—42no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22C2×#88411yescharacteristic
635C22C4×D3×D411yescharacteristic
636C22#554811yescharacteristic
637C22#577311yescharacteristic
638C22D3×#13711yescharacteristic
639C22D3×#13711yescharacteristic
640C22#577711yescharacteristic
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—62no—
646C4—62no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—122no—
651C4—122no—
652C4—241no—
653C4—241no—
654C4C4×#23511yescharacteristic
655C4#552811yescharacteristic
656C4C4×D3×D411yescharacteristic
657C4#554711yescharacteristic
658C3#2153911yescharacteristic, complemented, Sylow
659C2—122no—
660C2C4×#88411yescharacteristic
661C2#7888911yescharacteristic
662C2#7896211yescharacteristic
663C2#7977411yescharacteristic
664C2#7979311yescharacteristic
665C2#7982911yescharacteristic
666C2#7983811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1763D3
hypocenterC3#21539
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
6. C3×#1763C2semidirect
8. #70258C2semidirect
9. #70253C2semidirect
10. #69813C2semidirect
11. #69481C2semidirect
14. #68858C2semidirect
15. #68850C2semidirect
53. #4816C4semidirect
77. #1763D3semidirect
98. C2×#216C2×C4semidirect
131. C3×#103C2×C4semidirect
145. #816C2×C4semidirect
150. #808C2×C4semidirect
345. #103C4×D3semidirect
658. C3#21539semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
C2×#50302
C2×#48162
C2×#47072
C4×#8972
C3×#17632
#702712
#702582
#702532
#698132
#694812
#688842
#688702
#688582
#688502
#215393