Finite Group Databasebeta

← #10845430

Subgroups — #10845430

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4875C211yesmaximal, characteristic, complemented
2C2×#4729C211yesmaximal, characteristic
3#73942C211yesmaximal, characteristic
4#73936C211yesmaximal, characteristic
5C3×#2075C211yesmaximal, characteristic, complemented
6#70631C211yesmaximal, characteristic, complemented
7#69936C211yesmaximal, characteristic, complemented
8#69516C211yesmaximal, characteristic, complemented
9#69513C211yesmaximal, characteristic, complemented
10#68889C211yesmaximal, characteristic, complemented
11#68884C211yesmaximal, characteristic, complemented
12#68824C211yesmaximal, characteristic, complemented
13#68803C211yesmaximal, characteristic, complemented
14#68796C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#25349—31nomaximal, Sylow
17C2×C4×#67C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C2×#897C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4957—21no—
45#4957—21no—
46#4948—21no—
47#4947—21no—
48#4875—24no—
49#4729—24no—
50#4676—21no—
51#4673—21no—
52#4650—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#477—31no—
63C2×#436—31no—
64#2563—31no—
65#2538—31no—
66#2075—31no—
67#2075D311yescharacteristic, complemented
68#2035—31no—
69#2035—31no—
70#1990—31no—
71#1948—31no—
72#1894—31no—
73#1894—31no—
74#1809—31no—
75#1809—31no—
76C4×#103—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67D411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217D411yescharacteristic
93C4×#67—22no—
94C4×#67—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—41no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×#208—22no—
116C2×#208—22no—
117C2×#208—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—41no—
123#897—22no—
124#897—41no—
125#888—22no—
126#888—22no—
127#887—22no—
128#887—22no—
129C3×#107—22no—
130C3×C22⋊C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138(C2×D3)⋊C8—22no—
139#817—41no—
140#816—22no—
141#816—22no—
142Dic3⋊C8D414yes—
143C8×Dic3—22no—
144#810—22no—
145#810—22no—
146#808—41no—
147#807—22no—
148C22×M16—31no—
149C22×M16—31no—
150C2×C4⋊Q8—31no—
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#477—64no—
156#436—64no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8—31no—
167C2×C22⋊C8C2×D311yescharacteristic
168C2×C4×C8—31no—
169#396—61no—
170#396—61no—
171#393—61no—
172#393—61no—
173#390—61no—
174#390—61no—
175#389—61no—
176#384—31no—
177#382—61no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380C2×D311yescharacteristic
183#380C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—22no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—42no—
229C2×Dic6—81no—
230C2×Dic6—81no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248#217—22no—
249#217—42no—
250#217—81no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263Dic3⋊C4—22no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273#208—44no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C2×D412yes—
279C2×C3⋊C8#4112yes—
280ΓL(1,𝔽25)—44no—
281C22×Q8—31no—
282C22×Q8—61no—
283C22×D4—31no—
284C23×C4—31no—
285C23×C4C22×D311yescharacteristic
286C2×M16—62no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C2×M16—121no—
291C2×M16—121no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C4⋊Q8—62no—
297C4⋊Q8—62no—
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—121no—
301C22⋊Q8—121no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—62no—
305C4×D4—62no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42—61no—
333#107—22no—
334#107—62no—
335#106—121no—
336#106—121no—
337C4⋊C8—34no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103—62no—
342C22⋊C8—22no—
343C22⋊C8—62no—
344C4×C8—62no—
345#95—21no—
346#95—21no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3C22×D411yescharacteristic
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
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—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392#67—44no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3#12112yes—
411C2×Dic3#12112yes—
412C2×Dic3#13712yes—
413C2×Dic3#13712yes—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—41no—
418Dic6—22no—
419Dic6—22no—
420Dic6—41no—
421Dic6—41no—
422Dic6—84no—
423C3×C8—22no—
424C3×C8—41no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C24—31no—
428C24D3×D411yescharacteristic
429C2×Q8—32no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—122no—
436C2×Q8—241no—
437C2×Q8—241no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—62no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466M16—124no—
467M16—124no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—32no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8D3×D412yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—62no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—241no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C22⋊C4—241no—
511C42—32no—
512C42—32no—
513C42—122no—
514C42—122no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3C2×#12111yescharacteristic
523C22×C3#54611yescharacteristic
524C22×C3#54711yescharacteristic
525C22×C3C2×#13711yescharacteristic
526C22×C3C2×#13711yescharacteristic
527C22×C3#57611yescharacteristic
528C22×C3#57711yescharacteristic
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C2×D3—41no—
535C2×D3—41no—
536C3×C4—41no—
537C3×C4—81no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4#52911yescharacteristic
542C3×C4#54211yescharacteristic
543C3×C4#54611yescharacteristic
544C3×C4#54811yescharacteristic
545Dic3—22no—
546Dic3—42no—
547Dic3—41no—
548Dic3—81no—
549Dic3—81no—
550Dic3—81no—
551Dic3#47014yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23C2×D3×D411yescharacteristic
567Q8—62no—
568Q8—62no—
569Q8—121no—
570Q8—121no—
571Q8—244no—
572D4—124no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4C2×D3×D411yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C2×C4D3×#4111yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4#100911yescharacteristic
622C2×C4#101011yescharacteristic
623C8—22no—
624C8—41no—
625C8—62no—
626C8—62no—
627C2×C3—81no—
628C2×C3#242711yescharacteristic
629C2×C3C2×#47011yescharacteristic
630C2×C3#306211yescharacteristic
631C2×C3#332911yescharacteristic
632C2×C3#333311yescharacteristic
633C2×C3#335611yescharacteristic
634C2×C3#335711yescharacteristic
635D3—81no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22—121no—
647C22—121no—
648C22—121no—
649C22—121no—
650C22D3×#12111yescharacteristic
651C22#565411yescharacteristic
652C22#565511yescharacteristic
653C22#575711yescharacteristic
654C22#577011yescharacteristic
655C22D3×#13711yescharacteristic
656C22D3×#13711yescharacteristic
657C4—34no—
658C4—41no—
659C4—62no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—121no—
665C4—122no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4#565411yescharacteristic
670C4#567711yescharacteristic
671C4#570611yescharacteristic
672C4#571011yescharacteristic
673C3#2534911yescharacteristic, complemented, Sylow
674C2—81no—
675C2—241no—
676C2#7746611yescharacteristic
677C2#8071811yescharacteristic
678C2D3×#47011yescharacteristic
679C2#8129811yescharacteristic
680C2#8130711yescharacteristic
681C2#8151211yescharacteristic
682C2#8155511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2075C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2075D3
hypocenterC3#25349
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
1. C2×#4875C2semidirect
5. C3×#2075C2semidirect
6. #70631C2semidirect
7. #69936C2semidirect
8. #69516C2semidirect
9. #69513C2semidirect
10. #68889C2semidirect
11. #68884C2semidirect
12. #68824C2semidirect
13. #68803C2semidirect
14. #68796C2semidirect
15. #68781C2semidirect
67. #2075D3semidirect
673. C3#25349semidirect

Maximal subgroups

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

subgroupindex
C2×#48752
C2×#47292
#739422
#739362
C3×#20752
#706312
#699362
#695162
#695132
#688892
#688842
#688242
#688032
#687962
#687812
#253493