Finite Group Databasebeta

← #10855124

Subgroups — #10855124

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4888C211yesmaximal, characteristic, complemented
2C2×#4881C211yesmaximal, characteristic, complemented
3#76212C211yesmaximal, characteristic
4#73638C211yesmaximal, characteristic
5C3×#1921C211yesmaximal, characteristic, complemented
6#69608C211yesmaximal, characteristic, complemented
7#69465C211yesmaximal, characteristic, complemented
8#68884C211yesmaximal, characteristic, complemented
9#68879C211yesmaximal, characteristic, complemented
10#68876C211yesmaximal, characteristic, complemented
11#68875C211yesmaximal, characteristic, complemented
12#68832C211yesmaximal, characteristic, complemented
13#68824C211yesmaximal, characteristic, complemented
14#68692C211yesmaximal, characteristic, complemented
15#68669C211yesmaximal, characteristic
16#26155—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#897C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4994—21no—
35#4987—21no—
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4939—21no—
45#4935C2211yescharacteristic
46C4×C4⋊Dic3—21no—
47#4929C2211yescharacteristic
48#4888—24no—
49#4881—24no—
50#4670—21no—
51#4670—21no—
52#4665—21no—
53#4661—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#485—31no—
63C2×#428—31no—
64#2417—31no—
65#2354—31no—
66#2096—31no—
67#2084—31no—
68#2077—31no—
69#2035—31no—
70#1928—31no—
71#1921D311yescharacteristic, complemented
72#1894—31no—
73#1873—31no—
74#1870—31no—
75#1854—31no—
76#1853—31no—
77C22×D12D411yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×ΓL(1,𝔽25)—22no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—22no—
116#897—22no—
117#897—22no—
118#897—22no—
119#897—41no—
120#892—22no—
121#892—22no—
122#892—41no—
123#892—41no—
124D3×C4⋊C4—22no—
125D3×C4⋊C4—41no—
126C4×D12—22no—
127C4×D12—22no—
128C3×#107D414yescomplemented
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136(C2×D3)⋊C8—22no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#817—22no—
140#817—41no—
141#816—22no—
142#816—41no—
143#810—22no—
144#810—22no—
145#808—22no—
146#808—22no—
147#807—22no—
148C22×M16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4—31no—
155#485—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#398—61no—
171#396—61no—
172#391—31no—
173#390—61no—
174#389C2×D311yescharacteristic
175#388—61no—
176#388—61no—
177#386—61no—
178#384—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—61no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—41no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—81no—
224C2×C4×D3#4112yes—
225C2×C4×D3#4112yes—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
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—81no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268C4⋊Dic3—42no—
269C4⋊Dic3—81no—
270Dic3⋊C4—42no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8#4112yes—
279ΓL(1,𝔽25)—28no—
280C22×Q8—31no—
281C22×D4—31no—
282C23×C4—31no—
283C23×C4—31no—
284C23×C4—61no—
285C2×M16—62no—
286C2×M16—62no—
287C2×M16—62no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22⋊Q8—62no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—121no—
296C4×D4—62no—
297C4×D4—62no—
298#117—62no—
299#117—62no—
300#117—121no—
301#117—121no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—121no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—21no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—62no—
332#107D1214yescomplemented
333#106—62no—
334#106—62no—
335#106—62no—
336#106—121no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—121no—
341C22⋊C8—62no—
342C22⋊C8—62no—
343C22⋊C8—62no—
344#95—21no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351C23×C3C22×D411yescharacteristic
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3#12312yes—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
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#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
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#12312yes—
408D12—44no—
409C4×D3—22no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—84no—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422C3×C8—22no—
423C3×C8—41no—
424C3⋊C8—22no—
425C3⋊C8—41no—
426C24—31no—
427C24—31no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—121no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—31no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—241no—
462C22×C4D3×D411yescharacteristic
463C22×C4D3×D411yescharacteristic
464C22×C4C23×D311yescharacteristic
465M16—68no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—32no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8C2×D1212yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—62no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—81no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4—122no—
496C4⋊C4—241no—
497C4⋊C4C2×D1212yes—
498C4⋊C4C2×D1212yes—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—21no—
509C42—21no—
510C42—42no—
511C42—122no—
512C42—122no—
513C22×C3C2×#12311yescharacteristic
514C22×C3D4211yescharacteristic
515C22×C3#54711yescharacteristic
516C22×C3C2×#13711yescharacteristic
517C22×C3#57511yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57711yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C2×D3—41no—
533C2×D3—41no—
534C3×C4—22no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4#52511yescharacteristic
541C3×C4#53911yescharacteristic
542C3×C4#54811yescharacteristic
543C3×C4#55311yescharacteristic
544Dic3—22no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—32no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23C2×D3×D411yescharacteristic
565Q8—121no—
566Q8—121no—
567Q8—121no—
568Q8—121no—
569D4—124no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—32no—
578C2×C4—41no—
579C2×C4—41no—
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—61no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
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—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—244no—
614C2×C4#91812yes—
615C2×C4C22×D1211yescharacteristic
616C2×C4#100711yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C2×C4D3×#4111yescharacteristic
620C2×C4#100911yescharacteristic
621C8—22no—
622C8—41no—
623C8—62no—
624C8—121no—
625C2×C3#244311yescharacteristic
626C2×C3#309511yescharacteristic
627C2×C3#309911yescharacteristic
628C2×C3#326711yescharacteristic
629C2×C3#327811yescharacteristic
630C2×C3#334811yescharacteristic
631C2×C3#335611yescharacteristic
632D3—42no—
633D3—81no—
634C22—62no—
635C22—62no—
636C22—62no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22—121no—
647C22—121no—
648C22D4×D1211yescharacteristic
649C22#563111yescharacteristic
650C22D3×#12311yescharacteristic
651C22C2×#91811yescharacteristic
652C22#577011yescharacteristic
653C22#577311yescharacteristic
654C22#577511yescharacteristic
655C4—22no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—62no—
660C4—81no—
661C4—81no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4#552411yescharacteristic
668C4#557611yescharacteristic
669C4#569211yescharacteristic
670C4#570911yescharacteristic
671C3#2615511yescharacteristic, complemented, Sylow
672C2—122no—
673C2—241no—
674C2#7531911yescharacteristic
675C2#7973111yescharacteristic
676C2#7987911yescharacteristic
677C2#8112911yescharacteristic
678C2#8119911yescharacteristic
679C2#8137711yescharacteristic
680C2#8146911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1921D3
hypocenterC3#26155
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
1. C2×#4888C2semidirect
2. C2×#4881C2semidirect
5. C3×#1921C2semidirect
6. #69608C2semidirect
7. #69465C2semidirect
8. #68884C2semidirect
9. #68879C2semidirect
10. #68876C2semidirect
11. #68875C2semidirect
12. #68832C2semidirect
13. #68824C2semidirect
14. #68692C2semidirect
71. #1921D3semidirect
128. C3×#107D4semidirect
332. #107D12semidirect
671. C3#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#48882
C2×#48812
#762122
#736382
C3×#19212
#696082
#694652
#688842
#688792
#688762
#688752
#688322
#688242
#686922
#686692
#261553