Finite Group Databasebeta

← #10855141

Subgroups — #10855141

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic, complemented
2C2×#4779C211yesmaximal, characteristic, complemented
3#76438C211yesmaximal, characteristic
4#73771C211yesmaximal, characteristic, complemented
5C3×#1926C211yesmaximal, characteristic, complemented
6#70580C211yesmaximal, characteristic, complemented
7#70203C211yesmaximal, characteristic, complemented
8#70200C211yesmaximal, characteristic, complemented
9#70175C211yesmaximal, characteristic, complemented
10#69654C211yesmaximal, characteristic, complemented
11#69458C211yesmaximal, characteristic, complemented
12#68891C211yesmaximal, characteristic, complemented
13#68876C211yesmaximal, characteristic, complemented
14#68669C211yesmaximal, characteristic, complemented
15#68641C211yesmaximal, characteristic, complemented
16#26156—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4960—21no—
43#4959—21no—
44#4957—21no—
45#4951—21no—
46#4940—21no—
47C4×(C2×D3)⋊C4—21no—
48#4936C2211yescharacteristic
49#4928C2211yescharacteristic
50#4861—24no—
51#4779—24no—
52#4665—21no—
53#4653—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2414—31no—
65#2352—31no—
66#2092—31no—
67#2085—31no—
68#2077—31no—
69#2033—31no—
70#1926D311yescharacteristic, complemented
71#1918—31no—
72#1895—31no—
73#1874—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C22×#67—21no—
80C22×D12D411yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#938—22no—
96#938—22no—
97#938—22no—
98#938—41no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C2×ΓL(1,𝔽25)—22no—
118C2×ΓL(1,𝔽25)—22no—
119C2×ΓL(1,𝔽25)—22no—
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×#106—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#95—21no—
132#832—22no—
133#832—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#818—22no—
144#817—22no—
145#808—22no—
146#807—41no—
147#807D414yescomplemented
148C22×M16—31no—
149C2×#121—31no—
150C2×#121—31no—
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4—31no—
155#482—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#394—61no—
171#392—61no—
172#390—61no—
173#390C2×D311yescharacteristic
174#389—31no—
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C23×D3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×#67—42no—
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—81no—
212C2×#67—81no—
213C2×#67—81no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—22no—
221C2×D12—41no—
222C2×D12—41no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—81no—
236C2×C4×D3#4112yes—
237C2×C4×D3#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
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⋊C4C2×D412yes—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—81no—
255#217—42no—
256#217—42no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—21no—
265C4⋊Dic3—22no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3C2×D412yes—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283ΓL(1,𝔽25)—28no—
284C22×D4—31no—
285C22×D4—61no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—61no—
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#121—62no—
299#121—62no—
300#121—62no—
301#121—121no—
302C4×D4—62no—
303C4×D4—62no—
304#117—62no—
305#117—62no—
306#117—121no—
307#117—121no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—121no—
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×C22⋊C4—61no—
331C2×C42—21no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#107—34no—
336#107—62no—
337#107—121no—
338#106—22no—
339#106—62no—
340#106—62no—
341#102—22no—
342#102—22no—
343#102—62no—
344#102—121no—
345C22⋊C8—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3C22×D411yescharacteristic
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3#12312yes—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402#67—84no—
403#67—84no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3#12312yes—
422D12—44no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—84no—
432C3×C8—22no—
433C3×C8—41no—
434C3⋊C8—22no—
435C3⋊C8—41no—
436C24—21no—
437C24—31no—
438C24—31no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—62no—
446C2×D4—121no—
447C2×D4—121no—
448C2×D4—122no—
449C2×D4—122no—
450C2×D4—122no—
451C2×D4—241no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—31no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—31no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—241no—
478C22×C4D3×D411yescharacteristic
479C22×C4D3×D411yescharacteristic
480C22×C4C23×D311yescharacteristic
481M16—68no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—32no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8#23512yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—62no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—122no—
509C4⋊C4—241no—
510C4⋊C4D3×D412yes—
511C22⋊C4—42no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—81no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—122no—
526C42—122no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3C2×#12311yescharacteristic
535C22×C3D4211yescharacteristic
536C22×C3#54711yescharacteristic
537C22×C3#57511yescharacteristic
538C22×C3#57511yescharacteristic
539C22×C3#57611yescharacteristic
540C22×C3#57711yescharacteristic
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—22no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—42no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4#52511yescharacteristic
561C3×C4#53911yescharacteristic
562C3×C4#54811yescharacteristic
563C3×C4#55311yescharacteristic
564Dic3—22no—
565Dic3—41no—
566Dic3—42no—
567Dic3—81no—
568Dic3—81no—
569Dic3—81no—
570C23—32no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23C2×D3×D411yescharacteristic
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—124no—
597D4—244no—
598D4—244no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—32no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—62no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—244no—
641C2×C4C2×#23511yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C2×C4#100911yescharacteristic
646C2×C4#101011yescharacteristic
647C8—22no—
648C8—41no—
649C8—62no—
650C8—121no—
651C2×C3—161no—
652C2×C3#244311yescharacteristic
653C2×C3#309311yescharacteristic
654C2×C3#309611yescharacteristic
655C2×C3#327111yescharacteristic
656C2×C3#327311yescharacteristic
657C2×C3#334711yescharacteristic
658C2×C3#335111yescharacteristic
659D3—42no—
660D3—81no—
661C22—62no—
662C22—62no—
663C22—62no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—121no—
672C22—121no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22#555111yescharacteristic
683C22#560911yescharacteristic
684C22D3×#12311yescharacteristic
685C22#576211yescharacteristic
686C22#576811yescharacteristic
687C22#577311yescharacteristic
688C22#577411yescharacteristic
689C4—42no—
690C4—41no—
691C4—41no—
692C4—62no—
693C4—81no—
694C4—81no—
695C4—121no—
696C4—122no—
697C4—241no—
698C4—241no—
699C4—241no—
700C4#553011yescharacteristic
701C4#554711yescharacteristic
702C4#567711yescharacteristic
703C4#570711yescharacteristic
704C3#2615611yescharacteristic, complemented, Sylow
705C2—122no—
706C2—161no—
707C2—241no—
708C2#7584811yescharacteristic
709C2#7969111yescharacteristic
710C2#7988511yescharacteristic
711C2#8040711yescharacteristic
712C2#8055511yescharacteristic
713C2#8132711yescharacteristic
714C2#8137211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1926C2
socleC23×C3C22×D4
prime radical#76438C2
hypercenter#1926D3
hypocenterC3#26156
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
1. C2×#4861C2semidirect
2. C2×#4779C2semidirect
4. #73771C2semidirect
5. C3×#1926C2semidirect
6. #70580C2semidirect
7. #70203C2semidirect
8. #70200C2semidirect
9. #70175C2semidirect
10. #69654C2semidirect
11. #69458C2semidirect
12. #68891C2semidirect
13. #68876C2semidirect
14. #68669C2semidirect
15. #68641C2semidirect
70. #1926D3semidirect
147. #807D4semidirect
704. C3#26156semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
C2×#47792
#764382
#737712
C3×#19262
#705802
#702032
#702002
#701752
#696542
#694582
#688912
#688762
#686692
#686412
#261563