Finite Group Databasebeta

← #10845517

Subgroups — #10845517

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic, complemented
2#83063C211yesmaximal, characteristic, complemented
3#82417C211yesmaximal, characteristic
4C2×#4836C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#71609C211yesmaximal, characteristic, complemented
7#70636C211yesmaximal, characteristic, complemented
8#70631C211yesmaximal, characteristic, complemented
9#70583C211yesmaximal, characteristic, complemented
10#70577C211yesmaximal, characteristic, complemented
11#70454C211yesmaximal, characteristic, complemented
12#70203C211yesmaximal, characteristic, complemented
13#70202C211yesmaximal, characteristic, complemented
14#70176C211yesmaximal, characteristic, complemented
15#70176C211yesmaximal, characteristic, complemented
16#25355—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×#930—21no—
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#478—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5212—21no—
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38#4991—21no—
39#4990—21no—
40#4990—21no—
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4987C2211yescharacteristic
45#4987C2211yescharacteristic
46#4987C2211yescharacteristic
47#4986—21no—
48#4961—21no—
49#4961—21no—
50#4935C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4836—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478D311yescharacteristic, complemented
63C2×#435—31no—
64#2562—31no—
65#2537—31no—
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2034—31no—
70#1992—31no—
71#1948—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×#67—21no—
80C22×#67—21no—
81C22×D12D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99#938—22no—
100#938—22no—
101#938—41no—
102#938—41no—
103#930—44no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×(C2×D3)⋊C4D411yescharacteristic
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4×Dic3—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124C2×ΓL(1,𝔽25)—41no—
125C2×D3×C8—22no—
126C2×D3×C8—22no—
127C2×D3×C8—41no—
128C4×D12—22no—
129C4×D12—22no—
130C3×#107—22no—
131C3×#106—22no—
132C3×C4⋊C8D414yescomplemented
133C3×#102—22no—
134C3×#102—22no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144(C2×D3)⋊C8—22no—
145(C2×D3)⋊C8—22no—
146#817—41no—
147#817—41no—
148#803—22no—
149C22×M16—31no—
150C23×C8—31no—
151C2×#125C2×D311yescharacteristic
152C2×#121—31no—
153C2×#121—31no—
154C2×#121—61no—
155C2×C4×D4—31no—
156C2×C4×D4C2×D311yescharacteristic
157#478—24no—
158#435—64no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170C2×#97—31no—
171#400—61no—
172#394—61no—
173#394—61no—
174#394—61no—
175#388—61no—
176#388—61no—
177#384—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380—61no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C23×D3C2×D411yescharacteristic
196C23×D3C2×D411yescharacteristic
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4#4112yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—81no—
217C2×#67—81no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—22no—
225C2×D12—22no—
226C2×D12—41no—
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—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256#217—42no—
257#217—42no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—42no—
276C4⋊Dic3—81no—
277C4×Dic3—81no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8#4112yes—
284ΓL(1,𝔽25)—44no—
285D3×C8—44no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4C22×D311yescharacteristic
290C23×C4—21no—
291C23×C4—31no—
292C23×C4—31no—
293C2×M16—62no—
294C2×M16—62no—
295C2×M16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—121no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#125—22no—
304#125—22no—
305#121—62no—
306#121—62no—
307#121—121no—
308#121—121no—
309#121—124no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—62no—
313C4×D4—62no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—22no—
340#106—22no—
341#106—121no—
342#106—121no—
343C4⋊C8#6714yescomplemented
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348C22⋊C8—62no—
349C22⋊C8—62no—
350#97—62no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3C22×D411yescharacteristic
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—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3AGL(1,ℤ/8)12yes—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
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×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4#12112yes—
399C2×C3×C4#12112yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4H(5,𝔽2)11yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406#67—84no—
407#67—84no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426D12—44no—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—41no—
435C3×C8—22no—
436C3×C8—22no—
437C3⋊C8—22no—
438C3⋊C8—41no—
439C24—21no—
440C24—31no—
441C24—31no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—121no—
450C2×D4—122no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—241no—
455C2×D4—241no—
456C2×D4#23912yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
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—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4D3×D411yescharacteristic
482C22×C4D3×D411yescharacteristic
483C22×C4C23×D311yescharacteristic
484M16—124no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—124no—
496C2×C8C2×#6712yes—
497C2×C8C2×#6712yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—62no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—122no—
516C4⋊C4—241no—
517C22⋊C4—42no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—241no—
529C42#23712yes—
530C42C2×#6712yes—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3C2×#12111yescharacteristic
539C22×C3#54611yescharacteristic
540C22×C3#54711yescharacteristic
541C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57711yescharacteristic
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4#52911yescharacteristic
565C3×C4#54211yescharacteristic
566C3×C4#54611yescharacteristic
567C3×C4#54811yescharacteristic
568Dic3—41no—
569Dic3—41no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574C23—22no—
575C23—32no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23C2×D3×D411yescharacteristic
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600D4—124no—
601D4—244no—
602D4—244no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
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—41no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—84no—
623C2×C4—121no—
624C2×C4—121no—
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—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4#93812yes—
644C2×C4#93812yes—
645C2×C4C2×#23711yescharacteristic
646C2×C4C2×#23911yescharacteristic
647C2×C4#100711yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4#100911yescharacteristic
650C2×C4C22×#6711yescharacteristic
651C8—22no—
652C8—22no—
653C8—62no—
654C8—121no—
655C2×C3—82no—
656C2×C3#242711yescharacteristic
657C2×C3#305511yescharacteristic
658C2×C3#306011yescharacteristic
659C2×C3#330711yescharacteristic
660C2×C3#331011yescharacteristic
661C2×C3#334511yescharacteristic
662C2×C3#334811yescharacteristic
663D3—42no—
664D3—81no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—62no—
669C22—62no—
670C22—62no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22#563011yescharacteristic
687C22#563111yescharacteristic
688C22#577011yescharacteristic
689C22#577011yescharacteristic
690C22D3×AGL(1,ℤ/8)11yescharacteristic
691C22#577311yescharacteristic
692C22C2×#93811yescharacteristic
693C4—22no—
694C4—22no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—121no—
699C4—121no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#562411yescharacteristic
705C4#564311yescharacteristic
706C4#581711yescharacteristic
707C4#582911yescharacteristic
708C3#2535511yescharacteristic, complemented, Sylow
709C2—82no—
710C2—122no—
711C2—241no—
712C2#8118811yescharacteristic
713C2#8118911yescharacteristic
714C2#8119711yescharacteristic
715C2#8119911yescharacteristic
716C2#8275611yescharacteristic
717C2#8391611yescharacteristic
718C2#8395011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×#478D3
hypocenterC3#25355
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×C3×#478C2semidirect
2. #83063C2semidirect
4. C2×#4836C2semidirect
5. #71741C2semidirect
6. #71609C2semidirect
7. #70636C2semidirect
8. #70631C2semidirect
9. #70583C2semidirect
10. #70577C2semidirect
11. #70454C2semidirect
12. #70203C2semidirect
13. #70202C2semidirect
14. #70176C2semidirect
15. #70176C2semidirect
62. C2×#478D3semidirect
132. C3×C4⋊C8D4semidirect
343. C4⋊C8#67semidirect
708. C3#25355semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4782
#830632
#824172
C2×#48362
#717412
#716092
#706362
#706312
#705832
#705772
#704542
#702032
#702022
#701762
#253553