Finite Group Databasebeta

← #10813353

Subgroups — #10813353

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4881C211yesmaximal, characteristic, complemented
2C2×#4860C211yesmaximal, characteristic, complemented
3C2×#4706C211yesmaximal, characteristic
4#74324C211yesmaximal, characteristic
5#74229C211yesmaximal, characteristic
6C3×#1847C211yesmaximal, characteristic, complemented
7#70633C211yesmaximal, characteristic, complemented
8#70632C211yesmaximal, characteristic, complemented
9#70630C211yesmaximal, characteristic, complemented
10#70580C211yesmaximal, characteristic, complemented
11#70580C211yesmaximal, characteristic, complemented
12#70578C211yesmaximal, characteristic, complemented
13#69814C211yesmaximal, characteristic, complemented
14#69567C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#22660—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C3×#382C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4990—21no—
36#4990—21no—
37#4988—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42Dic3×C4⋊C4—21no—
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4881—24no—
51#4860—24no—
52#4706C418yescomplemented
53#4669—21no—
54#4666—21no—
55#4665—21no—
56#4654—21no—
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#435—31no—
64C2×#428—31no—
65C2×#427—31no—
66#2354—31no—
67#2352—31no—
68#1949—31no—
69#1947—31no—
70#1854—31no—
71#1854—31no—
72#1854—31no—
73#1852—31no—
74#1847D311yescharacteristic, complemented
75#1809—31no—
76#1808—31no—
77#1776—31no—
78C22×D12D411yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
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⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—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)C2×C414yes—
116C2×D3×C8—22no—
117C2×D3×C8—22no—
118C2×D3×C8C2×C414yescomplemented
119#892—22no—
120#892—22no—
121#892—22no—
122#892—41no—
123#892—41no—
124#892—41no—
125D3×C4⋊C4—22no—
126D3×C4⋊C4—41no—
127C4×D12C2×C414yes—
128C4×D12Q814yes—
129C3×#107—22no—
130C3×#107—41no—
131C3×#106—22no—
132C3×#106—41no—
133C3×#97C2×C414yescomplemented
134C3×#95—21no—
135C3×#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142(C2×D3)⋊C8C2×C414yes—
143(C2×D3)⋊C8C2×C414yescomplemented
144#818—22no—
145#817D414yescomplemented
146#808—22no—
147#807—22no—
148#807—22no—
149#807—22no—
150C4⋊(C3⋊C8)C2×C414yescomplemented
151C22×M16—31no—
152C23×C8—31no—
153C2×C4×D4—31no—
154C2×#117—31no—
155C2×#117—31no—
156C2×#117—31no—
157C22×C4⋊C4—31no—
158#435—38no—
159#428—64no—
160#427—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107C2×D311yescharacteristic
166C2×#107C2×D311yescharacteristic
167C2×#106—31no—
168C2×#106—31no—
169C2×#106C2×D311yescharacteristic
170C2×#106C2×D311yescharacteristic
171C2×C4⋊C8—31no—
172C2×C22⋊C8—31no—
173C2×C22⋊C8—31no—
174C2×#97C2×D311yescharacteristic
175#390—61no—
176#388—61no—
177#388—61no—
178#388—61no—
179#387—61no—
180#386—61no—
181#384C2×D311yescharacteristic
182#382C2×D311yescharacteristic
183#380—31no—
184#380—31no—
185#380—31no—
186#380—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197C23×D3C2×D411yescharacteristic
198C23×D3C2×D411yescharacteristic
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C2×D12—41no—
213C2×D12—41no—
214C2×D12C4⋊C414yes—
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—21no—
222C2×C4×D3—21no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—81no—
226C2×C4×D3C22×C412yes—
227C2×C4×D3C22×C412yes—
228C2×C4×D3C2×Q812yes—
229C2×C4×D3C2×Q812yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C22×C412yes—
233C2×C3×C8C22×C412yes—
234C2×C3×C8C2×D412yes—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
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—42no—
245C3×C4⋊C4—81no—
246C3×C42C22×C412yes—
247C3×C42C2×Q812yes—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4C4⋊C414yes—
256(C2×D3)⋊C4C4⋊C414yes—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C4⋊C414yes—
264C4⋊Dic3C2×D412yes—
265C4⋊Dic3C2×D412yes—
266Dic3⋊C4—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3—81no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C22×C412yes—
277C2×C3⋊C8C22×C412yes—
278ΓL(1,𝔽25)—28no—
279D3×C8—28no—
280C22×D4—31no—
281C23×C4—31no—
282C23×C4—31no—
283C23×C4—61no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—34no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4×D4—34no—
295C4×D4—34no—
296#117—62no—
297#117—62no—
298#117—62no—
299#117—121no—
300#117—121no—
301#117—121no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—121no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—22no—
332#107—41no—
333#107—62no—
334#107—62no—
335#107—62no—
336#107—62no—
337#106—22no—
338#106—34no—
339#106—41no—
340#106—62no—
341C4⋊C8—34no—
342C22⋊C8—34no—
343C22⋊C8—34no—
344#97C4×D314yescomplemented
345#95—21no—
346#95—21no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353C23×C3C22×D411yescharacteristic
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—21no—
366C22×D3C2×C4⋊C412yes—
367C22×D3C2×C4⋊C412yes—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
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×C4C2×C4⋊C412yes—
385C2×C3×C4C2×C4⋊C412yes—
386C2×C3×C4C23×C411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×Q811yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
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—41no—
408C2×Dic3C2×C4⋊C412yes—
409C2×Dic3C2×C4⋊C412yes—
410D12—44no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—84no—
420C3×C8—22no—
421C3×C8C4×D414yes—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C24—31no—
425C24—31no—
426C2×D4—34no—
427C2×D4—121no—
428C2×D4—121no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—241no—
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D411yescharacteristic
456C22×C4C23×D311yescharacteristic
457M16—68no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—32no—
461C2×C8—32no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—68no—
467C2×C8C2×C4×D312yes—
468C2×C8C2×C4×D312yes—
469C2×C8D3×D412yes—
470C2×C8#23912yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—34no—
480C4⋊C4—32no—
481C4⋊C4—32no—
482C4⋊C4—42no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—81no—
490C4⋊C4—122no—
491C22⋊C4—34no—
492C22⋊C4—34no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C42—122no—
501C42—122no—
502C42—122no—
503C42—122no—
504C42—241no—
505C42C2×C4×D312yes—
506C42D3×Q812yes—
507C22×C3C22×C4⋊C411yescharacteristic
508C22×C3D4211yescharacteristic
509C22×C3#54711yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57511yescharacteristic
512C22×C3#57611yescharacteristic
513C22×C3#57811yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4C2×C4×D411yescharacteristic
535C3×C4#52011yescharacteristic
536C3×C4D4×Q811yescharacteristic
537C3×C4#55411yescharacteristic
538Dic3—22no—
539Dic3—22no—
540Dic3—42no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—32no—
545C23—32no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23C2×D3×D411yescharacteristic
559D4—124no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—244no—
603C2×C4D3×C4⋊C412yes—
604C2×C4D3×C4⋊C412yes—
605C2×C4C22×C4×D311yescharacteristic
606C2×C4C2×D3×D411yescharacteristic
607C2×C4#100411yescharacteristic
608C2×C4C2×D3×Q811yescharacteristic
609C2×C4C2×#23911yescharacteristic
610C2×C4#100711yescharacteristic
611C8—22no—
612C8—62no—
613C8—62no—
614C8#89314yes—
615C2×C3D4×C4⋊C411yescharacteristic
616C2×C3#291111yescharacteristic
617C2×C3#291311yescharacteristic
618C2×C3#326811yescharacteristic
619C2×C3#326911yescharacteristic
620C2×C3#334711yescharacteristic
621C2×C3#334911yescharacteristic
622D3—42no—
623D3—42no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22C2×D3×C4⋊C411yescharacteristic
639C22#571711yescharacteristic
640C22#572611yescharacteristic
641C22#577311yescharacteristic
642C22#577411yescharacteristic
643C22#577711yescharacteristic
644C22#577911yescharacteristic
645C4—22no—
646C4—22no—
647C4—62no—
648C4—62no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—122no—
654C4—122no—
655C4—241no—
656C4—241no—
657C4C2×#89311yescharacteristic
658C4#550411yescharacteristic
659C4#572711yescharacteristic
660C4#573011yescharacteristic
661C3#2266011yescharacteristic, complemented, Sylow
662C2—122no—
663C2—122no—
664C2#7436011yescharacteristic
665C2#7920811yescharacteristic
666C2#7921011yescharacteristic
667C2#8185811yescharacteristic
668C2#8188311yescharacteristic
669C2#8199411yescharacteristic
670C2#8199611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1847C2
socleC23×C3C22×D4
prime radicalC22×D12D4
cosocleC22×C3×C4C24
hypercenter#1847D3
hypocenterC3#22660
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
1. C2×#4881C2semidirect
2. C2×#4860C2semidirect
6. C3×#1847C2semidirect
7. #70633C2semidirect
8. #70632C2semidirect
9. #70630C2semidirect
10. #70580C2semidirect
11. #70580C2semidirect
12. #70578C2semidirect
13. #69814C2semidirect
14. #69567C2semidirect
15. #69540C2semidirect
52. #4706C4semidirect
74. #1847D3semidirect
118. C2×D3×C8C2×C4semidirect
133. C3×#97C2×C4semidirect
143. (C2×D3)⋊C8C2×C4semidirect
145. #817D4semidirect
150. C4⋊(C3⋊C8)C2×C4semidirect
344. #97C4×D3semidirect
661. C3#22660semidirect

Maximal subgroups

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

subgroupindex
C2×#48812
C2×#48602
C2×#47062
#743242
#742292
C3×#18472
#706332
#706322
#706302
#705802
#705782
#698142
#695672
#695402
#226603