Finite Group Databasebeta

← #10813405

Subgroups — #10813405

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5128C211yesmaximal, characteristic
2C2×#5121C211yesmaximal, characteristic, complemented
3C2×#4861C211yesmaximal, characteristic, complemented
4#74012C211yesmaximal, characteristic
5#73883C211yesmaximal, characteristic
6C3×#1809C211yesmaximal, characteristic, complemented
7#70631C211yesmaximal, characteristic, complemented
8Dic3×#107C211yesmaximal, characteristic, complemented
9#70555C211yesmaximal, characteristic, complemented
10#69506C211yesmaximal, characteristic, complemented
11#69504C211yesmaximal, characteristic, complemented
12#68876C211yesmaximal, characteristic, complemented
13#68874C211yesmaximal, characteristic, complemented
14#68798C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#22664—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C2×#892C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C2×#830C2211yescharacteristic
29#5128C418yescomplemented
30#5121—24no—
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4977C2211yescharacteristic
41#4977C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4958—21no—
49#4956—21no—
50#4944—21no—
51Dic3×C22⋊C4—21no—
52#4861—24no—
53#4670—21no—
54#4666—21no—
55#4655—21no—
56#4652—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#436—31no—
64C2×#428—31no—
65C2×#428—31no—
66#2352—31no—
67#2352—31no—
68#1949—31no—
69#1949—31no—
70#1854—31no—
71#1854—31no—
72#1852—31no—
73#1852—31no—
74#1846—31no—
75#1809—31no—
76#1809D311yescharacteristic, complemented
77C4×#107—31no—
78C23×C3×C4C2311yescharacteristic
79C22×#67D411yescharacteristic
80C23×Dic3—21no—
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16C2×C414yescomplemented
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×#217—21no—
95C2×#217D411yescharacteristic
96C4×#67C2×C414yes—
97C4×#67Q814yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102#926—22no—
103#926—41no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—41no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)C2×C414yescomplemented
123#892—22no—
124#892—41no—
125D3×C4⋊C4—22no—
126D3×C4⋊C4—41no—
127C3×#107—22no—
128C3×#106—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#830C2×C414yescomplemented
140(C2×D3)⋊C8C2×C414yescomplemented
141#818—22no—
142#818D414yescomplemented
143#817—22no—
144#817—22no—
145Dic3⋊C8C2×C414yescomplemented
146C8×Dic3C2×C414yescomplemented
147#808—22no—
148#807—22no—
149#807—41no—
150#807—41no—
151C22×M16—31no—
152C22×M16C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×#117—31no—
155C2×#117—31no—
156C22×C4⋊C4—31no—
157C22×C4⋊C4—31no—
158#436—38no—
159#428—64no—
160#428—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107—31no—
166C2×#107C2×D311yescharacteristic
167C2×#106—31no—
168C2×#106—31no—
169C2×#106—31no—
170C2×#106C2×D311yescharacteristic
171C2×C4⋊C8—31no—
172C2×C22⋊C8—31no—
173C2×C22⋊C8—31no—
174C2×C4×C8—31no—
175#388—61no—
176#388—61no—
177#387—61no—
178#387—61no—
179#386—61no—
180#386—61no—
181#382—31no—
182#382—31no—
183#380—31no—
184#380—31no—
185#380C2×D311yescharacteristic
186#380C2×D311yescharacteristic
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C22⋊C4—61no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#340C2×D311yescharacteristic
196#340C2×D311yescharacteristic
197C24×C3C2×D411yescharacteristic
198C23×D3C2×D411yescharacteristic
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×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C22×C412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×Q812yes—
210C22×C3×C4C2411yescharacteristic
211C2×#67—41no—
212C2×#67—41no—
213C2×#67C4⋊C414yes—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—81no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3C22×C412yes—
232C2×C4×D3C2×Q812yes—
233C3×M16—28no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C22×C412yes—
237C2×C3×C8C22×C412yes—
238C2×C3×C8C2×D412yes—
239C2×C3×C8#4112yes—
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×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—81no—
250#217—42no—
251#217C4⋊C414yes—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4C4⋊C414yes—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3C2×D412yes—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—42no—
266Dic3⋊C4C4⋊C414yes—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—81no—
270C4×Dic3—81no—
271C4×Dic3C22×C412yes—
272C4×Dic3C2×Q812yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8C22×C412yes—
278C2×C3⋊C8C22×C412yes—
279ΓL(1,𝔽25)—28no—
280C22×D4—31no—
281C23×C4—31no—
282C23×C4—61no—
283C23×C4—61no—
284C23×C4C22×D311yescharacteristic
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—34no—
288C2×M16—62no—
289C2×M16—62no—
290C2×M16C4×D314yescomplemented
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4×D4—34no—
296C4×D4—34no—
297#117—62no—
298#117—62no—
299#117—121no—
300#117—121no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4—121no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
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×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329#107—22no—
330#107—34no—
331#107—62no—
332#107—62no—
333#107—121no—
334#107—121no—
335#106—22no—
336#106—62no—
337#106—62no—
338#106—62no—
339C4⋊C8—34no—
340C22⋊C8—34no—
341C22⋊C8—34no—
342C4×C8—34no—
343#95—21no—
344#95—21no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3C2×C4⋊C412yes—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3C2×C4⋊C412yes—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
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×C4C2×C4⋊C412yes—
387C2×C3×C4C23×C411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×Q811yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393#67—44no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—22no—
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—41no—
411C2×Dic3—41no—
412C2×Dic3—84no—
413C2×Dic3C2×C4⋊C412yes—
414C2×Dic3C2×C4⋊C412yes—
415C2×Dic3C2×C4⋊C412yes—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—84no—
421C3×C8—22no—
422C3×C8C4×D414yes—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—31no—
426C24D3×D411yescharacteristic
427C2×D4—34no—
428C2×D4—121no—
429C2×D4—121no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—32no—
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—122no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4—241no—
457C22×C4C2×C4×D312yes—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×Q812yes—
460C22×C4C23×D311yescharacteristic
461M16—28no—
462M16—68no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8C2×C4×D312yes—
472C2×C8C2×C4×D312yes—
473C2×C8D3×D412yes—
474C2×C8#23712yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—34no—
484C4⋊C4—32no—
485C4⋊C4—32no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C22⋊C4—34no—
495C22⋊C4—34no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C42—32no—
503C42—32no—
504C42—122no—
505C42—122no—
506C42—241no—
507C42—241no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3C22×C4⋊C411yescharacteristic
516C22×C3D4211yescharacteristic
517C22×C3#54711yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57511yescharacteristic
520C22×C3#57611yescharacteristic
521C22×C3#57811yescharacteristic
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
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—22no—
541Dic3—42no—
542Dic3—42no—
543Dic3—81no—
544Dic3—81no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—32no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23D3×C4⋊C412yes—
559C23C2×D3×D411yescharacteristic
560D4—124no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—32no—
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—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
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—121no—
603C2×C4—121no—
604C2×C4—244no—
605C2×C4—244no—
606C2×C4D3×C4⋊C412yes—
607C2×C4C22×C4×D311yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×#23711yescharacteristic
610C2×C4C2×D3×Q811yescharacteristic
611C2×C4#100911yescharacteristic
612C2×C4#101011yescharacteristic
613C8—22no—
614C8—62no—
615C8—62no—
616C8#67⋊C414yes—
617C2×C3—42no—
618C2×C3D4×C4⋊C411yescharacteristic
619C2×C3#291111yescharacteristic
620C2×C3#291311yescharacteristic
621C2×C3#327311yescharacteristic
622C2×C3#327711yescharacteristic
623C2×C3#335211yescharacteristic
624C2×C3#335511yescharacteristic
625D3—42no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—62no—
634C22—62no—
635C22—62no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22C2×D3×C4⋊C411yescharacteristic
641C22#560911yescharacteristic
642C22#561111yescharacteristic
643C22#577311yescharacteristic
644C22#577411yescharacteristic
645C22#577711yescharacteristic
646C22#577911yescharacteristic
647C4—22no—
648C4—62no—
649C4—62no—
650C4—62no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—122no—
656C4—122no—
657C4—241no—
658C4—241no—
659C4C2×#67⋊C411yescharacteristic
660C4#552311yescharacteristic
661C4#563211yescharacteristic
662C4#563511yescharacteristic
663C3#2266411yescharacteristic, complemented, Sylow
664C2—42no—
665C2—122no—
666C2#7401011yescharacteristic
667C2#7920811yescharacteristic
668C2#7921011yescharacteristic
669C2#8083911yescharacteristic
670C2#8084311yescharacteristic
671C2#8086111yescharacteristic
672C2#8090011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1809C2
socleC23×C3C22×D4
prime radicalC22×#67D4
cosocleC22×C3×C4C24
hypercenter#1809D3
hypocenterC3#22664
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#5121C2semidirect
3. C2×#4861C2semidirect
6. C3×#1809C2semidirect
7. #70631C2semidirect
8. Dic3×#107C2semidirect
9. #70555C2semidirect
10. #69506C2semidirect
11. #69504C2semidirect
12. #68876C2semidirect
13. #68874C2semidirect
14. #68798C2semidirect
15. #68797C2semidirect
29. #5128C4semidirect
76. #1809D3semidirect
85. C2×C3×M16C2×C4semidirect
122. C2×ΓL(1,𝔽25)C2×C4semidirect
139. #830C2×C4semidirect
140. (C2×D3)⋊C8C2×C4semidirect
142. #818D4semidirect
145. Dic3⋊C8C2×C4semidirect
146. C8×Dic3C2×C4semidirect
290. C2×M16C4×D3semidirect
663. C3#22664semidirect

Maximal subgroups

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

subgroupindex
C2×#51282
C2×#51212
C2×#48612
#740122
#738832
C3×#18092
#706312
Dic3×#1072
#705552
#695062
#695042
#688762
#688742
#687982
#687972
#226643