Finite Group Databasebeta

← #10813356

Subgroups — #10813356

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×C3×#435C211yesmaximal, characteristic
2C2×#5121C211yesmaximal, characteristic, complemented
3C2×#5106C211yesmaximal, characteristic, complemented
4#82337C211yesmaximal, characteristic
5#82335C211yesmaximal, characteristic
6#71988C211yesmaximal, characteristic, complemented
7#71971C211yesmaximal, characteristic, complemented
8#70455C211yesmaximal, characteristic, complemented
9#70453C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic, complemented
11#69929C211yesmaximal, characteristic, complemented
12#69929C211yesmaximal, characteristic, complemented
13#69927C211yesmaximal, characteristic, complemented
14#69816C211yesmaximal, characteristic, complemented
15#69814C211yesmaximal, characteristic, complemented
16#22660—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C3×#435C418yescomplemented
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5221—21no—
30#5220—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5121—24no—
36#5106—24no—
37C2×#818C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#817C2211yescharacteristic
44C2×#817C2211yescharacteristic
45C2×#817C2211yescharacteristic
46C2×#817C2211yescharacteristic
47#4980—21no—
48Dic3×C4⋊C4—21no—
49#4951—21no—
50#4951—21no—
51#4951—21no—
52Dic3×C22⋊C4—21no—
53#4936C2211yescharacteristic
54#4935C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62#4551C2211yescharacteristic
63C2×#435D311yescharacteristic
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#1847—31no—
75#1809—31no—
76#1808—31no—
77#1776—31no—
78C22×C3×D4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16C2×C414yes—
85C22×C3×C8—22no—
86C22×C3×C8—22no—
87C22×C3×C8C2×C414yescomplemented
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×C4×D4C2×C414yes—
93C3×C4×D4Q814yes—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105#926—22no—
106#926—22no—
107#926—22no—
108#926—41no—
109#926—41no—
110#926—41no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—41no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4×Dic3—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C3×C4⋊C8C2×C414yescomplemented
130C3×C22⋊C8C2×C414yes—
131C3×C22⋊C8C2×C414yescomplemented
132C3×#97C2×C414yescomplemented
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#818—22no—
142#818—22no—
143#818—22no—
144#818—22no—
145#818—22no—
146#818—41no—
147#817—22no—
148#817—22no—
149#817—41no—
150#817D414yescomplemented
151C22×M16C2×D311yescharacteristic
152C23×C8C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154C2×#117—31no—
155C2×#117—31no—
156C2×#117—31no—
157C22×C4⋊C4—31no—
158#435Dic318yescomplemented
159#428—64no—
160#427—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107—31no—
166C2×#107—31no—
167C2×#106—31no—
168C2×#106—31no—
169C2×#106—31no—
170C2×#106—31no—
171C2×C4⋊C8C2×D311yescharacteristic
172C2×C22⋊C8C2×D311yescharacteristic
173C2×C22⋊C8C2×D311yescharacteristic
174C2×#97C2×D311yescharacteristic
175#390—61no—
176#388—61no—
177#388—61no—
178#388—61no—
179#387—61no—
180#386—61no—
181#384—31no—
182#382—31no—
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—
197C24×C3C2×D411yescharacteristic
198C24×C3C2×D411yescharacteristic
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4C4⋊C414yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C22×C412yes—
211C22×C3×C4C22×C412yes—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×Q812yes—
217C22×C3×C4C2×Q812yes—
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3—81no—
230C3×M16—28no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—28no—
238C2×C3×C8C22×C412yes—
239C2×C3×C8C22×C412yes—
240C2×C3×C8C22×C412yes—
241C2×C3×C8C22×C412yes—
242C2×C3×C8C2×D412yes—
243C2×C3×C8#4112yes—
244C3×C4⋊C4C4⋊C414yes—
245C3×C22⋊C4C4⋊C414yes—
246C3×C22⋊C4C4⋊C414yes—
247C3×C42C22×C412yes—
248C3×C42C2×Q812yes—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—81no—
254#217—81no—
255#217—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3—42no—
272C4⋊Dic3—81no—
273C4⋊Dic3C2×D412yes—
274C4⋊Dic3C2×D412yes—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C4×Dic3—81no—
280C22×D4D1211yescharacteristic
281C23×C4—61no—
282C23×C4C22×D311yescharacteristic
283C23×C4C22×D311yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16C2×Dic314yes—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8C2×Dic314yescomplemented
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4×D4Dic614yes—
295C4×D4C2×Dic314yes—
296#117—62no—
297#117—62no—
298#117—62no—
299#117—121no—
300#117—121no—
301#117—121no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—121no—
315C2×C4⋊C4D1211yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4D1211yescharacteristic
324C2×C22⋊C4D1211yescharacteristic
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—62no—
332#107—62no—
333#107—62no—
334#107—62no—
335#107—62no—
336#107—121no—
337#106—34no—
338#106—62no—
339#106—62no—
340#106—121no—
341C4⋊C8C2×Dic314yescomplemented
342C22⋊C8C2×Dic314yes—
343C22⋊C8C2×Dic314yescomplemented
344#97C2×Dic314yescomplemented
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×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3C2×C4⋊C412yes—
366C23×C3C2×C4⋊C412yes—
367C23×C3C22×D411yescharacteristic
368C3×D4—44no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4C2×C4⋊C412yes—
386C2×C3×C4C2×C4⋊C412yes—
387C2×C3×C4C2×C4⋊C412yes—
388C2×C3×C4C2×C4⋊C412yes—
389C2×C3×C4C23×C411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×Q811yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C2×Dic3—22no—
396C2×Dic3—22no—
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—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
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—84no—
420C3×C8—22no—
421C3×C8—22no—
422C3×C8—22no—
423C3×C8C4×D414yes—
424C24C2×D1211yescharacteristic
425C24C2×D1211yescharacteristic
426C2×D4—41no—
427C2×D4—41no—
428C2×D4C4⋊Dic314yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—241no—
448C22×C4C2×Dic612yes—
449C22×C4C2×Dic612yes—
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4C2×D1211yescharacteristic
453C22×C4C2×D1211yescharacteristic
454C22×C4C22×Dic312yes—
455C22×C4C22×Dic312yes—
456C22×C4C23×D311yescharacteristic
457M16—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—28no—
465C2×C8D3×D412yes—
466C2×C8#23712yes—
467C2×C8C22×Dic312yes—
468C2×C8C22×Dic312yes—
469C2×C8C22×Dic312yes—
470C2×C8C22×Dic312yes—
471C4⋊C4—32no—
472C4⋊C4—32no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4C4⋊Dic314yes—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—241no—
496C22⋊C4—241no—
497C22⋊C4—241no—
498C22⋊C4C4⋊Dic314yes—
499C22⋊C4C4⋊Dic314yes—
500C42—122no—
501C42—122no—
502C42—122no—
503C42—122no—
504C42—241no—
505C42C2×Dic612yes—
506C42C22×Dic312yes—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3C22×C4⋊C411yescharacteristic
522C22×C3D4211yescharacteristic
523C22×C3#54711yescharacteristic
524C22×C3#57511yescharacteristic
525C22×C3#57511yescharacteristic
526C22×C3#57611yescharacteristic
527C22×C3#57811yescharacteristic
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4C2×C4×D411yescharacteristic
533C3×C4#52011yescharacteristic
534C3×C4D4×Q811yescharacteristic
535C3×C4#55411yescharacteristic
536Dic3—42no—
537Dic3—42no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23C2×C4⋊Dic312yes—
557C23C2×C4⋊Dic312yes—
558C23C22×D1211yescharacteristic
559D4—44no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
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—244no—
601C2×C4C2×C4⋊Dic312yes—
602C2×C4C2×C4⋊Dic312yes—
603C2×C4C2×C4⋊Dic312yes—
604C2×C4C2×C4⋊Dic312yes—
605C2×C4C22×Dic611yescharacteristic
606C2×C4C2×D3×D411yescharacteristic
607C2×C4C2×#23711yescharacteristic
608C2×C4#100911yescharacteristic
609C2×C4#101011yescharacteristic
610C2×C4C23×Dic311yescharacteristic
611C8—22no—
612C8—22no—
613C8—22no—
614C8D4×Dic314yes—
615C2×C3—42no—
616C2×C3—42no—
617C2×C3D4×C4⋊C411yescharacteristic
618C2×C3#291111yescharacteristic
619C2×C3#291311yescharacteristic
620C2×C3#326811yescharacteristic
621C2×C3#326911yescharacteristic
622C2×C3#334711yescharacteristic
623C2×C3#334911yescharacteristic
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22D4×D1211yescharacteristic
639C22#555511yescharacteristic
640C22#574811yescharacteristic
641C22#574811yescharacteristic
642C22#575211yescharacteristic
643C22#575311yescharacteristic
644C22C22×C4⋊Dic311yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—122no—
650C4—122no—
651C4—241no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4—241no—
657C4D4×Dic611yescharacteristic
658C4#554311yescharacteristic
659C4C2×D4×Dic311yescharacteristic
660C4#582711yescharacteristic
661C3#2266011yescharacteristic, complemented, Sylow
662C2—42no—
663C2—42no—
664C2#7947311yescharacteristic
665C2#7947511yescharacteristic
666C2#7949711yescharacteristic
667C2#7949811yescharacteristic
668C2D4×C4⋊Dic311yescharacteristic
669C2#8375211yescharacteristic
670C2#8378511yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#435C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
cosocleC22×C3×C4C24
hypercenterC2×#435D3
hypocenterC3#22660
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
2. C2×#5121C2semidirect
3. C2×#5106C2semidirect
6. #71988C2semidirect
7. #71971C2semidirect
8. #70455C2semidirect
9. #70453C2semidirect
10. #69929C2semidirect
11. #69929C2semidirect
12. #69929C2semidirect
13. #69927C2semidirect
14. #69816C2semidirect
15. #69814C2semidirect
24. C3×#435C4semidirect
87. C22×C3×C8C2×C4semidirect
129. C3×C4⋊C8C2×C4semidirect
131. C3×C22⋊C8C2×C4semidirect
132. C3×#97C2×C4semidirect
150. #817D4semidirect
158. #435Dic3semidirect
289. C22×C8C2×Dic3semidirect
341. C4⋊C8C2×Dic3semidirect
343. C22⋊C8C2×Dic3semidirect
344. #97C2×Dic3semidirect
661. C3#22660semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4352
C2×#51212
C2×#51062
#823372
#823352
#719882
#719712
#704552
#704532
#699292
#699272
#698162
#698142
#226603