Finite Group Databasebeta

← #25119064

Subgroups — #25119064

Order 11008.

982 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583000C211yesmaximal, characteristic
2C2×SD16×Dic43C211yesmaximal, characteristic
3#19137655C211yesmaximal, characteristic, complemented
4#19137505C211yesmaximal, characteristic
5#19135416C211yesmaximal, characteristic, complemented
6#19135365C211yesmaximal, characteristic, complemented
7#19135287C211yesmaximal, characteristic, complemented
8#19135273C211yesmaximal, characteristic, complemented
9#19134931C211yesmaximal, characteristic
10#19134931C211yesmaximal, characteristic
11#19134865C211yesmaximal, characteristic, complemented
12#19134851C211yesmaximal, characteristic, complemented
13#19134472C211yesmaximal, characteristic, complemented
14#19134420C211yesmaximal, characteristic
15C2×C43×#461C211yesmaximal, characteristic
16C2×Q8×Dic43C2211yescharacteristic
17C2×Q8×Dic43C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20#14583000—24no—
21SD16×Dic43—24no—
22C2×#13063259C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×#13063249C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063246C2211yescharacteristic
29C2×#13063246C2211yescharacteristic
30C2×#13063236C2211yescharacteristic
31C2×#13063235C2211yescharacteristic
32C2×C8×Dic43C2211yescharacteristic
33#14582806C2211yescharacteristic
34#14582806C2211yescharacteristic
35#14582798—21no—
36#14582798C2211yescharacteristic
37#14582798C2211yescharacteristic
38#14582785—21no—
39#14582785—21no—
40#14582774—21no—
41#14582774—21no—
42#14582765C2211yescharacteristic
43#14582759—21no—
44C4⋊C4×Dic43—21no—
45C4×C4⋊Dic43C2211yescharacteristic
46C4×C4⋊Dic43C2211yescharacteristic
47C2×C43⋊#95—21no—
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C22×SD16×C43C2211yescharacteristic
55C22×SD16×C43C2211yescharacteristic
56C2×C4⋊Q8×C43C2211yescharacteristic
57C2×C4×D4×C43C2211yescharacteristic
58C43×#461C418yescomplemented
59C2×C4⋊C8×C43C2211yescharacteristic
60C2×#103×C43C2211yescharacteristic
61C2×#103×C43C2211yescharacteristic
62C23×Dic43—21no—
63C23×Dic43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66Q8×Dic43—22no—
67Q8×Dic43—22no—
68Q8×Dic43—41no—
69Q8×Dic43—41no—
70D4×Dic43—22no—
71D4×Dic43—22no—
72D4×Dic43—41no—
73D4×Dic43—41no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43D411yescharacteristic
79C2×C4⋊Dic43D411yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43C2311yescharacteristic
90C2×C4×Dic43C2311yescharacteristic
91#13063259—22no—
92#13063257—22no—
93#13063249—22no—
94#13063249—22no—
95#13063249—41no—
96#13063249—41no—
97#13063246—22no—
98#13063246D414yes—
99#13063236—22no—
100#13063235—22no—
101C8×Dic43—22no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95D411yescharacteristic
112C43⋊#95D411yescharacteristic
113C22×Q8×C43C2311yescharacteristic
114C22×Q8×C43C2311yescharacteristic
115C22×D4×C43C2311yescharacteristic
116C23×C4×C43—21no—
117C2×SD16×C43—22no—
118C2×SD16×C43—22no—
119C2×SD16×C43—22no—
120C2×SD16×C43—22no—
121C2×SD16×C43C2×C414yes—
122C2×SD16×C43C2×C414yes—
123C22×C8×C43C2311yescharacteristic
124C22×C8×C43C2311yescharacteristic
125C4⋊Q8×C43—22no—
126C4⋊Q8×C43C2×C414yescomplemented
127C4×D4×C43—22no—
128C4×D4×C43C2×C414yes—
129C2×C4⋊C4×C43—21no—
130C2×C4⋊C4×C43—21no—
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C22⋊C4×C43—21no—
133C2×C42×C43C2311yescharacteristic
134C4⋊C8×C43C2×C414yescomplemented
135#103×C43C2×C414yescomplemented
136#103×C43C2×C414yescomplemented
137C24×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43C22×C412yes—
149C2×Q8×C43C22×C412yes—
150C2×D4×C43—21no—
151C2×D4×C43—21no—
152C2×D4×C43—21no—
153C2×D4×C43C22×C412yes—
154C2×D4×C43C2×D412yes—
155C2×D4×C43#4112yes—
156C22×C4×C43—24no—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43—81no—
161C22×C4×C43—81no—
162C22×C4×C43C2×D411yescharacteristic
163C22×C4×C43C2×D411yescharacteristic
164C22×C4×C43C2411yescharacteristic
165C22×Dic43—21no—
166C22×Dic43—21no—
167C22×Dic43—21no—
168C22×Dic43—21no—
169C22×Dic43—42no—
170C22×Dic43—42no—
171C22×Dic43—42no—
172C22×Dic43—42no—
173C22×Dic43—81no—
174C22×Dic43—81no—
175C22×Dic43C2×D411yescharacteristic
176C22×Dic43C2×D411yescharacteristic
177C22×Dic43C2×D411yescharacteristic
178C22×Dic43C2×D411yescharacteristic
179SD16×C43—28no—
180SD16×C43—28no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43C22×C412yes—
186C2×C8×C43C22×C412yes—
187C4⋊C4×C43—24no—
188C4⋊C4×C43—24no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43C22×C412yes—
191C22⋊C4×C43—24no—
192C42×C43C22×C412yes—
193C42×C43#4112yes—
194C22⋊Dic43—42no—
195C22⋊Dic43—42no—
196C4⋊Dic43—21no—
197C4⋊Dic43—22no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—41no—
205C4⋊Dic43—42no—
206C4⋊Dic43—42no—
207C4⋊Dic43—81no—
208C4⋊Dic43—81no—
209C4⋊Dic43C2×D412yes—
210C4×Dic43—21no—
211C4×Dic43—21no—
212C4×Dic43—21no—
213C4×Dic43—21no—
214C4×Dic43—42no—
215C4×Dic43—42no—
216C4×Dic43—42no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8C2×D412yes—
223C23×C43—22no—
224C23×C43—22no—
225C23×C43—22no—
226C23×C43—41no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43—41no—
230C23×C43C22×D411yescharacteristic
231Q8×C43—22no—
232Q8×C43—22no—
233Q8×C43—22no—
234Q8×C43—22no—
235Q8×C43—41no—
236Q8×C43—41no—
237Q8×C43—41no—
238Q8×C43—41no—
239D4×C43—22no—
240D4×C43—22no—
241D4×C43—22no—
242D4×C43C4×D414yes—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—22no—
248C2×C4×C43—22no—
249C2×C4×C43—22no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—84no—
257C2×C4×C43C4×D412yes—
258C2×C4×C43C4×D412yes—
259C2×C4×C43C23×C411yescharacteristic
260C2×C4×C43C22×D411yescharacteristic
261C2×C4×C43C2×#4111yescharacteristic
262C2×C4×C43C2×#4111yescharacteristic
263C2×C4×C43H(5,𝔽2)11yescharacteristic
264C2×C4×C432−1+411yescharacteristic
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—22no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—21no—
274C2×Dic43—21no—
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287C2×Dic43—41no—
288C2×Dic43—84no—
289C2×Dic43—84no—
290C2×Dic43#13712yes—
291C8×C43—22no—
292C8×C43—22no—
293C43⋊C8—22no—
294C43⋊C8—41no—
295#23012—431nomaximal, Sylow
296C22×C43—24no—
297C22×C43—42no—
298C22×C43—42no—
299C22×C43—42no—
300C22×C43—42no—
301C22×C43—42no—
302C22×C43—42no—
303C22×C43C2×C4×D411yescharacteristic
304C22×C43D4211yescharacteristic
305C22×C43#54811yescharacteristic
306C22×C43C2×#13711yescharacteristic
307C22×C43#57511yescharacteristic
308C22×C43#57711yescharacteristic
309C22×C43#57711yescharacteristic
310C4×C43—22no—
311C4×C43—22no—
312C4×C43—42no—
313C4×C43—42no—
314C4×C43—42no—
315C4×C43C2×C4×D411yescharacteristic
316C4×C43#52011yescharacteristic
317C4×C43#54711yescharacteristic
318C4×C43#55311yescharacteristic
319Dic43—22no—
320Dic43—41no—
321Dic43—41no—
322Dic43—41no—
323Dic43—42no—
324Dic43—81no—
325Dic43—81no—
326Dic43—81no—
327C2×#461D4311yescharacteristic
328C2×#440—431no—
329C2×C4×SD16—431no—
330#2365—431no—
331#2349—431no—
332#1960—431no—
333#1949—431no—
334#1935—431no—
335#1933—431no—
336#1877—431no—
337#1877—431no—
338#1865—431no—
339#1863—431no—
340#1771—431no—
341#1762—431no—
342C2×C43—44no—
343C2×C43#236111yescharacteristic
344C2×C43C4×#13711yescharacteristic
345C2×C43#294911yescharacteristic
346C2×C43#326611yescharacteristic
347C2×C43#326711yescharacteristic
348C2×C43#337211yescharacteristic
349C2×C43#337611yescharacteristic
350C22×SD16C2×D4311yescharacteristic
351C22×SD16C2×D4311yescharacteristic
352C2×C4⋊Q8C2×D4311yescharacteristic
353C2×C4×Q8—431no—
354C2×C4×Q8—431no—
355C2×C4×D4—431no—
356C2×C4×D4—431no—
357C2×C4×D4C2×D4311yescharacteristic
358#461Dic4318yescomplemented
359#440—864no—
360C4×SD16—864no—
361C2×#107—431no—
362C2×#106—431no—
363C2×C4⋊C8C2×D4311yescharacteristic
364C2×#103—431no—
365C2×#103—431no—
366C2×#103—431no—
367C2×#103—431no—
368C2×#103C2×D4311yescharacteristic
369C2×#103C2×D4311yescharacteristic
370C2×#102—431no—
371C2×#102—431no—
372C2×#97—431no—
373C2×#97—431no—
374C2×C4×C8—431no—
375#391—431no—
376#391—431no—
377#389—431no—
378#389—431no—
379#389—861no—
380#387—861no—
381#387—861no—
382#385—861no—
383#385—861no—
384#382—431no—
385#380—861no—
386C4×C4⋊C4—431no—
387C4×C4⋊C4—431no—
388C4×C4⋊C4—861no—
389C2×#95—861no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#336—431no—
395#336—431no—
396C43#2301211yescharacteristic, complemented, Sylow
397C22×Q8C22×D4311yescharacteristic
398C22×Q8C22×D4311yescharacteristic
399C22×D4C22×D4311yescharacteristic
400C23×C4—21no—
401C23×C4—861no—
402C23×C4—861no—
403C2×SD16—22no—
404C2×SD16—22no—
405C2×SD16—22no—
406C2×SD16—22no—
407C2×SD16C2×Dic4314yes—
408C2×SD16C2×Dic4314yes—
409C22×C8—431no—
410C22×C8—431no—
411C22×C8C22×D4311yescharacteristic
412C22×C8C22×D4311yescharacteristic
413C4⋊Q8—22no—
414C4⋊Q8C2×Dic4314yescomplemented
415C4×Q8—862no—
416C4×Q8—862no—
417C4×Q8—1721no—
418C4×Q8—1721no—
419C4×D4—22no—
420C4×D4—862no—
421C4×D4—862no—
422C4×D4—1721no—
423C4×D4—1721no—
424C4×D4C2×Dic4314yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—431no—
430C2×C4⋊C4—431no—
431C2×C4⋊C4—431no—
432C2×C4⋊C4—431no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4—861no—
436C2×C4⋊C4—861no—
437C2×C4⋊C4C22×D4311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C42—431no—
442C2×C42—431no—
443C2×C42—861no—
444C2×C42—861no—
445C2×C42—861no—
446C2×C42C22×D4311yescharacteristic
447#107—862no—
448#106—862no—
449C4⋊C8C2×Dic4314yescomplemented
450#103—862no—
451#103—862no—
452#103—1721no—
453#103—1721no—
454#103C2×Dic4314yescomplemented
455#103C2×Dic4314yescomplemented
456#102—434no—
457#102—862no—
458#97—862no—
459#97—862no—
460C4×C8—862no—
461#95—431no—
462#95—431no—
463#95—861no—
464#95—861no—
465#95—861no—
466#95—861no—
467#95—861no—
468#95—861no—
469#95—861no—
470#95—861no—
471#95—861no—
472C24—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8C22×Dic4312yes—
484C2×Q8C22×Dic4312yes—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4D4×D4312yes—
489C2×D4#1062190012yes—
490C2×D4C22×Dic4312yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—24no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—431no—
498C22×C4—431no—
499C22×C4—431no—
500C22×C4—431no—
501C22×C4—861no—
502C22×C4—861no—
503C22×C4—861no—
504C22×C4—861no—
505C22×C4—3441no—
506C22×C4—3441no—
507C22×C4—1722no—
508C22×C4—1722no—
509C22×C4—1722no—
510C22×C4—1722no—
511C22×C4D4×D4311yescharacteristic
512C22×C4D4×D4311yescharacteristic
513C22×C4C23×D4311yescharacteristic
514SD16—28no—
515SD16—28no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—432no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8—861no—
525C2×C8—861no—
526C2×C8C22×Dic4312yes—
527C2×C8C22×Dic4312yes—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—432no—
532C4⋊C4—862no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—861no—
538C4⋊C4—861no—
539C4⋊C4—861no—
540C4⋊C4—3441no—
541C4⋊C4—3441no—
542C4⋊C4—1721no—
543C4⋊C4—1722no—
544C4⋊C4—1722no—
545C4⋊C4C22×Dic4312yes—
546C22⋊C4—24no—
547C22⋊C4—1722no—
548C22⋊C4—1722no—
549C42—861no—
550C42—861no—
551C42—861no—
552C42—861no—
553C42—1722no—
554C42—1722no—
555C42—1722no—
556C42#1062190012yes—
557C42C22×Dic4312yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×D4×D4311yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4D4×Dic4314yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—432no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—861no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—862no—
605C2×C4—3444no—
606C2×C4—3444no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4D4×Dic4312yes—
619C2×C4D4×Dic4312yes—
620C2×C4C23×Dic4311yescharacteristic
621C2×C4C2×D4×D4311yescharacteristic
622C2×C4C2×#1062190011yescharacteristic
623C2×C4C2×#1062190011yescharacteristic
624C2×C4#1306337111yescharacteristic
625C2×C4#1306337211yescharacteristic
626C8—22no—
627C8—22no—
628C8—862no—
629C8—1721no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×D4×Dic4311yescharacteristic
638C22#1458353311yescharacteristic
639C22#1458355511yescharacteristic
640C22#137×D4311yescharacteristic
641C22#1458369811yescharacteristic
642C22#1458370511yescharacteristic
643C22#1458370511yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—862no—
650C4—3441no—
651C4—3441no—
652C4—3441no—
653C4—1721no—
654C4—1721no—
655C4—1721no—
656C4—1722no—
657C4C2×D4×Dic4311yescharacteristic
658C4#1458338211yescharacteristic
659C4#1458354911yescharacteristic
660C4#1458360111yescharacteristic
661C2—44no—
662C2#1913761511yescharacteristic
663C2#137×Dic4311yescharacteristic
664C2#1914345511yescharacteristic
665C2#1914611111yescharacteristic
666C2#1914612611yescharacteristic
667C2#1914737111yescharacteristic
668C2#1914741511yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#461C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
cosocleC22×C4×C43C24
hypercenterC2×#461D43
hypocenterC43#23012
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #19137655C2semidirect
5. #19135416C2semidirect
6. #19135365C2semidirect
7. #19135287C2semidirect
8. #19135273C2semidirect
11. #19134865C2semidirect
12. #19134851C2semidirect
13. #19134472C2semidirect
58. C43×#461C4semidirect
126. C4⋊Q8×C43C2×C4semidirect
134. C4⋊C8×C43C2×C4semidirect
135. #103×C43C2×C4semidirect
136. #103×C43C2×C4semidirect
358. #461Dic43semidirect
396. C43#23012semidirect
414. C4⋊Q8C2×Dic43semidirect
449. C4⋊C8C2×Dic43semidirect
454. #103C2×Dic43semidirect
455. #103C2×Dic43semidirect

Maximal subgroups

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

subgroupindex
C2×#145830002
C2×SD16×Dic432
#191376552
#191375052
#191354162
#191353652
#191352872
#191352732
#191349312
#191348652
#191348512
#191344722
#191344202
C2×C43×#4612
#2301243