Finite Group Databasebeta

← #25156691

Subgroups — #25156691

Order 11008.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583101C211yesmaximal, characteristic
2C2×C4×#10621884C211yesmaximal, characteristic
3C4⋊C4×D172C211yesmaximal, characteristic
4#19136138C211yesmaximal, characteristic, complemented
5#19136097C211yesmaximal, characteristic, complemented
6#19136003C211yesmaximal, characteristic, complemented
7#19135838C211yesmaximal, characteristic, complemented
8#19135274C211yesmaximal, characteristic, complemented
9#19135154C211yesmaximal, characteristic, complemented
10#19134932C211yesmaximal, characteristic
11#19134932C211yesmaximal, characteristic
12#19134932C211yesmaximal, characteristic
13#19134863C211yesmaximal, characteristic, complemented
14#19134855C211yesmaximal, characteristic
15C43×#2446C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C22×#10621884C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×C4⋊C4×D43—21no—
22#14583101—24no—
23C4×#10621884—24no—
24C2×#13063256C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063249C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063249C2211yescharacteristic
32C2×#13063249C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×#13063245C2211yescharacteristic
35C2×C4×C43⋊C8C2211yescharacteristic
36#14582797C2211yescharacteristic
37#14582797C2211yescharacteristic
38#14582773—21no—
39#14582773—21no—
40#14582769C2211yescharacteristic
41#14582756—21no—
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582654C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49#14582654C2211yescharacteristic
50C2×C4⋊Q8×C43C2211yescharacteristic
51C2×C4×Q8×C43C2211yescharacteristic
52C43×#400—21no—
53C43×#398—21no—
54C43×#391C2211yescharacteristic
55C43×#391C2211yescharacteristic
56C43×#391C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#387—21no—
59C43×#387—21no—
60C4×C4⋊C4×C43C2211yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C2×#10621884—22no—
66C2×#10621884—22no—
67C2×#10621884—22no—
68C2×#10621884—22no—
69C2×#10621884—22no—
70C2×#10621884D414yes—
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75C4×D172—22no—
76C4×D172—22no—
77C4×D172—22no—
78C4×D172—22no—
79C4×D172—41no—
80C4×D172—41no—
81C2×C4⋊Dic43—21no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C4⋊C4×D43—44no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92#13063256—22no—
93#13063256—41no—
94C4⋊(C43⋊C8)—22no—
95#13063249—22no—
96#13063249—22no—
97#13063249—22no—
98#13063249—22no—
99#13063249—22no—
100#13063249—41no—
101#13063245—22no—
102#13063245—22no—
103C4×C43⋊C8—22no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C22×Q8×C43C2311yescharacteristic
107C22×Q8×C43C2311yescharacteristic
108C4⋊Q8×C43—22no—
109C4⋊Q8×C43—41no—
110C4×Q8×C43—22no—
111C4×Q8×C43—41no—
112C2×C4⋊C4×C43—21no—
113C2×C4⋊C4×C43—21no—
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C4⋊C4×C43D411yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C42×C43—21no—
121C2×C42×C43C2311yescharacteristic
122C2×C42×C43C2311yescharacteristic
123C2×C42×C43C2311yescharacteristic
124#95×C43—21no—
125#95×C43—21no—
126#95×C43—21no—
127#95×C43—21no—
128#95×C43—21no—
129#95×C43—21no—
130#95×C43—21no—
131#95×C43D411yescharacteristic
132#95×C43D411yescharacteristic
133C23×D43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×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×C43C2×D412yes—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2411yescharacteristic
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×D172C2×D412yes—
164C2×D172#4112yes—
165C2×D172#4112yes—
166C2×C4×D43—24no—
167C2×C4×D43—42no—
168C2×C4×D43—42no—
169C2×C4×D43—42no—
170C2×C4×D43—42no—
171C2×C4×D43—42no—
172C2×C4×D43—42no—
173C2×C4×D43—81no—
174C2×C4×D43—81no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—41no—
180C4⋊C4×C43—42no—
181C4⋊C4×C43—42no—
182C4⋊C4×C43—42no—
183C4⋊C4×C43—42no—
184C4⋊C4×C43SD1614yescomplemented
185C42×C43—21no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—21no—
189C42×C43—21no—
190C42×C43—21no—
191C42×C43—42no—
192#10621884—28no—
193#10621884—44no—
194(C2×D43)⋊C4—24no—
195(C2×D43)⋊C4—42no—
196(C2×D43)⋊C4—42no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—42no—
203C4⋊Dic43—81no—
204C4⋊Dic43#4112yes—
205Dic43⋊C4—42no—
206Dic43⋊C4—42no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8C2×D412yes—
219C23×C43C22×D411yescharacteristic
220C22×D43—22no—
221C22×D43—22no—
222C22×D43—22no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227Q8×C43—22no—
228Q8×C43—22no—
229Q8×C43—41no—
230Q8×C43—41no—
231Q8×C43—41no—
232Q8×C43—41no—
233Q8×C43—41no—
234Q8×C43—41no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—22no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43C2×SD1612yes—
263C2×C4×C43C2×SD1612yes—
264C2×C4×C43C2×SD1612yes—
265C2×C4×C43C22×D411yescharacteristic
266C2×C4×C43C2×#4111yescharacteristic
267C2×C4×C43C2×#4111yescharacteristic
268C2×C4×C43C2×#4111yescharacteristic
269C2×C4×C43H(5,𝔽2)11yescharacteristic
270C2×C4×C432−1+411yescharacteristic
271C2×Dic43—22no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280D172—22no—
281D172—22no—
282D172—22no—
283D172#12314yes—
284C4×D43—48no—
285C4×D43—84no—
286C4×D43—84no—
287C43⋊C8—22no—
288C43⋊C8—41no—
289C43⋊C8—41no—
290C43⋊C8—41no—
291#26166—431nomaximal, Sylow
292C22×C43#52511yescharacteristic
293C22×C43D4211yescharacteristic
294C22×C43#54811yescharacteristic
295C22×C43C22×SD1611yescharacteristic
296C22×C43#57511yescharacteristic
297C22×C43#57711yescharacteristic
298C22×C43#57811yescharacteristic
299C2×D43—24no—
300C2×D43—42no—
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—42no—
305C2×D43—42no—
306C4×C43—22no—
307C4×C43—22no—
308C4×C43—22no—
309C4×C43—41no—
310C4×C43—42no—
311C4×C43—41no—
312C4×C43—41no—
313C4×C43—81no—
314C4×C43—81no—
315C4×C43—81no—
316C4×C43C2×#12311yescharacteristic
317C4×C43#53911yescharacteristic
318C4×C43#54711yescharacteristic
319C4×C43#55311yescharacteristic
320Dic43—42no—
321Dic43—81no—
322Dic43—81no—
323C2×#461—431no—
324C2×C4×SD16—431no—
325#2446D4311yescharacteristic, complemented
326D4×C4⋊C4—431no—
327#2096—431no—
328#2088—431no—
329#2072—431no—
330#2043—431no—
331#1933—431no—
332#1916—431no—
333#1877—431no—
334#1877—431no—
335#1877—431no—
336#1865—431no—
337#1864—431no—
338C2×C43#244311yescharacteristic
339C2×C43#310111yescharacteristic
340C2×C43#312011yescharacteristic
341C2×C43#326911yescharacteristic
342C2×C43D4×SD1611yescharacteristic
343C2×C43#335911yescharacteristic
344C2×C43#337611yescharacteristic
345D43—44no—
346C22×SD16—431no—
347C22×SD16—431no—
348C2×C4⋊Q8C2×D4311yescharacteristic
349C2×C4×Q8C2×D4311yescharacteristic
350C2×C4×D4—431no—
351C2×C4×D4—431no—
352C2×C4×D4—431no—
353C22×C4⋊C4—861no—
354#461—864no—
355C4×SD16—864no—
356C2×#106—431no—
357C2×#106—431no—
358C2×C4⋊C8—431no—
359C2×#103—431no—
360C2×#103—431no—
361C2×#103—431no—
362C2×#103—431no—
363C2×#103—431no—
364C2×#103—431no—
365C2×#102—431no—
366C2×#102—431no—
367C2×C4×C8—431no—
368#400—21no—
369#398—21no—
370#391C2×D4311yescharacteristic
371#391C2×D4311yescharacteristic
372#391C2×D4311yescharacteristic
373#389—431no—
374#389—431no—
375#389C2×D4311yescharacteristic
376#387—21no—
377#387—21no—
378#385—861no—
379#385—861no—
380#384—431no—
381#380—861no—
382C4×C4⋊C4C2×D4311yescharacteristic
383#340—431no—
384#340—431no—
385#340—431no—
386#340—431no—
387#336—431no—
388#336—431no—
389#336—431no—
390#336—431no—
391C43#2616611yescharacteristic, complemented, Sylow
392C22×Q8C22×D4311yescharacteristic
393C22×Q8C22×D4311yescharacteristic
394C22×D4—431no—
395C23×C4—861no—
396C23×C4—861no—
397C23×C4—861no—
398C2×SD16—434no—
399C2×SD16—862no—
400C2×SD16—862no—
401C2×SD16—862no—
402C2×SD16—862no—
403C2×SD16—862no—
404C22×C8—431no—
405C22×C8—431no—
406C22×C8—431no—
407C22×C8—431no—
408C4⋊Q8—22no—
409C4⋊Q8—41no—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×D4—862no—
413C4×D4—862no—
414C4×D4—862no—
415C4×D4—862no—
416C4×D4—1721no—
417C4×D4—1721no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—431no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4—1724no—
430C2×C4⋊C4C22⋊D4311yescharacteristic
431C2×C4⋊C4C22⋊D4311yescharacteristic
432C2×C4⋊C4C22×D4311yescharacteristic
433C2×C4⋊C4C22×D4311yescharacteristic
434C2×C22⋊C4—861no—
435C2×C22⋊C4—861no—
436C2×C22⋊C4—861no—
437C2×C42—21no—
438C2×C42C22×D4311yescharacteristic
439C2×C42C22×D4311yescharacteristic
440C2×C42C22×D4311yescharacteristic
441#106—862no—
442#106—1721no—
443C4⋊C8—862no—
444#103—862no—
445#103—862no—
446#103—862no—
447#103—862no—
448#103—862no—
449#103—1721no—
450#102—862no—
451#102—862no—
452C4×C8—862no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—861no—
461#95—861no—
462#95C22⋊D4311yescharacteristic
463#95C22⋊D4311yescharacteristic
464C24—861no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8D4×D4312yes—
477C2×D4—432no—
478C2×D4—432no—
479C2×D4—432no—
480C2×D4—861no—
481C2×D4—861no—
482C2×D4—861no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—864no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—3441no—
492C22×C4—3441no—
493C22×C4—1722no—
494C22×C4—1722no—
495C22×C4—1722no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4—1722no—
499C22×C4C2×C22⋊D4311yescharacteristic
500C22×C4C2×C22⋊D4311yescharacteristic
501C22×C4C2×C22⋊D4311yescharacteristic
502C22×C4C2×C22⋊D4311yescharacteristic
503C22×C4C2×C22⋊D4311yescharacteristic
504C22×C4C2×C22⋊D4311yescharacteristic
505C22×C4C23×D4311yescharacteristic
506SD16—868no—
507SD16—1724no—
508C2×C8—432no—
509C2×C8—861no—
510C2×C8—861no—
511C2×C8—861no—
512C2×C8—861no—
513C2×C8—861no—
514C2×C8—861no—
515C2×C8—861no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—432no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—3441no—
536C4⋊C4—1722no—
537C4⋊C4—1722no—
538C4⋊C4—1722no—
539C4⋊C4#1062188414yescomplemented
540C22⋊C4—864no—
541C22⋊C4—1722no—
542C22⋊C4—1722no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C23—862no—
551C23—862no—
552C23—862no—
553C23—1721no—
554C23—1721no—
555C23—1721no—
556C23—1721no—
557C23C22×C22⋊D4311yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566D4—434no—
567D4—862no—
568D4—862no—
569D4—862no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—862no—
598C2×C4—3444no—
599C2×C4—3444no—
600C2×C4—1728no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4C2×#1062188412yes—
610C2×C4C2×#1062188412yes—
611C2×C4C2×#1062188412yes—
612C2×C4C2×D4×D4311yescharacteristic
613C2×C4#41×D4311yescharacteristic
614C2×C4#41×D4311yescharacteristic
615C2×C4#41×D4311yescharacteristic
616C2×C4#1306337011yescharacteristic
617C2×C4#1306337211yescharacteristic
618C8—862no—
619C8—1721no—
620C8—1721no—
621C8—1721no—
622C22—864no—
623C22—1722no—
624C22—1722no—
625C22—1722no—
626C22—1722no—
627C22—1722no—
628C22—1722no—
629C22#1458340611yescharacteristic
630C22D4×C22⋊D4311yescharacteristic
631C22#1458355611yescharacteristic
632C22C22×#1062188411yescharacteristic
633C22#1458369711yescharacteristic
634C22#1458370711yescharacteristic
635C22#1458371411yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—3441no—
647C4—3441no—
648C4—1722no—
649C4#123×D4311yescharacteristic
650C4#1458349211yescharacteristic
651C4#1458354711yescharacteristic
652C4#1458359411yescharacteristic
653C2—1724no—
654C2#1913832711yescharacteristic
655C2#1914459911yescharacteristic
656C2#1914480311yescharacteristic
657C2#1914615111yescharacteristic
658C2D4×#1062188411yescharacteristic
659C2#1914723311yescharacteristic
660C2#1914741611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2446C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2446D43
hypocenterC43#26166
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2446

Complements

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

NH ≅ G/Ntype
4. #19136138C2semidirect
5. #19136097C2semidirect
6. #19136003C2semidirect
7. #19135838C2semidirect
8. #19135274C2semidirect
9. #19135154C2semidirect
13. #19134863C2semidirect
15. C43×#2446C2semidirect
184. C4⋊C4×C43SD16semidirect
325. #2446D43semidirect
391. C43#26166semidirect
539. C4⋊C4#10621884semidirect

Maximal subgroups

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

subgroupindex
C2×#145831012
C2×C4×#106218842
C4⋊C4×D1722
#191361382
#191360972
#191360032
#191358382
#191352742
#191351542
#191349322
#191348632
#191348552
C43×#24462
#2616643