Finite Group Databasebeta

← #25118444

Subgroups — #25118444

Order 11008.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583200C211yesmaximal, characteristic
2#19137689C211yesmaximal, characteristic, complemented
3#19137506C211yesmaximal, characteristic
4#19135560C211yesmaximal, characteristic, complemented
5#19135365C211yesmaximal, characteristic, complemented
6#19135328C211yesmaximal, characteristic, complemented
7#19135323C211yesmaximal, characteristic, complemented
8#19135188C211yesmaximal, characteristic, complemented
9#19135181C211yesmaximal, characteristic, complemented
10#19134859C211yesmaximal, characteristic
11#19134845C211yesmaximal, characteristic
12C4×#13063247C211yesmaximal, characteristic
13C4×#13063247C211yesmaximal, characteristic
14C4×#13063247C211yesmaximal, characteristic
15C8×C4⋊C4×C43C211yesmaximal, characteristic, complemented
16C2×#13063317C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20#14583200C418yescomplemented
21C2×#13063259C2211yescharacteristic
22C2×#13063259C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063257C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29#14582797C2211yescharacteristic
30#14582797C2211yescharacteristic
31#14582797C2211yescharacteristic
32#14582797C2211yescharacteristic
33#14582794—21no—
34#14582781—21no—
35#14582780—21no—
36#14582770—21no—
37#14582770—21no—
38#14582765C2211yescharacteristic
39#14582758—21no—
40C4×C4⋊Dic43C2211yescharacteristic
41C4×C4⋊Dic43C2211yescharacteristic
42C4×C4⋊Dic43C2211yescharacteristic
43#14582750—21no—
44#95×D43—21no—
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53C2×C4⋊C8×C43C2211yescharacteristic
54C2×C4×C8×C43C2211yescharacteristic
55C2×C4×C8×C43C2211yescharacteristic
56C2×C4×C8×C43C2211yescharacteristic
57C4×C4⋊C4×C43C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64#13063317—22no—
65#13063317C2×C414yescomplemented
66C4×D172—22no—
67C4×D172—22no—
68C4×D172—22no—
69C4×D172—41no—
70C4×D172—41no—
71C4×D172C2×C414yes—
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43—21no—
91#13063259—41no—
92#13063259C2×C414yescomplemented
93#13063257—41no—
94#13063257C2×C414yescomplemented
95#13063247—22no—
96#13063247C2×C414yes—
97#13063247C2×C414yes—
98#13063247Q814yes—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C22×C8×C43C2311yescharacteristic
109C22×C8×C43C2311yescharacteristic
110C22×C8×C43C2311yescharacteristic
111C22×C8×C43C2311yescharacteristic
112C2×C4⋊C4×C43D411yescharacteristic
113C2×C4⋊C4×C43D411yescharacteristic
114C2×C42×C43C2311yescharacteristic
115C2×C42×C43C2311yescharacteristic
116C2×C42×C43C2311yescharacteristic
117C4⋊C8×C43C2×C414yescomplemented
118C4×C8×C43—22no—
119C4×C8×C43—41no—
120C4×C8×C43—41no—
121#95×C43D411yescharacteristic
122#95×C43D411yescharacteristic
123C23×D43—21no—
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2411yescharacteristic
131C22×Dic43—21no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C2×D172—21no—
139C2×D172—21no—
140C2×D172—21no—
141C2×D172C22×C412yes—
142C2×D172C2×Q812yes—
143C2×D172#4112yes—
144C2×C4×D43—24no—
145C2×C4×D43—42no—
146C2×C4×D43—42no—
147C2×C4×D43—42no—
148C2×C4×D43—42no—
149C2×C4×D43—81no—
150C2×C4×D43—81no—
151C2×C4×D43—81no—
152C2×C4×D43—81no—
153C2×C8×C43—21no—
154C2×C8×C43—21no—
155C2×C8×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43C22×C412yes—
163C2×C8×C43C22×C412yes—
164C2×C8×C43C2×Q812yes—
165C4⋊C4×C43—22no—
166C4⋊C4×C43—22no—
167C42×C43—21no—
168C42×C43—21no—
169C42×C43—21no—
170C42×C43—21no—
171C42×C43C22×C412yes—
172C42×C43#4112yes—
173(C2×D43)⋊C4—24no—
174(C2×D43)⋊C4—42no—
175(C2×D43)⋊C4—42no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43—21no—
186C4⋊Dic43C22×C412yes—
187C4⋊Dic43C22×C412yes—
188C4⋊Dic43C22×C412yes—
189C4⋊Dic43C2×Q812yes—
190Dic43⋊C4—24no—
191Dic43⋊C4—24no—
192Dic43⋊C4—42no—
193Dic43⋊C4—81no—
194Dic43⋊C4—81no—
195Dic43⋊C4—81no—
196C4×Dic43—42no—
197C4×Dic43—81no—
198C4×Dic43—81no—
199C23×C43C22×D411yescharacteristic
200C22×D43—22no—
201C22×D43—22no—
202C22×D43—22no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43—41no—
206C22×D43—41no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—21no—
221C2×C4×C43C4×D412yes—
222C2×C4×C43C4×D412yes—
223C2×C4×C43#13512yes—
224C2×C4×C43#13512yes—
225C2×C4×C43C23×C411yescharacteristic
226C2×C4×C43C22×Q811yescharacteristic
227C2×C4×C43C2×#4111yescharacteristic
228C2×C4×C43C2×#4111yescharacteristic
229C2×C4×C43H(5,𝔽2)11yescharacteristic
230C2×C4×C432−1+411yescharacteristic
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—22no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252D172—22no—
253D172—22no—
254D172—22no—
255D172C4×Q814yes—
256C4×D43—84no—
257C4×D43—84no—
258C4×D43—84no—
259C8×C43—22no—
260C8×C43—22no—
261C8×C43—22no—
262C8×C43—41no—
263#22966—431nomaximal, Sylow
264C22×C43C2×C4×D411yescharacteristic
265C22×C43#54611yescharacteristic
266C22×C43D4×Q811yescharacteristic
267C22×C43C2×#13511yescharacteristic
268C22×C43C2×#13511yescharacteristic
269C22×C43#57611yescharacteristic
270C22×C43#57711yescharacteristic
271C2×D43—24no—
272C2×D43—42no—
273C2×D43—42no—
274C2×D43—42no—
275C2×D43—42no—
276C2×D43—42no—
277C2×D43—42no—
278C4×C43—22no—
279C4×C43—22no—
280C4×C43—22no—
281C4×C43—22no—
282C4×C43—41no—
283C4×C43—41no—
284C4×C43C2×C4×Q811yescharacteristic
285C4×C43#52011yescharacteristic
286C4×C43#55211yescharacteristic
287C4×C43#55411yescharacteristic
288Dic43—42no—
289Dic43—42no—
290Dic43—42no—
291Dic43—42no—
292Dic43—81no—
293Dic43—81no—
294Dic43—81no—
295C2×#478—431no—
296#2370—431no—
297#2349—431no—
298#1991—431no—
299#1949—431no—
300#1942—431no—
301#1941—431no—
302#1921—431no—
303#1920—431no—
304#1864—431no—
305#1862—431no—
306C8×C4⋊C4D4311yescharacteristic, complemented
307C4×#102—431no—
308C4×#102—431no—
309C4×#102—431no—
310C2×C43#236011yescharacteristic
311C2×C43C4×#13511yescharacteristic
312C2×C43#294611yescharacteristic
313C2×C43#329611yescharacteristic
314C2×C43#329711yescharacteristic
315C2×C43#338011yescharacteristic
316C2×C43#338111yescharacteristic
317D43—44no—
318C2×#125—431no—
319C2×C4×D4—431no—
320C2×C4×D4—431no—
321C2×C4×D4—431no—
322#478—438no—
323C2×#107—431no—
324C2×#107—431no—
325C2×#106—431no—
326C2×#106—431no—
327C2×C4⋊C8C2×D4311yescharacteristic
328C2×#102—431no—
329C2×#102—431no—
330C2×#102—431no—
331C2×#102—431no—
332C2×C4×C8C2×D4311yescharacteristic
333C2×C4×C8C2×D4311yescharacteristic
334C2×C4×C8C2×D4311yescharacteristic
335#389—431no—
336#389—431no—
337#389—431no—
338#389—431no—
339#389—861no—
340#387—861no—
341#387—861no—
342#385—861no—
343#385—861no—
344#382—431no—
345#380—861no—
346C4×C4⋊C4—431no—
347C4×C4⋊C4—431no—
348C4×C4⋊C4—431no—
349C4×C4⋊C4—861no—
350C4×C4⋊C4C2×D4311yescharacteristic
351C2×#95—861no—
352#340—431no—
353#340—431no—
354#340—431no—
355#340—431no—
356#340—431no—
357#340—431no—
358#340—431no—
359#340—431no—
360#336C2×D4311yescharacteristic
361#336C2×D4311yescharacteristic
362C43#2296611yescharacteristic, complemented, Sylow
363C22×D4—431no—
364C23×C4—861no—
365C23×C4—861no—
366C23×C4—861no—
367C22×C8C22×D4311yescharacteristic
368C22×C8C22×D4311yescharacteristic
369C22×C8C22×D4311yescharacteristic
370C22×C8C22×D4311yescharacteristic
371#125—434no—
372#125—862no—
373C4×D4—434no—
374C4×D4—862no—
375C4×D4—862no—
376C4×D4—862no—
377C4×D4—1721no—
378C4×D4—1721no—
379C2×C4⋊C4—431no—
380C2×C4⋊C4—431no—
381C2×C4⋊C4—431no—
382C2×C4⋊C4—431no—
383C2×C4⋊C4—431no—
384C2×C4⋊C4—431no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—861no—
387C2×C4⋊C4—861no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4—861no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4D17211yescharacteristic
393C2×C4⋊C4D17211yescharacteristic
394C2×C22⋊C4—861no—
395C2×C22⋊C4—861no—
396C2×C22⋊C4—861no—
397C2×C42—861no—
398C2×C42—861no—
399C2×C42—861no—
400C2×C42C22×D4311yescharacteristic
401C2×C42C22×D4311yescharacteristic
402C2×C42C22×D4311yescharacteristic
403#107—434no—
404#107—1721no—
405#106—434no—
406#106—1721no—
407C4⋊C8C4×D4314yescomplemented
408#102—434no—
409#102—434no—
410#102—434no—
411#102—862no—
412C4×C8—22no—
413C4×C8—41no—
414C4×C8—41no—
415#95—861no—
416#95—861no—
417#95—861no—
418#95—861no—
419#95—861no—
420#95—861no—
421#95—861no—
422#95—861no—
423#95—861no—
424#95D17211yescharacteristic
425#95D17211yescharacteristic
426C24—861no—
427C2×D4—432no—
428C2×D4—432no—
429C2×D4—432no—
430C2×D4—861no—
431C2×D4—861no—
432C2×D4—861no—
433C22×C4—861no—
434C22×C4—861no—
435C22×C4—861no—
436C22×C4—861no—
437C22×C4—864no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—861no—
441C22×C4—3441no—
442C22×C4—3441no—
443C22×C4—3441no—
444C22×C4—3441no—
445C22×C4—1722no—
446C22×C4—1722no—
447C22×C4—1722no—
448C22×C4—1722no—
449C22×C4C2×D17211yescharacteristic
450C22×C4C2×D17211yescharacteristic
451C22×C4C2×D17211yescharacteristic
452C22×C4C2×D17211yescharacteristic
453C22×C4C2×D17211yescharacteristic
454C22×C4C2×D17211yescharacteristic
455C22×C4C23×D4311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8C2×C4×D4312yes—
466C2×C8C2×C4×D4312yes—
467C2×C8Q8×D4312yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—432no—
471C4⋊C4—432no—
472C4⋊C4—432no—
473C4⋊C4—432no—
474C4⋊C4—864no—
475C4⋊C4—864no—
476C4⋊C4—861no—
477C4⋊C4—861no—
478C4⋊C4—861no—
479C4⋊C4—861no—
480C4⋊C4—861no—
481C4⋊C4—861no—
482C4⋊C4—861no—
483C4⋊C4—861no—
484C4⋊C4—861no—
485C4⋊C4—861no—
486C4⋊C4—3441no—
487C4⋊C4—3441no—
488C4⋊C4—3441no—
489C4⋊C4—1722no—
490C22⋊C4—864no—
491C22⋊C4—1722no—
492C22⋊C4—1722no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—3441no—
498C42—3441no—
499C42—1722no—
500C42C2×C4×D4312yes—
501C42#1062189812yes—
502C23—862no—
503C23—862no—
504C23—862no—
505C23—1721no—
506C23—1721no—
507C23—1721no—
508C23—1721no—
509C23C22×D17211yescharacteristic
510D4—434no—
511D4—862no—
512D4—862no—
513D4—862no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—862no—
532C2×C4—3444no—
533C2×C4—3444no—
534C2×C4—3444no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1721no—
543C2×C4—1721no—
544C2×C4—1721no—
545C2×C4—1721no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4C4×D17212yes—
553C2×C4C4×D17212yes—
554C2×C4#1306334712yes—
555C2×C4#1306334712yes—
556C2×C4C22×C4×D4311yescharacteristic
557C2×C4C2×Q8×D4311yescharacteristic
558C2×C4#41×D4311yescharacteristic
559C2×C4C2×#1062189811yescharacteristic
560C2×C4#1306337011yescharacteristic
561C2×C4#1306337211yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—864no—
567C22—1722no—
568C22—1722no—
569C22—1722no—
570C22—1722no—
571C22—1722no—
572C22—1722no—
573C22C2×C4×D17211yescharacteristic
574C22#1458354411yescharacteristic
575C22Q8×D17211yescharacteristic
576C22C2×#1306334711yescharacteristic
577C22C2×#1306334711yescharacteristic
578C22#1458370411yescharacteristic
579C22#1458371111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—3441no—
587C4—3441no—
588C4—3441no—
589C4—1722no—
590C4—1722no—
591C4—1722no—
592C4—1722no—
593C4C4×Q8×D4311yescharacteristic
594C4#1458337811yescharacteristic
595C4#1458358811yescharacteristic
596C4#1458360411yescharacteristic
597C2—1724no—
598C2#1913760211yescharacteristic
599C2C4×#1306334711yescharacteristic
600C2#1914342911yescharacteristic
601C2#1914643211yescharacteristic
602C2#1914644711yescharacteristic
603C2#1914746311yescharacteristic
604C2#1914747211yescharacteristic
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC8×C4⋊C4×C43C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenterC8×C4⋊C4D43
hypocenterC43#22966
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #19137689C2semidirect
4. #19135560C2semidirect
5. #19135365C2semidirect
6. #19135328C2semidirect
7. #19135323C2semidirect
8. #19135188C2semidirect
9. #19135181C2semidirect
15. C8×C4⋊C4×C43C2semidirect
20. #14583200C4semidirect
65. #13063317C2×C4semidirect
92. #13063259C2×C4semidirect
94. #13063257C2×C4semidirect
117. C4⋊C8×C43C2×C4semidirect
306. C8×C4⋊C4D43semidirect
362. C43#22966semidirect
407. C4⋊C8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832002
#191376892
#191375062
#191355602
#191353652
#191353282
#191353232
#191351882
#191351812
#191348592
#191348452
C4×#130632472
C8×C4⋊C4×C432
#2296643