Finite Group Databasebeta

← #25148386

Subgroups — #25148386

Order 11008.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583307C211yesmaximal, characteristic, complemented
2C2×#14583201C211yesmaximal, characteristic
3#19139546C211yesmaximal, characteristic
4#19137693C211yesmaximal, characteristic, complemented
5#19136073C211yesmaximal, characteristic, complemented
6#19136069C211yesmaximal, characteristic, complemented
7#19135979C211yesmaximal, characteristic
8#19135969C211yesmaximal, characteristic
9#19135541C211yesmaximal, characteristic, complemented
10#19135357C211yesmaximal, characteristic, complemented
11#19135353C211yesmaximal, characteristic, complemented
12#19135317C211yesmaximal, characteristic, complemented
13#19135224C211yesmaximal, characteristic
14#19134422C211yesmaximal, characteristic
15C43×#1925C211yesmaximal, characteristic
16C2×#13063324C2211yescharacteristic
17C2×#13063316C2211yescharacteristic
18C2×#13063302—21no—
19C2×#13063302—21no—
20C2×D4×Dic43C2211yescharacteristic
21#14583307—24no—
22#14583201—24no—
23C2×#13063258C2211yescharacteristic
24C2×#13063258C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27C2×#13063256C2211yescharacteristic
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×Dic43⋊C8C2211yescharacteristic
30C2×Dic43⋊C8C2211yescharacteristic
31C2×#13063246C2211yescharacteristic
32C2×#13063246C2211yescharacteristic
33C2×#13063235C2211yescharacteristic
34#14582835—21no—
35#14582835—21no—
36#14582802C2211yescharacteristic
37#14582798C2211yescharacteristic
38#14582797C2211yescharacteristic
39#14582796—21no—
40#14582783—21no—
41#14582782—21no—
42#14582769C2211yescharacteristic
43#14582768C2211yescharacteristic
44C4⋊C4×Dic43C2211yescharacteristic
45C4⋊C4×Dic43C2211yescharacteristic
46C4×Dic43⋊C4C2211yescharacteristic
47C4×C4⋊Dic43—21no—
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C2×#102×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C43×#390C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86—21no—
62C23×Dic43—21no—
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65#13063324—22no—
66#13063324—22no—
67#13063316—22no—
68#13063316D414yes—
69#13063302—44no—
70#13063302—44no—
71D4×Dic43—22no—
72D4×Dic43—22no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4D411yescharacteristic
87C2×Dic43⋊C4D411yescharacteristic
88C2×C22⋊Dic43—21no—
89C2×C22⋊Dic43—21no—
90C2×C22⋊Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43C2311yescharacteristic
94C2×C4×Dic43C2311yescharacteristic
95#13063258—22no—
96#13063258—22no—
97#13063257—22no—
98#13063257—22no—
99#13063256—22no—
100C4⋊(C43⋊C8)—22no—
101Dic43⋊C8—22no—
102Dic43⋊C8—22no—
103#13063246—22no—
104#13063246—22no—
105#13063235—22no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95D411yescharacteristic
111C43⋊#95D411yescharacteristic
112C22×D4×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C22⋊C4×C43—21no—
119C2×C22⋊C4×C43—21no—
120C2×C42×C43—21no—
121C2×C42×C43C2311yescharacteristic
122#107×C43—22no—
123#102×C43—22no—
124#102×C43Q814yescomplemented
125#95×C43—21no—
126C24×C43—21no—
127C2×D4×C43—21no—
128C2×D4×C43—21no—
129C2×D4×C43—21no—
130C2×D4×C43—21no—
131C2×D4×C43—21no—
132C2×D4×C43C2×Q812yes—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2411yescharacteristic
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—21no—
143C22×Dic43—42no—
144C22×Dic43—42no—
145C22×Dic43—42no—
146C22×Dic43C2×D411yescharacteristic
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150C2×Dic86—42no—
151C2×Dic86—42no—
152C2×Dic86—42no—
153C2×C8×C43—21no—
154C2×C8×C43—21no—
155C2×C8×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43C2×Q812yes—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43C2×D412yes—
164C4⋊C4×C43C2×Q812yes—
165C22⋊C4×C43—42no—
166C22⋊C4×C43—42no—
167C42×C43—21no—
168C42×C43—21no—
169C42×C43—42no—
170C22⋊Dic43—42no—
171C22⋊Dic43—42no—
172C22⋊Dic43—42no—
173C4⋊Dic43—21no—
174C4⋊Dic43—24no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—21no—
180C4⋊Dic43—21no—
181C4⋊Dic43—42no—
182C4⋊Dic43—42no—
183C4⋊Dic43—42no—
184C4⋊Dic43C2×D412yes—
185Dic43⋊C4—22no—
186Dic43⋊C4—24no—
187Dic43⋊C4—22no—
188Dic43⋊C4—42no—
189Dic43⋊C4—42no—
190Dic43⋊C4—42no—
191Dic43⋊C4—81no—
192C4×Dic43—21no—
193C4×Dic43—21no—
194C4×Dic43—81no—
195C4×Dic43—81no—
196C4×Dic43C2×D412yes—
197C4×Dic43#4112yes—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8#4112yes—
204C23×C43—22no—
205C23×C43—41no—
206C23×C43—41no—
207C23×C43—41no—
208C23×C43—41no—
209C23×C43—41no—
210C23×C43—41no—
211C23×C43C22×D411yescharacteristic
212D4×C43—22no—
213D4×C43—22no—
214D4×C43—41no—
215D4×C43—41no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—22no—
219C2×C4×C43—21no—
220C2×C4×C43—22no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43AGL(1,ℤ/8)12yes—
231C2×C4×C43C22×D411yescharacteristic
232C2×C4×C43C22×Q811yescharacteristic
233C2×C4×C43C2×#4111yescharacteristic
234C2×C4×C43C2×#4111yescharacteristic
235C2×C4×C432−1+411yescharacteristic
236C2×C4×C432−1+411yescharacteristic
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—22no—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—21no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—84no—
259C2×Dic43#12112yes—
260C2×Dic43#12112yes—
261C2×Dic43AGL(1,ℤ/8)12yes—
262Dic86—84no—
263C8×C43—22no—
264C8×C43—41no—
265C43⋊C8—22no—
266C43⋊C8—41no—
267#25465—431nomaximal, Sylow
268C22×C43—42no—
269C22×C43—42no—
270C22×C43—42no—
271C22×C43—81no—
272C22×C43—81no—
273C22×C43—81no—
274C22×C43—81no—
275C22×C43C2×#12111yescharacteristic
276C22×C43#54711yescharacteristic
277C22×C43D4×Q811yescharacteristic
278C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C43#57711yescharacteristic
281C22×C43#57811yescharacteristic
282C4×C43—22no—
283C4×C43—41no—
284C4×C43—42no—
285C4×C43—42no—
286C4×C43—81no—
287C4×C43#53311yescharacteristic
288C4×C43#54111yescharacteristic
289C4×C43#54811yescharacteristic
290C4×C43D4×Q811yescharacteristic
291Dic43—22no—
292Dic43—22no—
293Dic43—22no—
294Dic43—41no—
295Dic43—42no—
296Dic43—81no—
297Dic43—81no—
298Dic43—81no—
299C2×#501—431no—
300C2×#478—431no—
301#2542—431no—
302#2370—431no—
303#2084—431no—
304#2083—431no—
305#2067—431no—
306#2065—431no—
307#1987—431no—
308#1948—431no—
309#1947—431no—
310#1940—431no—
311#1926—431no—
312#1925D4311yescharacteristic
313#1762—431no—
314C2×C43—82no—
315C2×C43#242911yescharacteristic
316C2×C43#303811yescharacteristic
317C2×C43#304611yescharacteristic
318C2×C43#334311yescharacteristic
319C2×C43#334511yescharacteristic
320C2×C43#338411yescharacteristic
321C2×C43#338611yescharacteristic
322C2×C4⋊Q8—431no—
323C2×#125—431no—
324C2×C22⋊Q8—861no—
325C2×C22⋊Q8—861no—
326C2×C4×D4—431no—
327#501—864no—
328#478—864no—
329C2×#107—431no—
330C2×#107—431no—
331C2×#107C2×D4311yescharacteristic
332C2×#106—431no—
333C2×#106—431no—
334C2×#106—431no—
335C2×C4⋊C8—431no—
336C2×C4⋊C8—431no—
337C2×C4⋊C8—431no—
338C2×#102—431no—
339C2×#102—431no—
340C2×#102C2×D4311yescharacteristic
341C2×#102C2×D4311yescharacteristic
342C2×#97—431no—
343#399—861no—
344#399—861no—
345#390—431no—
346#390C2×D4311yescharacteristic
347#389—431no—
348#389—431no—
349#389—861no—
350#387—861no—
351#387—861no—
352#384—431no—
353#384—431no—
354C4×C4⋊C4—431no—
355C4×C4⋊C4—431no—
356C4×C4⋊C4—431no—
357C4×C4⋊C4—861no—
358C4×C4⋊C4C2×D4311yescharacteristic
359#340—431no—
360#340—431no—
361#340—431no—
362#340—431no—
363#340—431no—
364#340C2×D4311yescharacteristic
365#336—431no—
366#336C2×D4311yescharacteristic
367C43#2546511yescharacteristic, complemented, Sylow
368C22×Q8—861no—
369C22×D4C22×D4311yescharacteristic
370C23×C4—861no—
371C22×C8—431no—
372C22×C8—431no—
373C22×C8C22×D4311yescharacteristic
374C22×C8C22×D4311yescharacteristic
375C4⋊Q8—862no—
376C4⋊Q8—862no—
377#125—434no—
378#125—862no—
379C22⋊Q8—1724no—
380C22⋊Q8—1724no—
381C4×D4—862no—
382C4×D4—862no—
383C2×C4⋊C4—431no—
384C2×C4⋊C4—431no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4—861no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4—861no—
397C2×C4⋊C4—861no—
398C2×C4⋊C4C22×D4311yescharacteristic
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C4⋊C4C22×D4311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—861no—
404C2×C22⋊C4—861no—
405C2×C22⋊C4—861no—
406C2×C42—21no—
407C2×C42—431no—
408C2×C42—431no—
409C2×C42—861no—
410C2×C42—861no—
411C2×C42C22×D4311yescharacteristic
412#107—22no—
413#107—862no—
414#107—862no—
415#106—862no—
416#106—862no—
417#106—862no—
418C4⋊C8—862no—
419C4⋊C8—862no—
420C4⋊C8—862no—
421#102—22no—
422#102—862no—
423#102—862no—
424#102Dic8614yescomplemented
425#97—862no—
426#95—21no—
427#95—431no—
428#95—431no—
429#95—861no—
430#95—861no—
431#95—861no—
432#95—861no—
433C24—21no—
434C2×Q8—1722no—
435C2×Q8—1722no—
436C2×Q8—1722no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×Dic8612yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—431no—
447C22×C4—431no—
448C22×C4—431no—
449C22×C4—431no—
450C22×C4—861no—
451C22×C4—861no—
452C22×C4—861no—
453C22×C4—861no—
454C22×C4—1722no—
455C22×C4—1722no—
456C22×C4—1722no—
457C22×C4D4×D4311yescharacteristic
458C22×C4D4×D4311yescharacteristic
459C22×C4C23×D4311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—432no—
466C2×C8—861no—
467C2×C8—861no—
468C2×C8—861no—
469C2×C8—861no—
470C2×C8—861no—
471C2×C8C2×Dic8612yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—432no—
477C4⋊C4—862no—
478C4⋊C4—864no—
479C4⋊C4—864no—
480C4⋊C4—862no—
481C4⋊C4—861no—
482C4⋊C4—861no—
483C4⋊C4—861no—
484C4⋊C4—861no—
485C4⋊C4—861no—
486C4⋊C4—861no—
487C4⋊C4—861no—
488C4⋊C4—3441no—
489C4⋊C4—1722no—
490C4⋊C4—1722no—
491C4⋊C4—1722no—
492C4⋊C4—1722no—
493C4⋊C4—1722no—
494C4⋊C4—1722no—
495C4⋊C4C2×Dic8612yes—
496C4⋊C4D4×D4312yes—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—1722no—
500C22⋊C4—1722no—
501C22⋊C4—1722no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—432no—
506C42—432no—
507C42—861no—
508C42—861no—
509C42—3441no—
510C42—3441no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D4311yescharacteristic
519Q8—3444no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—432no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—861no—
542C2×C4—861no—
543C2×C4—861no—
544C2×C4—861no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—862no—
548C2×C4—861no—
549C2×C4—861no—
550C2×C4—861no—
551C2×C4—3444no—
552C2×C4—1721no—
553C2×C4—1721no—
554C2×C4—1721no—
555C2×C4—1721no—
556C2×C4—1721no—
557C2×C4—1721no—
558C2×C4—1721no—
559C2×C4—1721no—
560C2×C4—1721no—
561C2×C4—1721no—
562C2×C4—1721no—
563C2×C4#1306334912yes—
564C2×C4C2×D4×D4311yescharacteristic
565C2×C4C22×Dic8611yescharacteristic
566C2×C4#41×D4311yescharacteristic
567C2×C4#41×D4311yescharacteristic
568C2×C4#1306337211yescharacteristic
569C2×C4#1306337211yescharacteristic
570C8—22no—
571C8—41no—
572C8—862no—
573C8—1721no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#121×D4311yescharacteristic
582C22#1458354711yescharacteristic
583C22D4×Dic8611yescharacteristic
584C22AGL(1,ℤ/8)×D4311yescharacteristic
585C22C2×#1306334911yescharacteristic
586C22#1458370611yescharacteristic
587C22#1458371511yescharacteristic
588C4—22no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—862no—
594C4—862no—
595C4—862no—
596C4—3441no—
597C4—3441no—
598C4—3441no—
599C4—1721no—
600C4—1722no—
601C4#1458345211yescharacteristic
602C4#1458350611yescharacteristic
603C4#1458355711yescharacteristic
604C4D4×Dic8611yescharacteristic
605C2—82no—
606C2#1913819111yescharacteristic
607C2#1914407011yescharacteristic
608C2#1914414811yescharacteristic
609C2#1914703911yescharacteristic
610C2#1914706811yescharacteristic
611C2#1914749211yescharacteristic
612C2#1914751711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1925C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#1925D43
hypocenterC43#25465
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
1. C2×#14583307C2semidirect
4. #19137693C2semidirect
5. #19136073C2semidirect
6. #19136069C2semidirect
9. #19135541C2semidirect
10. #19135357C2semidirect
11. #19135353C2semidirect
12. #19135317C2semidirect
124. #102×C43Q8semidirect
367. C43#25465semidirect
424. #102Dic86semidirect

Maximal subgroups

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

subgroupindex
C2×#145833072
C2×#145832012
#191395462
#191376932
#191360732
#191360692
#191359792
#191359692
#191355412
#191353572
#191353532
#191353172
#191352242
#191344222
C43×#19252
#2546543