Finite Group Databasebeta

← #25148652

Subgroups — #25148652

Order 11008.

790 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×#14583188C211yesmaximal, characteristic
2C2×#14583026C211yesmaximal, characteristic, complemented
3#19139955C211yesmaximal, characteristic
4#19139582C211yesmaximal, characteristic, complemented
5#19136098C211yesmaximal, characteristic, complemented
6#19135982C211yesmaximal, characteristic
7#19135970C211yesmaximal, characteristic
8#19135566C211yesmaximal, characteristic, complemented
9#19135366C211yesmaximal, characteristic, complemented
10#19135331C211yesmaximal, characteristic, complemented
11#19135320C211yesmaximal, characteristic, complemented
12#19135293C211yesmaximal, characteristic
13#19135277C211yesmaximal, characteristic
14#19134419C211yesmaximal, characteristic
15C43×#2088C211yesmaximal, characteristic, complemented
16C2×#13063324—21no—
17C2×#13063324C2211yescharacteristic
18C2×C4×Dic86C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20#14583188—24no—
21#14583026—24no—
22C2×#13063259C2211yescharacteristic
23C2×#13063256C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×Dic43⋊C8C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063245C2211yescharacteristic
30C2×#13063245C2211yescharacteristic
31C2×#13063235C2211yescharacteristic
32C2×#13063235C2211yescharacteristic
33C2×C4×C43⋊C8C2211yescharacteristic
34#14582834—21no—
35#14582833—21no—
36#14582832—21no—
37#14582812—21no—
38#14582803—21no—
39#14582801C2211yescharacteristic
40#14582800C2211yescharacteristic
41#14582798C2211yescharacteristic
42#14582797C2211yescharacteristic
43#14582795C2211yescharacteristic
44#14582795C2211yescharacteristic
45#14582773—21no—
46#14582770—21no—
47#14582764C2211yescharacteristic
48C4⋊C4×Dic43C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582661C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×C4⋊C8×C43C2211yescharacteristic
56C43×#389C2211yescharacteristic
57C43×#389C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C22×Dic86—21no—
62C22×Dic86D411yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C22×C43⋊C8C2311yescharacteristic
66C22×C43⋊C8C2311yescharacteristic
67#13063324—22no—
68#13063324—22no—
69#13063324—44no—
70C4×Dic86—22no—
71C4×Dic86—22no—
72#13063287—22no—
73#13063287—22no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43D411yescharacteristic
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⋊C4D411yescharacteristic
84C2×Dic43⋊C4D411yescharacteristic
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43C2311yescharacteristic
92C2×C4×Dic43C2311yescharacteristic
93#13063259—22no—
94#13063256—22no—
95#13063256—22no—
96Dic43⋊C8—22no—
97Dic43⋊C8D414yes—
98#13063247—22no—
99#13063247—22no—
100#13063245—22no—
101#13063245Q814yescomplemented
102#13063235—22no—
103#13063235—22no—
104C4×C43⋊C8—22no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C42×C43C2311yescharacteristic
121#106×C43—22no—
122C4⋊C8×C43—22no—
123#95×C43—21no—
124#95×C43—21no—
125C23×D43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2411yescharacteristic
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43—21no—
136C22×Dic43C2×D411yescharacteristic
137C22×Dic43C2×D411yescharacteristic
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C2×D172—21no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172C2×Q812yes—
146C2×C4×D43—42no—
147C2×C4×D43—81no—
148C2×C4×D43—81no—
149C2×Dic86—22no—
150C2×Dic86—22no—
151C2×Dic86—41no—
152C2×Dic86—42no—
153C2×Dic86—42no—
154C2×Dic86—42no—
155C2×C8×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43C2×D412yes—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—42no—
169C4⋊C4×C43—81no—
170C4⋊C4×C43C2×Q812yes—
171C42×C43—21no—
172C42×C43—21no—
173(C2×D43)⋊C4—42no—
174(C2×D43)⋊C4—42no—
175(C2×D43)⋊C4—42no—
176(C2×D43)⋊C4—81no—
177(C2×D43)⋊C4—81no—
178C4⋊Dic43—21no—
179C4⋊Dic43—22no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43—42no—
185C4⋊Dic43#4112yes—
186Dic43⋊C4—22no—
187Dic43⋊C4—22no—
188Dic43⋊C4—42no—
189Dic43⋊C4—42no—
190Dic43⋊C4—42no—
191Dic43⋊C4—81no—
192C4×Dic43—21no—
193C4×Dic43—21no—
194C4×Dic43—42no—
195C4×Dic43C2×D412yes—
196C4×Dic43#4112yes—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8C2×D412yes—
202C2×C43⋊C8C2×Q812yes—
203C23×C43C22×D411yescharacteristic
204C22×D43—22no—
205C22×D43—41no—
206C22×D43—41no—
207C22×D43—41no—
208C22×D43—41no—
209C22×D43—41no—
210C22×D43—41no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—22no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43AGL(1,ℤ/8)12yes—
229C2×C4×C43C22×D411yescharacteristic
230C2×C4×C43C22×Q811yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C2×C4×C432−1+411yescharacteristic
234C2×C4×C432−1+411yescharacteristic
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—22no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—21no—
245C2×Dic43—21no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43#12112yes—
255C2×Dic43#12112yes—
256D172—22no—
257D172—22no—
258D172—41no—
259D172—41no—
260C4×D43—84no—
261Dic86—44no—
262Dic86—84no—
263C8×C43—22no—
264C8×C43—41no—
265C43⋊C8—22no—
266C43⋊C8—22no—
267#25485—431nomaximal, Sylow
268C22×C43C2×#12111yescharacteristic
269C22×C43#54811yescharacteristic
270C22×C43D4×Q811yescharacteristic
271C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C43#57511yescharacteristic
273C22×C43#57711yescharacteristic
274C22×C43#57711yescharacteristic
275C2×D43—42no—
276C2×D43—42no—
277C2×D43—42no—
278C2×D43—81no—
279C2×D43—81no—
280C2×D43—81no—
281C2×D43—81no—
282C4×C43—22no—
283C4×C43—42no—
284C4×C43—41no—
285C4×C43—81no—
286C4×C43—81no—
287C4×C43—81no—
288C4×C43#53311yescharacteristic
289C4×C43#54111yescharacteristic
290C4×C43#54711yescharacteristic
291C4×C43D4×Q811yescharacteristic
292Dic43—22no—
293Dic43—22no—
294Dic43—42no—
295Dic43—41no—
296Dic43—41no—
297Dic43—81no—
298Dic43—81no—
299C2×#476—431no—
300C2×#446—431no—
301#2580—431no—
302#2546—431no—
303#2088—431no—
304#2088D4311yescharacteristic, complemented
305#2067—431no—
306#2065—431no—
307#1993—431no—
308#1949—431no—
309#1943—431no—
310#1941—431no—
311#1936—431no—
312#1934—431no—
313#1762—431no—
314C2×C43#242911yescharacteristic
315C2×C43#305511yescharacteristic
316C2×C43#305711yescharacteristic
317C2×C43#337111yescharacteristic
318C2×C43#337311yescharacteristic
319C2×C43#338411yescharacteristic
320C2×C43#338811yescharacteristic
321D43—82no—
322C2×C4⋊Q8—431no—
323C2×C4⋊Q8—861no—
324C2×C4×Q8—431no—
325C2×C4×D4—431no—
326#476—864no—
327#446—864no—
328C2×#107—431no—
329C2×#106—431no—
330C2×#106—431no—
331C2×#106C2×D4311yescharacteristic
332C2×C4⋊C8—431no—
333C2×C4⋊C8—431no—
334C2×C4⋊C8C2×D4311yescharacteristic
335C2×#102—431no—
336C2×#102—431no—
337C2×#102—431no—
338C2×#102—431no—
339C2×#97—431no—
340C2×#97—431no—
341C2×C4×C8—431no—
342#399—861no—
343#398—861no—
344#398—861no—
345#393—861no—
346#391—861no—
347#390—431no—
348#390—431no—
349#389—431no—
350#389—431no—
351#389—431no—
352#389—431no—
353#389C2×D4311yescharacteristic
354#389C2×D4311yescharacteristic
355#385—861no—
356#385—861no—
357#382—431no—
358C4×C4⋊C4—431no—
359#340—431no—
360#340—431no—
361#340—431no—
362#340—431no—
363#340—431no—
364#340C2×D4311yescharacteristic
365#340C2×D4311yescharacteristic
366#340C2×D4311yescharacteristic
367C43#2548511yescharacteristic, complemented, Sylow
368C22×Q8—431no—
369C22×Q8—861no—
370C22×D4—431no—
371C23×C4—861no—
372C22×C8—431no—
373C22×C8—431no—
374C22×C8C22×D4311yescharacteristic
375C22×C8C22×D4311yescharacteristic
376C4⋊Q8—862no—
377C4⋊Q8—862no—
378C4⋊Q8—1724no—
379C4×Q8—862no—
380C4×Q8—862no—
381C4×D4—862no—
382C4×D4—862no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—431no—
390C2×C4⋊C4—431no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4C22×D4311yescharacteristic
397C2×C4⋊C4C22×D4311yescharacteristic
398C2×C4⋊C4C22×D4311yescharacteristic
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C22⋊C4—861no—
401C2×C22⋊C4—861no—
402C2×C22⋊C4—861no—
403C2×C22⋊C4—861no—
404C2×C22⋊C4—861no—
405C2×C42—431no—
406C2×C42—431no—
407C2×C42—861no—
408C2×C42C22×D4311yescharacteristic
409#107—862no—
410#106—22no—
411#106—862no—
412#106—862no—
413C4⋊C8—22no—
414C4⋊C8—434no—
415C4⋊C8—862no—
416#102—434no—
417#102—862no—
418#102—862no—
419#102—862no—
420#97—862no—
421#97—862no—
422C4×C8—862no—
423#95—21no—
424#95—21no—
425#95—861no—
426#95—861no—
427#95—861no—
428#95—861no—
429#95—861no—
430#95—861no—
431#95—861no—
432C24—861no—
433C2×Q8—862no—
434C2×Q8—862no—
435C2×Q8—1721no—
436C2×Q8—1722no—
437C2×Q8—1722no—
438C2×Q8—1722no—
439C2×D4—432no—
440C2×D4—861no—
441C2×D4—861no—
442C2×D4—861no—
443C2×D4—861no—
444C2×D4—861no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—431no—
450C22×C4—431no—
451C22×C4—431no—
452C22×C4—431no—
453C22×C4—861no—
454C22×C4—861no—
455C22×C4—861no—
456C22×C4—3441no—
457C22×C4—3441no—
458C22×C4—1722no—
459C22×C4D4×D4311yescharacteristic
460C22×C4D4×D4311yescharacteristic
461C22×C4C23×D4311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—432no—
468C2×C8—432no—
469C2×C8—861no—
470C2×C8—861no—
471C2×C8—861no—
472C2×C8—861no—
473C2×C8D4×D4312yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—432no—
484C4⋊C4—862no—
485C4⋊C4—862no—
486C4⋊C4—862no—
487C4⋊C4—861no—
488C4⋊C4—861no—
489C4⋊C4—861no—
490C4⋊C4—861no—
491C4⋊C4—861no—
492C4⋊C4—3441no—
493C4⋊C4—1722no—
494C4⋊C4—1722no—
495C4⋊C4—1722no—
496C4⋊C4—1722no—
497C4⋊C4Q8×D4312yes—
498C22⋊C4—3441no—
499C22⋊C4—3441no—
500C22⋊C4—1722no—
501C22⋊C4—1722no—
502C22⋊C4—1722no—
503C42—21no—
504C42—21no—
505C42—432no—
506C42—432no—
507C42—861no—
508C42—861no—
509C42—1722no—
510C23—862no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23—1721no—
515C23—1721no—
516C23—1721no—
517C23C2×D4×D4311yescharacteristic
518Q8—3444no—
519Q8—1724no—
520D4—862no—
521D4—862no—
522D4—1721no—
523D4—1721no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—432no—
542C2×C4—432no—
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—861no—
552C2×C4—861no—
553C2×C4—861no—
554C2×C4—3444no—
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#1306335212yes—
564C2×C4C2×D4×D4311yescharacteristic
565C2×C4C2×Q8×D4311yescharacteristic
566C2×C4#41×D4311yescharacteristic
567C2×C4#41×D4311yescharacteristic
568C2×C4#1306337211yescharacteristic
569C2×C4#1306337311yescharacteristic
570C8—22no—
571C8—41no—
572C8—862no—
573C8—862no—
574C22—3441no—
575C22—3441no—
576C22—3441no—
577C22—3441no—
578C22—1722no—
579C22—1722no—
580C22—1722no—
581C22#121×D4311yescharacteristic
582C22#1458355911yescharacteristic
583C22#1458356211yescharacteristic
584C22C2×#1306335211yescharacteristic
585C22#1458369611yescharacteristic
586C22#1458370811yescharacteristic
587C22#1458370911yescharacteristic
588C4—22no—
589C4—41no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—862no—
595C4—862no—
596C4—3441no—
597C4—3441no—
598C4—1721no—
599C4—1721no—
600C4—1722no—
601C4#1458345511yescharacteristic
602C4#1458350811yescharacteristic
603C4#1458354711yescharacteristic
604C4#1458356411yescharacteristic
605C2—3442no—
606C2#1913818911yescharacteristic
607C2#1914420711yescharacteristic
608C2#1914423211yescharacteristic
609C2#1914736111yescharacteristic
610C2#1914738611yescharacteristic
611C2#1914749311yescharacteristic
612C2#1914753811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2088C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2088D43
hypocenterC43#25485
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2088

Complements

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

NH ≅ G/Ntype
2. C2×#14583026C2semidirect
4. #19139582C2semidirect
5. #19136098C2semidirect
8. #19135566C2semidirect
9. #19135366C2semidirect
10. #19135331C2semidirect
11. #19135320C2semidirect
15. C43×#2088C2semidirect
101. #13063245Q8semidirect
304. #2088D43semidirect
367. C43#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#145831882
C2×#145830262
#191399552
#191395822
#191360982
#191359822
#191359702
#191355662
#191353662
#191353312
#191353202
#191352932
#191352772
#191344192
C43×#20882
#2548543