Finite Group Databasebeta

← #25159518

Subgroups — #25159518

Order 11008.

896 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×#14583202C211yesmaximal, characteristic, complemented
2#19138302C211yesmaximal, characteristic, complemented
3#19137611C211yesmaximal, characteristic
4#19136125C211yesmaximal, characteristic, complemented
5#19136094C211yesmaximal, characteristic, complemented
6#19135978C211yesmaximal, characteristic, complemented
7#19135931C211yesmaximal, characteristic, complemented
8#19135355C211yesmaximal, characteristic, complemented
9#19135168C211yesmaximal, characteristic, complemented
10#19135033C211yesmaximal, characteristic
11#19134917C211yesmaximal, characteristic
12#19134853C211yesmaximal, characteristic, complemented
13#19134788C211yesmaximal, characteristic, complemented
14#19134778C211yesmaximal, characteristic, complemented
15C2×C43×#428C211yesmaximal, characteristic, complemented
16C2×#13063316C2211yescharacteristic
17C2×#13063299C2211yescharacteristic
18C2×#13063287C2211yescharacteristic
19C2×C4×C22⋊D43—21no—
20#14583202—24no—
21C2×#13063256C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×Dic43⋊C8C2211yescharacteristic
24C2×Dic43⋊C8C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×C22⋊(C43⋊C8)C2211yescharacteristic
30#14582835—21no—
31#14582822—21no—
32#14582800C2211yescharacteristic
33#14582798C2211yescharacteristic
34#14582797—21no—
35#14582795C2211yescharacteristic
36#14582795C2211yescharacteristic
37#14582789—21no—
38#14582783—21no—
39#14582778—21no—
40#14582770—21no—
41#14582768C2211yescharacteristic
42#14582759C2211yescharacteristic
43C4⋊C4×Dic43C2211yescharacteristic
44C4×C22⋊Dic43C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C22×M16×C43C2211yescharacteristic
54C2×#117×C43C2211yescharacteristic
55C22×C4⋊C4×C43C2211yescharacteristic
56C43×#428—24no—
57C2×#107×C43C2211yescharacteristic
58C2×#107×C43C2211yescharacteristic
59C2×#106×C43C2211yescharacteristic
60C2×#106×C43C2211yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C22⋊D43—21no—
63C22×C4×D43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66#13063316—22no—
67#13063316—41no—
68#13063299—22no—
69#13063299Q814yes—
70#13063287—22no—
71#13063287—41no—
72C4×C22⋊D43—44no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×C22⋊Dic43—21no—
84C2×C22⋊Dic43D411yescharacteristic
85C2×C22⋊Dic43D411yescharacteristic
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#13063256—22no—
94#13063256—41no—
95Dic43⋊C8—22no—
96Dic43⋊C8—22no—
97#13063247—22no—
98#13063247—22no—
99#13063245—22no—
100#13063245—41no—
101C22⋊(C43⋊C8)—22no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95D411yescharacteristic
108C43⋊#95D411yescharacteristic
109C23×C4×C43—21no—
110C23×C4×C43C2311yescharacteristic
111C2×M16×C43—22no—
112C2×M16×C43—22no—
113C2×M16×C43—22no—
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116#117×C43—22no—
117#117×C43—41no—
118C2×C4⋊C4×C43—22no—
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43—41no—
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C22⋊C4×C43—21no—
126C2×C42×C43—21no—
127#107×C43—22no—
128#107×C43D414yescomplemented
129#106×C43—22no—
130#106×C43—22no—
131C24×C43C2×D411yescharacteristic
132C23×D43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—42no—
142C22×C4×C43—42no—
143C22×C4×C43—81no—
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2×Q812yes—
146C22×C4×C43#4112yes—
147C22×C4×C43C2411yescharacteristic
148C2×C22⋊D43—24no—
149C2×C22⋊D43—42no—
150C2×C22⋊D43—42no—
151C22×Dic43—21no—
152C22×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43C2×D411yescharacteristic
155C22×Dic43C2×D411yescharacteristic
156C22×Dic43C2×D411yescharacteristic
157C22×Dic43C2×D411yescharacteristic
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×D172C2×Q812yes—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C4×D43—81no—
167M16×C43—28no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43C2×D412yes—
173C2×C8×C43#4112yes—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—42no—
181C4⋊C4×C43C2×D412yes—
182C4⋊C4×C43C2×D412yes—
183C22⋊C4×C43—42no—
184C42×C43—42no—
185C22⋊Dic43—22no—
186C22⋊Dic43—41no—
187C22⋊Dic43—81no—
188(C2×D43)⋊C4—24no—
189(C2×D43)⋊C4—42no—
190(C2×D43)⋊C4—42no—
191(C2×D43)⋊C4—81no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—42no—
198C4⋊Dic43—42no—
199C4⋊Dic43C2×Q812yes—
200Dic43⋊C4—42no—
201Dic43⋊C4—42no—
202Dic43⋊C4—42no—
203Dic43⋊C4—81no—
204Dic43⋊C4—81no—
205C4×Dic43—21no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C4×Dic43—42no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C23×C43—21no—
217C23×C43—21no—
218C23×C43—21no—
219C23×C43—21no—
220C23×C43—21no—
221C23×C43—21no—
222C23×C43C22⋊Q812yes—
223C23×C43C22×D411yescharacteristic
224C22×D43—22no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43—22no—
236C2×C4×C43—22no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—84no—
250C2×C4×C43C22⋊Q812yes—
251C2×C4×C43C22×D411yescharacteristic
252C2×C4×C43C22×Q811yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43H(5,𝔽2)11yescharacteristic
256C2×C4×C432−1+411yescharacteristic
257C22⋊D43—48no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—22no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43AGL(1,ℤ/8)12yes—
279D172—22no—
280D172—22no—
281D172—41no—
282D172—41no—
283C4×D43—84no—
284C8×C43—22no—
285C8×C43—22no—
286C43⋊C8—41no—
287C43⋊C8—41no—
288#26366—431nomaximal, Sylow
289C22×C43—22no—
290C22×C43—22no—
291C22×C43—22no—
292C22×C43—41no—
293C22×C43—41no—
294C22×C43—41no—
295C22×C43—41no—
296C22×C43C2×C22⋊Q811yescharacteristic
297C22×C43D4211yescharacteristic
298C22×C43#54711yescharacteristic
299C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C43#57511yescharacteristic
301C22×C43#57711yescharacteristic
302C22×C43#57711yescharacteristic
303C2×D43—42no—
304C2×D43—42no—
305C2×D43—42no—
306C2×D43—81no—
307C2×D43—81no—
308C2×D43—81no—
309C2×D43—81no—
310C4×C43—22no—
311C4×C43—42no—
312C4×C43—42no—
313C4×C43—81no—
314C4×C43—81no—
315C4×C43#52511yescharacteristic
316C4×C43#53911yescharacteristic
317C4×C43D4×Q811yescharacteristic
318C4×C43#55411yescharacteristic
319Dic43—22no—
320Dic43—42no—
321Dic43—41no—
322Dic43—41no—
323Dic43—41no—
324Dic43—81no—
325Dic43—81no—
326C2×#478—431no—
327C2×#428D4311yescharacteristic, complemented
328#2441—431no—
329#2361—431no—
330#2094—431no—
331#2088—431no—
332#2067—431no—
333#2059—431no—
334#1948—431no—
335#1918—431no—
336#1895—431no—
337#1875—431no—
338#1864—431no—
339#1853—431no—
340#1851—431no—
341C2×C43—42no—
342C2×C43#244911yescharacteristic
343C2×C43#307711yescharacteristic
344C2×C43#308411yescharacteristic
345C2×C43#327111yescharacteristic
346C2×C43#327211yescharacteristic
347C2×C43#334511yescharacteristic
348C2×C43#335111yescharacteristic
349D43—82no—
350C22×M16C2×D4311yescharacteristic
351C2×#125—431no—
352C2×#121—431no—
353C2×C4×D4—431no—
354C2×C4×D4—861no—
355C2×#117C2×D4311yescharacteristic
356C22×C4⋊C4C2×D4311yescharacteristic
357#478—864no—
358#428—24no—
359C2×#107C2×D4311yescharacteristic
360C2×#107C2×D4311yescharacteristic
361C2×#106—431no—
362C2×#106—431no—
363C2×#106C2×D4311yescharacteristic
364C2×#106C2×D4311yescharacteristic
365C2×C4⋊C8—431no—
366C2×C4⋊C8—431no—
367C2×#102—431no—
368C2×#102—431no—
369C2×#102—431no—
370C2×#102—431no—
371C2×C22⋊C8—431no—
372#399—861no—
373#396—861no—
374#390—431no—
375#389—431no—
376#389—431no—
377#389—431no—
378#389—861no—
379#388—861no—
380#387—861no—
381#386—861no—
382#385—861no—
383#384—431no—
384#380—431no—
385C4×C4⋊C4—431no—
386C4×C22⋊C4—431no—
387#340—431no—
388#340—431no—
389#340—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#336—431no—
394#336—431no—
395C43#2636611yescharacteristic, complemented, Sylow
396C22×D4—431no—
397C22×D4—861no—
398C23×C4—21no—
399C23×C4—861no—
400C23×C4C22×D4311yescharacteristic
401C2×M16—22no—
402C2×M16—22no—
403C2×M16—22no—
404C22×C8—431no—
405C22×C8—431no—
406C22×C8C22×D4311yescharacteristic
407C22×C8C22×D4311yescharacteristic
408#125—862no—
409#125—1721no—
410#121—434no—
411#121—862no—
412C4×D4—862no—
413C4×D4—1721no—
414C4×D4—1724no—
415#117—22no—
416#117—41no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4C22×D4311yescharacteristic
431C2×C4⋊C4C22×D4311yescharacteristic
432C2×C4⋊C4C22×D4311yescharacteristic
433C2×C4⋊C4C22×D4311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—431no—
436C2×C22⋊C4—431no—
437C2×C22⋊C4—861no—
438C2×C22⋊C4—861no—
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C22⋊C4—861no—
442C2×C42—21no—
443C2×C42—431no—
444C2×C42—431no—
445C2×C42—861no—
446#107—22no—
447#107C22⋊D4314yescomplemented
448#106—22no—
449#106—22no—
450#106—862no—
451#106—1721no—
452C4⋊C8—862no—
453C4⋊C8—862no—
454#102—862no—
455#102—862no—
456#102—862no—
457#102—1721no—
458C22⋊C8—862no—
459#95—431no—
460#95—431no—
461#95—861no—
462#95—861no—
463#95—861no—
464#95—861no—
465#95—861no—
466C24—861no—
467C24D4×D4311yescharacteristic
468C2×D4—432no—
469C2×D4—864no—
470C2×D4—861no—
471C2×D4—861no—
472C2×D4—861no—
473C2×D4—861no—
474C2×D4—861no—
475C2×D4—1722no—
476C2×D4—1722no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—431no—
489C22×C4—431no—
490C22×C4—431no—
491C22×C4—431no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—861no—
495C22×C4—3441no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4#1062189812yes—
499C22×C4D4×D4311yescharacteristic
500C22×C4Q8×D4312yes—
501C22×C4C23×D4311yescharacteristic
502M16—28no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8—861no—
510C2×C8—861no—
511C2×C8—861no—
512C2×C8—861no—
513C2×C8#1062189812yes—
514C2×C8C2×C22⋊D4312yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—432no—
523C4⋊C4—861no—
524C4⋊C4—861no—
525C4⋊C4—861no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—3441no—
529C4⋊C4—3441no—
530C4⋊C4—1722no—
531C4⋊C4—1722no—
532C4⋊C4—1722no—
533C4⋊C4—1722no—
534C4⋊C4—1722no—
535C4⋊C4C2×C22⋊D4312yes—
536C4⋊C4C2×C22⋊D4312yes—
537C22⋊C4—42no—
538C22⋊C4—862no—
539C22⋊C4—864no—
540C22⋊C4—3441no—
541C22⋊C4—3441no—
542C22⋊C4—1722no—
543C22⋊C4—1722no—
544C22⋊C4—1721no—
545C42—42no—
546C42—861no—
547C42—861no—
548C42—861no—
549C42—861no—
550C42—1722no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—862no—
558C23—1721no—
559C23—1721no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23—1721no—
564C23#1306330312yes—
565C23C2×D4×D4311yescharacteristic
566D4—862no—
567D4—862no—
568D4—1728no—
569D4—1721no—
570D4—1721no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—432no—
591C2×C4—861no—
592C2×C4—861no—
593C2×C4—862no—
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—3444no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4#1306330312yes—
613C2×C4C22×C22⋊D4311yescharacteristic
614C2×C4C2×Q8×D4311yescharacteristic
615C2×C4C2×#1062189811yescharacteristic
616C2×C4C2×#1062189811yescharacteristic
617C2×C4#1306337011yescharacteristic
618C2×C4#1306337311yescharacteristic
619C8—22no—
620C8—22no—
621C8—1721no—
622C8—1721no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—3441no—
631C22—3441no—
632C22—3441no—
633C22—3441no—
634C22—1722no—
635C22—1722no—
636C22—1722no—
637C22C2×#1306330311yescharacteristic
638C22D4×C22⋊D4311yescharacteristic
639C22#1458355011yescharacteristic
640C22AGL(1,ℤ/8)×D4311yescharacteristic
641C22#1458369811yescharacteristic
642C22#1458370811yescharacteristic
643C22#1458370911yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—862no—
650C4—3441no—
651C4—3441no—
652C4—1721no—
653C4—1721no—
654C4—1722no—
655C4—1721no—
656C4#1458341011yescharacteristic
657C4#1458349611yescharacteristic
658C4Q8×C22⋊D4311yescharacteristic
659C4#1458360411yescharacteristic
660C2—42no—
661C2—3442no—
662C2#1913840411yescharacteristic
663C2#1914439111yescharacteristic
664C2#1914445611yescharacteristic
665C2#1914617411yescharacteristic
666C2#1914618911yescharacteristic
667C2#1914706611yescharacteristic
668C2#1914715611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#428C2
socleC23×C43C22×D4
prime radicalC2×#13063299C22
hypercenterC2×#428D43
hypocenterC43#26366
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#14583202C2semidirect
2. #19138302C2semidirect
4. #19136125C2semidirect
5. #19136094C2semidirect
6. #19135978C2semidirect
7. #19135931C2semidirect
8. #19135355C2semidirect
9. #19135168C2semidirect
12. #19134853C2semidirect
13. #19134788C2semidirect
14. #19134778C2semidirect
15. C2×C43×#428C2semidirect
128. #107×C43D4semidirect
327. C2×#428D43semidirect
395. C43#26366semidirect
447. #107C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832022
#191383022
#191376112
#191361252
#191360942
#191359782
#191359312
#191353552
#191351682
#191350332
#191349172
#191348532
#191347882
#191347782
C2×C43×#4282
#2636643