Finite Group Databasebeta

← #25164691

Subgroups — #25164691

Order 11008.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19138961C211yesmaximal, characteristic
2#19137605C211yesmaximal, characteristic, complemented
3#19135910C211yesmaximal, characteristic, complemented
4#19135790C211yesmaximal, characteristic, complemented
5#19135597C211yesmaximal, characteristic
6#19135548C211yesmaximal, characteristic, complemented
7#19135276C211yesmaximal, characteristic, complemented
8#19135223C211yesmaximal, characteristic, complemented
9#19135202C211yesmaximal, characteristic, complemented
10#19135179C211yesmaximal, characteristic, complemented
11#19134948C211yesmaximal, characteristic, complemented
12#19134911C211yesmaximal, characteristic
13#19134684C211yesmaximal, characteristic, complemented
14#103×Dic43C211yesmaximal, characteristic, complemented
15C43×#2024C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C2×#13063314—21no—
18C2×#13063301C2211yescharacteristic
19C2×#13063300C2211yescharacteristic
20C2×Q8×Dic43C2211yescharacteristic
21C2×C4⋊C4×D43C2211yescharacteristic
22C2×#13063257C2211yescharacteristic
23C2×C4⋊(C43⋊C8)C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×#13063247C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×(C2×D43)⋊C8C2211yescharacteristic
29C2×(C2×D43)⋊C8C2211yescharacteristic
30C2×C8×Dic43C2211yescharacteristic
31#14582816—21no—
32#14582810—21no—
33#14582801C2211yescharacteristic
34#14582800C2211yescharacteristic
35#14582796—21no—
36#14582795C2211yescharacteristic
37#14582789—21no—
38#14582781—21no—
39#14582773—21no—
40#14582756C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42C4×C4⋊Dic43C2211yescharacteristic
43C4×(C2×D43)⋊C4C2211yescharacteristic
44#14582746—21no—
45#14582662C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52C2×C4⋊Q8×C43C2211yescharacteristic
53C2×#103×C43C2211yescharacteristic
54C2×#103×C43C2211yescharacteristic
55C43×#391C2211yescharacteristic
56C43×#340C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C22×D172—21no—
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C4×D43C2311yescharacteristic
63C2×#10621884—22no—
64C2×#10621884—41no—
65C2×#10621884—41no—
66C22×C43⋊C8C2311yescharacteristic
67C22×C43⋊C8C2311yescharacteristic
68#13063314—44no—
69#13063301—22no—
70#13063301—22no—
71#13063300—22no—
72#13063300—41no—
73Q8×Dic43—22no—
74Q8×Dic43—41no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C4⋊C4×D43—22no—
83C4⋊C4×D43—41no—
84C2×(C2×D43)⋊C4—21no—
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×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4D411yescharacteristic
92C2×(C2×D43)⋊C4D411yescharacteristic
93C2×C4×Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43C2311yescharacteristic
97#13063257—22no—
98C4⋊(C43⋊C8)—41no—
99#13063249—22no—
100#13063247—22no—
101#13063247—22no—
102#13063245—41no—
103(C2×D43)⋊C8—41no—
104(C2×D43)⋊C8—41no—
105C8×Dic43—41no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95D411yescharacteristic
111C43⋊#95D411yescharacteristic
112C22×Q8×C43—21no—
113C22×Q8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C4⋊Q8×C43—22no—
117C4⋊Q8×C43—41no—
118C2×C4⋊C4×C43—21no—
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C4⋊C4×C43C2311yescharacteristic
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C42×C43C2311yescharacteristic
123#103×C43—22no—
124#103×C43—41no—
125#95×C43—21no—
126#95×C43—21no—
127C23×D43—21no—
128C23×D43C2×D411yescharacteristic
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C2×Q8×C43—21no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—42no—
135C2×Q8×C43—42no—
136C2×Q8×C43—81no—
137C2×Q8×C43#4112yes—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2411yescharacteristic
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150C22×Dic43C2×D411yescharacteristic
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×D172—21no—
156C2×D172—21no—
157C2×D172—42no—
158C2×D172—42no—
159C2×D172—81no—
160C2×C4×D43—21no—
161C2×C4×D43—21no—
162C2×C4×D43—21no—
163C2×C4×D43—21no—
164C2×C4×D43—21no—
165C2×C4×D43—21no—
166C2×C4×D43—42no—
167C2×C4×D43—42no—
168C2×C4×D43—81no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×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×C43#4112yes—
182C42×C43—21no—
183C42×C43—21no—
184#10621884—44no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—41no—
187(C2×D43)⋊C4—42no—
188(C2×D43)⋊C4—42no—
189(C2×D43)⋊C4—42no—
190(C2×D43)⋊C4—42no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—81no—
193(C2×D43)⋊C4SD1614yescomplemented
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—42no—
199C4⋊Dic43#4112yes—
200C4⋊Dic43#4112yes—
201Dic43⋊C4—24no—
202Dic43⋊C4—42no—
203Dic43⋊C4—81no—
204C4×Dic43—21no—
205C4×Dic43—21no—
206C4×Dic43—42no—
207C4×Dic43—42no—
208C4×Dic43—81no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C23×C43C22×D411yescharacteristic
216C22×D43—21no—
217C22×D43—21no—
218C22×D43—21no—
219C22×D43—21no—
220C22×D43—21no—
221C22×D43—21no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43C2×SD1612yes—
230Q8×C43—22no—
231Q8×C43—22no—
232Q8×C43—41no—
233Q8×C43—41no—
234Q8×C43—84no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
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—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43C2×SD1612yes—
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43H(5,𝔽2)11yescharacteristic
258C2×C4×C43H(5,𝔽2)11yescharacteristic
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—22no—
263C2×Dic43—21no—
264C2×Dic43—22no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43C2×SD1612yes—
277D172—41no—
278D172—41no—
279D172—41no—
280D172—41no—
281D172—84no—
282C4×D43—41no—
283C4×D43—41no—
284C4×D43—41no—
285C4×D43—41no—
286C4×D43—84no—
287C8×C43—41no—
288C8×C43—41no—
289C43⋊C8—41no—
290C43⋊C8—41no—
291#26724—431nomaximal, Sylow
292C22×C43#52511yescharacteristic
293C22×C43#54611yescharacteristic
294C22×C43#54611yescharacteristic
295C22×C43C22×SD1611yescharacteristic
296C22×C43#57511yescharacteristic
297C22×C43#57611yescharacteristic
298C22×C43#57811yescharacteristic
299C2×D43—22no—
300C2×D43—22no—
301C2×D43—22no—
302C2×D43—41no—
303C2×D43—41no—
304C2×D43—41no—
305C2×D43—41no—
306C2×D43—81no—
307C2×D43—81no—
308C2×D43—81no—
309C2×D43—81no—
310C2×D43—81no—
311C2×D43—81no—
312C2×D43—81no—
313C4×C43—22no—
314C4×C43—42no—
315C4×C43—42no—
316C4×C43—41no—
317C4×C43—81no—
318C4×C43—81no—
319C4×C43#52911yescharacteristic
320C4×C43#53811yescharacteristic
321C4×C43#55511yescharacteristic
322C4×C43#55911yescharacteristic
323Dic43—22no—
324Dic43—42no—
325Dic43—41no—
326Dic43—42no—
327Dic43—41no—
328Dic43—81no—
329#2491—431no—
330#2360—431no—
331#2055—431no—
332#2035—431no—
333#2024D4311yescharacteristic, complemented
334#2000—431no—
335#1988—431no—
336#1934—431no—
337#1926—431no—
338#1923—431no—
339#1920—431no—
340#1880—431no—
341#1874—431no—
342#1824—431no—
343C4×#103—431no—
344C2×C43#245911yescharacteristic
345C2×C43#310311yescharacteristic
346C2×C43#312411yescharacteristic
347C2×C43#329511yescharacteristic
348C2×C43#329811yescharacteristic
349C2×C43#331911yescharacteristic
350C2×C43#332711yescharacteristic
351D43—42no—
352D43—161no—
353C22×SD16—431no—
354C2×C4⋊Q8C2×D4311yescharacteristic
355C2×#124—861no—
356C2×C22⋊Q8—431no—
357C2×#121—431no—
358C2×C4×Q8—431no—
359C22×C4⋊C4—431no—
360C2×#106—431no—
361C2×C4⋊C8—431no—
362C2×#103—431no—
363C2×#103C2×D4311yescharacteristic
364C2×#103C2×D4311yescharacteristic
365C2×#102—431no—
366C2×#102—431no—
367C2×#102—431no—
368C2×C22⋊C8—431no—
369C2×C22⋊C8—431no—
370C2×C4×C8—431no—
371#394—861no—
372#393—861no—
373#391C2×D4311yescharacteristic
374#390—431no—
375#390—431no—
376#389—431no—
377#389—861no—
378#388—861no—
379#387—861no—
380#385—861no—
381#380—431no—
382C4×C4⋊C4—431no—
383C4×C4⋊C4—431no—
384C4×C22⋊C4—431no—
385C4×C22⋊C4—861no—
386#340—431no—
387#340—431no—
388#340—431no—
389#340—431no—
390#340—431no—
391#340—431no—
392#340C2×D4311yescharacteristic
393#340C2×D4311yescharacteristic
394#336—431no—
395#336C2×D4311yescharacteristic
396C43#2672411yescharacteristic, complemented, Sylow
397C22×Q8—21no—
398C22×Q8C22×D4311yescharacteristic
399C22×D4—431no—
400C22×D4—861no—
401C23×C4—431no—
402C23×C4—861no—
403C2×SD16—862no—
404C2×SD16—1721no—
405C2×SD16—1721no—
406C22×C8—431no—
407C22×C8—431no—
408C22×C8C22×D4311yescharacteristic
409C22×C8C22×D4311yescharacteristic
410C4⋊Q8—22no—
411C4⋊Q8—41no—
412#124—1724no—
413C22⋊Q8—862no—
414C22⋊Q8—862no—
415#121—862no—
416#121—1721no—
417C4×Q8—862no—
418C4×Q8—1721no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—862no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—1721no—
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C4⋊C4C22×D4311yescharacteristic
431C2×C4⋊C4C22×D4311yescharacteristic
432C2×C22⋊C4—431no—
433C2×C22⋊C4—431no—
434C2×C22⋊C4—861no—
435C2×C22⋊C4—861no—
436C2×C22⋊C4—861no—
437C2×C22⋊C4—861no—
438C2×C22⋊C4—861no—
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C42—431no—
442C2×C42—861no—
443C2×C42—861no—
444C2×C42—861no—
445C2×C42C22×D4311yescharacteristic
446#106—862no—
447C4⋊C8—1721no—
448#103—22no—
449#103—41no—
450#103—862no—
451#102—862no—
452#102—862no—
453#102—1721no—
454C22⋊C8—1721no—
455C22⋊C8—1721no—
456C4×C8—1721no—
457#95—21no—
458#95—21no—
459#95—431no—
460#95—431no—
461#95—861no—
462#95—861no—
463#95—861no—
464#95—861no—
465C24—431no—
466C24—861no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—81no—
475C2×Q8#1062190212yes—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4—861no—
481C2×D4—861no—
482C2×D4—3441no—
483C2×D4—1722no—
484C2×D4—1722no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—431no—
490C22×C4—431no—
491C22×C4—431no—
492C22×C4—861no—
493C22×C4—861no—
494C22×C4—861no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—861no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—3441no—
502C22×C4—1722no—
503C22×C4—1722no—
504C22×C4D4×D4311yescharacteristic
505C22×C4D4×D4311yescharacteristic
506C22×C4C23×D4311yescharacteristic
507SD16—1724no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
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—21no—
525C4⋊C4—42no—
526C4⋊C4—432no—
527C4⋊C4—432no—
528C4⋊C4—864no—
529C4⋊C4—861no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—3441no—
534C4⋊C4—1722no—
535C4⋊C4—1722no—
536C4⋊C4#1062190012yes—
537C22⋊C4—434no—
538C22⋊C4—864no—
539C22⋊C4—3441no—
540C22⋊C4—1721no—
541C22⋊C4—1722no—
542C22⋊C4—1722no—
543C22⋊C4—1722no—
544C22⋊C4—1722no—
545C22⋊C4—1722no—
546C42—21no—
547C42—21no—
548C42—861no—
549C42—861no—
550C42—3441no—
551C42—1722no—
552C42—1722no—
553C23—432no—
554C23—861no—
555C23—861no—
556C23—861no—
557C23—861no—
558C23—861no—
559C23—861no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23—1721no—
564C23—1721no—
565C23—1721no—
566C23—1721no—
567C23C2×D4×D4311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—84no—
573D4—3444no—
574D4—1721no—
575D4—1721no—
576D4—1721no—
577D4—1721no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
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—432no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—862no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—862no—
605C2×C4—861no—
606C2×C4—3444no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4SD16×D4312yes—
619C2×C4#41×D4311yescharacteristic
620C2×C4#41×D4311yescharacteristic
621C2×C4C2×#1062190011yescharacteristic
622C2×C4C2×#1062190211yescharacteristic
623C2×C4#1306337011yescharacteristic
624C2×C4#1306337111yescharacteristic
625C8—41no—
626C8—41no—
627C8—1721no—
628C8—1721no—
629C22—862no—
630C22—862no—
631C22—862no—
632C22—3441no—
633C22—3441no—
634C22—3441no—
635C22—3441no—
636C22—3441no—
637C22—3441no—
638C22—3441no—
639C22—1721no—
640C22—1721no—
641C22—1721no—
642C22—1721no—
643C22#1458341211yescharacteristic
644C22#1458353611yescharacteristic
645C22#1458354211yescharacteristic
646C22C2×SD16×D4311yescharacteristic
647C22#1458369811yescharacteristic
648C22#1458370311yescharacteristic
649C22#1458371211yescharacteristic
650C4—22no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—862no—
657C4—3441no—
658C4—1722no—
659C4—1721no—
660C4—1721no—
661C4—1722no—
662C4#1458343511yescharacteristic
663C4#1458348911yescharacteristic
664C4#1458361111yescharacteristic
665C4#1458363111yescharacteristic
666C2—1722no—
667C2—6881no—
668C2#1913853511yescharacteristic
669C2#1914462811yescharacteristic
670C2#1914486411yescharacteristic
671C2#1914641211yescharacteristic
672C2#1914645911yescharacteristic
673C2#1914675011yescharacteristic
674C2#1914686711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2024C2
socleC23×C43C22×D4
prime radicalC2×#13063300C22
hypercenter#2024D43
hypocenterC43#26724
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
2. #19137605C2semidirect
3. #19135910C2semidirect
4. #19135790C2semidirect
6. #19135548C2semidirect
7. #19135276C2semidirect
8. #19135223C2semidirect
9. #19135202C2semidirect
10. #19135179C2semidirect
11. #19134948C2semidirect
13. #19134684C2semidirect
14. #103×Dic43C2semidirect
15. C43×#2024C2semidirect
193. (C2×D43)⋊C4SD16semidirect
333. #2024D43semidirect
396. C43#26724semidirect

Maximal subgroups

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

subgroupindex
#191389612
#191376052
#191359102
#191357902
#191355972
#191355482
#191352762
#191352232
#191352022
#191351792
#191349482
#191349112
#191346842
#103×Dic432
C43×#20242
#2672443