Finite Group Databasebeta

← #25259417

Subgroups — #25259417

Order 11008.

2238 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063324C211yesmaximal, characteristic
2#19137773C214yesmaximal, complemented
3#19137682C212yesmaximal, complemented
4#19137681C212yesmaximal, complemented
5#19137466C211yesmaximal, characteristic
6#19137465C212yesmaximal
7#19137464C211yesmaximal, characteristic
8#19137462C211yesmaximal, characteristic
9#19137459C212yesmaximal
10#19137342C212yesmaximal, complemented
11#19137340C214yesmaximal, complemented
12#19137339C212yesmaximal, complemented
13#19137180C212yesmaximal
14#19137179C212yesmaximal
15#19136990C211yesmaximal, characteristic
16#19136989C211yesmaximal, characteristic
17C43×#2344C211yesmaximal, characteristic
18C23×Dic86C2212yes—
19C2×#13063324C4116yescomplemented
20C2×#13063324C2211yescharacteristic
21C2×#13063324C2212yes—
22C2×#13063324C2211yescharacteristic
23C2×#13063324C2211yescharacteristic
24C2×#13063324C2211yescharacteristic
25C2×#13063324C2212yes—
26C2×#13063284C2212yes—
27C2×#13063281C2212yes—
28C2×#13063281C2212yes—
29C22×C4⋊Dic43C2211yescharacteristic
30C22×Dic43⋊C4C2212yes—
31C22×C4×Dic43C2211yescharacteristic
32#14582805C2214yes—
33#14582805C2214yes—
34#14582804C2214yes—
35#14582804C2214yes—
36#14582803C2214yes—
37#14582803C2214yes—
38#14582803C2214yes—
39#14582803C2214yes—
40#14582798C2212yes—
41#14582797C2212yes—
42#14582796C2214yes—
43#14582795C2212yes—
44#14582794C2214yes—
45#14582786C2214yes—
46#14582785C2214yes—
47#14582784C2214yes—
48#14582783C2214yes—
49#14582782C2214yes—
50#14582781C2214yes—
51#14582780C2214yes—
52#14582766C2212yes—
53#14582766C2212yes—
54#14582764C2212yes—
55#14582763C2212yes—
56#14582762C2212yes—
57#14582760C2211yescharacteristic
58#14582760C2212yes—
59#14582760C2212yes—
60#14582760C2212yes—
61#14582760C2211yescharacteristic
62#14582759C2211yescharacteristic
63#14582757C2211yescharacteristic
64#14582757C2211yescharacteristic
65C4⋊C4×Dic43C2212yes—
66C4⋊C4×Dic43C2212yes—
67C4⋊C4×Dic43C2212yes—
68C4×Dic43⋊C4C2214yes—
69C4×Dic43⋊C4C2212yes—
70C4×C4⋊Dic43C2212yes—
71#14582750C2214yes—
72C22⋊C4×Dic43C2212yes—
73C4×C22⋊Dic43C2211yescharacteristic
74#14582745C2212yes—
75#14582744C2212yes—
76C2×#117×C43C2212yes—
77C22×C4⋊C4×C43C2211yescharacteristic
78C43×#389C2212yes—
79C43×#387C2214yes—
80C43×#381C2212yes—
81C43×#380C2211yescharacteristic
82C4×C4⋊C4×C43C2212yes—
83C4×C22⋊C4×C43C2211yescharacteristic
84C22×Dic86—44no—
85C22×Dic86—44no—
86C22×Dic86C2×C4116yes—
87C22×Dic86C2312yes—
88C22×Dic86C2312yes—
89C22×Dic86C2312yes—
90C22×Dic86C2312yes—
91C23×Dic43C2311yescharacteristic
92C23×Dic43C2311yescharacteristic
93C23×Dic43C2312yes—
94#13063324C2×C418yes—
95#13063324C2×C4116yes—
96#13063324C2×C418yes—
97#13063324C2×C418yes—
98#13063324C2×C418yes—
99#13063324C2×C4116yes—
100#13063284—44no—
101#13063281—44no—
102#13063281—44no—
103C2×C4⋊Dic43C2×C418yes—
104C2×C4⋊Dic43C2312yes—
105C2×C4⋊Dic43C2314yes—
106C2×C4⋊Dic43C2312yes—
107C2×C4⋊Dic43C2311yescharacteristic
108C2×C4⋊Dic43C2311yescharacteristic
109C2×Dic43⋊C4C2×C4116yes—
110C2×Dic43⋊C4C2312yes—
111C2×Dic43⋊C4C2312yes—
112C2×Dic43⋊C4C2314yes—
113C2×Dic43⋊C4C2314yes—
114C2×Dic43⋊C4C2312yes—
115C2×Dic43⋊C4C2312yes—
116C2×Dic43⋊C4C2312yes—
117C2×Dic43⋊C4C2312yes—
118C2×C22⋊Dic43C2311yescharacteristic
119C2×C22⋊Dic43C2312yes—
120C2×C22⋊Dic43C2311yescharacteristic
121C2×C4×Dic43C2×C418yes—
122C2×C4×Dic43C2314yes—
123C2×C4×Dic43C2312yes—
124C2×C4×Dic43C2314yes—
125C2×C4×Dic43C2314yes—
126C2×C4×Dic43C2314yes—
127C2×C4×Dic43C2311yescharacteristic
128C2×C4×Dic43C2311yescharacteristic
129C2×C4×Dic43C2311yescharacteristic
130C2×C4×Dic43C2311yescharacteristic
131C43⋊#95C2312yes—
132C43⋊#95C2314yes—
133C43⋊#95C2314yes—
134C43⋊#95C2312yes—
135C43⋊#95C2314yes—
136C43⋊#95C2314yes—
137C43⋊#95C2312yes—
138C43⋊#95C2314yes—
139C43⋊#95C2314yes—
140C43⋊#95C2314yes—
141C43⋊#95C2312yes—
142C43⋊#95C2312yes—
143C43⋊#95C2314yes—
144C43⋊#95C2314yes—
145C43⋊#95C2312yes—
146C43⋊#95C2312yes—
147C43⋊#95C2312yes—
148C23×C4×C43C2312yes—
149C23×C4×C43C2311yescharacteristic
150#117×C43—44no—
151C2×C4⋊C4×C43C2×C418yes—
152C2×C4⋊C4×C43C2312yes—
153C2×C4⋊C4×C43C2314yes—
154C2×C4⋊C4×C43C2312yes—
155C2×C4⋊C4×C43C2311yescharacteristic
156C2×C4⋊C4×C43C2311yescharacteristic
157C2×C22⋊C4×C43C2311yescharacteristic
158C2×C22⋊C4×C43C2312yes—
159C2×C22⋊C4×C43C2311yescharacteristic
160C2×C42×C43C2312yes—
161C2×C42×C43C2314yes—
162#95×C43C2312yes—
163#95×C43C2314yes—
164#95×C43C2314yes—
165#95×C43C2312yes—
166C24×C43C2411yescharacteristic
167C22×C4×C43—24no—
168C22×C4×C43—24no—
169C22×C4×C43—24no—
170C22×C4×C43C22×C414yes—
171C22×C4×C43C22×C418yes—
172C22×C4×C43C2412yes—
173C22×C4×C43C2414yes—
174C22×C4×C43C2412yes—
175C22×C4×C43C2411yescharacteristic
176C22×C4×C43C2412yes—
177C22×C4×C43C2412yes—
178C22×C4×C43C2411yescharacteristic
179C22×Dic43—24no—
180C22×Dic43—24no—
181C22×Dic43—24no—
182C22×Dic43—22no—
183C22×Dic43—22no—
184C22×Dic43C22×C414yes—
185C22×Dic43C22×C414yes—
186C22×Dic43C22×C418yes—
187C22×Dic43C2412yes—
188C22×Dic43C2414yes—
189C22×Dic43C2412yes—
190C22×Dic43C2411yescharacteristic
191C22×Dic43C2411yescharacteristic
192C22×Dic43C2411yescharacteristic
193C22×Dic43C2412yes—
194C22×Dic43C2412yes—
195C22×Dic43C2411yescharacteristic
196C2×Dic86—432no—
197C2×Dic86—42no—
198C2×Dic86—42no—
199C2×Dic86—42no—
200C2×Dic86—42no—
201C2×Dic86—42no—
202C2×Dic86—42no—
203C2×Dic86—42no—
204C2×Dic86—42no—
205C2×Dic86C22×C418yes—
206C2×Dic86C22×C418yes—
207C2×Dic86C22×C418yes—
208C2×Dic86C22×C418yes—
209C4⋊C4×C43—24no—
210C4⋊C4×C43—44no—
211C4⋊C4×C43C22×C418yes—
212C4⋊C4×C43C22×C414yes—
213C4⋊C4×C43C22×C414yes—
214C22⋊C4×C43—22no—
215C22⋊C4×C43—24no—
216C22⋊C4×C43—22no—
217C42×C43—24no—
218C42×C43—44no—
219C22⋊Dic43—22no—
220C22⋊Dic43—24no—
221C22⋊Dic43—22no—
222C4⋊Dic43—24no—
223C4⋊Dic43—44no—
224C4⋊Dic43C22×C418yes—
225C4⋊Dic43C22×C414yes—
226C4⋊Dic43C22×C414yes—
227Dic43⋊C4—24no—
228Dic43⋊C4—24no—
229Dic43⋊C4—44no—
230Dic43⋊C4—44no—
231Dic43⋊C4C22×C418yes—
232Dic43⋊C4C22×C418yes—
233Dic43⋊C4C22×C418yes—
234Dic43⋊C4C22×C418yes—
235C4×Dic43—24no—
236C4×Dic43—28no—
237C4×Dic43—44no—
238C4×Dic43—44no—
239C4×Dic43—44no—
240C4×Dic43C22×C414yes—
241C4×Dic43C22×C414yes—
242C4×Dic43C22×C414yes—
243C4×Dic43C22×C414yes—
244C23×C43—22no—
245C23×C43—22no—
246C23×C43C23×C412yes—
247C23×C43H(5,𝔽2)12yes—
248C23×C43H(5,𝔽2)12yes—
249C23×C43C2511yescharacteristic
250C2×C4×C43—216no—
251C2×C4×C43—22no—
252C2×C4×C43—22no—
253C2×C4×C43—22no—
254C2×C4×C43—22no—
255C2×C4×C43—22no—
256C2×C4×C43—28no—
257C2×C4×C43—22no—
258C2×C4×C43—24no—
259C2×C4×C43—22no—
260C2×C4×C43—24no—
261C2×C4×C43—22no—
262C2×C4×C43—22no—
263C2×C4×C43—22no—
264C2×C4×C43—22no—
265C2×C4×C43—24no—
266C2×C4×C43C23×C412yes—
267C2×C4×C43C23×C414yes—
268C2×C4×C43C23×C414yes—
269C2×C4×C43C23×C412yes—
270C2×C4×C432−1+414yes—
271C2×Dic43—28no—
272C2×Dic43—216no—
273C2×Dic43—22no—
274C2×Dic43—22no—
275C2×Dic43—22no—
276C2×Dic43—22no—
277C2×Dic43—21no—
278C2×Dic43—21no—
279C2×Dic43—21no—
280C2×Dic43—21no—
281C2×Dic43—22no—
282C2×Dic43—28no—
283C2×Dic43—22no—
284C2×Dic43—24no—
285C2×Dic43—22no—
286C2×Dic43—24no—
287C2×Dic43—22no—
288C2×Dic43—22no—
289C2×Dic43—22no—
290C2×Dic43—22no—
291C2×Dic43—24no—
292C2×Dic43C23×C412yes—
293C2×Dic43C23×C412yes—
294C2×Dic43C23×C412yes—
295C2×Dic43C23×C414yes—
296C2×Dic43C23×C414yes—
297C2×Dic43C23×C412yes—
298C2×Dic432−1+414yes—
299Dic86—48no—
300Dic86—48no—
301Dic86—48no—
302Dic86—48no—
303#34365—431nomaximal, Sylow
304C22×C43—22no—
305C22×C43—22no—
306C22×C43—24no—
307C22×C43—24no—
308C22×C43#51814yes—
309C22×C43C24×C411yescharacteristic
310C22×C43C2×H(5,𝔽2)11yescharacteristic
311C22×C43C2×H(5,𝔽2)11yescharacteristic
312C22×C43C2×2−1+412yes—
313C22×C43C2×2−1+412yes—
314C4×C43—24no—
315C4×C43—24no—
316C4×C43—22no—
317C4×C43—44no—
318C4×C43—42no—
319C4×C43—42no—
320C4×C43#51914yes—
321Dic43—22no—
322Dic43—22no—
323Dic43—24no—
324Dic43—24no—
325Dic43—22no—
326Dic43—42no—
327Dic43—44no—
328Dic43—42no—
329Dic43#51914yes—
330C22×C4⋊Q8—431no—
331#2381—434no—
332#2369—432no—
333#2369—432no—
334#2344—432no—
335#2344—431no—
336#2344—431no—
337#2344—432no—
338#2344—431no—
339#2344D4311yescharacteristic
340#2331—432no—
341#2331—434no—
342#2331—432no—
343#2310—432no—
344#2310—432no—
345#2283—431no—
346#2283—431no—
347C2×C43—24no—
348C2×C43C2×#51811yescharacteristic
349C2×C43C2×#51911yescharacteristic
350C2×C43C2×#51911yescharacteristic
351C2×C43#354412yes—
352C2×C43#354412yes—
353C23×Q8—432no—
354C2×C4⋊Q8—432no—
355C2×C4⋊Q8—431no—
356C2×C4⋊Q8—431no—
357C2×C4⋊Q8—431no—
358C2×C4⋊Q8—431no—
359C2×C4⋊Q8—432no—
360C2×C4⋊Q8—4316no—
361C2×#117—432no—
362C2×#117—432no—
363C2×#117—432no—
364C2×#117C2×D4312yes—
365C22×C4⋊C4—432no—
366C22×C4⋊C4—431no—
367C22×C4⋊C4C2×D4311yescharacteristic
368C22×C42—431no—
369#391—434no—
370#391—434no—
371#391—434no—
372#391—434no—
373#391—434no—
374#391—434no—
375#391—434no—
376#391—434no—
377#389—434no—
378#389—432no—
379#389—432no—
380#389—432no—
381#389—434no—
382#389C2×D4312yes—
383#387—434no—
384#387—434no—
385#387—434no—
386#387—434no—
387#387—434no—
388#387—434no—
389#387—434no—
390#387C2×D4314yes—
391#383—432no—
392#383—432no—
393#382—432no—
394#382—432no—
395#381—432no—
396#381—431no—
397#381—432no—
398#381—431no—
399#381—432no—
400#381—432no—
401#381C2×D4312yes—
402#380—431no—
403#380—431no—
404#380—431no—
405#380C2×D4311yescharacteristic
406C4×C4⋊C4—432no—
407C4×C4⋊C4—432no—
408C4×C4⋊C4—432no—
409C4×C4⋊C4—432no—
410C4×C4⋊C4—434no—
411C4×C4⋊C4—432no—
412C4×C4⋊C4—434no—
413C4×C4⋊C4C2×D4312yes—
414C4×C22⋊C4—431no—
415C4×C22⋊C4—432no—
416C4×C22⋊C4C2×D4311yescharacteristic
417#376—432no—
418#376—432no—
419C43#3436511yescharacteristic, complemented, Sylow
420C22×Q8—4316no—
421C22×Q8—432no—
422C22×Q8—432no—
423C22×Q8—432no—
424C22×Q8—432no—
425C22×Q8—1724no—
426C22×Q8—1724no—
427C23×C4—432no—
428C23×C4—431no—
429C23×C4—431no—
430C23×C4C22×D4312yes—
431C23×C4C22×D4311yescharacteristic
432C4⋊Q8—4316no—
433C4⋊Q8—438no—
434C4⋊Q8—438no—
435C4⋊Q8—438no—
436C4⋊Q8—438no—
437C4⋊Q8—4316no—
438#117—44no—
439#117—1724no—
440#117—1724no—
441#117—1724no—
442C2×C4⋊C4—4316no—
443C2×C4⋊C4—438no—
444C2×C4⋊C4—432no—
445C2×C4⋊C4—432no—
446C2×C4⋊C4—432no—
447C2×C4⋊C4—432no—
448C2×C4⋊C4—431no—
449C2×C4⋊C4—431no—
450C2×C4⋊C4—432no—
451C2×C4⋊C4—432no—
452C2×C4⋊C4—432no—
453C2×C4⋊C4—432no—
454C2×C4⋊C4—434no—
455C2×C4⋊C4—434no—
456C2×C4⋊C4—434no—
457C2×C4⋊C4C4×D4318yes—
458C2×C4⋊C4C22×D4312yes—
459C2×C4⋊C4C22×D4314yes—
460C2×C4⋊C4C22×D4311yescharacteristic
461C2×C4⋊C4C22×D4311yescharacteristic
462C2×C4⋊C4C22×D4312yes—
463C2×C22⋊C4—431no—
464C2×C22⋊C4—431no—
465C2×C22⋊C4—432no—
466C2×C22⋊C4C22×D4311yescharacteristic
467C2×C22⋊C4C22×D4312yes—
468C2×C22⋊C4C22×D4311yescharacteristic
469C2×C42—438no—
470C2×C42—431no—
471C2×C42—431no—
472C2×C42—431no—
473C2×C42—431no—
474C2×C42—434no—
475C2×C42—432no—
476C2×C42—434no—
477C2×C42—434no—
478C2×C42—434no—
479C2×C42C22×D4312yes—
480C2×C42C22×D4314yes—
481#95—432no—
482#95—432no—
483#95—432no—
484#95—432no—
485#95—434no—
486#95—432no—
487#95—434no—
488#95—432no—
489#95—432no—
490#95—434no—
491#95—432no—
492#95—434no—
493#95—434no—
494#95—434no—
495#95—434no—
496#95—434no—
497#95—434no—
498#95C22×D4312yes—
499#95C22×D4312yes—
500#95C22×D4314yes—
501#95C22×D4314yes—
502C24C23×D4311yescharacteristic
503C2×Q8—438no—
504C2×Q8—438no—
505C2×Q8—438no—
506C2×Q8—438no—
507C2×Q8—17232no—
508C2×Q8—1722no—
509C2×Q8—1722no—
510C2×Q8—1722no—
511C2×Q8—1722no—
512C2×Q8—1722no—
513C2×Q8—1722no—
514C2×Q8—1722no—
515C2×Q8—1722no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—431no—
520C22×C4—434no—
521C22×C4—434no—
522C22×C4—438no—
523C22×C4—431no—
524C22×C4—431no—
525C22×C4—431no—
526C22×C4—432no—
527C22×C4—432no—
528C22×C4—432no—
529C22×C4—434no—
530C22×C4—432no—
531C22×C4—864no—
532C22×C4—864no—
533C22×C4—862no—
534C22×C4—864no—
535C22×C4—862no—
536C22×C4C2×C4×D4314yes—
537C22×C4C2×C4×D4318yes—
538C22×C4C23×D4312yes—
539C22×C4C23×D4314yes—
540C22×C4C23×D4312yes—
541C22×C4C23×D4311yescharacteristic
542C22×C4C23×D4312yes—
543C22×C4C23×D4312yes—
544C22×C4C23×D4311yescharacteristic
545C4⋊C4—24no—
546C4⋊C4—44no—
547C4⋊C4—438no—
548C4⋊C4—438no—
549C4⋊C4—438no—
550C4⋊C4—438no—
551C4⋊C4—438no—
552C4⋊C4—434no—
553C4⋊C4—434no—
554C4⋊C4—864no—
555C4⋊C4—864no—
556C4⋊C4—864no—
557C4⋊C4—1724no—
558C4⋊C4—1724no—
559C4⋊C4—1724no—
560C4⋊C4C2×C4×D4318yes—
561C4⋊C4C2×C4×D4314yes—
562C4⋊C4C2×C4×D4314yes—
563C22⋊C4—22no—
564C22⋊C4—24no—
565C22⋊C4—22no—
566C22⋊C4—862no—
567C22⋊C4—864no—
568C22⋊C4—862no—
569C42—24no—
570C42—44no—
571C42—434no—
572C42—434no—
573C42—434no—
574C42—434no—
575C42—868no—
576C42—864no—
577C42—1724no—
578C42—1724no—
579C42—1724no—
580C23—22no—
581C23—22no—
582C23C22×C4×D4312yes—
583C23#1306337012yes—
584C23#1306337112yes—
585C23C24×D4311yescharacteristic
586Q8—1728no—
587Q8—1728no—
588Q8—1728no—
589Q8—1728no—
590C2×C4—216no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—432no—
607C2×C4—434no—
608C2×C4—432no—
609C2×C4—432no—
610C2×C4—432no—
611C2×C4—434no—
612C2×C4—434no—
613C2×C4—868no—
614C2×C4—868no—
615C2×C4—8616no—
616C2×C4—862no—
617C2×C4—862no—
618C2×C4—861no—
619C2×C4—861no—
620C2×C4—862no—
621C2×C4—862no—
622C2×C4—861no—
623C2×C4—861no—
624C2×C4—862no—
625C2×C4—862no—
626C2×C4—864no—
627C2×C4—862no—
628C2×C4—864no—
629C2×C4—862no—
630C2×C4—862no—
631C2×C4—862no—
632C2×C4—862no—
633C2×C4—864no—
634C2×C4C22×C4×D4312yes—
635C2×C4C22×C4×D4314yes—
636C2×C4C22×C4×D4314yes—
637C2×C4C22×C4×D4312yes—
638C2×C4#1306337214yes—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22#1458337114yes—
644C22C23×C4×D4311yescharacteristic
645C22C2×#1306337011yescharacteristic
646C22C2×#1306337111yescharacteristic
647C222−1+4×D4312yes—
648C22C2×#1306337212yes—
649C4—24no—
650C4—24no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—434no—
656C4—862no—
657C4—862no—
658C4—864no—
659C4—864no—
660C4—862no—
661C4—1722no—
662C4—1724no—
663C4—1722no—
664C4#1458337614yes—
665C2—24no—
666C2C2×#1458337111yescharacteristic
667C2#519×D4311yescharacteristic
668C2C2×#1458337611yescharacteristic
669C2#1914893112yes—
670C2#1914893712yes—
subgroupG/H
center, FrattiniC23C24×D43
commutatorC22×C43C24×C4
FittingC43×#2344C2
socle, cosocleC23×C43C25
prime radicalC24×C43C24
hypercenter#2344D43
hypocenterC43#34365
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ #2344

Complements

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

NH ≅ G/Ntype
2. #19137773C2semidirect
3. #19137682C2semidirect
4. #19137681C2semidirect
10. #19137342C2semidirect
11. #19137340C2semidirect
12. #19137339C2semidirect
19. C2×#13063324C4semidirect
419. C43#34365semidirect

Maximal subgroups

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

subgroupindex
C22×#130633242
#191377732
#191376822
#191376812
#191374662
#191374652
#191374642
#191374622
#191374592
#191373422
#191373402
#191373392
#191371802
#191371792
#191369902
#191369892
C43×#23442
#3436543