Finite Group Databasebeta

← #25133105

Subgroups — #25133105

Order 11008.

1070 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
1C2×#14583609C211yesmaximal, characteristic
2C2×#14583206C211yesmaximal, characteristic, complemented
3C2×C4×#10621884C211yesmaximal, characteristic
4#19135933C211yesmaximal, characteristic, complemented
5#19135928C211yesmaximal, characteristic, complemented
6#19135746C211yesmaximal, characteristic, complemented
7#19135736C211yesmaximal, characteristic, complemented
8#19135559C211yesmaximal, characteristic, complemented
9#19135532C211yesmaximal, characteristic, complemented
10#19134932C211yesmaximal, characteristic
11#19134932C211yesmaximal, characteristic
12#19134855C211yesmaximal, characteristic
13#19134841C211yesmaximal, characteristic
14#19134602C211yesmaximal, characteristic
15C43×#2581C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17#14583609—24no—
18#14583609—24no—
19C2×#13063325C2211yescharacteristic
20C2×#13063317C2211yescharacteristic
21C2×#13063317C2211yescharacteristic
22C2×#13063303—21no—
23C2×#13063303—21no—
24C2×#13063301—21no—
25C2×C4×D172C2211yescharacteristic
26C2×C4×D172C2211yescharacteristic
27C2×C4×D172C2211yescharacteristic
28C2×C4⋊C4×D43—21no—
29#14583206—24no—
30C4×#10621884—24no—
31C2×#13063258C2211yescharacteristic
32C2×#13063256C2211yescharacteristic
33C2×#13063256C2211yescharacteristic
34C2×C4⋊(C43⋊C8)C2211yescharacteristic
35C2×#13063249C2211yescharacteristic
36C2×#13063249C2211yescharacteristic
37C2×#13063249C2211yescharacteristic
38C2×#13063245C2211yescharacteristic
39C2×#13063245C2211yescharacteristic
40C2×#13063245C2211yescharacteristic
41C2×C4×C43⋊C8C2211yescharacteristic
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582654C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582654C2211yescharacteristic
51#14582654C2211yescharacteristic
52C2×C4×Q8×C43C2211yescharacteristic
53C2×C4×Q8×C43C2211yescharacteristic
54C43×#400—21no—
55C43×#395—21no—
56C43×#391C2211yescharacteristic
57C43×#391C2211yescharacteristic
58C43×#389C2211yescharacteristic
59C43×#389C2211yescharacteristic
60C43×#387—21no—
61C43×#387—21no—
62C43×#382C2211yescharacteristic
63C22×Dic86—21no—
64C22×D172C2311yescharacteristic
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C22×C4×D43—21no—
68C2×#10621884—22no—
69C2×#10621884—22no—
70C2×#10621884Q814yescomplemented
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75#13063325—22no—
76#13063325—22no—
77#13063317—22no—
78#13063317—22no—
79#13063317—22no—
80#13063317D414yes—
81#13063303—28no—
82#13063303—28no—
83#13063301—28no—
84C4×D172—22no—
85C4×D172—22no—
86C4×D172—22no—
87C4×D172—22no—
88C4×D172—22no—
89C4×D172D414yes—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43C2311yescharacteristic
92C2×C4⋊Dic43C2311yescharacteristic
93C2×C4⋊Dic43C2311yescharacteristic
94C2×Dic43⋊C4—21no—
95C2×Dic43⋊C4—21no—
96C2×Dic43⋊C4—21no—
97C2×Dic43⋊C4—21no—
98C4⋊C4×D43—28no—
99C2×(C2×D43)⋊C4—21no—
100C2×(C2×D43)⋊C4—21no—
101C2×(C2×D43)⋊C4—21no—
102#13063258—22no—
103#13063256—22no—
104#13063256—22no—
105C4⋊(C43⋊C8)—22no—
106#13063249—22no—
107#13063249—22no—
108#13063249—22no—
109#13063245—22no—
110#13063245—22no—
111#13063245—22no—
112C4×C43⋊C8—22no—
113C22×Q8×C43D411yescharacteristic
114C22×Q8×C43C2311yescharacteristic
115C4×Q8×C43—22no—
116C4×Q8×C43—22no—
117C4×Q8×C43—22no—
118C4×Q8×C43—22no—
119C2×C4⋊C4×C43—21no—
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43D411yescharacteristic
123C2×C4⋊C4×C43D411yescharacteristic
124C2×C4⋊C4×C43D411yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C4⋊C4×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C42×C43—21no—
129C2×C42×C43C2311yescharacteristic
130C2×C42×C43C2311yescharacteristic
131C2×C42×C43C2311yescharacteristic
132#95×C43—21no—
133#95×C43—21no—
134#95×C43—21no—
135#95×C43—21no—
136#95×C43—21no—
137#95×C43—21no—
138#95×C43—21no—
139#95×C43—21no—
140C23×D43—21no—
141C2×Q8×C43—22no—
142C2×Q8×C43—22no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43—41no—
149C2×Q8×C43C2×Q812yes—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2411yescharacteristic
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43—21no—
164C2×D172—21no—
165C2×D172—21no—
166C2×D172—21no—
167C2×D172C2×D412yes—
168C2×D172C2×Q812yes—
169C2×D172#4112yes—
170C2×C4×D43—24no—
171C2×C4×D43—24no—
172C2×C4×D43—24no—
173C2×C4×D43—42no—
174C2×C4×D43—42no—
175C2×C4×D43—81no—
176C2×C4×D43—81no—
177C2×C4×D43—81no—
178C2×C4×D43—81no—
179C2×Dic86—24no—
180C2×Dic86—81no—
181C2×Dic86—81no—
182C4⋊C4×C43—22no—
183C4⋊C4×C43—22no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—42no—
192C4⋊C4×C43—42no—
193C4⋊C4×C43—42no—
194C42×C43—21no—
195C42×C43—21no—
196C42×C43—21no—
197C42×C43—21no—
198C42×C43—42no—
199C42×C43C2×D412yes—
200C42×C43C2×D412yes—
201#10621884—28no—
202(C2×D43)⋊C4—24no—
203(C2×D43)⋊C4—24no—
204(C2×D43)⋊C4—24no—
205C4⋊Dic43—24no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43C2×D412yes—
210C4⋊Dic43C2×D412yes—
211C4⋊Dic43C2×D412yes—
212Dic43⋊C4—24no—
213Dic43⋊C4—24no—
214Dic43⋊C4—24no—
215Dic43⋊C4—24no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8C2×Q812yes—
228C23×C43C22×D411yescharacteristic
229C22×D43—22no—
230C22×D43—22no—
231C22×D43—22no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235C22×D43—41no—
236Q8×C43—22no—
237Q8×C43—22no—
238Q8×C43—44no—
239Q8×C43—41no—
240Q8×C43—41no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—22no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—21no—
255C2×C4×C43—21no—
256C2×C4×C43—21no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43—41no—
266C2×C4×C43—41no—
267C2×C4×C43—41no—
268C2×C4×C43#12012yes—
269C2×C4×C43#12012yes—
270C2×C4×C43#13512yes—
271C2×C4×C43C22×D411yescharacteristic
272C2×C4×C43C22×D411yescharacteristic
273C2×C4×C43C22×Q811yescharacteristic
274C2×C4×C43C2×#4111yescharacteristic
275C2×C4×C432−1+411yescharacteristic
276C2×C4×C432−1+411yescharacteristic
277C2×Dic43—22no—
278C2×Dic43—22no—
279C2×Dic43—22no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286D172—22no—
287D172—22no—
288D172—22no—
289D172C22⋊Q814yes—
290C4×D43—48no—
291C4×D43—84no—
292C4×D43—84no—
293Dic86—84no—
294C43⋊C8—22no—
295C43⋊C8—41no—
296C43⋊C8—41no—
297C43⋊C8—41no—
298#24337—431nomaximal, Sylow
299C22×C43C2×#12011yescharacteristic
300C22×C43#54811yescharacteristic
301C22×C43D4×Q811yescharacteristic
302C22×C43C2×#13511yescharacteristic
303C22×C43#57511yescharacteristic
304C22×C43#57711yescharacteristic
305C22×C43#57711yescharacteristic
306C2×D43—24no—
307C2×D43—42no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—42no—
311C2×D43—42no—
312C2×D43—42no—
313C4×C43—22no—
314C4×C43—22no—
315C4×C43—22no—
316C4×C43—42no—
317C4×C43—41no—
318C4×C43—41no—
319C4×C43—41no—
320C4×C43—81no—
321C4×C43—81no—
322C4×C43—81no—
323C4×C43C2×C22⋊Q811yescharacteristic
324C4×C43#53611yescharacteristic
325C4×C43#54811yescharacteristic
326C4×C43D4×Q811yescharacteristic
327Dic43—42no—
328Dic43—42no—
329Dic43—42no—
330C2×#554—431no—
331C2×#479—431no—
332C2×C4×SD16—431no—
333#2581D4311yescharacteristic, complemented
334#2059—431no—
335#2058—431no—
336#2026—431no—
337#2024—431no—
338#1991—431no—
339#1984—431no—
340#1877—431no—
341#1877—431no—
342#1864—431no—
343#1862—431no—
344#1804—431no—
345C2×C43#240911yescharacteristic
346C2×C43#300211yescharacteristic
347C2×C43#300911yescharacteristic
348C2×C43#336011yescharacteristic
349C2×C43#337611yescharacteristic
350C2×C43#338111yescharacteristic
351C2×C43#338811yescharacteristic
352D43—44no—
353C22×SD16—431no—
354#554—864no—
355#554—864no—
356C2×C4⋊Q8—431no—
357C2×#125—431no—
358C2×#125—431no—
359C2×C22⋊Q8—861no—
360C2×C22⋊Q8—861no—
361C2×C22⋊Q8—861no—
362C2×C4×Q8C2×D4311yescharacteristic
363C2×C4×Q8C2×D4311yescharacteristic
364C2×C4×D4—431no—
365C2×C4×D4—431no—
366C2×C4×D4—431no—
367C22×C4⋊C4—861no—
368#479—864no—
369C4×SD16—864no—
370C2×#107—431no—
371C2×#106—431no—
372C2×#106—431no—
373C2×C4⋊C8—431no—
374C2×#103—431no—
375C2×#103—431no—
376C2×#103—431no—
377C2×#102—431no—
378C2×#102—431no—
379C2×#102—431no—
380C2×C4×C8—431no—
381#400—21no—
382#395—21no—
383#391C2×D4311yescharacteristic
384#391C2×D4311yescharacteristic
385#389C2×D4311yescharacteristic
386#389C2×D4311yescharacteristic
387#387—21no—
388#387—21no—
389#382C2×D4311yescharacteristic
390#340—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#340—431no—
395#340—431no—
396#336—431no—
397#336—431no—
398#336—431no—
399#336—431no—
400C43#2433711yescharacteristic, complemented, Sylow
401C22×Q8—861no—
402C22×Q8C22⋊D4311yescharacteristic
403C22×Q8C22×D4311yescharacteristic
404C22×D4—431no—
405C23×C4—861no—
406C23×C4—861no—
407C23×C4—861no—
408C2×SD16—434no—
409C2×SD16—862no—
410C2×SD16—862no—
411C22×C8—431no—
412C22×C8—431no—
413C22×C8—431no—
414C22×C8—431no—
415C4⋊Q8—862no—
416C4⋊Q8—862no—
417#125—434no—
418#125—862no—
419#125—862no—
420#125—862no—
421C22⋊Q8—868no—
422C22⋊Q8—868no—
423C22⋊Q8—868no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×D4—434no—
429C4×D4—862no—
430C4×D4—862no—
431C4×D4—862no—
432C4×D4—862no—
433C4×D4—862no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—431no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—868no—
441C2×C4⋊C4—861no—
442C2×C4⋊C4—861no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4—861no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4C22⋊D4311yescharacteristic
447C2×C4⋊C4C22⋊D4311yescharacteristic
448C2×C4⋊C4C22⋊D4311yescharacteristic
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C4⋊C4C22×D4311yescharacteristic
451C2×C4⋊C4C22×D4311yescharacteristic
452C2×C22⋊C4—861no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—21no—
456C2×C42C22×D4311yescharacteristic
457C2×C42C22×D4311yescharacteristic
458C2×C42C22×D4311yescharacteristic
459#107—862no—
460#106—862no—
461#106—862no—
462C4⋊C8—862no—
463#103—862no—
464#103—862no—
465#103—862no—
466#102—862no—
467#102—862no—
468#102—862no—
469C4×C8—862no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—21no—
477#95—21no—
478C24—861no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—41no—
487C2×Q8—864no—
488C2×Q8—3441no—
489C2×Q8—3441no—
490C2×Q8Q8×D4312yes—
491C2×D4—432no—
492C2×D4—432no—
493C2×D4—432no—
494C2×D4—861no—
495C2×D4—861no—
496C2×D4—861no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—864no—
502C22×C4—864no—
503C22×C4—861no—
504C22×C4—861no—
505C22×C4—861no—
506C22×C4—864no—
507C22×C4—3441no—
508C22×C4—3441no—
509C22×C4—3441no—
510C22×C4—3441no—
511C22×C4—1722no—
512C22×C4—1722no—
513C22×C4C2×C22⋊D4311yescharacteristic
514C22×C4C2×C22⋊D4311yescharacteristic
515C22×C4C2×C22⋊D4311yescharacteristic
516C22×C4C2×C22⋊D4311yescharacteristic
517C22×C4C2×C22⋊D4311yescharacteristic
518C22×C4C2×C22⋊D4311yescharacteristic
519C22×C4C23×D4311yescharacteristic
520SD16—868no—
521C2×C8—432no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8—861no—
525C2×C8—861no—
526C2×C8—861no—
527C2×C8—861no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—432no—
546C4⋊C4—432no—
547C4⋊C4—432no—
548C4⋊C4—864no—
549C4⋊C4—864no—
550C4⋊C4—864no—
551C4⋊C4—864no—
552C4⋊C4—864no—
553C4⋊C4—861no—
554C4⋊C4—861no—
555C4⋊C4—861no—
556C22⋊C4—864no—
557C22⋊C4—864no—
558C22⋊C4—864no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—42no—
564C42D4×D4312yes—
565C42D4×D4312yes—
566C23—862no—
567C23—862no—
568C23—862no—
569C23—1721no—
570C23—1721no—
571C23—1721no—
572C23—1721no—
573C23C22×C22⋊D4311yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—44no—
577Q8—41no—
578Q8—41no—
579Q8—3444no—
580D4—434no—
581D4—862no—
582D4—862no—
583D4—862no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—862no—
612C2×C4—862no—
613C2×C4—862no—
614C2×C4—3444no—
615C2×C4—3444no—
616C2×C4—1728no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4#1306329312yes—
624C2×C4#1306329312yes—
625C2×C4#1306334512yes—
626C2×C4C2×D4×D4311yescharacteristic
627C2×C4C2×D4×D4311yescharacteristic
628C2×C4C2×Q8×D4311yescharacteristic
629C2×C4#41×D4311yescharacteristic
630C2×C4#1306337211yescharacteristic
631C2×C4#1306337311yescharacteristic
632C8—862no—
633C8—1721no—
634C8—1721no—
635C8—1721no—
636C22—864no—
637C22—1722no—
638C22—1722no—
639C22—1722no—
640C22—1722no—
641C22—1722no—
642C22—1722no—
643C22C2×#1306329311yescharacteristic
644C22#1458355611yescharacteristic
645C22Q8×C22⋊D4311yescharacteristic
646C22C2×#1306334511yescharacteristic
647C22#1458369711yescharacteristic
648C22#1458370711yescharacteristic
649C22#1458370711yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—1722no—
661C4—1722no—
662C4—1722no—
663C4C22⋊Q8×D4311yescharacteristic
664C4#1458347211yescharacteristic
665C4#1458355911yescharacteristic
666C4#1458356211yescharacteristic
667C2—1724no—
668C2#1913795611yescharacteristic
669C2#1914381111yescharacteristic
670C2#1914387011yescharacteristic
671C2#1914724011yescharacteristic
672C2#1914741611yescharacteristic
673C2#1914746611yescharacteristic
674C2#1914753411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2581C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#2581D43
hypocenterC43#24337
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C2×#14583206C2semidirect
4. #19135933C2semidirect
5. #19135928C2semidirect
6. #19135746C2semidirect
7. #19135736C2semidirect
8. #19135559C2semidirect
9. #19135532C2semidirect
15. C43×#2581C2semidirect
70. C2×#10621884Q8semidirect
333. #2581D43semidirect
400. C43#24337semidirect

Maximal subgroups

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

subgroupindex
C2×#145836092
C2×#145832062
C2×C4×#106218842
#191359332
#191359282
#191357462
#191357362
#191355592
#191355322
#191349322
#191348552
#191348412
#191346022
C43×#25812
#2433743