Finite Group Databasebeta

← #25096069

Subgroups — #25096069

Order 11008.

1514 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583367C211yesmaximal, characteristic
2C22×Dic43⋊C8C211yesmaximal, characteristic, complemented
3C2×#14582896C211yesmaximal, characteristic
4C2×#14582893C211yesmaximal, characteristic
5#19136998C211yesmaximal, characteristic, complemented
6#19134609C212yesmaximal, complemented
7#19134604C212yesmaximal, complemented
8#19134599C212yesmaximal, complemented
9#19134594C212yesmaximal, complemented
10#19134311C211yesmaximal, characteristic, complemented
11C43×#1735C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13C23×C43⋊C8C2211yescharacteristic
14#14583367—24no—
15#14583367C418yescomplemented
16C2×#13063290C2211yescharacteristic
17C2×#13063290C2211yescharacteristic
18C2×#13063287C2211yescharacteristic
19C2×#13063287C2211yescharacteristic
20C2×(C22⋊D43)⋊C4—21no—
21C2×(C22⋊D43)⋊C4—21no—
22C2×#13063284C2211yescharacteristic
23C2×#13063282—21no—
24C22×C4×Dic43C2211yescharacteristic
25C2×C42×D43—21no—
26C2×Dic43⋊C8—24no—
27C2×Dic43⋊C8C2212yes—
28C2×Dic43⋊C8C2212yes—
29#14582896—24no—
30#14582893—24no—
31C2×#13063250C2212yes—
32C2×#13063248C2212yes—
33C2×#13063247C2212yes—
34C2×#13063245C2212yes—
35#14582768C2212yes—
36#14582764C2212yes—
37#14582757—21no—
38#14582757—21no—
39C4×Dic43⋊C4—21no—
40C4×Dic43⋊C4—21no—
41#14582661C2212yes—
42#14582661C2212yes—
43C23×C8×C43C2211yescharacteristic
44C2×#117×C43C2211yescharacteristic
45C2×#117×C43C2211yescharacteristic
46C43×#340C2212yes—
47C43×#340C2212yes—
48C2×#10621898—22no—
49C2×#10621898—22no—
50C2×#10621898—22no—
51C2×#10621898C2×C414yes—
52C2×#10621898D414yes—
53C2×#10621898Q814yes—
54C22×Dic86C2311yescharacteristic
55C22×D172C2311yescharacteristic
56C22×C22⋊D43—21no—
57C23×Dic43D411yescharacteristic
58C23×Dic43D411yescharacteristic
59C22×C4×D43—21no—
60C22×C4×D43—21no—
61C22×C4×D43—21no—
62C22×C43⋊C8—22no—
63C22×C43⋊C8—22no—
64C22×C43⋊C8—22no—
65C22×C43⋊C8C2312yes—
66#13063290—22no—
67#13063290—22no—
68#13063290C2×C414yes—
69#13063290C2×C414yes—
70#13063287—22no—
71#13063287—22no—
72#13063287C2×C414yes—
73#13063287C2×C414yes—
74(C22⋊D43)⋊C4—28no—
75(C22⋊D43)⋊C4—28no—
76#13063284—22no—
77#13063284—22no—
78#13063282—28no—
79C2×C4⋊Dic43C2312yes—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—22no—
83C2×Dic43⋊C4—22no—
84C2×C22⋊Dic43—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43C2×C414yes—
91C2×C4×Dic43D411yescharacteristic
92C2×C4×Dic43D411yescharacteristic
93C2×C4×Dic43D414yes—
94C2×C4×Dic43C2311yescharacteristic
95C2×C4×Dic43C2311yescharacteristic
96C42×D43—28no—
97Dic43⋊C8—24no—
98Dic43⋊C8—24no—
99#13063250—24no—
100#13063248—24no—
101#13063247—24no—
102#13063245—24no—
103C43⋊#95—22no—
104C43⋊#95—22no—
105C23×C4×C43C2311yescharacteristic
106C22×C8×C43—22no—
107C22×C8×C43—22no—
108C22×C8×C43—22no—
109C22×C8×C43C2312yes—
110#117×C43—22no—
111#117×C43—22no—
112#117×C43—22no—
113#117×C43C2×C414yes—
114C2×C4⋊C4×C43C2312yes—
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C22⋊C4×C43—21no—
118C2×C22⋊C4×C43—21no—
119C2×C42×C43—21no—
120C2×C42×C43—21no—
121C24×C43C2×D411yescharacteristic
122C23×D43—21no—
123C22×C4×C43—21no—
124C22×C4×C43—21no—
125C22×C4×C43—22no—
126C22×C4×C43C22×C412yes—
127C22×C4×C43C2×D412yes—
128C22×C4×C43C2×D412yes—
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×Q812yes—
131C22×C4×C43#4112yes—
132C22×C4×C43#4112yes—
133C22×C4×C43C2411yescharacteristic
134C2×C22⋊D43—24no—
135C2×C22⋊D43—24no—
136C2×C22⋊D43—24no—
137C22×Dic43—24no—
138C22×Dic43—21no—
139C22×Dic43—22no—
140C22×Dic43—42no—
141C22×Dic43C22⋊C414yes—
142C22×Dic43C22⋊C414yes—
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C22×Dic43C2×D411yescharacteristic
146C22×Dic43C2×D411yescharacteristic
147#10621898—28no—
148#10621898—28no—
149#10621898—28no—
150#10621898C4⋊C4116yescomplemented
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172C22×C412yes—
155C2×D172C2×D412yes—
156C2×D172C2×Q812yes—
157C2×C4×D43—24no—
158C2×C4×D43—24no—
159C2×C4×D43—24no—
160C2×C4×D43—24no—
161C2×C4×D43—24no—
162C2×C4×D43—44no—
163C2×C4×D43—44no—
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86C22×C412yes—
168C2×Dic86C2×D412yes—
169C2×Dic86C2×Q812yes—
170C2×C8×C43—22no—
171C2×C8×C43—22no—
172C2×C8×C43—22no—
173C2×C8×C43—44no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—22no—
177C4⋊C4×C43—22no—
178C4⋊C4×C43C22×C412yes—
179C4⋊C4×C43C22×C412yes—
180C22⋊C4×C43—24no—
181C22⋊C4×C43—42no—
182C42×C43—24no—
183C42×C43—42no—
184C22⋊Dic43—42no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—24no—
187C4⋊Dic43—22no—
188C4⋊Dic43—22no—
189Dic43⋊C4—24no—
190Dic43⋊C4—24no—
191Dic43⋊C4—44no—
192Dic43⋊C4—44no—
193C4×Dic43—24no—
194C4×Dic43—24no—
195C4×Dic43—42no—
196C4×Dic43C22⋊C414yes—
197C4×Dic43C22⋊C414yes—
198C4×Dic43C22×C412yes—
199C4×Dic43C22×C412yes—
200C4×Dic43C2×D412yes—
201C4×Dic43C2×D412yes—
202C2×C43⋊C8—22no—
203C2×C43⋊C8—22no—
204C2×C43⋊C8—22no—
205C2×C43⋊C8—44no—
206C23×C43—22no—
207C23×C43—22no—
208C23×C43C2×C22⋊C412yes—
209C23×C43#12112yes—
210C23×C43C22⋊Q812yes—
211C23×C43C22×D411yescharacteristic
212C22×D43—22no—
213C22×D43—22no—
214C22×D43—22no—
215C22×D43—42no—
216C22×D43—42no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—42no—
220C2×C4×C43—42no—
221C2×C4×C43—42no—
222C2×C4×C43—42no—
223C2×C4×C43—42no—
224C2×C4×C43C2×C22⋊C412yes—
225C2×C4×C43C2×C4⋊C414yes—
226C2×C4×C43#11714yes—
227C2×C4×C43#12114yes—
228C2×C4×C43#12112yes—
229C2×C4×C43C22⋊Q814yes—
230C2×C4×C43C22⋊Q812yes—
231C2×C4×C43C23×C411yescharacteristic
232C2×C4×C43C22×D411yescharacteristic
233C2×C4×C43C22×D411yescharacteristic
234C2×C4×C43C22×Q811yescharacteristic
235C2×C4×C43C2×#4111yescharacteristic
236C2×C4×C43C2×#4111yescharacteristic
237C22⋊D43—216no—
238C2×Dic43—22no—
239C2×Dic43—22no—
240C2×Dic43—22no—
241C2×Dic43—22no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—28no—
245C2×Dic43—44no—
246C2×Dic43—42no—
247C2×Dic43—42no—
248C2×Dic43—42no—
249C2×Dic43C2×C22⋊C412yes—
250C2×Dic43C2×C22⋊C412yes—
251C2×Dic43C2×C22⋊C412yes—
252C2×Dic43C2×C22⋊C412yes—
253C2×Dic43#13514yes—
254D172—22no—
255D172—22no—
256D172—22no—
257D172C2×C4⋊C414yes—
258C4×D43—216no—
259C4×D43—48no—
260C4×D43—48no—
261Dic86—22no—
262Dic86—22no—
263Dic86—22no—
264Dic86C2×C4⋊C414yes—
265C8×C43—42no—
266C43⋊C8—42no—
267#20645—431nomaximal, Sylow
268C22×C43—24no—
269C22×C43—22no—
270C22×C43—22no—
271C22×C43—24no—
272C22×C43#38014yes—
273C22×C43C22×C22⋊C411yescharacteristic
274C22×C43C2×#12111yescharacteristic
275C22×C43C2×C22⋊Q811yescharacteristic
276C22×C43C2×#13512yes—
277C22×C43#57512yes—
278C2×D43—24no—
279C2×D43—44no—
280C2×D43—42no—
281C2×D43—42no—
282C2×D43—44no—
283C4×C43—42no—
284C4×C43—42no—
285C4×C43—82no—
286C4×C43#38014yes—
287C4×C43C22×C4⋊C411yescharacteristic
288C4×C43C2×#11711yescharacteristic
289C4×C43C2×#12111yescharacteristic
290C4×C43C2×C22⋊Q811yescharacteristic
291Dic43—22no—
292Dic43—22no—
293Dic43—22no—
294Dic43—24no—
295Dic43—82no—
296Dic43#41614yescomplemented
297C2×C4×#41—431no—
298C22×C4⋊C8—431no—
299C2×#416—431no—
300C2×#416—431no—
301#2285—431no—
302#1805—432no—
303#1804—432no—
304#1803—432no—
305#1802—432no—
306#1735—431no—
307#1735D4311yescharacteristic, complemented
308C2×C43—24no—
309C2×C43C2×#38011yescharacteristic
310C2×C43C2×#41611yescharacteristic
311C2×C43#289611yescharacteristic
312C2×C43#303512yes—
313C2×C43#307612yes—
314D43—44no—
315C22×#41—431no—
316C23×C8—431no—
317C23×C8C2×D4311yescharacteristic
318C4×#41—438no—
319C4×#41—864no—
320C2×C4×Q8—431no—
321C2×C4×Q8—431no—
322C2×C4×D4—431no—
323C2×C4×D4—431no—
324C2×C4×D4—861no—
325C2×C4×D4—861no—
326C2×#117—431no—
327C2×#117—861no—
328C2×#117C2×D4311yescharacteristic
329C2×#117C2×D4311yescharacteristic
330C22×C42—431no—
331C22×C42—861no—
332C2×C4⋊C8—432no—
333C2×C4⋊C8—432no—
334C2×C4⋊C8—864no—
335#416—864no—
336#416—864no—
337C2×#103—432no—
338C2×#103—432no—
339C2×#102—432no—
340C2×#102—432no—
341#384—432no—
342#382—432no—
343#380—861no—
344#380—861no—
345C4×C4⋊C4—861no—
346C4×C4⋊C4—861no—
347#340—432no—
348#340—432no—
349#340C2×D4312yes—
350#340C2×D4312yes—
351C43#2064511yescharacteristic, complemented, Sylow
352C2×#41—434no—
353C2×#41—434no—
354C2×#41—434no—
355C2×#41—862no—
356C2×#41—862no—
357C2×#41—862no—
358C22×Q8—431no—
359C22×D4—431no—
360C22×D4—861no—
361C23×C4—431no—
362C23×C4—431no—
363C23×C4—861no—
364C23×C4—861no—
365C23×C4—861no—
366C23×C4C22×D4311yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—22no—
370C22×C8—432no—
371C22×C8—862no—
372C22×C8—862no—
373C22×C8—862no—
374C22×C8C22×D4312yes—
375C4×Q8—434no—
376C4×Q8—434no—
377C4×Q8—862no—
378C4×Q8—862no—
379C4×D4—434no—
380C4×D4—434no—
381C4×D4—862no—
382C4×D4—862no—
383C4×D4—868no—
384C4×D4—868no—
385#117—22no—
386#117—22no—
387#117—22no—
388#117—868no—
389#117—862no—
390#117—862no—
391#117C4×D4314yes—
392C2×C4⋊C4—432no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—861no—
395C2×C4⋊C4—862no—
396C2×C4⋊C4—862no—
397C2×C4⋊C4C22×D4312yes—
398C2×C4⋊C4C22×D4311yescharacteristic
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—861no—
403C2×C22⋊C4—861no—
404C2×C22⋊C4—861no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—434no—
408C2×C42—434no—
409C2×C42—431no—
410C2×C42—431no—
411C2×C42—431no—
412C2×C42—431no—
413C2×C42—868no—
414C2×C42—861no—
415C2×C42—861no—
416C2×C42—861no—
417C4⋊C8—864no—
418C4⋊C8—864no—
419#103—864no—
420#103—864no—
421#102—864no—
422#102—864no—
423#95—862no—
424#95—862no—
425C24—861no—
426C24D4×D4311yescharacteristic
427#41—4316no—
428#41—868no—
429#41—868no—
430#41—868no—
431C2×Q8—432no—
432C2×Q8—432no—
433C2×Q8—432no—
434C2×Q8—861no—
435C2×Q8—861no—
436C2×Q8—861no—
437C2×D4—432no—
438C2×D4—432no—
439C2×D4—432no—
440C2×D4—864no—
441C2×D4—864no—
442C2×D4—864no—
443C2×D4—861no—
444C2×D4—861no—
445C2×D4—861no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—431no—
450C22×C4—434no—
451C22×C4—431no—
452C22×C4—431no—
453C22×C4—431no—
454C22×C4—434no—
455C22×C4—864no—
456C22×C4—864no—
457C22×C4—864no—
458C22×C4—861no—
459C22×C4—864no—
460C22×C4—864no—
461C22×C4—864no—
462C22×C4—862no—
463C22×C4—1724no—
464C22×C4—1724no—
465C22×C4—1722no—
466C22×C4C2×C4×D4312yes—
467C22×C4#1062189812yes—
468C22×C4#1062189812yes—
469C22×C4D4×D4312yes—
470C22×C4D4×D4312yes—
471C22×C4D4×D4311yescharacteristic
472C22×C4Q8×D4312yes—
473C22×C4C23×D4311yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—862no—
479C2×C8—862no—
480C2×C8—862no—
481C2×C8—1724no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—864no—
487C4⋊C4—864no—
488C4⋊C4—862no—
489C4⋊C4—862no—
490C4⋊C4—1724no—
491C4⋊C4—1724no—
492C4⋊C4C2×C4×D4312yes—
493C4⋊C4C2×C4×D4312yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—864no—
497C22⋊C4—864no—
498C22⋊C4—1722no—
499C42—24no—
500C42—42no—
501C42—434no—
502C42—434no—
503C42—432no—
504C42—432no—
505C42—432no—
506C42—432no—
507C42—864no—
508C42—864no—
509C42—1722no—
510C23—22no—
511C23—22no—
512C23—862no—
513C23—862no—
514C23—862no—
515C23—1722no—
516C23—1722no—
517C23C22⋊C4×D4312yes—
518C23#1306329812yes—
519C23#1306330312yes—
520C23C2×D4×D4311yescharacteristic
521Q8—434no—
522Q8—862no—
523Q8—862no—
524Q8—862no—
525D4—434no—
526D4—8616no—
527D4—862no—
528D4—862no—
529D4—862no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—432no—
538C2×C4—434no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—432no—
542C2×C4—868no—
543C2×C4—862no—
544C2×C4—862no—
545C2×C4—862no—
546C2×C4—862no—
547C2×C4—862no—
548C2×C4—862no—
549C2×C4—8616no—
550C2×C4—1728no—
551C2×C4—1728no—
552C2×C4—1724no—
553C2×C4—1722no—
554C2×C4—1722no—
555C2×C4—1722no—
556C2×C4C22⋊C4×D4312yes—
557C2×C4C4⋊C4×D4314yes—
558C2×C4#1306328214yes—
559C2×C4#1306329814yes—
560C2×C4#1306329812yes—
561C2×C4#1306330314yes—
562C2×C4#1306330312yes—
563C2×C4C22×C4×D4311yescharacteristic
564C2×C4C2×D4×D4311yescharacteristic
565C2×C4C2×D4×D4311yescharacteristic
566C2×C4C2×Q8×D4311yescharacteristic
567C2×C4C2×#1062189811yescharacteristic
568C2×C4C2×#1062189811yescharacteristic
569C8—42no—
570C8—1722no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—864no—
576C22—1724no—
577C22—1722no—
578C22—1724no—
579C22—1722no—
580C22#1458275814yes—
581C22C2×C22⋊C4×D4311yescharacteristic
582C22C2×#1306329811yescharacteristic
583C22C2×#1306330311yescharacteristic
584C22#135×D4312yes—
585C22#1458369612yes—
586C4—42no—
587C4—42no—
588C4—82no—
589C4—434no—
590C4—862no—
591C4—862no—
592C4—862no—
593C4—864no—
594C4—3442no—
595C4#1458275814yes—
596C4C2×C4⋊C4×D4311yescharacteristic
597C4C2×#1306328211yescharacteristic
598C4C2×#1306329811yescharacteristic
599C4C2×#1306330311yescharacteristic
600C2—24no—
601C2—1724no—
602C2C2×#1458275811yescharacteristic
603C2#416×D4311yescharacteristic
604C2#1914311011yescharacteristic
605C2#1914404812yes—
606C2#1914438112yes—
subgroupG/H
centerC22×C4D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1735C2
socleC23×C43C22×D4
prime radicalC22×#10621898C22
cosocleC22×C4×C43C24
hypercenter#1735D43
hypocenterC43#20645
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C22×Dic43⋊C8C2semidirect
5. #19136998C2semidirect
6. #19134609C2semidirect
7. #19134604C2semidirect
8. #19134599C2semidirect
9. #19134594C2semidirect
10. #19134311C2semidirect
11. C43×#1735C2semidirect
15. #14583367C4semidirect
150. #10621898C4⋊C4semidirect
296. Dic43#416semidirect
307. #1735D43semidirect
351. C43#20645semidirect

Maximal subgroups

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

subgroupindex
C2×#145833672
C22×Dic43⋊C82
C2×#145828962
C2×#145828932
#191369982
#191346092
#191346042
#191345992
#191345942
#191343112
C43×#17352
#2064543