Finite Group Databasebeta

← #29650105

Subgroups — #29650105

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454930C211yesmaximal, characteristic, complemented
2#21454895C211yesmaximal, characteristic, complemented
3#21444698C211yesmaximal, characteristic, complemented
4#21444684C211yesmaximal, characteristic, complemented
5#21444663C211yesmaximal, characteristic, complemented
6#21444649C211yesmaximal, characteristic, complemented
7#21444643C211yesmaximal, characteristic, complemented
8SD16⋊#92418C211yesmaximal, characteristic, complemented
9SD16×#92418C211yesmaximal, characteristic, complemented
10D8×#92418C211yesmaximal, characteristic, complemented
11#21444604C211yesmaximal, characteristic, complemented
12#21444253C211yesmaximal, characteristic, complemented
13#21444218C211yesmaximal, characteristic, complemented
14#21444119C211yesmaximal, characteristic, complemented
15C61×#3106C211yesmaximal, characteristic
16Q8×#92418C2211yescharacteristic
17Q8×#92418C2211yescharacteristic
18#16839840C2211yescharacteristic
19#16839840C2211yescharacteristic
20#16839834—21no—
21#16839834—21no—
22D4×#92418—21no—
23D4×#92418—21no—
24D4×#92418C2211yescharacteristic
25D4×#92418C2211yescharacteristic
26D4×#92418C2211yescharacteristic
27D4×#92418C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839815—21no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32C2×C4⋊#92418—21no—
33C2×C4⋊#92418C2211yescharacteristic
34#16839729—21no—
35#16839729—21no—
36#16839515C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839513C2211yescharacteristic
39#16839513C2211yescharacteristic
40Q8⋊#92418C2211yescharacteristic
41Q8⋊#92418C2211yescharacteristic
42D4⋊#92418C2211yescharacteristic
43D4⋊#92418C2211yescharacteristic
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46D4⋊#92418C2211yescharacteristic
47D4⋊#92418C2211yescharacteristic
48#16839484C2211yescharacteristic
49#16839484C2211yescharacteristic
50#16839483C2211yescharacteristic
51C8×#92418C2211yescharacteristic
52C61×#532C2211yescharacteristic
53C4×#41×C61C2211yescharacteristic
54C61×#482C411yescharacteristic
55C61×#482C411yescharacteristic
56C61×#477C411yescharacteristic
57C61×#476C411yescharacteristic
58C61×#460C411yescharacteristic
59C61×#459C411yescharacteristic
60C61×#449C411yescharacteristic
61C61×#449C411yescharacteristic
62C4×SD16×C61C2211yescharacteristic
63C4×SD16×C61C2211yescharacteristic
64C4×D8×C61C2211yescharacteristic
65C4×D8×C61C2211yescharacteristic
66C61×#431C2211yescharacteristic
67C2×#11758194—21no—
68C2×#11758194—21no—
69C22×#92418—21no—
70C22×#92418—21no—
71C22×#92418—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418D411yescharacteristic
79C4⋊#92418D411yescharacteristic
80C4⋊#92418C2311yescharacteristic
81C4⋊#92418C2311yescharacteristic
82C4⋊#92418C2311yescharacteristic
83C4⋊#92418C2311yescharacteristic
84C22⋊#92418—21no—
85C22⋊#92418—21no—
86C22⋊#92418—21no—
87C22⋊#92418—21no—
88C22⋊#92418—21no—
89C22⋊#92418—21no—
90C22⋊#92418—21no—
91C22⋊#92418—21no—
92C22⋊#92418D411yescharacteristic
93C22⋊#92418D411yescharacteristic
94#14519688—21no—
95C4×#92418—21no—
96C4×#92418—21no—
97C4×#92418C2311yescharacteristic
98C4×#92418C2311yescharacteristic
99C4×#92418C2311yescharacteristic
100C4×#92418C2311yescharacteristic
101C2×#41×C61—21no—
102C2×#41×C61C2×C411yescharacteristic
103C2×SD16×C61C2×C411yescharacteristic
104C2×SD16×C61C2×C411yescharacteristic
105C2×D8×C61C2×C411yescharacteristic
106C2×D8×C61C2×C411yescharacteristic
107C4⋊Q8×C61C2×C411yescharacteristic
108#127×C61C2×C411yescharacteristic
109#124×C61—21no—
110#121×C61—21no—
111#121×C61C2×C411yescharacteristic
112#121×C61C2×C411yescharacteristic
113C4×Q8×C61C2×C411yescharacteristic
114C4×Q8×C61C2311yescharacteristic
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61C2×C411yescharacteristic
119C4×D4×C61C2311yescharacteristic
120C4×D4×C61C2311yescharacteristic
121C4×D4×C61C2311yescharacteristic
122#117×C61—21no—
123#117×C61C2×C411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61C2×C411yescharacteristic
127#107×C61C2×C411yescharacteristic
128#106×C61C2×C411yescharacteristic
129#106×C61C2×C411yescharacteristic
130C4⋊C8×C61C2×C411yescharacteristic
131C4⋊C8×C61C2×C411yescharacteristic
132#103×C61C2×C411yescharacteristic
133#103×C61C2×C411yescharacteristic
134#102×C61C2×C411yescharacteristic
135#102×C61C2×C411yescharacteristic
136#102×C61C2×C411yescharacteristic
137#102×C61C2×C411yescharacteristic
138#102×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61C2×C411yescharacteristic
141C22⋊C8×C61C2×C411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—22no—
145#41×C61—22no—
146#41×C61—81no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C22×C411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61C22×C411yescharacteristic
156C2×D4×C61C22×C411yescharacteristic
157C2×D4×C61C22×C411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61C22×C411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61#4111yescharacteristic
166SD16×C61—22no—
167SD16×C61—22no—
168D8×C61—22no—
169D8×C61—22no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C22×C411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C411yescharacteristic
178C4⋊C4×C61C22×C411yescharacteristic
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61#4111yescharacteristic
194C42×C61C2411yescharacteristic
195#11758194—81no—
196#11758194—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—81no—
204C2×#92418—81no—
205C2×#92418—81no—
206C2×#92418—81no—
207C2×#92418—81no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61C4×D411yescharacteristic
216Q8×C61—21no—
217Q8×C61—81no—
218Q8×C61#13112yes—
219D4×C61—24no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—81no—
226D4×C61—81no—
227D4×C61—81no—
228D4×C61—81no—
229D4×C61#13112yes—
230C2×C4×C61—24no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—81no—
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C4×D411yescharacteristic
250C2×C4×C61C4×D411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—21no—
258#92418—21no—
259#92418—21no—
260#92418—21no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#17920—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—22no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61C2×C4×D411yescharacteristic
282C4×C61—21no—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61C2×#13111yescharacteristic
292C4×C61C2×#13111yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295#3106D6111yescharacteristic
296#2988—611no—
297#2981—611no—
298#1588—611no—
299#1586—611no—
300#1583—611no—
301#1581—611no—
302#1580—611no—
303SD16⋊C8—611no—
304C8×SD16—611no—
305C8×D8—611no—
306#1573—611no—
307#1488—611no—
308#1480—611no—
309#1454—611no—
310C2×C61—41no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61—81no—
314C2×C61#292911yescharacteristic
315C2×C61#294611yescharacteristic
316C2×C61#295711yescharacteristic
317#532C2×D6111yescharacteristic
318C4×#41C2×D6111yescharacteristic
319#482Dic6111yescharacteristic
320#482Dic6111yescharacteristic
321#477Dic6111yescharacteristic
322#476Dic6111yescharacteristic
323#460Dic6111yescharacteristic
324#459Dic6111yescharacteristic
325#449Dic6111yescharacteristic
326#449Dic6111yescharacteristic
327C8×Q8—611no—
328C8×Q8—611no—
329C4×SD16C2×D6111yescharacteristic
330C4×SD16C2×D6111yescharacteristic
331C4×D8C2×D6111yescharacteristic
332C4×D8C2×D6111yescharacteristic
333#436—611no—
334#436—611no—
335#435—1221no—
336#435—1221no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—611no—
340C8×D4—611no—
341C8×D4—1221no—
342C8×D4—1221no—
343#432—611no—
344#431—611no—
345#431—611no—
346#431—1221no—
347#431C2×D6111yescharacteristic
348C2×C4⋊C8—611no—
349C2×C4⋊C8—1221no—
350#408—1221no—
351#408—1221no—
352#335—611no—
353#335—611no—
354#334—611no—
355#334—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365#322—611no—
366#322—611no—
367C82—611no—
368C61#1792011yescharacteristic, complemented, Sylow
369C2×#41—21no—
370C2×#41C2×Dic6111yescharacteristic
371C2×SD16C2×Dic6111yescharacteristic
372C2×SD16C2×Dic6111yescharacteristic
373C2×D8C2×Dic6111yescharacteristic
374C2×D8C2×Dic6111yescharacteristic
375C2×M16—1221no—
376C2×M16—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C4⋊Q8C2×Dic6111yescharacteristic
383#127C2×Dic6111yescharacteristic
384#124—21no—
385#121—21no—
386#121C2×Dic6111yescharacteristic
387#121C2×Dic6111yescharacteristic
388C4×Q8C2×Dic6111yescharacteristic
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4C2×Dic6111yescharacteristic
394C4×D4C22×D6111yescharacteristic
395C4×D4C22×D6111yescharacteristic
396C4×D4C22×D6111yescharacteristic
397#117—21no—
398#117C2×Dic6111yescharacteristic
399C2×C42—21no—
400C2×C42C22×D6111yescharacteristic
401#107C2×Dic6111yescharacteristic
402#107C2×Dic6111yescharacteristic
403#106C2×Dic6111yescharacteristic
404#106C2×Dic6111yescharacteristic
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8C2×Dic6111yescharacteristic
416C4⋊C8C2×Dic6111yescharacteristic
417#103C2×Dic6111yescharacteristic
418#103C2×Dic6111yescharacteristic
419#102C2×Dic6111yescharacteristic
420#102C2×Dic6111yescharacteristic
421#102C2×Dic6111yescharacteristic
422#102C2×Dic6111yescharacteristic
423#102C2×Dic6111yescharacteristic
424#102C2×Dic6111yescharacteristic
425C22⋊C8—611no—
426C22⋊C8—611no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8C2×Dic6111yescharacteristic
436C22⋊C8C2×Dic6111yescharacteristic
437#97—1221no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446#41—22no—
447#41—22no—
448#41—81no—
449#41—81no—
450C2×Q8—21no—
451C2×Q8C22×Dic6111yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4C22×Dic6111yescharacteristic
458C2×D4C22×Dic6111yescharacteristic
459C2×D4C22×Dic6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4D4×D6111yescharacteristic
466C22×C4#1175822411yescharacteristic
467C22×C4C22×Dic6111yescharacteristic
468SD16—22no—
469SD16—22no—
470D8—22no—
471D8—22no—
472M16—4881no—
473M16—4881no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—4881no—
485C2×C8—4881no—
486C2×C8—4881no—
487C2×C8—4881no—
488C2×C8—4881no—
489C2×C8C22×Dic6111yescharacteristic
490C2×C8C22×Dic6111yescharacteristic
491C2×C8C22×Dic6111yescharacteristic
492C2×C8C22×Dic6111yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic6111yescharacteristic
497C4⋊C4C22×Dic6111yescharacteristic
498C4⋊C4C22×Dic6111yescharacteristic
499C4⋊C4C22×Dic6111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42D4×D6111yescharacteristic
511C42#1175822411yescharacteristic
512C42C22×Dic6111yescharacteristic
513C42C23×D6111yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic6111yescharacteristic
518Q8—21no—
519Q8—81no—
520Q8#1451978412yes—
521D4—24no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4#1451978412yes—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4D4×Dic6111yescharacteristic
551C2×C4D4×Dic6111yescharacteristic
552C2×C4D4×Dic6111yescharacteristic
553C2×C4C23×Dic6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—1221no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×D4×Dic6111yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4C2×#1451978411yescharacteristic
593C4C2×#1451978411yescharacteristic
594C4#1684056611yescharacteristic
595C4#1684057211yescharacteristic
596C2—41no—
597C2—42no—
598C2—81no—
599C2—81no—
600C2#2145451411yescharacteristic
601C2#2145465211yescharacteristic
602C2#2145473611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
Fitting, prime radicalC61×#3106C2
socleC22×C61C2×C4×D4
cosocleC42×C61C24
hypercenter#3106D61
hypocenterC61#17920
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #21454930C2semidirect
2. #21454895C2semidirect
3. #21444698C2semidirect
4. #21444684C2semidirect
5. #21444663C2semidirect
6. #21444649C2semidirect
7. #21444643C2semidirect
8. SD16⋊#92418C2semidirect
9. SD16×#92418C2semidirect
10. D8×#92418C2semidirect
11. #21444604C2semidirect
12. #21444253C2semidirect
13. #21444218C2semidirect
14. #21444119C2semidirect
368. C61#17920semidirect

Maximal subgroups

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

subgroupindex
#214549302
#214548952
#214446982
#214446842
#214446632
#214446492
#214446432
SD16⋊#924182
SD16×#924182
D8×#924182
#214446042
#214442532
#214442182
#214441192
C61×#31062
#1792061