Finite Group Databasebeta

← #30606288

Subgroups — #30606288

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840749C211yesmaximal, characteristic
2#21462403C211yesmaximal, characteristic
3#21460909C214yesmaximal, complemented
4C2×#16839743C211yesmaximal, characteristic
5C22×#11758217—21no—
6C22×#11758217C411yescharacteristic
7C22×#11758217C411yescharacteristic
8#16840749—21no—
9#16840749—21no—
10#16840749C412yes—
11#16840749C2212yes—
12C2×#14519838—21no—
13C2×#14519837C412yes—
14#16840735—21no—
15#16840735—21no—
16#16840735C412yes—
17#16840735C2212yes—
18#16840734—21no—
19#16840732—21no—
20C22×#11758224C2211yescharacteristic
21C2×M16×D61—21no—
22#16839743—21no—
23#16839743—21no—
24#16839743C412yes—
25#16839743C2212yes—
26C2×#14519695—21no—
27C2×#14519694C411yescharacteristic
28C2×#100×C61C411yescharacteristic
29C2×#11758217—21no—
30C2×#11758217—21no—
31C2×#11758217—21no—
32C2×#11758217—22no—
33C2×#11758217—22no—
34C2×#11758217—22no—
35C2×#11758217—41no—
36C2×#11758217—41no—
37C2×#11758217—41no—
38C2×#11758217—41no—
39C2×#11758217—41no—
40C2×#11758217—41no—
41C2×#11758217—41no—
42C2×#11758217C2×C412yes—
43C22×#92420—21no—
44C22×#92420—41no—
45C22×#92420C2×C412yes—
46#14519838—22no—
47#14519838—22no—
48#14519837—22no—
49#14519837C2×C414yes—
50#14519834—22no—
51#14519834—22no—
52C22⋊#92420—22no—
53C22⋊#92420—22no—
54C2×#11758224—41no—
55C2×#11758224—41no—
56C2×#11758224—41no—
57C2×#11758224—41no—
58C2×#11758224C2×C411yescharacteristic
59C2×#11758224C2×C411yescharacteristic
60C2×#11758224C2×C411yescharacteristic
61C2×#11758224D411yescharacteristic
62C2×#11758224D411yescharacteristic
63C2×#11758224D411yescharacteristic
64C2×#11758224Q811yescharacteristic
65C2×#11758224C2311yescharacteristic
66C22×Dic122D411yescharacteristic
67C22×C22⋊D61C2×C411yescharacteristic
68C22×C22⋊D61Q811yescharacteristic
69C23×Dic61C2×C411yescharacteristic
70C23×Dic61D411yescharacteristic
71C22×C4×D61D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758189—41no—
74M16×D61—22no—
75M16×D61—22no—
76M16×D61—41no—
77M16×D61—41no—
78C2×C8×D61—41no—
79#14519695—22no—
80#14519695—22no—
81#14519694—21no—
82#14519694C2×C412yes—
83C22×D4×C61C2×C411yescharacteristic
84C2×M16×C61—21no—
85#100×C61—21no—
86#100×C61C2×C412yes—
87C24×C61C4211yescharacteristic
88C24×C61C4⋊C411yescharacteristic
89C23×D61C4⋊C411yescharacteristic
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61C4211yescharacteristic
93C2×D4×C61C22⋊C411yescharacteristic
94C2×D4×C61C4⋊C411yescharacteristic
95C2×D4×C61C22×C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61C4211yescharacteristic
102C2×C22⋊D61C22⋊C411yescharacteristic
103C2×C22⋊D61C4⋊C411yescharacteristic
104C2×C22⋊D61C4⋊C411yescharacteristic
105C2×C22⋊D61C4⋊C411yescharacteristic
106C2×C22⋊D61C4⋊C411yescharacteristic
107C2×C22⋊D61C22×C411yescharacteristic
108C2×C22⋊D61C2×Q811yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61C4211yescharacteristic
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C22⋊C411yescharacteristic
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C4⋊C411yescharacteristic
124C22×Dic61C4⋊C411yescharacteristic
125C22×Dic61C4⋊C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128#11758224—41no—
129#11758224—41no—
130#11758224—41no—
131#11758224—41no—
132#11758224—41no—
133#11758224—41no—
134#11758224—41no—
135#11758224—41no—
136#11758224—41no—
137#11758224—41no—
138#11758224—41no—
139#11758224—41no—
140#11758224—41no—
141#11758224—41no—
142#11758224—41no—
143#11758224—41no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C4⋊C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×Dic122—41no—
153C2×Dic122—41no—
154C2×Dic122C22⋊C411yescharacteristic
155C2×Dic122C22⋊C411yescharacteristic
156C2×Dic122C4⋊C411yescharacteristic
157C2×Dic122C2×D411yescharacteristic
158#11758217—24no—
159#11758217—42no—
160#11758217—42no—
161#11758217—42no—
162#11758217—42no—
163#11758217—41no—
164#11758217—41no—
165#11758217—41no—
166#11758217—41no—
167#11758217—42no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—42no—
171C2×#92420—42no—
172C2×#92420—42no—
173C2×#92420—42no—
174C2×#92420—42no—
175M16×C61—22no—
176M16×C61—41no—
177C2×C8×C61—41no—
178#11758194—22no—
179#11758194—41no—
180C2×#92418—41no—
181#11758189—42no—
182#11758189—42no—
183C8×D61—42no—
184C8×D61—42no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—81no—
190C23×C61—81no—
191C23×C61#9511yescharacteristic
192C23×C61#9511yescharacteristic
193C23×C61#9511yescharacteristic
194C23×C61C2×C4211yescharacteristic
195C23×C61C2×C4⋊C411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—81no—
199C22×D61#9511yescharacteristic
200C22×D61C2×C4⋊C411yescharacteristic
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—41no—
204D4×C61—41no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61#9511yescharacteristic
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C22⋊D61—41no—
210C22⋊D61—41no—
211C22⋊D61—41no—
212C22⋊D61—41no—
213C22⋊D61—41no—
214C22⋊D61—41no—
215C22⋊D61—41no—
216C22⋊D61—41no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61#9511yescharacteristic
238C2×Dic61#9511yescharacteristic
239C2×Dic61#9511yescharacteristic
240C2×Dic61#9911yescharacteristic
241C2×Dic61#9911yescharacteristic
242C2×Dic61#9911yescharacteristic
243C2×Dic61#9911yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C4⋊C411yescharacteristic
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259#92420—44no—
260#92420—42no—
261#92420—42no—
262C8×C61—42no—
263#92418—42no—
264#12579—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61#34211yescharacteristic
279C22×C61#34211yescharacteristic
280C22×C61C2×#9511yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—42no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C4×C61—21no—
288C4×C61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—21no—
293Dic61#34211yescharacteristic
294Dic61#34211yescharacteristic
295Dic61C2×#9911yescharacteristic
296Dic61C2×#9911yescharacteristic
297C2×#413—611no—
298C2×#413—611no—
299#2879—611no—
300#1392—614no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—41no—
305C2×C61—41no—
306C2×C61—41no—
307C2×C61—41no—
308C2×C61C2×#34211yescharacteristic
309D61—41no—
310D61—41no—
311C22×#41—611no—
312C22×M16—611no—
313C22×M16—611no—
314C22×M16—1221no—
315C22×M16—1221no—
316#413—612no—
317#413—612no—
318#413—612no—
319#413—612no—
320#413—1221no—
321#413—1221no—
322#413—1221no—
323#413—1221no—
324C2×#101—1221no—
325C2×#101—1221no—
326C2×#100—612no—
327C2×#100—611no—
328C2×#100#692311yescharacteristic
329#408—612no—
330#408—612no—
331#408—1221no—
332#408—1221no—
333#407—1221no—
334#407—1221no—
335C61#1257911yescharacteristic, complemented, Sylow
336C2×#41—611no—
337C2×#41—611no—
338C2×#41—611no—
339C2×#41—611no—
340C2×#41—611no—
341C2×#41—611no—
342C2×#41—611no—
343C2×#41—611no—
344C2×#41—2441no—
345C2×#41—2441no—
346C2×#41—2441no—
347C2×#41—2441no—
348C22×Q8—611no—
349C22×D4—611no—
350C22×D4—611no—
351C22×D4C2×#692311yescharacteristic
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C2×M16—21no—
356C2×M16—612no—
357C2×M16—1221no—
358C2×M16—1221no—
359C2×M16—1221no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16—1222no—
364C2×M16—1222no—
365C2×M16—1221no—
366C2×M16—2441no—
367C2×M16—2441no—
368C2×M16—2441no—
369C2×M16—2441no—
370C2×M16—2441no—
371C2×M16—2441no—
372C2×M16—2441no—
373C2×M16—2441no—
374C2×M16—2441no—
375C2×M16—2441no—
376C22×C8—612no—
377C22×C8—1221no—
378C22×C8—2441no—
379C22×C8—2441no—
380#101—1222no—
381#101—1222no—
382#101—1222no—
383#101—1222no—
384#100—21no—
385#100—614no—
386#100—612no—
387#100—1222no—
388#100—1221no—
389#100C2×#692312yes—
390C22⋊C8—1222no—
391C22⋊C8—1222no—
392C22⋊C8—1222no—
393C22⋊C8—1222no—
394C24—611no—
395C24C4×#692311yescharacteristic
396C24C4⋊#692311yescharacteristic
397#41—2441no—
398#41—2441no—
399#41—2441no—
400#41—2441no—
401#41—2441no—
402#41—2441no—
403#41—2441no—
404#41—2441no—
405#41—2441no—
406#41—2441no—
407#41—2441no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413C2×Q8—611no—
414C2×Q8—611no—
415C2×Q8—611no—
416C2×Q8—611no—
417C2×Q8—2441no—
418C2×Q8—2441no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—611no—
422C2×D4—611no—
423C2×D4—611no—
424C2×D4—611no—
425C2×D4—611no—
426C2×D4—611no—
427C2×D4—611no—
428C2×D4—611no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4C4×#692311yescharacteristic
434C2×D4C4⋊#692311yescharacteristic
435C2×D4C22⋊#692311yescharacteristic
436C2×D4C22×#692311yescharacteristic
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4C22⋊#692311yescharacteristic
465M16—22no—
466M16—41no—
467M16—1224no—
468M16—1222no—
469M16—2442no—
470M16—2442no—
471M16—2441no—
472M16—2441no—
473M16—2441no—
474M16—2441no—
475M16—2441no—
476M16—2442no—
477M16—2442no—
478M16—2442no—
479M16—2442no—
480M16—2442no—
481C2×C8—41no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—2442no—
485C2×C8—2442no—
486C2×C8—2441no—
487C2×C8—2442no—
488C2×C8—2442no—
489C2×C8—2442no—
490C2×C8—2442no—
491C2×C8—2442no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—81no—
497C23—81no—
498C23—611no—
499C23—611no—
500C23—1221no—
501C23—1221no—
502C23—4881no—
503C23(C2×#6923)⋊C411yescharacteristic
504C23(C2×#6923)⋊C411yescharacteristic
505C23(C2×#6923)⋊C411yescharacteristic
506C23C2×C4×#692311yescharacteristic
507C23C2×C4⋊#692311yescharacteristic
508Q8—2441no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520D4—2441no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4(C2×#6923)⋊C411yescharacteristic
565C2×C4C2×C22⋊#692311yescharacteristic
566C8—42no—
567C8—2442no—
568C8—2442no—
569C8—2442no—
570C8—2444no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—1221no—
585C22—1221no—
586C22—4881no—
587C22—2441no—
588C22—2441no—
589C22—2442no—
590C22#1684063911yescharacteristic
591C22#1684063911yescharacteristic
592C22C2×(C2×#6923)⋊C411yescharacteristic
593C4—21no—
594C4—21no—
595C4—611no—
596C4—611no—
597C4—611no—
598C4—611no—
599C4—1221no—
600C4—1221no—
601C4—1221no—
602C4—1221no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—41no—
607C2—41no—
608C2—41no—
609C2—41no—
610C2—2441no—
611C2—2441no—
612C2C2×#1684063911yescharacteristic
subgroupG/H
centerC2C2×#16840639
commutatorC23×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC2×#100×C61C4
socleC2×C61C2×#342
prime radicalC22×#11758224C22
cosocleC2×#11758224C23
hypercenterC2×#100#6923
hypocenterC61#12579
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
3. #21460909C2semidirect
335. C61#12579semidirect

Maximal subgroups

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

subgroupindex
C2×#168407492
#214624032
#214609092
C2×#168397432
#1257961