Finite Group Databasebeta

← #29585457

Subgroups — #29585457

Order 15616.

1148 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840528C211yesmaximal, characteristic
2C2×#16839744C211yesmaximal, characteristic
3C2×#16839744C211yesmaximal, characteristic
4#21445546C211yesmaximal, characteristic, complemented
5#21445545C211yesmaximal, characteristic, complemented
6#21445545C211yesmaximal, characteristic, complemented
7C61×#1738C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9#16840528—22no—
10#16840528—22no—
11#16840528—22no—
12#16840528—22no—
13C2×#14519785—22no—
14C2×#14519783—22no—
15C22×#11758194C2211yescharacteristic
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×M16×D61C414yes—
19#16839744—22no—
20#16839744—22no—
21#16839744C414yes—
22#16839744C414yes—
23C2×#14519695—21no—
24C2×#14519695—21no—
25C2×#14519693—21no—
26C2×#14519693—21no—
27#16839527—22no—
28#16839526—22no—
29#16839526—22no—
30C22×M16×C61C2211yescharacteristic
31C22×M16×C61C2211yescharacteristic
32C22×M16×C61C2211yescharacteristic
33C61×#341—22no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222C2×C412yes—
44C2×#11758222D412yes—
45C2×#11758222D412yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61Q812yes—
49C22×C4×D61C2×C412yes—
50#14519785—28no—
51#14519785—28no—
52#14519783—28no—
53#14519783—28no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758194—21no—
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194—21no—
66C2×#11758194—21no—
67C2×#11758194C2×C412yes—
68C2×#11758194C2×C412yes—
69C2×#11758194C2×C412yes—
70C2×#11758194D412yes—
71C2×#11758194D412yes—
72C2×#11758189—24no—
73M16×D61—28no—
74M16×D61—28no—
75C22×#92418—21no—
76C22×#92418—21no—
77C22×#92418—21no—
78C2×C8×D61—24no—
79#14519695—24no—
80#14519695—24no—
81#14519693—24no—
82#14519693—24no—
83C23×C4×C61C2311yescharacteristic
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61—21no—
97C2×M16×C61C2×C412yes—
98C2×M16×C61C2×C412yes—
99C2×M16×C61C2×C412yes—
100C2×M16×C61D412yes—
101C2×M16×C61D412yes—
102C22×C8×C61—21no—
103C22×C8×C61—21no—
104C22×C8×C61—21no—
105C24×C61C2×Q811yescharacteristic
106C23×D61C4⋊C412yes—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61C4⋊C414yes—
126C2×C22⋊D61C4⋊C414yes—
127C22×Dic61C4⋊C412yes—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244C22⋊C412yes—
149C2×D244C22⋊C412yes—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61C4214yes—
155C2×C4×D61C22⋊C414yes—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22⋊C412yes—
161C2×Dic122C22⋊C412yes—
162M16×C61—22no—
163M16×C61—22no—
164M16×C61—22no—
165M16×C61—22no—
166M16×C61—22no—
167M16×C61—22no—
168M16×C61—22no—
169M16×C61—41no—
170M16×C61—41no—
171M16×C61—41no—
172M16×C61—41no—
173M16×C61C22⋊C414yes—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—41no—
180C2×C8×C61—41no—
181C2×C8×C61—41no—
182C2×C8×C61—41no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194—41no—
191#11758194—41no—
192#11758194—41no—
193#11758194—41no—
194#11758194C22⋊C414yes—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—41no—
201C2×#92418—41no—
202C2×#92418—41no—
203C2×#92418—41no—
204#11758189—216no—
205C8×D61—216no—
206C23×C61—81no—
207C23×C61C2×C4⋊C411yescharacteristic
208C23×C61C2×C4⋊C411yescharacteristic
209C23×C61C2×C4⋊C411yescharacteristic
210C23×C61C22⋊Q811yescharacteristic
211C23×C61C22⋊Q811yescharacteristic
212C23×C61C22⋊Q811yescharacteristic
213C23×C61C22⋊Q811yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—82no—
217C22×D61#9514yes—
218C2×C4×C61C2×C4211yescharacteristic
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61#11711yescharacteristic
225C2×C4×C61#11711yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61#12311yescharacteristic
243C2×C4×C61#12311yescharacteristic
244C2×C4×C61#12311yescharacteristic
245C2×C4×C61#12311yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—24no—
248C22⋊D61—24no—
249C22⋊D61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61#9514yes—
253D244—22no—
254D244—22no—
255D244—22no—
256D244—22no—
257C4×D61—24no—
258C4×D61—24no—
259C4×D61—24no—
260C4×D61—24no—
261Dic122—22no—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265C8×C61—24no—
266C8×C61—42no—
267C8×C61—42no—
268#92418—24no—
269#92418—42no—
270#92418—42no—
271#10767—611nomaximal, Sylow
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×#9511yescharacteristic
280C22×C61#38011yescharacteristic
281C22×C61#38011yescharacteristic
282C22×C61#38511yescharacteristic
283C22×C61#38511yescharacteristic
284C22×C61#38511yescharacteristic
285C22×C61#38511yescharacteristic
286C2×D61—24no—
287C2×D61—44no—
288C2×D61—82no—
289C2×D61—82no—
290C4×C61C4×C22⋊C411yescharacteristic
291C4×C61C4×C22⋊C411yescharacteristic
292C4×C61#37911yescharacteristic
293C4×C61#38111yescharacteristic
294C4×C61#38611yescharacteristic
295C4×C61#38611yescharacteristic
296C4×C61#38611yescharacteristic
297C4×C61#38611yescharacteristic
298Dic61—24no—
299C2×#568—611no—
300C2×#413—611no—
301C2×#413—611no—
302#1738—611no—
303#1738—611no—
304#1738—611no—
305#1738D6111yescharacteristic, complemented
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61C22⋊#9511yescharacteristic
314D61—44no—
315C22×#41—611no—
316#568—1222no—
317#568—1222no—
318#568—1222no—
319#568—1222no—
320C2×#131—1222no—
321C2×#131—1222no—
322C22×M16—611no—
323C22×M16—611no—
324C22×M16—611no—
325C22×M16—614no—
326C22×M16C2×D6111yescharacteristic
327C22×M16C2×D6111yescharacteristic
328C22×M16C2×D6111yescharacteristic
329#413—614no—
330#413—614no—
331#413—1222no—
332#413—1222no—
333C2×#101—1221no—
334C2×#101—1221no—
335C2×#100—1221no—
336C2×#100—1221no—
337#341—22no—
338#341—1222no—
339#341—1222no—
340#341—1222no—
341C61#1076711yescharacteristic, complemented, Sylow
342C2×#41—612no—
343C2×#41—612no—
344C2×#41—612no—
345C2×#41—612no—
346C2×#41—1221no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41—1221no—
351C2×#41—1221no—
352C2×#41—1221no—
353C2×#41—1221no—
354C22×Q8—611no—
355C22×D4—612no—
356C22×D4—611no—
357C23×C4—612no—
358C23×C4C22×D6111yescharacteristic
359#131—1228no—
360#131—1228no—
361#131—1228no—
362#131—1228no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—612no—
377C2×M16—612no—
378C2×M16—612no—
379C2×M16—612no—
380C2×M16—612no—
381C2×M16—1224no—
382C2×M16—1228no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1228no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C2×M16—1221no—
394C2×M16—1221no—
395C2×M16—1221no—
396C2×M16—1221no—
397C2×M16C4×D6112yes—
398C2×M16C4×D6112yes—
399C2×M16C4×D6112yes—
400C2×M16D24412yes—
401C2×M16C22⋊D6112yes—
402C22×C8—21no—
403C22×C8—21no—
404C22×C8—21no—
405C22×C8—1224no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—1221no—
409#101—1224no—
410#101—1224no—
411#100—1224no—
412#100—1224no—
413C24—612no—
414C24Q8×D6111yescharacteristic
415#41—1222no—
416#41—1222no—
417#41—1222no—
418#41—1222no—
419#41—1222no—
420#41—1222no—
421#41—1222no—
422#41—1222no—
423#41—1222no—
424#41—1222no—
425#41—1222no—
426#41—1222no—
427#41—1222no—
428#41—1222no—
429#41—1222no—
430#41—1222no—
431C2×Q8—612no—
432C2×Q8—612no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—614no—
438C2×D4—614no—
439C2×D4—612no—
440C2×D4—612no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—614no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4C2×C4×D6111yescharacteristic
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4C2×C4×D6111yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4#1175822211yescharacteristic
461C22×C4#1175822211yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4#1175822411yescharacteristic
468C22×C4#1175822611yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16—41no—
481M16—614no—
482M16—12216no—
483M16—1222no—
484M16—1222no—
485M16—1222no—
486M16—1222no—
487M16—1222no—
488M16—1222no—
489M16—1222no—
490M16—2441no—
491M16—2441no—
492M16—2441no—
493M16—2441no—
494M16#1175819814yes—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—12216no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2441no—
511C2×C8—2441no—
512C2×C8—2441no—
513C2×C8—2441no—
514C23—81no—
515C23—614no—
516C23—1222no—
517C23—1222no—
518C23—4882no—
519C23C4⋊C4×D6111yescharacteristic
520C23C4⋊C4×D6111yescharacteristic
521C23C4⋊C4×D6111yescharacteristic
522C23#1451975311yescharacteristic
523C23#1451975511yescharacteristic
524C23#1451975511yescharacteristic
525C23#1451975711yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—1222no—
529Q8—1222no—
530D4—1224no—
531D4—1224no—
532D4—1224no—
533D4—1224no—
534D4—1222no—
535D4—1222no—
536D4—1222no—
537D4—1222no—
538C2×C4—614no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4C42×D6111yescharacteristic
546C2×C4C22⋊C4×D6111yescharacteristic
547C2×C4C22⋊C4×D6111yescharacteristic
548C2×C4C22⋊C4×D6111yescharacteristic
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C2×#1175819811yescharacteristic
551C2×C4#1451973411yescharacteristic
552C2×C4#1451973511yescharacteristic
553C2×C4#1451973711yescharacteristic
554C2×C4#1451973711yescharacteristic
555C2×C4C4×C22⋊D6111yescharacteristic
556C2×C4C4×C22⋊D6111yescharacteristic
557C2×C4#1451973911yescharacteristic
558C2×C4#1451973911yescharacteristic
559C2×C4C4×D24411yescharacteristic
560C2×C4C4×D24411yescharacteristic
561C2×C4#1451974511yescharacteristic
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451974711yescharacteristic
565C2×C4#1451974811yescharacteristic
566C2×C4#1451975011yescharacteristic
567C2×C4#1451975011yescharacteristic
568C2×C4#1451975211yescharacteristic
569C2×C4#1451975911yescharacteristic
570C2×C4#1451975911yescharacteristic
571C2×C4#1451976211yescharacteristic
572C2×C4#1451976211yescharacteristic
573C8—24no—
574C8—42no—
575C8—42no—
576C8—1224no—
577C8—2442no—
578C8—2442no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—1224no—
587C22—4882no—
588C22—4882no—
589C22—2444no—
590C22#95×D6111yescharacteristic
591C22#1683961911yescharacteristic
592C22#1683962111yescharacteristic
593C22#1683963311yescharacteristic
594C22#1683963311yescharacteristic
595C22#1683963611yescharacteristic
596C22#1683963611yescharacteristic
597C4—1224no—
598C4#1683960911yescharacteristic
599C4C4×#1175819811yescharacteristic
600C4#1683961711yescharacteristic
601C4#1683962411yescharacteristic
602C4#1683963811yescharacteristic
603C4#1683963811yescharacteristic
604C4#1683964011yescharacteristic
605C4#1683964011yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—2444no—
614C2#2144404011yescharacteristic
subgroupG/H
centerC4#16839617
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1738C2
socleC2×C61C22⋊#95
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1738D61
hypocenterC61#10767
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
4. #21445546C2semidirect
5. #21445545C2semidirect
6. #21445545C2semidirect
7. C61×#1738C2semidirect
305. #1738D61semidirect
341. C61#10767semidirect

Maximal subgroups

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

subgroupindex
C2×#168405282
C2×#168397442
#214455462
#214455452
C61×#17382
#1076761