Finite Group Databasebeta

← #29840959

Subgroups — #29840959

Order 15616.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840244C211yesmaximal, characteristic
2C2×#16840244C211yesmaximal, characteristic
3C2×C4×#11758226C211yesmaximal, characteristic
4#21448737C211yesmaximal, characteristic, complemented
5Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
6Q8×#11758196C211yesmaximal, characteristic, complemented
7#21448728C213yesmaximal, complemented
8#21448727C213yesmaximal, complemented
9C4⋊C4×D244C213yesmaximal, complemented
10#21448720C213yesmaximal, complemented
11#21448444C213yesmaximal, complemented
12#21448434C213yesmaximal, complemented
13#21448299C213yesmaximal, complemented
14#389×D61C213yesmaximal, complemented
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17#16840244—28no—
18#16840244C4116yescomplemented
19C4×#11758226—28no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×C4×D244C2213yes—
24C2×C4×D244C2213yes—
25C2×C4×D244C2213yes—
26C2×#14519739C2213yes—
27C2×#14519739C2213yes—
28C2×#14519739C2213yes—
29C2×#14519735C2213yes—
30C2×#14519734C2213yes—
31C2×#14519734C2213yes—
32C2×C4⋊C4×D61C2213yes—
33C2×C4⋊C4×D61C2216yes—
34C2×C4⋊C4×D61C2213yes—
35C2×C42×D61C2213yes—
36#16839661C2213yes—
37#16839661C2213yes—
38#16839660C2213yes—
39#16839660C2213yes—
40#16839659C2216yes—
41#16839659C2211yescharacteristic
42#16839659C2213yes—
43#16839658C2213yes—
44#16839658C2213yes—
45#16839657C2213yes—
46#16839657C2213yes—
47#16839636C2216yes—
48#16839636C2213yes—
49#16839636C2216yes—
50#16839636C2213yes—
51#16839633C2213yes—
52#16839633C2213yes—
53#16839632C2213yes—
54#16839631C2213yes—
55#16839627C2213yes—
56#16839621C2216yes—
57#16839619C2216yes—
58C4×#11758196C2213yes—
59C4×#11758196C2213yes—
60C4×C4⋊Dic61C2213yes—
61#95×D61C2216yes—
62C2×C4×Q8×C61C2211yescharacteristic
63C2×C4×Q8×C61C2211yescharacteristic
64C2×C4×Q8×C61C2211yescharacteristic
65C61×#389C2213yes—
66C61×#389C2213yes—
67C61×#382C2213yes—
68C4×C4⋊C4×C61C2213yes—
69C2×#11758226C2×C418yes—
70C2×#11758226D418yes—
71C2×#11758226Q818yes—
72C22×D244C2313yes—
73C22×C4×D61C2313yes—
74C22×C4×D61C2313yes—
75C22×C4×D61C2316yes—
76C22×C4×D61C2316yes—
77C22×C4×D61C2313yes—
78Q8×Dic61—24no—
79Q8×Dic61—24no—
80Q8×Dic61C2×C418yescomplemented
81C4×D244—212no—
82C4×D244—212no—
83C4×D244C2×C4124yescomplemented
84#14519739—212no—
85#14519739—212no—
86#14519739C2×C4124yescomplemented
87#14519735—212no—
88#14519734—212no—
89#14519734C2×C4124yescomplemented
90C2×C4⋊Dic61C2313yes—
91C2×C4⋊Dic61C2313yes—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2316yes—
94C2×C4⋊Dic61C2313yes—
95C2×C4⋊Dic61C2313yes—
96C2×#11758196C2313yes—
97C2×#11758196C2313yes—
98C2×#11758196C2311yescharacteristic
99C2×#11758196C2311yescharacteristic
100C2×#11758196C2316yes—
101C2×#11758196C2313yes—
102C2×#11758196C2313yes—
103C2×#11758196C2313yes—
104C2×#11758196C2311yescharacteristic
105C4⋊C4×D61—212no—
106C4⋊C4×D61—412no—
107C4⋊C4×D61C2×C4124yescomplemented
108C2×#11758198C2313yes—
109C2×#11758198C2313yes—
110C2×#11758198C2316yes—
111C2×#11758198C2313yes—
112C2×#11758198C2313yes—
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2313yes—
115C2×C4×Dic61C2313yes—
116C2×C4×Dic61C2313yes—
117C2×C4×Dic61C2313yes—
118C42×D61—212no—
119#14519684C2313yes—
120#14519684C2313yes—
121#14519684C2316yes—
122#14519684C2313yes—
123#14519684C2311yescharacteristic
124#14519684C2313yes—
125#14519684C2313yes—
126C22×Q8×C61C2311yescharacteristic
127C4×Q8×C61—24no—
128C4×Q8×C61—24no—
129C4×Q8×C61C2×C418yescomplemented
130C2×C4⋊C4×C61C2313yes—
131C2×C4⋊C4×C61C2313yes—
132C2×C4⋊C4×C61C2313yes—
133C2×C4⋊C4×C61C2313yes—
134C2×C4⋊C4×C61C2313yes—
135C2×C42×C61C2313yes—
136C2×C42×C61C2313yes—
137C2×C42×C61C2313yes—
138#95×C61C2311yescharacteristic
139#95×C61C2313yes—
140#95×C61C2316yes—
141C23×D61C2413yes—
142C2×Q8×C61C22×C414yes—
143C2×Q8×C61C2×D414yes—
144C2×Q8×C61C2×Q814yes—
145C22×C4×C61C2413yes—
146C22×C4×C61C2413yes—
147C22×C4×C61C2411yescharacteristic
148C22×C4×C61C2413yes—
149C22×C4×C61C2411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×C4×C61C2413yes—
152C22×Dic61C2413yes—
153C22×Dic61C2413yes—
154C22×Dic61C2411yescharacteristic
155C22×Dic61C2413yes—
156C22×Dic61C2411yescharacteristic
157C22×Dic61C2411yescharacteristic
158C22×Dic61C2411yescharacteristic
159#11758226C4⋊C4164yescomplemented
160C2×D244C22×C4112yes—
161C2×D244C2×D4112yes—
162C2×D244C2×Q8112yes—
163C2×C4×D61—26no—
164C2×C4×D61—26no—
165C2×C4×D61—26no—
166C2×C4×D61—26no—
167C2×C4×D61—212no—
168C2×C4×D61—46no—
169C2×C4×D61—43no—
170C2×C4×D61—43no—
171C2×C4×D61—46no—
172C2×C4×D61—46no—
173C2×C4×D61—46no—
174C2×C4×D61C22×C4124yes—
175C2×C4×D61C22×C4112yes—
176C2×C4×D61C2×D4112yes—
177C2×C4×D61C2×Q8112yes—
178C4⋊C4×C61—26no—
179C4⋊C4×C61—26no—
180C4⋊C4×C61—26no—
181C4⋊C4×C61—43no—
182C4⋊C4×C61C22×C4112yes—
183C42×C61—26no—
184C42×C61—26no—
185C42×C61C22×C4112yes—
186#11758198—26no—
187#11758198—26no—
188#11758198—212no—
189#11758198C22×C4112yes—
190#11758198C22×C4112yes—
191C4⋊Dic61—26no—
192C4⋊Dic61—26no—
193C4⋊Dic61—22no—
194C4⋊Dic61—46no—
195C4⋊Dic61—43no—
196C4⋊Dic61C22×C4112yes—
197#11758196—26no—
198#11758196—26no—
199#11758196—22no—
200#11758196—22no—
201#11758196—46no—
202#11758196—43no—
203#11758196—43no—
204#11758196C22×C4112yes—
205#11758196C22×C414yes—
206C4×Dic61—26no—
207C4×Dic61—26no—
208C4×Dic61—22no—
209C4×Dic61—26no—
210C4×Dic61C22×C4112yes—
211C23×C61C2511yescharacteristic
212C22×D61—23no—
213C22×D61—23no—
214C22×D61—23no—
215C22×D61—23no—
216C22×D61C23×C416yes—
217C22×D61C22×D416yes—
218C22×D61C22×Q816yes—
219Q8×C61C2×C4⋊C4116yescomplemented
220C2×C4×C61—21no—
221C2×C4×C61—23no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—23no—
225C2×C4×C61—21no—
226C2×C4×C61—23no—
227C2×C4×C61—21no—
228C2×C4×C61—23no—
229C2×C4×C61—23no—
230C2×C4×C61—21no—
231C2×C4×C61—23no—
232C2×C4×C61—23no—
233C2×C4×C61—23no—
234C2×C4×C61—21no—
235C2×C4×C61C23×C412yes—
236C2×C4×C61C23×C416yes—
237C2×C4×C61C23×C416yes—
238C2×C4×C61C22×D416yes—
239C2×C4×C61C22×Q816yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C2×Dic61—21no—
242C2×Dic61—23no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—23no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—23no—
249C2×Dic61—23no—
250C2×Dic61—23no—
251C2×Dic61—23no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—23no—
255C2×Dic61—23no—
256C2×Dic61C23×C412yes—
257C2×Dic61C23×C416yes—
258C2×Dic61C23×C412yes—
259C2×Dic61C22×D412yes—
260C2×Dic61C22×Q812yes—
261C2×Dic612−1+412yes—
262D244C2×C4⋊C4148yescomplemented
263C4×D61—424no—
264C4×D61—412no—
265C4×D61—412no—
266C4×D61—812no—
267C4×D61C2×C4⋊C4148yescomplemented
268#34250—611nomaximal, Sylow
269C22×C61C24×C411yescharacteristic
270C22×C61C23×D411yescharacteristic
271C22×C61C23×Q811yescharacteristic
272C22×C61C2×H(5,𝔽2)11yescharacteristic
273C22×C61C2×2−1+411yescharacteristic
274C22×C61#58511yescharacteristic
275C22×C61#58511yescharacteristic
276C2×D61—26no—
277C2×D61—26no—
278C2×D61—26no—
279C2×D61—26no—
280C2×D61—26no—
281C2×D61—26no—
282C2×D61C22×C4⋊C4112yes—
283C4×C61—26no—
284C4×C61—22no—
285C4×C61—22no—
286C4×C61—43no—
287C4×C61—43no—
288C4×C61—41no—
289C4×C61C22×C4⋊C4112yes—
290Dic61—26no—
291Dic61—22no—
292Dic61—22no—
293Dic61—43no—
294Dic61—43no—
295Dic61—41no—
296Dic61C22×C4⋊C414yes—
297C2×#520—611no—
298C2×#520—611no—
299C2×C4×#41—611no—
300#2351—611no—
301#2351D6111yescharacteristic, complemented
302Q8×C4⋊C4—611no—
303Q8×C4⋊C4—611no—
304#2349—613no—
305#2349—613no—
306D4×C4⋊C4—613no—
307D4×C4⋊C4—613no—
308#2317—613no—
309#2316—613no—
310#2296—613no—
311C2×#389—613no—
312C2×C61C23×C4⋊C411yescharacteristic
313C2×C61C4×H(5,𝔽2)11yescharacteristic
314C2×C61C4×2−1+411yescharacteristic
315C2×C61#348811yescharacteristic
316C2×C61#348911yescharacteristic
317C2×C61#350311yescharacteristic
318C2×C61#350411yescharacteristic
319D61—212no—
320C22×#41—611no—
321#520—6116no—
322#520—1228no—
323C4×#41—1228no—
324C2×C4×Q8—611no—
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×Q8C2×D6111yescharacteristic
329C2×C4×Q8C2×D6111yescharacteristic
330C2×C4×D4—613no—
331C2×C4×D4—613no—
332C2×C4×D4—613no—
333C2×C4×D4—613no—
334C2×C4×D4—613no—
335C2×C4×D4—613no—
336C2×#117—613no—
337C2×#117—613no—
338C2×#117—613no—
339C22×C4⋊C4—613no—
340C22×C4⋊C4—613no—
341C22×C4⋊C4—616no—
342C22×C42—613no—
343#389—613no—
344#389—613no—
345#389—613no—
346#389—613no—
347#389—613no—
348#389—613no—
349#389—613no—
350#389—613no—
351#389—616no—
352#389—613no—
353#389—611no—
354#389C2×D6113yes—
355#389C2×D6113yes—
356#385—613no—
357#385—613no—
358#385—616no—
359#385—616no—
360#385—613no—
361#385—613no—
362#384—613no—
363#384—613no—
364#382—613no—
365#382C2×D6113yes—
366#380—616no—
367#380—616no—
368C4×C4⋊C4—613no—
369C4×C4⋊C4—613no—
370C4×C4⋊C4—613no—
371C4×C4⋊C4C2×D6113yes—
372C2×#95—616no—
373C61#3425011yescharacteristic, complemented, Sylow
374C2×#41—618no—
375C2×#41—618no—
376C2×#41—618no—
377C22×Q8C22×D6111yescharacteristic
378C22×D4—613no—
379C23×C4—613no—
380C23×C4—616no—
381C23×C4—613no—
382C23×C4—613no—
383C23×C4—616no—
384C4×Q8—24no—
385C4×Q8—24no—
386C4×Q8—618no—
387C4×Q8—1224no—
388C4×Q8—1224no—
389C4×Q8C4×D6118yescomplemented
390C4×D4—6124no—
391C4×D4—6124no—
392C4×D4—12212no—
393C4×D4—12212no—
394C4×D4—12212no—
395C4×D4—12212no—
396#117—6124no—
397#117—12212no—
398#117—12212no—
399C2×C4⋊C4—613no—
400C2×C4⋊C4—613no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—6124no—
403C2×C4⋊C4—613no—
404C2×C4⋊C4—613no—
405C2×C4⋊C4—613no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—613no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—616no—
411C2×C4⋊C4—616no—
412C2×C4⋊C4—613no—
413C2×C4⋊C4—613no—
414C2×C4⋊C4—613no—
415C2×C4⋊C4—12212no—
416C2×C4⋊C4—24412no—
417C2×C4⋊C4C22×D6113yes—
418C2×C4⋊C4C22×D6113yes—
419C2×C4⋊C4C22×D6113yes—
420C2×C4⋊C4C22×D6113yes—
421C2×C4⋊C4C22×D6113yes—
422C2×C22⋊C4—613no—
423C2×C22⋊C4—613no—
424C2×C22⋊C4—613no—
425C2×C22⋊C4—613no—
426C2×C22⋊C4—616no—
427C2×C42—613no—
428C2×C42—613no—
429C2×C42—613no—
430C2×C42—613no—
431C2×C42—611no—
432C2×C42—12212no—
433C2×C42C22×D6113yes—
434C2×C42C22×D6113yes—
435C2×C42C22×D6113yes—
436#95—613no—
437#95—613no—
438#95—616no—
439#95—611no—
440#95—613no—
441#95—613no—
442#95—613no—
443#95C22×D6116yes—
444#95C22×D6111yescharacteristic
445#95C22×D6113yes—
446C24—613no—
447#41—6164no—
448C2×Q8C2×C4×D6114yes—
449C2×Q8D4×D6114yes—
450C2×Q8Q8×D6114yes—
451C2×D4—6112no—
452C2×D4—6112no—
453C2×D4—6112no—
454C22×C4—611no—
455C22×C4—6112no—
456C22×C4—6112no—
457C22×C4—6112no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—613no—
461C22×C4—6124no—
462C22×C4—611no—
463C22×C4—613no—
464C22×C4—613no—
465C22×C4—1226no—
466C22×C4—12212no—
467C22×C4—1226no—
468C22×C4—1226no—
469C22×C4—1226no—
470C22×C4—2446no—
471C22×C4—2446no—
472C22×C4—2446no—
473C22×C4—2446no—
474C22×C4—2443no—
475C22×C4—2443no—
476C22×C4C23×D6113yes—
477C22×C4C23×D6111yescharacteristic
478C22×C4C23×D6113yes—
479C22×C4C23×D6113yes—
480C22×C4C23×D6111yescharacteristic
481C22×C4C23×D6111yescharacteristic
482C22×C4C23×D6113yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—26no—
486C4⋊C4—43no—
487C4⋊C4—6112no—
488C4⋊C4—6112no—
489C4⋊C4—614no—
490C4⋊C4—1226no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1226no—
495C4⋊C4—1226no—
496C4⋊C4—1226no—
497C4⋊C4—2446no—
498C4⋊C4—2446no—
499C4⋊C4—2443no—
500C4⋊C4—2443no—
501C4⋊C4—2443no—
502C4⋊C4C2×C4×D61112yes—
503C22⋊C4—6112no—
504C22⋊C4—6112no—
505C22⋊C4—1226no—
506C22⋊C4—12212no—
507C22⋊C4—1226no—
508C42—26no—
509C42—26no—
510C42—6112no—
511C42—1222no—
512C42—1226no—
513C42—1226no—
514C42—1226no—
515C42C2×C4×D61112yes—
516C23—616no—
517C23—616no—
518C23—616no—
519C23—1223no—
520C23—1223no—
521C23—1223no—
522C23—1223no—
523C23C24×D6111yescharacteristic
524Q8C4⋊C4×D61116yescomplemented
525D4—6148no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—21no—
537C2×C4—23no—
538C2×C4—21no—
539C2×C4—23no—
540C2×C4—23no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—6148no—
545C2×C4—612no—
546C2×C4—616no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1223no—
550C2×C4—1221no—
551C2×C4—1223no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1223no—
556C2×C4—1223no—
557C2×C4—1223no—
558C2×C4—1223no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1223no—
562C2×C4—1223no—
563C2×C4—48812no—
564C2×C4—24424no—
565C2×C4—24412no—
566C2×C4—24412no—
567C2×C4C22×C4×D6112yes—
568C2×C4C22×C4×D6116yes—
569C2×C4C22×C4×D6116yes—
570C2×C4C2×D4×D6116yes—
571C2×C4C2×Q8×D6116yes—
572C2×C4#1451982312yes—
573C22—6112no—
574C22—1226no—
575C22—1226no—
576C22—1226no—
577C22—1226no—
578C22—1226no—
579C22—1226no—
580C22C23×C4×D6111yescharacteristic
581C22C22×D4×D6111yescharacteristic
582C22C22×Q8×D6111yescharacteristic
583C22C2×#1451982311yescharacteristic
584C222−1+4×D6111yescharacteristic
585C22#1684059811yescharacteristic
586C22#1684059911yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—41no—
592C4—43no—
593C4—614no—
594C4—1222no—
595C4—1226no—
596C4—1222no—
597C4—2443no—
598C4—2443no—
599C4—2441no—
600C4C2×C4⋊C4×D61112yes—
601C2—12212no—
602C2C22×C4⋊C4×D6111yescharacteristic
603C2C4×#1451982311yescharacteristic
604C2#2145902811yescharacteristic
605C2#2145948511yescharacteristic
606C2#2145949811yescharacteristic
607C2#2145963611yescharacteristic
608C2#2145964411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2351C2
socle, cosocleC23×C61C25
prime radicalC22×#11758226C22
hypercenter#2351D61
hypocenterC61#34250
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #21448737C2semidirect
5. Q8×C4⋊Dic61C2semidirect
6. Q8×#11758196C2semidirect
7. #21448728C2semidirect
8. #21448727C2semidirect
9. C4⋊C4×D244C2semidirect
10. #21448720C2semidirect
11. #21448444C2semidirect
12. #21448434C2semidirect
13. #21448299C2semidirect
14. #389×D61C2semidirect
15. C61×#2351C2semidirect
18. #16840244C4semidirect
80. Q8×Dic61C2×C4semidirect
83. C4×D244C2×C4semidirect
86. #14519739C2×C4semidirect
89. #14519734C2×C4semidirect
107. C4⋊C4×D61C2×C4semidirect
129. C4×Q8×C61C2×C4semidirect
159. #11758226C4⋊C4semidirect
219. Q8×C61C2×C4⋊C4semidirect
262. D244C2×C4⋊C4semidirect
267. C4×D61C2×C4⋊C4semidirect
301. #2351D61semidirect
373. C61#34250semidirect
389. C4×Q8C4×D61semidirect
524. Q8C4⋊C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402442
C2×C4×#117582262
#214487372
Q8×C4⋊Dic612
Q8×#117581962
#214487282
#214487272
C4⋊C4×D2442
#214487202
#214484442
#214484342
#214482992
#389×D612
C61×#23512
#3425061