Finite Group Databasebeta

← #29845213

Subgroups — #29845213

Order 15616.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2#21448981C214yesmaximal, complemented
3#21448915C212yesmaximal, complemented
4#21448908C211yesmaximal, characteristic, complemented
5#21448842C212yesmaximal, complemented
6#21448770C212yesmaximal
7#21448770C211yesmaximal, characteristic
8#21448687C212yesmaximal, complemented
9#21448558C212yesmaximal, complemented
10#21448496C214yesmaximal, complemented
11#21448495C212yesmaximal, complemented
12#21448457C212yesmaximal
13#21448434C212yesmaximal, complemented
14#21448386C211yesmaximal, characteristic, complemented
15#21448297C211yesmaximal, characteristic, complemented
16#21448280C211yesmaximal, characteristic
17Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
18#16840472C4116yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519769C2212yes—
21C2×#14519755C2214yes—
22C2×#14519753C2212yes—
23C2×#14519742C2212yes—
24C2×C4×D244C2212yes—
25C2×C4×D244C2211yescharacteristic
26C2×#14519735C2214yes—
27C2×#14519735C2212yes—
28C2×#14519734C2212yes—
29C2×C4⋊C4×D61C2212yes—
30#16839669C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839667C2212yes—
33#16839666C2214yes—
34#16839663C2211yescharacteristic
35#16839663C2212yes—
36#16839663C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2212yes—
40#16839659C2214yes—
41#16839658C2212yes—
42#16839658C2212yes—
43#16839658C2212yes—
44#16839658C2211yescharacteristic
45#16839658C2211yescharacteristic
46#16839657C2214yes—
47#16839657C2214yes—
48#16839653C2214yes—
49#16839653C2214yes—
50#16839652C2214yes—
51#16839652C2214yes—
52#16839652C2214yes—
53#16839651C2212yes—
54#16839651C2212yes—
55#16839649C2214yes—
56#16839648C2212yes—
57#16839648C2212yes—
58#16839646C2214yes—
59#16839643C2214yes—
60#16839643C2214yes—
61#16839624C2214yes—
62#16839621C2214yes—
63#16839619C2214yes—
64#16839619C2212yes—
65C4⋊C4×Dic61C2212yes—
66C4⋊C4×Dic61C2212yes—
67C4⋊C4×Dic61C2212yes—
68C4⋊C4×Dic61C2212yes—
69C4×#11758196C2214yes—
70C4×C4⋊Dic61C2212yes—
71C4×C4⋊Dic61C2211yescharacteristic
72C4×C4⋊Dic61C2212yes—
73C4×#11758198C2212yes—
74#16839608C2212yes—
75C42×Dic61C2211yescharacteristic
76C2×C4×Q8×C61C2212yes—
77C2×C4×Q8×C61C2211yescharacteristic
78C61×#389C2214yes—
79C61×#389C2212yes—
80C61×#384C2212yes—
81C61×#384C2211yescharacteristic
82C4×C4⋊C4×C61C2212yes—
83C4×C4⋊C4×C61C2211yescharacteristic
84C22×Dic122C2312yes—
85C22×D244C2311yescharacteristic
86C22×C4×D61C2312yes—
87C22×C4×D61C2314yes—
88#14519777C2×C418yes—
89#14519769C2×C4116yes—
90#14519755C2×C4132yes—
91#14519753C2×C4116yes—
92#14519742—44no—
93C4×D244C2×C418yes—
94C4×D244C2×C4116yes—
95#14519735—48no—
96#14519735—44no—
97#14519734—44no—
98C2×C4⋊Dic61C2312yes—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×C4⋊Dic61C2311yescharacteristic
101C2×C4⋊Dic61C2312yes—
102C2×C4⋊Dic61C2312yes—
103C2×C4⋊Dic61C2311yescharacteristic
104C2×C4⋊Dic61C2312yes—
105C2×#11758196C2312yes—
106C2×#11758196C2312yes—
107C2×#11758196C2314yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2314yes—
111C4⋊C4×D61C2×C4116yes—
112C2×#11758198C2312yes—
113C2×#11758198C2312yes—
114C2×#11758198C2312yes—
115C2×#11758198C2314yes—
116C2×#11758198C2314yes—
117C2×#11758198C2312yes—
118C2×#11758198C2314yes—
119C2×#11758198C2312yes—
120C2×C4×Dic61C2311yescharacteristic
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124C2×C4×Dic61C2311yescharacteristic
125C2×C4×Dic61C2311yescharacteristic
126C2×C4×Dic61C2312yes—
127C2×C4×Dic61C2314yes—
128C2×C4×Dic61C2314yes—
129C2×C4×Dic61C2311yescharacteristic
130#14519684C2312yes—
131#14519684C2312yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2312yes—
135#14519684C2312yes—
136#14519684C2312yes—
137#14519684C2314yes—
138#14519684C2312yes—
139#14519684C2312yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144C22×Q8×C61C2311yescharacteristic
145C4×Q8×C61—44no—
146C4×Q8×C61C2×C418yes—
147C2×C4⋊C4×C61C2312yes—
148C2×C4⋊C4×C61C2311yescharacteristic
149C2×C4⋊C4×C61C2312yes—
150C2×C4⋊C4×C61C2311yescharacteristic
151C2×C4⋊C4×C61C2314yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2314yes—
154C2×C4⋊C4×C61C2311yescharacteristic
155C2×C4⋊C4×C61C2312yes—
156C2×C42×C61C2312yes—
157C2×C42×C61C2314yes—
158C2×C42×C61C2312yes—
159C2×C42×C61C2311yescharacteristic
160#95×C61C2314yes—
161#95×C61C2312yes—
162C23×D61C2412yes—
163C2×Q8×C61—22no—
164C2×Q8×C61—22no—
165C2×Q8×C61C22×C414yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2414yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2412yes—
170C22×C4×C61C2411yescharacteristic
171C22×C4×C61C2412yes—
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C22×Dic61C2411yescharacteristic
175C22×Dic61C2412yes—
176C22×Dic61C2411yescharacteristic
177C22×Dic61C2412yes—
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2414yes—
180C22×Dic61C2412yes—
181C22×Dic61C2411yescharacteristic
182C2×D244—22no—
183C2×D244—22no—
184C2×D244C22×C414yes—
185C2×C4×D61—24no—
186C2×C4×D61—24no—
187C2×C4×D61—44no—
188C2×C4×D61—44no—
189C2×C4×D61C22×C418yes—
190C2×C4×D61C22×C4116yes—
191C2×Dic122—42no—
192C2×Dic122—42no—
193C2×Dic122C22×C418yes—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—44no—
199C4⋊C4×C61—44no—
200C4⋊C4×C61—42no—
201C4⋊C4×C61C22×C414yes—
202C4⋊C4×C61C22×C418yes—
203C42×C61—24no—
204C42×C61—44no—
205C42×C61C22×C418yes—
206C42×C61C22×C414yes—
207#11758198—24no—
208#11758198—42no—
209#11758198—42no—
210#11758198—42no—
211#11758198—44no—
212#11758198—44no—
213#11758198C22×C4116yes—
214#11758198C22×C418yes—
215C4⋊Dic61—22no—
216C4⋊Dic61—22no—
217C4⋊Dic61—24no—
218C4⋊Dic61—42no—
219C4⋊Dic61C22×C418yes—
220C4⋊Dic61C22×C414yes—
221C4⋊Dic61C22×C418yes—
222#11758196—44no—
223#11758196—42no—
224#11758196—42no—
225#11758196—44no—
226#11758196C22×C4116yes—
227#11758196C22×C4116yes—
228C4×Dic61—22no—
229C4×Dic61—22no—
230C4×Dic61—22no—
231C4×Dic61—22no—
232C4×Dic61—24no—
233C4×Dic61—28no—
234C4×Dic61—22no—
235C4×Dic61—41no—
236C4×Dic61—44no—
237C4×Dic61—44no—
238C23×C61C2511yescharacteristic
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—22no—
245C22×D61C23×C414yes—
246Q8×C61—28no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—24no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61C23×C414yes—
263C2×C4×C61C23×C414yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61C23×C412yes—
266C2×C4×C61C23×C412yes—
267C2×C4×C61H(5,𝔽2)12yes—
268C2×C4×C612−1+412yes—
269C2×C4×C612−1+412yes—
270C2×C4×C612−1+412yes—
271C2×Dic61—24no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—24no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—21no—
284C2×Dic61—22no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—22no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—22no—
292C2×Dic61C23×C414yes—
293C2×Dic61C23×C418yes—
294C2×Dic61H(5,𝔽2)12yes—
295D244—28no—
296C4×D61—216no—
297C4×D61—416no—
298Dic122—48no—
299#34484—611nomaximal, Sylow
300C22×C61C24×C411yescharacteristic
301C22×C61C2×H(5,𝔽2)11yescharacteristic
302C22×C61C2×H(5,𝔽2)11yescharacteristic
303C22×C61C2×2−1+411yescharacteristic
304C22×C61C2×2−1+411yescharacteristic
305C22×C61C2×2−1+411yescharacteristic
306C22×C61#58511yescharacteristic
307C2×D61—24no—
308C2×D61—24no—
309C2×D61—24no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C4×C61—24no—
315C4×C61—22no—
316C4×C61—24no—
317C4×C61—42no—
318C4×C61—44no—
319C4×C61—42no—
320C4×C61#51914yes—
321C4×C61#52014yes—
322Dic61—24no—
323Dic61—28no—
324Dic61—22no—
325Dic61—42no—
326Dic61—41no—
327Dic61—42no—
328Dic61—41no—
329Dic61—41no—
330C2×#554—611no—
331#2378—614no—
332#2370—612no—
333#2369—611no—
334#2362—612no—
335#2354—611no—
336#2354—612no—
337Q8×C4⋊C4D6111yescharacteristic, complemented
338#2344—612no—
339#2330—612no—
340#2323—614no—
341#2323—612no—
342#2319—612no—
343#2316—612no—
344C4×C4⋊Q8—611no—
345#2295—611no—
346#2293—611no—
347C2×C61C2×#51911yescharacteristic
348C2×C61C2×#52011yescharacteristic
349C2×C61C4×H(5,𝔽2)11yescharacteristic
350C2×C61#354411yescharacteristic
351C2×C61#354711yescharacteristic
352C2×C61#354811yescharacteristic
353C2×C61#355511yescharacteristic
354D61—44no—
355#554—6116no—
356C2×C4⋊Q8—611no—
357C2×#125—612no—
358C2×C22⋊Q8—612no—
359C2×C22⋊Q8—614no—
360C2×C4×Q8—612no—
361C2×C4×Q8C2×D6112yes—
362C2×C4×Q8C2×D6111yescharacteristic
363C2×C4×D4—612no—
364C2×C4×D4—611no—
365C2×#117—612no—
366C2×#117—614no—
367C2×#117—612no—
368C22×C4⋊C4—612no—
369#391—611no—
370#391—611no—
371#391—612no—
372#391—614no—
373#390—611no—
374#390—612no—
375#390—611no—
376#389—612no—
377#389—612no—
378#389—611no—
379#389—614no—
380#389—612no—
381#389—614no—
382#389—611no—
383#389—611no—
384#389—611no—
385#389—612no—
386#389—614no—
387#389C2×D6112yes—
388#389C2×D6114yes—
389#388—614no—
390#388—614no—
391#388—614no—
392#388—614no—
393#388—614no—
394#388—612no—
395#388—612no—
396#387—612no—
397#387—614no—
398#387—614no—
399#387—612no—
400#387—614no—
401#387—614no—
402#384C2×D6111yescharacteristic
403#384C2×D6112yes—
404#381—614no—
405#380—614no—
406#380—612no—
407#380—614no—
408C4×C4⋊C4—612no—
409C4×C4⋊C4—612no—
410C4×C4⋊C4—612no—
411C4×C4⋊C4—612no—
412C4×C4⋊C4—612no—
413C4×C4⋊C4—614no—
414C4×C4⋊C4—611no—
415C4×C4⋊C4—612no—
416C4×C4⋊C4C2×D6112yes—
417C4×C4⋊C4C2×D6111yescharacteristic
418C4×C22⋊C4—612no—
419#376—612no—
420C43—611no—
421C61#3448411yescharacteristic, complemented, Sylow
422C22×Q8—612no—
423C22×Q8C22×D6111yescharacteristic
424C22×D4—611no—
425C23×C4—612no—
426C23×C4—614no—
427C4⋊Q8—618no—
428#125—6116no—
429C22⋊Q8—6116no—
430C22⋊Q8—6132no—
431C4×Q8—44no—
432C4×Q8—2444no—
433C4×Q8C4×D6118yes—
434C4×D4—6116no—
435C4×D4—618no—
436#117—2444no—
437#117—2448no—
438#117—2444no—
439C2×C4⋊C4—614no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—6116no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—614no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4C22×D6112yes—
454C2×C4⋊C4C22×D6112yes—
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6114yes—
458C2×C4⋊C4C22×D6112yes—
459C2×C4⋊C4C22×D6114yes—
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6112yes—
462C2×C22⋊C4—612no—
463C2×C22⋊C4—614no—
464C2×C22⋊C4—612no—
465C2×C22⋊C4—614no—
466C2×C22⋊C4—612no—
467C2×C22⋊C4—612no—
468C2×C22⋊C4—614no—
469C2×C22⋊C4—612no—
470C2×C42—611no—
471C2×C42—614no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—612no—
477C2×C42—614no—
478C2×C42—614no—
479C2×C42—611no—
480C2×C42C22×D6112yes—
481C2×C42C22×D6114yes—
482C2×C42C22×D6112yes—
483C2×C42C22×D6111yescharacteristic
484#95—612no—
485#95—612no—
486#95—612no—
487#95—612no—
488#95—612no—
489#95—612no—
490#95—612no—
491#95—614no—
492#95—614no—
493#95—614no—
494#95—614no—
495#95—614no—
496#95—612no—
497#95—612no—
498#95C22×D6114yes—
499#95C22×D6112yes—
500C24—612no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—618no—
504C2×Q8—2442no—
505C2×Q8—2442no—
506C2×Q8C2×C4×D6114yes—
507C2×D4—614no—
508C2×D4—1222no—
509C2×D4—1222no—
510C22×C4—611no—
511C22×C4—612no—
512C22×C4—614no—
513C22×C4—618no—
514C22×C4—6116no—
515C22×C4—611no—
516C22×C4—612no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—611no—
520C22×C4—1224no—
521C22×C4—1224no—
522C22×C4—2444no—
523C22×C4—2444no—
524C22×C4C23×D6112yes—
525C22×C4C23×D6114yes—
526C22×C4C23×D6112yes—
527C22×C4C23×D6112yes—
528C22×C4C23×D6111yescharacteristic
529C22×C4C23×D6112yes—
530C22×C4C23×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—6116no—
540C4⋊C4—618no—
541C4⋊C4—6116no—
542C4⋊C4—614no—
543C4⋊C4—618no—
544C4⋊C4—1224no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—2444no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4—2444no—
551C4⋊C4—2442no—
552C4⋊C4C2×C4×D6114yes—
553C4⋊C4C2×C4×D6118yes—
554C22⋊C4—618no—
555C22⋊C4—6116no—
556C22⋊C4—1224no—
557C22⋊C4—2444no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2444no—
561C22⋊C4—2442no—
562C42—24no—
563C42—44no—
564C42—1228no—
565C42—1222no—
566C42—1222no—
567C42—1222no—
568C42—1222no—
569C42—1222no—
570C42—1224no—
571C42—2441no—
572C42—2444no—
573C42—2444no—
574C42C2×C4×D6118yes—
575C42C2×C4×D6114yes—
576C23—614no—
577C23—1222no—
578C23—1222no—
579C23—1222no—
580C23—1222no—
581C23—1222no—
582C23—1222no—
583C23C24×D6111yescharacteristic
584Q8—28no—
585Q8—2448no—
586D4—1228no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—614no—
603C2×C4—618no—
604C2×C4—612no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1224no—
611C2×C4—12216no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1221no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1222no—
627C2×C4—24416no—
628C2×C4C22×C4×D6114yes—
629C2×C4C22×C4×D6112yes—
630C2×C4C22×C4×D6114yes—
631C2×C4C22×C4×D6112yes—
632C2×C4C22×C4×D6112yes—
633C2×C4#1451982212yes—
634C2×C4#1451982412yes—
635C2×C4#1451982512yes—
636C2×C4#1451982512yes—
637C22—1224no—
638C22—1224no—
639C22—1224no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22C23×C4×D6111yescharacteristic
645C22H(5,𝔽2)×D6111yescharacteristic
646C22C2×#1451982211yescharacteristic
647C22C2×#1451982411yescharacteristic
648C22C2×#1451982511yescharacteristic
649C22C2×#1451982511yescharacteristic
650C22#1684059911yescharacteristic
651C4—24no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—1224no—
658C4—1228no—
659C4—1222no—
660C4—2441no—
661C4—2442no—
662C4—2441no—
663C4—2442no—
664C4—2441no—
665C4#1684023814yes—
666C4#1684024314yes—
667C2—2444no—
668C2C2×#1684023811yescharacteristic
669C2C2×#1684024311yescharacteristic
670C2#2145902511yescharacteristic
671C2#2146016411yescharacteristic
672C2#2146020411yescharacteristic
673C2#2146021211yescharacteristic
674C2#2146030311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingQ8×C4⋊C4×C61C2
socle, cosocleC23×C61C25
prime radicalC22×D244C23
hypercenterQ8×C4⋊C4D61
hypocenterC61#34484
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21448981C2semidirect
3. #21448915C2semidirect
4. #21448908C2semidirect
5. #21448842C2semidirect
8. #21448687C2semidirect
9. #21448558C2semidirect
10. #21448496C2semidirect
11. #21448495C2semidirect
13. #21448434C2semidirect
14. #21448386C2semidirect
15. #21448297C2semidirect
17. Q8×C4⋊C4×C61C2semidirect
18. #16840472C4semidirect
337. Q8×C4⋊C4D61semidirect
421. C61#34484semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214489812
#214489152
#214489082
#214488422
#214487702
#214486872
#214485582
#214484962
#214484952
#214484572
#214484342
#214483862
#214482972
#214482802
Q8×C4⋊C4×C612
#3448461