Finite Group Databasebeta

← #29842917

Subgroups — #29842917

Order 15616.

2098 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21449002C214yesmaximal, complemented
3#21448879C211yesmaximal, characteristic, complemented
4#21448877C211yesmaximal, characteristic, complemented
5#21448875C211yesmaximal, characteristic, complemented
6#21448775C212yesmaximal, complemented
7#21448772C212yesmaximal, complemented
8#21448733C212yesmaximal, complemented
9Q8×#11758196C212yesmaximal, complemented
10#21448504C212yesmaximal
11#21448502C212yesmaximal
12#21448499C212yesmaximal
13#21448495C212yesmaximal
14#21448425C214yesmaximal
15#21448223C211yesmaximal, characteristic
16#21448223C211yesmaximal, characteristic
17C61×#2365C211yesmaximal, characteristic, complemented
18#16840317C4116yescomplemented
19C2×#14519777C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519757C2212yes—
22C2×#14519753C2212yes—
23C2×Q8×Dic61C2212yes—
24C2×Q8×Dic61C2212yes—
25C2×C4×Dic122C2212yes—
26C2×#14519742C2212yes—
27C2×#14519742C2212yes—
28C2×#14519735C2212yes—
29C2×#14519734C2212yes—
30C2×#14519734C2212yes—
31C2×C42×D61C2211yescharacteristic
32#16839669C2212yes—
33#16839668C2212yes—
34#16839667C2212yes—
35#16839661C2212yes—
36#16839660C2214yes—
37#16839660C2212yes—
38#16839659C2214yes—
39#16839658C2214yes—
40#16839658C2212yes—
41#16839657C2214yes—
42#16839654C2214yes—
43#16839653C2214yes—
44#16839652C2214yes—
45#16839651C2214yes—
46#16839648C2214yes—
47#16839647C2214yes—
48#16839647C2214yes—
49#16839646C2214yes—
50#16839645C2214yes—
51#16839645C2214yes—
52#16839644C2214yes—
53#16839631C2211yescharacteristic
54#16839631C2214yes—
55#16839631C2211yescharacteristic
56#16839631C2211yescharacteristic
57#16839631C2211yescharacteristic
58#16839626C2212yes—
59#16839626C2212yes—
60#16839626C2214yes—
61#16839621C2212yes—
62#16839621C2212yes—
63#16839619C2211yescharacteristic
64#16839619C2211yescharacteristic
65#16839619C2211yescharacteristic
66#16839619C2211yescharacteristic
67C4⋊C4×Dic61C2212yes—
68C4⋊C4×Dic61C2212yes—
69C4×#11758196C2212yes—
70C4×#11758196C2214yes—
71C4×#11758196C2212yes—
72C4×C4⋊Dic61C2211yescharacteristic
73C4×C4⋊Dic61C2211yescharacteristic
74C4×C4⋊Dic61C2212yes—
75#16839613C2214yes—
76C2×C4⋊Q8×C61C2211yescharacteristic
77C2×C4×Q8×C61C2212yes—
78C61×#391C2212yes—
79C61×#389C2212yes—
80C61×#387C2214yes—
81C4×C4⋊C4×C61C2211yescharacteristic
82C4×C4⋊C4×C61C2211yescharacteristic
83C4×C4⋊C4×C61C2212yes—
84C22×Dic122C2312yes—
85C22×C4×D61C2311yescharacteristic
86C22×C4×D61C2311yescharacteristic
87C22×C4×D61C2311yescharacteristic
88#14519777C2×C418yes—
89#14519768C2×C4116yes—
90#14519757C2×C4116yes—
91#14519753C2×C4116yes—
92Q8×Dic61—44no—
93Q8×Dic61C2×C4116yes—
94C4×Dic122—44no—
95#14519742—44no—
96#14519742C2×C4116yes—
97#14519735C2×C4116yes—
98#14519734—44no—
99#14519734—44no—
100C2×C4⋊Dic61C2311yescharacteristic
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2311yescharacteristic
104C2×C4⋊Dic61C2314yes—
105C2×C4⋊Dic61C2312yes—
106C2×C4⋊Dic61C2312yes—
107C2×C4⋊Dic61C2314yes—
108C2×#11758196C2312yes—
109C2×#11758196C2312yes—
110C2×#11758196C2312yes—
111C2×#11758196C2312yes—
112C2×#11758196C2312yes—
113C2×#11758196C2312yes—
114C2×#11758196C2312yes—
115C2×#11758196C2312yes—
116C2×#11758196C2314yes—
117C2×#11758196C2314yes—
118C2×#11758196C2314yes—
119C2×#11758198C2312yes—
120C2×#11758198C2312yes—
121C2×#11758198C2312yes—
122C2×#11758198C2312yes—
123C2×#11758198C2314yes—
124C2×C4×Dic61C2312yes—
125C2×C4×Dic61C2312yes—
126C2×C4×Dic61C2314yes—
127C2×C4×Dic61C2314yes—
128C2×C4×Dic61C2314yes—
129C2×C4×Dic61C2311yescharacteristic
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2311yescharacteristic
132C2×C4×Dic61C2314yes—
133C42×D61C2×C418yes—
134#14519684C2314yes—
135#14519684C2312yes—
136#14519684C2312yes—
137#14519684C2314yes—
138#14519684C2312yes—
139#14519684C2312yes—
140#14519684C2312yes—
141#14519684C2312yes—
142#14519684C2314yes—
143#14519684C2314yes—
144#14519684C2314yes—
145#14519684C2314yes—
146C22×Q8×C61C2312yes—
147C4⋊Q8×C61C2×C418yes—
148C4×Q8×C61—44no—
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C4⋊C4×C61C2311yescharacteristic
151C2×C4⋊C4×C61C2311yescharacteristic
152C2×C4⋊C4×C61C2311yescharacteristic
153C2×C4⋊C4×C61C2314yes—
154C2×C4⋊C4×C61C2312yes—
155C2×C4⋊C4×C61C2312yes—
156C2×C42×C61C2312yes—
157C2×C42×C61C2312yes—
158C2×C42×C61C2314yes—
159C2×C42×C61C2311yescharacteristic
160#95×C61C2312yes—
161#95×C61C2312yes—
162#95×C61C2314yes—
163#95×C61C2314yes—
164C23×D61C2411yescharacteristic
165C2×Q8×C61—24no—
166C2×Q8×C61—24no—
167C2×Q8×C61C22×C418yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2412yes—
170C22×C4×C61C2412yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2414yes—
174C22×C4×C61C2411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×Dic61C2412yes—
177C22×Dic61C2412yes—
178C22×Dic61C2412yes—
179C22×Dic61C2412yes—
180C22×Dic61C2411yescharacteristic
181C22×Dic61C2411yescharacteristic
182C22×Dic61C2414yes—
183C22×Dic61C2411yescharacteristic
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—22no—
190C2×C4×D61C22×C414yes—
191C2×C4×D61C22×C414yes—
192C2×C4×D61C22×C414yes—
193C2×Dic122—24no—
194C2×Dic122—24no—
195C2×Dic122C22×C418yes—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—44no—
201C4⋊C4×C61C22×C418yes—
202C4⋊C4×C61C22×C418yes—
203C42×C61—24no—
204C42×C61—24no—
205C42×C61—44no—
206C42×C61C22×C414yes—
207#11758198—42no—
208#11758198—42no—
209#11758198—42no—
210#11758198—42no—
211#11758198C22×C4116yes—
212C4⋊Dic61—22no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215C4⋊Dic61—22no—
216C4⋊Dic61—44no—
217C4⋊Dic61C22×C418yes—
218C4⋊Dic61C22×C418yes—
219C4⋊Dic61C22×C4116yes—
220#11758196—24no—
221#11758196—24no—
222#11758196—24no—
223#11758196—24no—
224#11758196—42no—
225#11758196—42no—
226#11758196—42no—
227#11758196—42no—
228#11758196—44no—
229#11758196—44no—
230#11758196C22×C4116yes—
231C4×Dic61—24no—
232C4×Dic61—24no—
233C4×Dic61—44no—
234C4×Dic61—44no—
235C4×Dic61—44no—
236C4×Dic61C22×C414yes—
237C4×Dic61C22×C414yes—
238C4×Dic61C22×C414yes—
239C4×Dic61C22×C4116yes—
240C23×C61C2511yescharacteristic
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246C22×D61—21no—
247C22×D61C23×C412yes—
248Q8×C61—216no—
249C2×C4×C61—24no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—24no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61C23×C418yes—
266C2×C4×C61C23×C412yes—
267C2×C4×C61C23×C412yes—
268C2×C4×C61C23×C412yes—
269C2×C4×C61H(5,𝔽2)12yes—
270C2×C4×C61H(5,𝔽2)12yes—
271C2×C4×C612−1+412yes—
272C2×C4×C612−1+412yes—
273C2×Dic61—24no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—24no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—22no—
290C2×Dic61—22no—
291C2×Dic61C23×C412yes—
292C2×Dic61C23×C412yes—
293C2×Dic61C23×C418yes—
294C2×Dic61C23×C412yes—
295C2×Dic612−1+412yes—
296C2×Dic612−1+412yes—
297C4×D61—28no—
298C4×D61—28no—
299C4×D61—28no—
300Dic122—216no—
301#34374—611nomaximal, Sylow
302C22×C61C24×C411yescharacteristic
303C22×C61C2×H(5,𝔽2)11yescharacteristic
304C22×C61C2×H(5,𝔽2)11yescharacteristic
305C22×C61C2×2−1+411yescharacteristic
306C22×C61C2×2−1+411yescharacteristic
307C22×C61C2×2−1+411yescharacteristic
308C22×C61C2×2−1+411yescharacteristic
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—22no—
317C4×C61—28no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—42no—
322C4×C61#52014yes—
323C4×C61#52014yes—
324Dic61—28no—
325Dic61—22no—
326Dic61—22no—
327Dic61—42no—
328Dic61—42no—
329Dic61—42no—
330Dic61—42no—
331Dic61#51914yes—
332C2×#533—611no—
333#2382—614no—
334#2365—611no—
335#2365—611no—
336#2365—611no—
337#2365D6111yescharacteristic, complemented
338#2355—612no—
339#2355—612no—
340Q8×C4⋊C4—612no—
341Q8×C4⋊C4—612no—
342#2323—612no—
343#2323—612no—
344#2323—612no—
345#2323—612no—
346#2314—614no—
347#2285—611no—
348#2285—611no—
349C2×C61C2×#51911yescharacteristic
350C2×C61C2×#52011yescharacteristic
351C2×C61C2×#52011yescharacteristic
352C2×C61#354811yescharacteristic
353C2×C61#355011yescharacteristic
354C2×C61#355011yescharacteristic
355C2×C61#355611yescharacteristic
356D61—42no—
357#533—6116no—
358C2×C4⋊Q8—611no—
359C2×C4⋊Q8C2×D6111yescharacteristic
360C2×#125—612no—
361C2×C22⋊Q8—612no—
362C2×C22⋊Q8—612no—
363C2×C4×Q8—612no—
364C2×C4×Q8—612no—
365C2×C4×Q8—612no—
366C2×C4×Q8—612no—
367C2×C4×Q8—612no—
368C2×C4×Q8C2×D6112yes—
369C2×#117—612no—
370C2×#117—612no—
371C2×#117—612no—
372C22×C42—611no—
373#391—612no—
374#391—612no—
375#391—612no—
376#391C2×D6112yes—
377#389—612no—
378#389—612no—
379#389—614no—
380#389—614no—
381#389—612no—
382#389—614no—
383#389—614no—
384#389C2×D6112yes—
385#388—614no—
386#388—614no—
387#388—614no—
388#388—614no—
389#387—614no—
390#387—614no—
391#387—614no—
392#387—614no—
393#387—614no—
394#387—614no—
395#387—614no—
396#387C2×D6114yes—
397#384—614no—
398#384—611no—
399#384—611no—
400#384—611no—
401#384—611no—
402#382—614no—
403#382—612no—
404#382—612no—
405#380—611no—
406#380—611no—
407#380—611no—
408#380—611no—
409#380—612no—
410#380—612no—
411C4×C4⋊C4—612no—
412C4×C4⋊C4—612no—
413C4×C4⋊C4—614no—
414C4×C4⋊C4—612no—
415C4×C4⋊C4—612no—
416C4×C4⋊C4—612no—
417C4×C4⋊C4—611no—
418C4×C4⋊C4—611no—
419C4×C4⋊C4—614no—
420C4×C4⋊C4C2×D6112yes—
421C4×C4⋊C4C2×D6111yescharacteristic
422C4×C4⋊C4C2×D6111yescharacteristic
423C61#3437411yescharacteristic, complemented, Sylow
424C22×Q8—612no—
425C22×Q8C22×D6112yes—
426C23×C4—611no—
427C23×C4—611no—
428C23×C4—611no—
429C4⋊Q8—618no—
430C4⋊Q8C4×D6118yes—
431#125—6116no—
432C22⋊Q8—6116no—
433C22⋊Q8—6116no—
434C4×Q8—44no—
435C4×Q8—6116no—
436C4×Q8—6116no—
437C4×Q8—2444no—
438C4×Q8—2444no—
439C4×Q8—2444no—
440#117—6116no—
441#117—2444no—
442#117—2444no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—614no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—612no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—611no—
455C2×C4⋊C4—612no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—612no—
458C2×C4⋊C4—612no—
459C2×C4⋊C4—614no—
460C2×C4⋊C4—614no—
461C2×C4⋊C4—614no—
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C4⋊C4C22×D6111yescharacteristic
464C2×C4⋊C4C22×D6111yescharacteristic
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C4⋊C4C22×D6114yes—
467C2×C4⋊C4C22×D6112yes—
468C2×C4⋊C4C22×D6112yes—
469C2×C22⋊C4—614no—
470C2×C22⋊C4—612no—
471C2×C22⋊C4—612no—
472C2×C22⋊C4—612no—
473C2×C22⋊C4—612no—
474C2×C42—614no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—618no—
479C2×C42—612no—
480C2×C42—612no—
481C2×C42—614no—
482C2×C42—614no—
483C2×C42—614no—
484C2×C42C22×D6112yes—
485C2×C42C22×D6112yes—
486C2×C42C22×D6114yes—
487C2×C42C22×D6111yescharacteristic
488#95—612no—
489#95—612no—
490#95—612no—
491#95—612no—
492#95—612no—
493#95—612no—
494#95—614no—
495#95—614no—
496#95—614no—
497#95—614no—
498#95—614no—
499#95—614no—
500#95C22×D6114yes—
501#95C22×D6112yes—
502#95C22×D6112yes—
503#95C22×D6114yes—
504C24—611no—
505C2×Q8—24no—
506C2×Q8—24no—
507C2×Q8—618no—
508C2×Q8—1224no—
509C2×Q8—1224no—
510C2×Q8C2×C4×D6118yes—
511C22×C4—611no—
512C22×C4—614no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—614no—
516C22×C4—614no—
517C22×C4—614no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4C23×D6112yes—
529C22×C4C23×D6112yes—
530C22×C4C23×D6112yes—
531C22×C4C23×D6112yes—
532C22×C4C23×D6111yescharacteristic
533C22×C4C23×D6114yes—
534C22×C4C23×D6111yescharacteristic
535C22×C4C23×D6111yescharacteristic
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—6116no—
542C4⋊C4—618no—
543C4⋊C4—6116no—
544C4⋊C4—618no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4—2444no—
558C4⋊C4—2444no—
559C4⋊C4—2444no—
560C4⋊C4C2×C4×D6118yes—
561C4⋊C4C2×C4×D6118yes—
562C22⋊C4—6116no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C22⋊C4—2442no—
566C22⋊C4—2442no—
567C42—24no—
568C42—24no—
569C42—44no—
570C42—614no—
571C42—614no—
572C42—614no—
573C42—6116no—
574C42—1224no—
575C42—1224no—
576C42—2444no—
577C42—2444no—
578C42—2444no—
579C42C2×C4×D6114yes—
580C23—612no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23—1221no—
587C23C24×D6111yescharacteristic
588Q8—216no—
589Q8—12216no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—618no—
612C2×C4—1228no—
613C2×C4—1221no—
614C2×C4—1224no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1224no—
619C2×C4—1228no—
620C2×C4—1228no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4C22×C4×D6112yes—
634C2×C4C22×C4×D6118yes—
635C2×C4C22×C4×D6112yes—
636C2×C4C22×C4×D6112yes—
637C2×C4#1451982212yes—
638C2×C4#1451982312yes—
639C2×C4#1451982412yes—
640C2×C4#1451982512yes—
641C22—1222no—
642C22—1222no—
643C22—1222no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22C23×C4×D6111yescharacteristic
649C22C2×#1451982211yescharacteristic
650C22C2×#1451982311yescharacteristic
651C222−1+4×D6111yescharacteristic
652C222−1+4×D6111yescharacteristic
653C22C2×#1451982411yescharacteristic
654C22C2×#1451982511yescharacteristic
655C4—22no—
656C4—28no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—614no—
662C4—1222no—
663C4—1222no—
664C4—1228no—
665C4—2442no—
666C4—2442no—
667C4—2442no—
668C4—2442no—
669C4#1684024114yes—
670C4#1684024414yes—
671C2—2442no—
672C2#519×D6111yescharacteristic
673C2C2×#1684024111yescharacteristic
674C2C2×#1684024411yescharacteristic
675C2#2146021611yescharacteristic
676C2#2146022611yescharacteristic
677C2#2146023211yescharacteristic
678C2#2146031111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2365C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2365D61
hypocenterC61#34374
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #21449002C2semidirect
3. #21448879C2semidirect
4. #21448877C2semidirect
5. #21448875C2semidirect
6. #21448775C2semidirect
7. #21448772C2semidirect
8. #21448733C2semidirect
9. Q8×#11758196C2semidirect
17. C61×#2365C2semidirect
18. #16840317C4semidirect
337. #2365D61semidirect
423. C61#34374semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214490022
#214488792
#214488772
#214488752
#214487752
#214487722
#214487332
Q8×#117581962
#214485042
#214485022
#214484992
#214484952
#214484252
#214482232
C61×#23652
#3437461