Finite Group Databasebeta

← #29842913

Subgroups — #29842913

Order 15616.

1998 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×#16840316C211yesmaximal, characteristic
2#21448981C214yesmaximal
3#21448845C212yesmaximal
4#21448842C212yesmaximal
5#21448778C211yesmaximal, characteristic, complemented
6#21448778C212yesmaximal, complemented
7#21448777C212yesmaximal, complemented
8#21448776C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
10#21448617C214yesmaximal, complemented
11#21448566C212yesmaximal, complemented
12#21448535C212yesmaximal, complemented
13#21448427C212yesmaximal
14#21448424C212yesmaximal
15#21448224C211yesmaximal, characteristic
16#21448215C211yesmaximal, characteristic
17Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
18#16840316C4116yescomplemented
19C2×#14519776C2211yescharacteristic
20C2×#14519775C2211yescharacteristic
21C2×#14519767C2212yes—
22C2×#14519757C2211yescharacteristic
23C2×#14519755C2212yes—
24C2×#14519753C2211yescharacteristic
25C2×C4×Dic122C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C2×C4×Dic122C2212yes—
28C2×#14519735C2212yes—
29C2×#14519734C2212yes—
30C2×C42×D61C2211yescharacteristic
31#16839669C2212yes—
32#16839667C2214yes—
33#16839667C2212yes—
34#16839666C2214yes—
35#16839660C2212yes—
36#16839659C2214yes—
37#16839653C2214yes—
38#16839652C2214yes—
39#16839652C2214yes—
40#16839651C2214yes—
41#16839648C2214yes—
42#16839646C2214yes—
43#16839646C2214yes—
44#16839645C2214yes—
45#16839644C2214yes—
46#16839643C2214yes—
47#16839643C2214yes—
48#16839643C2214yes—
49#16839643C2214yes—
50#16839643C2214yes—
51#16839632C2211yescharacteristic
52#16839631C2212yes—
53#16839629C2212yes—
54#16839628C2212yes—
55#16839627C2211yescharacteristic
56#16839627C2211yescharacteristic
57#16839627C2212yes—
58#16839627C2212yes—
59#16839624C2212yes—
60#16839619C2212yes—
61#16839619C2211yescharacteristic
62#16839619C2211yescharacteristic
63#16839619C2212yes—
64C4⋊C4×Dic61C2212yes—
65C4⋊C4×Dic61C2212yes—
66C4⋊C4×Dic61C2212yes—
67C4⋊C4×Dic61C2212yes—
68C4⋊C4×Dic61C2212yes—
69C4⋊C4×Dic61C2212yes—
70C4⋊C4×Dic61C2214yes—
71C4×#11758196C2214yes—
72C4×C4⋊Dic61C2211yescharacteristic
73C4×C4⋊Dic61C2212yes—
74C4×#11758198C2212yes—
75#16839608C2212yes—
76C2×C4×Q8×C61C2212yes—
77C2×C4×Q8×C61C2211yescharacteristic
78C61×#389C2214yes—
79C61×#389C2212yes—
80C61×#384C2211yescharacteristic
81C61×#384C2212yes—
82C4×C4⋊C4×C61C2211yescharacteristic
83C4×C4⋊C4×C61C2212yes—
84C22×Dic122C2311yescharacteristic
85C22×Dic122C2312yes—
86C22×C4×D61C2311yescharacteristic
87C22×C4×D61C2311yescharacteristic
88C22×C4×D61C2311yescharacteristic
89#14519776C2×C418yes—
90#14519775C2×C418yes—
91#14519767C2×C4116yes—
92#14519757C2×C418yes—
93#14519755C2×C4116yes—
94#14519753C2×C418yes—
95C4×Dic122—44no—
96C4×Dic122C2×C4116yes—
97C4×Dic122C2×C418yes—
98#14519735—44no—
99#14519734C2×C4116yes—
100C2×C4⋊Dic61C2312yes—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2312yes—
104C2×C4⋊Dic61C2312yes—
105C2×C4⋊Dic61C2311yescharacteristic
106C2×C4⋊Dic61C2312yes—
107C2×#11758196C2314yes—
108C2×#11758196C2314yes—
109C2×#11758196C2312yes—
110C2×#11758196C2312yes—
111C2×#11758196C2312yes—
112C2×#11758196C2312yes—
113C2×#11758196C2312yes—
114C2×#11758198C2314yes—
115C2×#11758198C2312yes—
116C2×#11758198C2312yes—
117C2×#11758198C2312yes—
118C2×#11758198C2312yes—
119C2×C4×Dic61C2312yes—
120C2×C4×Dic61C2314yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2314yes—
124C2×C4×Dic61C2311yescharacteristic
125C2×C4×Dic61C2311yescharacteristic
126C2×C4×Dic61C2311yescharacteristic
127C2×C4×Dic61C2312yes—
128C42×D61C2×C418yes—
129#14519684C2312yes—
130#14519684C2312yes—
131#14519684C2312yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2312yes—
135#14519684C2312yes—
136#14519684C2314yes—
137#14519684C2314yes—
138#14519684C2314yes—
139#14519684C2314yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2314yes—
143#14519684C2314yes—
144#14519684C2314yes—
145C22×Q8×C61C2311yescharacteristic
146C4×Q8×C61—44no—
147C4×Q8×C61C2×C418yes—
148C2×C4⋊C4×C61C2312yes—
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C4⋊C4×C61C2311yescharacteristic
151C2×C4⋊C4×C61C2312yes—
152C2×C4⋊C4×C61C2312yes—
153C2×C4⋊C4×C61C2314yes—
154C2×C4⋊C4×C61C2314yes—
155C2×C4⋊C4×C61C2312yes—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C42×C61C2312yes—
158C2×C42×C61C2314yes—
159C2×C42×C61C2311yescharacteristic
160C2×C42×C61C2312yes—
161#95×C61C2312yes—
162#95×C61C2314yes—
163C23×D61C2411yescharacteristic
164C2×Q8×C61—22no—
165C2×Q8×C61—22no—
166C2×Q8×C61C22×C414yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2414yes—
170C22×C4×C61C2411yescharacteristic
171C22×C4×C61C2412yes—
172C22×C4×C61C2412yes—
173C22×C4×C61C2411yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×Dic61C2412yes—
176C22×Dic61C2412yes—
177C22×Dic61C2414yes—
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2411yescharacteristic
181C22×Dic61C2412yes—
182C22×Dic61C2412yes—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—41no—
188C2×C4×D61—41no—
189C2×C4×D61C22×C414yes—
190C2×C4×D61C22×C414yes—
191C2×C4×D61C22×C414yes—
192C2×Dic122—22no—
193C2×Dic122—22no—
194C2×Dic122—42no—
195C2×Dic122—42no—
196C2×Dic122C22×C414yes—
197C2×Dic122C22×C418yes—
198C4⋊C4×C61—24no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—44no—
203C4⋊C4×C61—42no—
204C4⋊C4×C61—44no—
205C4⋊C4×C61C22×C418yes—
206C4⋊C4×C61C22×C414yes—
207C42×C61—24no—
208C42×C61—44no—
209C42×C61C22×C414yes—
210C42×C61C22×C418yes—
211#11758198—42no—
212#11758198—42no—
213#11758198—44no—
214#11758198C22×C418yes—
215#11758198C22×C418yes—
216C4⋊Dic61—24no—
217C4⋊Dic61—22no—
218C4⋊Dic61—22no—
219C4⋊Dic61—24no—
220C4⋊Dic61—42no—
221C4⋊Dic61C22×C414yes—
222C4⋊Dic61C22×C418yes—
223#11758196—44no—
224#11758196—44no—
225#11758196C22×C418yes—
226#11758196C22×C418yes—
227#11758196C22×C418yes—
228#11758196C22×C418yes—
229#11758196C22×C418yes—
230C4×Dic61—24no—
231C4×Dic61—44no—
232C4×Dic61—44no—
233C4×Dic61—44no—
234C4×Dic61—44no—
235C4×Dic61C22×C414yes—
236C4×Dic61C22×C414yes—
237C4×Dic61C22×C414yes—
238C4×Dic61C22×C418yes—
239C23×C61C2511yescharacteristic
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246C22×D61C23×C412yes—
247Q8×C61—28no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—24no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—24no—
259C2×C4×C61—22no—
260C2×C4×C61—24no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61C23×C412yes—
264C2×C4×C61C23×C414yes—
265C2×C4×C61C23×C414yes—
266C2×C4×C61C23×C412yes—
267C2×C4×C61C23×C412yes—
268C2×C4×C61H(5,𝔽2)12yes—
269C2×C4×C612−1+412yes—
270C2×C4×C612−1+412yes—
271C2×C4×C612−1+412yes—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—24no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—22no—
290C2×Dic61—24no—
291C2×Dic61C23×C412yes—
292C2×Dic61C23×C412yes—
293C2×Dic61C23×C412yes—
294C2×Dic61C23×C414yes—
295C2×Dic61C23×C414yes—
296C4×D61—28no—
297C4×D61—28no—
298C4×D61—44no—
299Dic122—28no—
300Dic122—48no—
301#34373—611nomaximal, Sylow
302C22×C61C24×C411yescharacteristic
303C22×C61C2×H(5,𝔽2)11yescharacteristic
304C22×C61C2×2−1+411yescharacteristic
305C22×C61C2×2−1+411yescharacteristic
306C22×C61C2×2−1+411yescharacteristic
307C22×C61#58511yescharacteristic
308C22×C61#58511yescharacteristic
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—24no—
317C4×C61—24no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61—42no—
322C4×C61#51914yes—
323C4×C61#52014yes—
324Dic61—22no—
325Dic61—24no—
326Dic61—22no—
327Dic61—22no—
328Dic61—24no—
329Dic61—44no—
330Dic61—42no—
331Dic61—42no—
332C2×#533—611no—
333#2378—614no—
334#2362—612no—
335#2362—612no—
336#2355—611no—
337#2355—612no—
338#2355—612no—
339#2355—611no—
340Q8×C4⋊C4—611no—
341Q8×C4⋊C4D6111yescharacteristic, complemented
342#2336—614no—
343#2331—612no—
344#2327—612no—
345#2314—612no—
346#2314—612no—
347#2285—611no—
348#2283—611no—
349C2×C61C2×#51911yescharacteristic
350C2×C61C2×#52011yescharacteristic
351C2×C61#342811yescharacteristic
352C2×C61#355411yescharacteristic
353C2×C61#355411yescharacteristic
354C2×C61#355511yescharacteristic
355C2×C61#355511yescharacteristic
356D61—42no—
357#533—6116no—
358C2×C4⋊Q8—611no—
359C2×C4⋊Q8—611no—
360C2×#125—612no—
361C2×C22⋊Q8—611no—
362C2×C22⋊Q8—611no—
363C2×C22⋊Q8—612no—
364C2×C4×Q8—611no—
365C2×C4×Q8—612no—
366C2×C4×Q8—612no—
367C2×C4×Q8C2×D6112yes—
368C2×C4×Q8C2×D6111yescharacteristic
369C2×#117—612no—
370C2×#117—612no—
371C22×C42—611no—
372#391—612no—
373#391—612no—
374#391—614no—
375#391—614no—
376#389—612no—
377#389—614no—
378#389C2×D6112yes—
379#389C2×D6114yes—
380#388—614no—
381#388—614no—
382#388—614no—
383#388—614no—
384#387—614no—
385#387—614no—
386#387—614no—
387#387—614no—
388#387—614no—
389#387—614no—
390#387—614no—
391#387—614no—
392#387—614no—
393#387—614no—
394#384—612no—
395#384—611no—
396#384C2×D6112yes—
397#384C2×D6111yescharacteristic
398#383—612no—
399#382—612no—
400#382—612no—
401#382—611no—
402#382—611no—
403#382—612no—
404#381—612no—
405#380—611no—
406#380—611no—
407#380—612no—
408#380—612no—
409C4×C4⋊C4—612no—
410C4×C4⋊C4—612no—
411C4×C4⋊C4—612no—
412C4×C4⋊C4—612no—
413C4×C4⋊C4—612no—
414C4×C4⋊C4—614no—
415C4×C4⋊C4—612no—
416C4×C4⋊C4—612no—
417C4×C4⋊C4—614no—
418C4×C4⋊C4—611no—
419C4×C4⋊C4C2×D6112yes—
420C4×C4⋊C4C2×D6111yescharacteristic
421C4×C22⋊C4—612no—
422#376—612no—
423C61#3437311yescharacteristic, complemented, Sylow
424C22×Q8—611no—
425C22×Q8—612no—
426C22×Q8C22×D6111yescharacteristic
427C23×C4—611no—
428C23×C4—611no—
429C23×C4—611no—
430C4⋊Q8—618no—
431C4⋊Q8—618no—
432#125—6116no—
433C22⋊Q8—618no—
434C22⋊Q8—6116no—
435C22⋊Q8—618no—
436C4×Q8—44no—
437C4×Q8—6116no—
438C4×Q8—618no—
439C4×Q8—2444no—
440C4×Q8C4×D6118yes—
441#117—6116no—
442#117—2444no—
443C2×C4⋊C4—612no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—612no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—612no—
454C2×C4⋊C4—614no—
455C2×C4⋊C4—612no—
456C2×C4⋊C4—614no—
457C2×C4⋊C4C22×D6112yes—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6112yes—
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6114yes—
462C2×C4⋊C4C22×D6112yes—
463C2×C4⋊C4C22×D6114yes—
464C2×C4⋊C4C22×D6112yes—
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—612no—
467C2×C22⋊C4—612no—
468C2×C22⋊C4—614no—
469C2×C22⋊C4—612no—
470C2×C22⋊C4—612no—
471C2×C42—612no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—618no—
476C2×C42—612no—
477C2×C42—614no—
478C2×C42—614no—
479C2×C42—614no—
480C2×C42—614no—
481C2×C42C22×D6112yes—
482C2×C42C22×D6114yes—
483C2×C42C22×D6112yes—
484C2×C42C22×D6111yescharacteristic
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—612no—
494#95—614no—
495#95—614no—
496#95—614no—
497#95—614no—
498#95—614no—
499#95—614no—
500#95—614no—
501#95C22×D6112yes—
502#95C22×D6114yes—
503C24—611no—
504C2×Q8—22no—
505C2×Q8—22no—
506C2×Q8—614no—
507C2×Q8—618no—
508C2×Q8—1222no—
509C2×Q8—1222no—
510C2×Q8—2442no—
511C2×Q8—2442no—
512C2×Q8C2×C4×D6114yes—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—614no—
519C22×C4—614no—
520C22×C4—614no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—614no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—2441no—
529C22×C4—2441no—
530C22×C4C23×D6112yes—
531C22×C4C23×D6114yes—
532C22×C4C23×D6112yes—
533C22×C4C23×D6112yes—
534C22×C4C23×D6112yes—
535C22×C4C23×D6111yescharacteristic
536C22×C4C23×D6111yescharacteristic
537C22×C4C23×D6111yescharacteristic
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—618no—
546C4⋊C4—618no—
547C4⋊C4—618no—
548C4⋊C4—618no—
549C4⋊C4—618no—
550C4⋊C4—618no—
551C4⋊C4—614no—
552C4⋊C4—1224no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1224no—
556C4⋊C4—2444no—
557C4⋊C4—2444no—
558C4⋊C4—2442no—
559C4⋊C4C2×C4×D6118yes—
560C4⋊C4C2×C4×D6114yes—
561C22⋊C4—618no—
562C22⋊C4—618no—
563C22⋊C4—2444no—
564C22⋊C4—2442no—
565C22⋊C4—2442no—
566C42—24no—
567C42—44no—
568C42—614no—
569C42—614no—
570C42—614no—
571C42—618no—
572C42—1224no—
573C42—2444no—
574C42—2444no—
575C42—2444no—
576C42—2444no—
577C42C2×C4×D6114yes—
578C42C2×C4×D6118yes—
579C23—612no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23C24×D6111yescharacteristic
587Q8—28no—
588Q8—1228no—
589Q8—2448no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—614no—
609C2×C4—614no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1228no—
621C2×C4—1228no—
622C2×C4—1224no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1224no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1224no—
631C2×C4—2444no—
632C2×C4C22×C4×D6112yes—
633C2×C4C22×C4×D6114yes—
634C2×C4C22×C4×D6114yes—
635C2×C4C22×C4×D6112yes—
636C2×C4C22×C4×D6112yes—
637C2×C4#1451982212yes—
638C2×C4#1451982412yes—
639C2×C4#1451982512yes—
640C2×C4#1451982512yes—
641C22—1222no—
642C22—1222no—
643C22—1222no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22C23×C4×D6111yescharacteristic
649C22C2×#1451982211yescharacteristic
650C22C2×#1451982411yescharacteristic
651C22C2×#1451982511yescharacteristic
652C22C2×#1451982511yescharacteristic
653C22#1684059711yescharacteristic
654C22#1684059911yescharacteristic
655C4—24no—
656C4—22no—
657C4—24no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—1222no—
662C4—1222no—
663C4—1224no—
664C4—1222no—
665C4—1224no—
666C4—2444no—
667C4—2442no—
668C4—2442no—
669C4#1684023814yes—
670C4#1684024314yes—
671C2—2442no—
672C2C2×#1684023811yescharacteristic
673C2C2×#1684024311yescharacteristic
674C2#2145901911yescharacteristic
675C2#2146028611yescharacteristic
676C2#2146029111yescharacteristic
677C2#2146030311yescharacteristic
678C2#2146030811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingQ8×C4⋊C4×C61C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenterQ8×C4⋊C4D61
hypocenterC61#34373
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #21448778C2semidirect
6. #21448778C2semidirect
7. #21448777C2semidirect
8. #21448776C2semidirect
9. C4⋊C4×Dic122C2semidirect
10. #21448617C2semidirect
11. #21448566C2semidirect
12. #21448535C2semidirect
17. Q8×C4⋊C4×C61C2semidirect
18. #16840316C4semidirect
341. Q8×C4⋊C4D61semidirect
423. C61#34373semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
#214489812
#214488452
#214488422
#214487782
#214487772
#214487762
C4⋊C4×Dic1222
#214486172
#214485662
#214485352
#214484272
#214484242
#214482242
#214482152
Q8×C4⋊C4×C612
#3437361