Finite Group Databasebeta

← #29842888

Subgroups — #29842888

Order 15616.

2238 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic
2#21448997C214yesmaximal, complemented
3#21448879C211yesmaximal, characteristic, complemented
4#21448877C211yesmaximal, characteristic, complemented
5#21448875C211yesmaximal, characteristic, complemented
6#21448695C212yesmaximal
7#21448692C212yesmaximal
8#21448653C212yesmaximal
9Q8×#11758198C212yesmaximal
10#21448566C212yesmaximal, complemented
11#21448563C212yesmaximal, complemented
12#21448537C212yesmaximal, complemented
13#21448535C212yesmaximal, complemented
14#21448418C214yesmaximal
15#21448215C211yesmaximal, characteristic
16#21448215C211yesmaximal, characteristic
17C61×#2365C211yesmaximal, characteristic, complemented
18C22×Q8×D61C2212yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519776C2212yes—
21C2×#14519776C2212yes—
22C2×#14519775C2212yes—
23C4⋊Q8×D61C4116yescomplemented
24C2×Q8×Dic61C2212yes—
25C2×C4×Dic122C2212yes—
26C2×#14519742C2212yes—
27C2×C4⋊C4×D61C2212yes—
28C2×C4⋊C4×D61C2212yes—
29C2×C42×D61C2211yescharacteristic
30#16839669C2212yes—
31#16839668C2214yes—
32#16839668C2212yes—
33#16839667C2214yes—
34#16839667C2212yes—
35#16839666C2214yes—
36#16839666C2214yes—
37#16839661C2212yes—
38#16839660C2212yes—
39#16839658C2212yes—
40#16839649C2214yes—
41#16839649C2214yes—
42#16839648C2214yes—
43#16839647C2214yes—
44#16839646C2214yes—
45#16839646C2214yes—
46#16839645C2214yes—
47#16839644C2214yes—
48#16839644C2214yes—
49#16839643C2214yes—
50#16839643C2214yes—
51#16839629C2212yes—
52#16839629C2212yes—
53#16839627C2211yescharacteristic
54#16839627C2211yescharacteristic
55#16839627C2211yescharacteristic
56#16839627C2211yescharacteristic
57#16839624C2212yes—
58#16839624C2212yes—
59#16839621C2214yes—
60#16839619C2211yescharacteristic
61#16839619C2211yescharacteristic
62#16839619C2211yescharacteristic
63#16839619C2211yescharacteristic
64#16839619C2214yes—
65C4⋊C4×Dic61C2212yes—
66C4⋊C4×Dic61C2212yes—
67C4×C4⋊Dic61C2211yescharacteristic
68C4×C4⋊Dic61C2211yescharacteristic
69C4×C4⋊Dic61C2212yes—
70C4×#11758198C2212yes—
71C4×#11758198C2212yes—
72C4×#11758198C2214yes—
73#16839609C2214yes—
74#16839608C2212yes—
75#16839608C2212yes—
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×Dic122C2314yes—
86C2×Q8×D61—44no—
87C2×Q8×D61—44no—
88C2×Q8×D61C2×C4116yes—
89C22×C4×D61C2311yescharacteristic
90C22×C4×D61C2311yescharacteristic
91C22×C4×D61C2311yescharacteristic
92C22×C4×D61C2314yes—
93#14519777C2×C418yes—
94#14519776C2×C4116yes—
95#14519776C2×C4116yes—
96#14519775C2×C4116yes—
97Q8×Dic61—44no—
98C4×Dic122—44no—
99#14519742—44no—
100C2×C4⋊Dic61C2311yescharacteristic
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2311yescharacteristic
104C2×C4⋊Dic61C2314yes—
105C2×C4⋊Dic61C2312yes—
106C2×C4⋊Dic61C2312yes—
107C2×#11758196C2314yes—
108C2×#11758196C2314yes—
109C2×#11758196C2314yes—
110C2×#11758196C2314yes—
111C4⋊C4×D61C2×C4116yes—
112C4⋊C4×D61C2×C4116yes—
113C2×#11758198C2312yes—
114C2×#11758198C2312yes—
115C2×#11758198C2312yes—
116C2×#11758198C2312yes—
117C2×C4×Dic61C2312yes—
118C2×C4×Dic61C2312yes—
119C2×C4×Dic61C2314yes—
120C2×C4×Dic61C2314yes—
121C2×C4×Dic61C2314yes—
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124C2×C4×Dic61C2311yescharacteristic
125C42×D61C2×C418yes—
126#14519684C2312yes—
127#14519684C2312yes—
128#14519684C2312yes—
129#14519684C2312yes—
130#14519684C2312yes—
131#14519684C2312yes—
132#14519684C2312yes—
133#14519684C2312yes—
134#14519684C2314yes—
135#14519684C2312yes—
136#14519684C2312yes—
137#14519684C2312yes—
138#14519684C2312yes—
139#14519684C2314yes—
140#14519684C2314yes—
141#14519684C2314yes—
142#14519684C2312yes—
143#14519684C2312yes—
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×C61C2414yes—
173C22×C4×C61C2411yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×Dic61C2412yes—
177C22×Dic61C2412yes—
178C22×Dic61C2412yes—
179C22×Dic61C2412yes—
180C22×Dic61C2411yescharacteristic
181C22×Dic61C2414yes—
182C22×Dic61C2411yescharacteristic
183C22×Dic61C2411yescharacteristic
184Q8×D61—432no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—22no—
188C2×C4×D61—22no—
189C2×C4×D61—22no—
190C2×C4×D61—22no—
191C2×C4×D61—44no—
192C2×C4×D61—44no—
193C2×C4×D61C22×C414yes—
194C2×C4×D61C22×C414yes—
195C2×C4×D61C22×C414yes—
196C2×C4×D61C22×C4116yes—
197C2×Dic122—24no—
198C2×Dic122—24no—
199C2×Dic122—44no—
200C2×Dic122—44no—
201C2×Dic122C22×C418yes—
202C2×Dic122C22×C4116yes—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—44no—
208C4⋊C4×C61C22×C418yes—
209C4⋊C4×C61C22×C418yes—
210C42×C61—24no—
211C42×C61—24no—
212C42×C61—44no—
213C42×C61C22×C414yes—
214#11758198—24no—
215#11758198—24no—
216#11758198—24no—
217#11758198—24no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—22no—
221C4⋊Dic61—22no—
222C4⋊Dic61—44no—
223C4⋊Dic61C22×C418yes—
224C4⋊Dic61C22×C418yes—
225#11758196—44no—
226#11758196—44no—
227#11758196C22×C4116yes—
228#11758196C22×C4116yes—
229C4×Dic61—24no—
230C4×Dic61—24no—
231C4×Dic61—44no—
232C4×Dic61—44no—
233C4×Dic61—44no—
234C4×Dic61C22×C414yes—
235C4×Dic61C22×C414yes—
236C4×Dic61C22×C414yes—
237C23×C61C2511yescharacteristic
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61C23×C412yes—
243C22×D612−1+412yes—
244C22×D612−1+412yes—
245Q8×C61—216no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
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×C61C23×C412yes—
263C2×C4×C61C23×C418yes—
264C2×C4×C61C23×C412yes—
265C2×C4×C61C23×C412yes—
266C2×C4×C61H(5,𝔽2)12yes—
267C2×C4×C61H(5,𝔽2)12yes—
268C2×C4×C612−1+412yes—
269C2×C4×C612−1+412yes—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—22no—
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×Dic61C23×C412yes—
291C2×Dic61C23×C412yes—
292C2×Dic61C23×C412yes—
293C2×Dic61C23×C418yes—
294C4×D61—28no—
295C4×D61—28no—
296C4×D61—28no—
297C4×D61—416no—
298Dic122—216no—
299Dic122—416no—
300#34371—611nomaximal, Sylow
301C22×C61C24×C411yescharacteristic
302C22×C61C2×H(5,𝔽2)11yescharacteristic
303C22×C61C2×H(5,𝔽2)11yescharacteristic
304C22×C61C2×2−1+411yescharacteristic
305C22×C61C2×2−1+411yescharacteristic
306C22×C61C2×2−1+411yescharacteristic
307C22×C61C2×2−1+411yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61#51914yes—
315C4×C61—28no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61#51814yes—
322C4×C61#51914yes—
323Dic61—28no—
324Dic61—22no—
325Dic61—22no—
326Dic61—22no—
327Dic61—42no—
328Dic61—42no—
329Dic61—42no—
330Dic61—42no—
331C22×C4⋊Q8—611no—
332#2381—614no—
333#2365—611no—
334#2365—611no—
335#2365—611no—
336#2365D6111yescharacteristic, complemented
337#2345—612no—
338#2345—612no—
339Q8×C22⋊C4—612no—
340Q8×C22⋊C4—612no—
341#2331—612no—
342#2331—612no—
343#2327—612no—
344#2327—612no—
345#2313—614no—
346#2283—611no—
347#2283—611no—
348C2×C61C2×#51811yescharacteristic
349C2×C61C2×#51911yescharacteristic
350C2×C61C2×#51911yescharacteristic
351C2×C61#355211yescharacteristic
352C2×C61#355211yescharacteristic
353C2×C61#355211yescharacteristic
354C2×C61#355211yescharacteristic
355D61—24no—
356C23×Q8—612no—
357C2×C4⋊Q8—612no—
358C2×C4⋊Q8—6116no—
359C2×C4⋊Q8—612no—
360C2×C4⋊Q8—612no—
361C2×C4⋊Q8—611no—
362C2×C4⋊Q8C2×D6111yescharacteristic
363C2×C4×Q8—612no—
364C2×C4×Q8—612no—
365C2×C4×Q8—612no—
366C2×C4×Q8C2×D6112yes—
367C22×C4⋊C4—612no—
368C22×C4⋊C4—612no—
369C22×C42—611no—
370#391—612no—
371#391—612no—
372#391—614no—
373#391—612no—
374#391—614no—
375#391—614no—
376#391—614no—
377#391C2×D6112yes—
378#389—612no—
379#389—612no—
380#389—612no—
381#389C2×D6112yes—
382#387—614no—
383#387—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#387C2×D6114yes—
394#383—612no—
395#383—612no—
396#382—611no—
397#382—611no—
398#382—611no—
399#382—611no—
400#381—612no—
401#381—612no—
402#380—614no—
403#380—611no—
404#380—611no—
405#380—611no—
406#380—611no—
407#380—614no—
408C4×C4⋊C4—612no—
409C4×C4⋊C4—612no—
410C4×C4⋊C4—612no—
411C4×C4⋊C4—611no—
412C4×C4⋊C4—611no—
413C4×C4⋊C4C2×D6112yes—
414C4×C4⋊C4C2×D6111yescharacteristic
415C4×C4⋊C4C2×D6111yescharacteristic
416C4×C22⋊C4—614no—
417C4×C22⋊C4—612no—
418C4×C22⋊C4—612no—
419C4×C22⋊C4—614no—
420#376—612no—
421#376—612no—
422C61#3437111yescharacteristic, complemented, Sylow
423C22×Q8—6116no—
424C22×Q8—612no—
425C22×Q8—614no—
426C22×Q8—2444no—
427C22×Q8—2444no—
428C22×Q8C22×D6112yes—
429C23×C4—611no—
430C23×C4—611no—
431C23×C4—611no—
432C23×C4—614no—
433C4⋊Q8—6116no—
434C4⋊Q8—6116no—
435C4⋊Q8—6116no—
436C4⋊Q8—618no—
437C4⋊Q8C4×D6118yes—
438C4×Q8—44no—
439C4×Q8—2444no—
440C4×Q8—2444no—
441C4×Q8—2444no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—614no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—6116no—
447C2×C4⋊C4—6116no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—614no—
453C2×C4⋊C4—614no—
454C2×C4⋊C4—614no—
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6114yes—
460C2×C4⋊C4C22×D6112yes—
461C2×C4⋊C4C22×D6112yes—
462C2×C22⋊C4—612no—
463C2×C22⋊C4—612no—
464C2×C22⋊C4—612no—
465C2×C22⋊C4—612no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—618no—
470C2×C42—612no—
471C2×C42—612no—
472C2×C42—614no—
473C2×C42—614no—
474C2×C42—614no—
475C2×C42C22×D6112yes—
476C2×C42C22×D6112yes—
477C2×C42C22×D6114yes—
478C2×C42C22×D6111yescharacteristic
479#95—612no—
480#95—612no—
481#95—612no—
482#95—612no—
483#95—612no—
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—614no—
497#95—612no—
498#95—612no—
499#95C22×D6114yes—
500#95C22×D6114yes—
501#95C22×D6112yes—
502#95C22×D6112yes—
503C24—611no—
504C2×Q8—24no—
505C2×Q8—24no—
506C2×Q8—618no—
507C2×Q8—6116no—
508C2×Q8—1224no—
509C2×Q8—1224no—
510C2×Q8—2444no—
511C2×Q8—2444no—
512C2×Q8—24432no—
513C2×Q8C2×C4×D6118yes—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—614no—
518C22×C4—614no—
519C22×C4—614no—
520C22×C4—614no—
521C22×C4—6116no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—612no—
525C22×C4—612no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—2444no—
533C22×C4—2444no—
534C22×C4C23×D6112yes—
535C22×C4C23×D6112yes—
536C22×C4C23×D6112yes—
537C22×C4C23×D6112yes—
538C22×C4C23×D6111yescharacteristic
539C22×C4C23×D6114yes—
540C22×C4C23×D6111yescharacteristic
541C22×C4C23×D6111yescharacteristic
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—44no—
547C4⋊C4—6116no—
548C4⋊C4—6116no—
549C4⋊C4—618no—
550C4⋊C4—618no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—2444no—
556C4⋊C4—2444no—
557C4⋊C4—2444no—
558C4⋊C4C2×C4×D6118yes—
559C4⋊C4C2×C4×D6118yes—
560C22⋊C4—1224no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C42—24no—
565C42—24no—
566C42—44no—
567C42—614no—
568C42—614no—
569C42—614no—
570C42—1224no—
571C42—1224no—
572C42—2444no—
573C42—2444no—
574C42—2444no—
575C42C2×C4×D6114yes—
576C23—612no—
577C23—612no—
578C23—612no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23C24×D6111yescharacteristic
584Q8—216no—
585Q8—12216no—
586Q8—24416no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
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—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—618no—
607C2×C4—1228no—
608C2×C4—1224no—
609C2×C4—1221no—
610C2×C4—1224no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1228no—
617C2×C4—1228no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
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—24416no—
631C2×C4C22×C4×D6112yes—
632C2×C4C22×C4×D6118yes—
633C2×C4C22×C4×D6112yes—
634C2×C4C22×C4×D6112yes—
635C2×C4#1451982212yes—
636C2×C4#1451982312yes—
637C2×C4#1451982412yes—
638C2×C4#1451982512yes—
639C22—614no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22C23×C4×D6111yescharacteristic
647C22C2×#1451982211yescharacteristic
648C22C2×#1451982311yescharacteristic
649C222−1+4×D6111yescharacteristic
650C222−1+4×D6111yescharacteristic
651C22C2×#1451982411yescharacteristic
652C22C2×#1451982511yescharacteristic
653C4—28no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—1222no—
660C4—1228no—
661C4—1222no—
662C4—1222no—
663C4—2442no—
664C4—2442no—
665C4—2442no—
666C4—2442no—
667C4#1684023414yes—
668C4#1684023814yes—
669C2—1224no—
670C2C2×#1684023411yescharacteristic
671C2#519×D6111yescharacteristic
672C2C2×#1684023811yescharacteristic
673C2#2146024911yescharacteristic
674C2#2146025711yescharacteristic
675C2#2146026111yescharacteristic
676C2#2146026511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2365C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2365D61
hypocenterC61#34371
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #21448997C2semidirect
3. #21448879C2semidirect
4. #21448877C2semidirect
5. #21448875C2semidirect
10. #21448566C2semidirect
11. #21448563C2semidirect
12. #21448537C2semidirect
13. #21448535C2semidirect
17. C61×#2365C2semidirect
23. C4⋊Q8×D61C4semidirect
336. #2365D61semidirect
422. C61#34371semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
#214489972
#214488792
#214488772
#214488752
#214486952
#214486922
#214486532
Q8×#117581982
#214485662
#214485632
#214485372
#214485352
#214484182
#214482152
C61×#23652
#3437161