Finite Group Databasebeta

← #29683769

Subgroups — #29683769

Order 15616.

1598 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840339C211yesmaximal, characteristic
2C2×#16839734C214yesmaximal, complemented
3#21448436C211yesmaximal, characteristic, complemented
4#21448174C211yesmaximal, characteristic, complemented
5#21446656C211yesmaximal, characteristic, complemented
6#21446655C211yesmaximal, characteristic, complemented
7#21446654C211yesmaximal, characteristic, complemented
8#21446653C214yesmaximal, complemented
9C61×#1964C211yesmaximal, characteristic, complemented
10C22×#11758224C2212yes—
11#16840339—22no—
12#16840339—22no—
13#16840339—22no—
14#16840339—22no—
15C2×#14519755C412yescomplemented
16C2×#14519754C412yescomplemented
17C2×#14519751C2211yescharacteristic
18C2×#14519750C412yescomplemented
19C2×#14519749C412yescomplemented
20C2×#14519749C2212yes—
21C2×#14519733C2212yes—
22C2×#14519733C2212yes—
23C22×#11758196C2211yescharacteristic
24#16839734—216no—
25C2×#14519692C2214yes—
26C2×#14519691C2214yes—
27C2×#14519691C2214yes—
28#16839659—22no—
29#16839657—22no—
30#16839622C2211yescharacteristic
31#16839620C2212yes—
32C4×#11758196—22no—
33#16839613—22no—
34C22⋊C4×Dic61C2212yes—
35C4×C22⋊Dic61C2211yescharacteristic
36C2×#121×C61C2211yescharacteristic
37C2×#99×C61C2214yes—
38C61×#380C2211yescharacteristic
39C4×C22⋊C4×C61C2211yescharacteristic
40C2×#11758224—28no—
41C2×#11758224—24no—
42C2×#11758224—24no—
43C2×#11758224—28no—
44C22×Dic122C2×C412yes—
45C22×D244C2×C411yescharacteristic
46C22×C22⋊D61C2×C412yes—
47C22×C22⋊D61C2×C412yes—
48C22×C22⋊D61C2312yes—
49C23×Dic61C2×C412yes—
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52C22×C4×D61C2×C412yes—
53#14519755—24no—
54#14519755—24no—
55#14519754—24no—
56#14519754—24no—
57#14519751—21no—
58#14519751—21no—
59#14519751—21no—
60#14519751—21no—
61#14519750—24no—
62#14519750—24no—
63#14519749—24no—
64#14519749—24no—
65#14519749—24no—
66#14519749—24no—
67#14519733—24no—
68#14519733—28no—
69#14519733D418yescomplemented
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758196C2×C412yes—
76C2×#11758196C2×C412yes—
77C2×#11758196C2×C412yes—
78C2×#11758196D412yes—
79C2×#11758196D412yes—
80C2×#11758196D412yes—
81C2×#11758196D412yes—
82C2×C22⋊Dic61C2×C412yes—
83C2×C22⋊Dic61C2312yes—
84C2×C22⋊Dic61C2312yes—
85C2×#11758198C2×C412yes—
86C2×#11758198C2×C412yes—
87C2×C4×Dic61—22no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—22no—
91#14519692—28no—
92#14519691—28no—
93#14519691—28no—
94#14519684—21no—
95#14519684—21no—
96#14519684—22no—
97#14519684—22no—
98C22×D4×C61C2×C411yescharacteristic
99C22×D4×C61C2312yes—
100C23×C4×C61C2311yescharacteristic
101#121×C61—21no—
102#121×C61—21no—
103#121×C61—21no—
104#121×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C22⋊C4×C61C2×C412yes—
108C2×C22⋊C4×C61C2312yes—
109C2×C22⋊C4×C61C2312yes—
110C2×C42×C61—21no—
111#99×C61—28no—
112#95×C61—21no—
113#95×C61—21no—
114C24×C61C22×C412yes—
115C24×C61C2411yescharacteristic
116C23×D61C22×C412yes—
117C2×D4×C61—24no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—81no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—81no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—24no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—28no—
141C2×C22⋊D61—82no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—28no—
145C22×Dic61—22no—
146C22×Dic61—81no—
147C22×Dic61C22×C412yes—
148C22×Dic61C22×C412yes—
149C22×Dic61C22×C412yes—
150C22×Dic61C2×D412yes—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D414yes—
155C22×Dic61#4114yes—
156#11758224—48no—
157#11758224—48no—
158#11758224—48no—
159#11758224—48no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—81no—
163C2×C4×D61—24no—
164C2×C4×D61—28no—
165C2×Dic122—24no—
166C2×Dic122—28no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—81no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61C2×D414yes—
175C42×C61—81no—
176C22⋊Dic61—22no—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C22⋊Dic61C2×D414yes—
180#11758198—22no—
181#11758198—22no—
182#11758198—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—81no—
186#11758196—24no—
187#11758196—28no—
188#11758196—44no—
189#11758196—82no—
190#11758196C22⋊C414yes—
191#11758196C22⋊C414yes—
192#11758196C22⋊C414yes—
193#11758196C22⋊C414yes—
194C4×Dic61—28no—
195C4×Dic61—44no—
196C4×Dic61—82no—
197C4×Dic61—81no—
198C23×C61—22no—
199C23×C61—24no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61C23×C411yescharacteristic
205C23×C61C22×D411yescharacteristic
206C23×C61C22×D411yescharacteristic
207C23×C61C22×D412yes—
208C23×C61C2×#4112yes—
209C22×D61—22no—
210C22×D61—24no—
211C22×D61—42no—
212C22×D61—42no—
213D4×C61—44no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—81no—
217D4×C61—81no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C22⋊D61—44no—
237C22⋊D61—42no—
238C22⋊D61—42no—
239C22⋊D61—48no—
240C22⋊D61—84no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—48no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C414yes—
254C2×Dic61C2×C22⋊C414yes—
255C2×Dic61C4×D418yes—
256C2×Dic61#12114yes—
257C2×Dic61#12114yes—
258C2×Dic61#12414yes—
259C2×Dic61#12714yes—
260D244—81no—
261D244—81no—
262C4×D61—48no—
263Dic122—48no—
264#21214—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—84no—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×C4×D412yes—
279C22×C61C2×#12112yes—
280C22×C61C2×#12411yescharacteristic
281C22×C61C2×#12711yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—44no—
291C4×C61—42no—
292C4×C61—82no—
293Dic61—24no—
294Dic61—44no—
295Dic61—42no—
296Dic61—82no—
297Dic61#39018yes—
298C2×#537—611no—
299C2×#410—614no—
300#2316—611no—
301#2274—611no—
302#1964—614no—
303#1964—611no—
304#1964—611no—
305#1964—611no—
306#1964D6111yescharacteristic, complemented
307C2×C61—22no—
308C2×C61—21no—
309C2×C61—22no—
310C2×C61—21no—
311C2×C61—82no—
312C2×C61C2×#39011yescharacteristic
313C2×C61#288311yescharacteristic
314C2×C61#288411yescharacteristic
315D61—82no—
316C22×#41—612no—
317#537—1222no—
318#537—1222no—
319#537—1222no—
320#537—1222no—
321C2×C22⋊Q8—612no—
322C2×C22⋊Q8—612no—
323C2×#121—612no—
324C2×#121—612no—
325C2×#121—611no—
326C2×#121—612no—
327C2×#121C2×D6111yescharacteristic
328C2×#117—612no—
329C2×#117—612no—
330C22×C4⋊C4—611no—
331#410—12216no—
332C2×#99—614no—
333C2×#99—614no—
334C2×#99—614no—
335C2×#99C2×D6114yes—
336#389—1222no—
337#389—1222no—
338#380—611no—
339#380—612no—
340#380C2×D6111yescharacteristic
341C4×C4⋊C4—1222no—
342C4×C4⋊C4—1222no—
343C4×C22⋊C4—611no—
344C4×C22⋊C4—612no—
345C4×C22⋊C4C2×D6111yescharacteristic
346C61#2121411yescharacteristic, complemented, Sylow
347C2×#41—1224no—
348C2×#41—1224no—
349C2×#41—1228no—
350C2×#41—1228no—
351C22×Q8—612no—
352C22×D4—612no—
353C22×D4—612no—
354C22×D4—612no—
355C22×D4—611no—
356C22×D4C4×D6111yescharacteristic
357C22×D4C22×D6112yes—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—612no—
361C23×C4—612no—
362C23×C4C22×D6111yescharacteristic
363C22⋊Q8—1224no—
364C22⋊Q8—1224no—
365C22⋊Q8—1224no—
366C22⋊Q8—1224no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121—21no—
371#121—1224no—
372#121—1224no—
373#121—1224no—
374#121—1224no—
375#121—1224no—
376#121—1224no—
377#121—1221no—
378#121—1221no—
379#121—1221no—
380#121—1221no—
381#117—618no—
382#117—1224no—
383#117—1228no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4C4×D6111yescharacteristic
398C2×C22⋊C4—612no—
399C2×C22⋊C4—612no—
400C2×C22⋊C4—612no—
401C2×C22⋊C4—612no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4C4×D6112yes—
404C2×C22⋊C4C22×D6112yes—
405C2×C22⋊C4C22×D6112yes—
406C2×C42—21no—
407C2×C42—1222no—
408C2×C42—1222no—
409C2×C42—1221no—
410C2×C42—1222no—
411#99—28no—
412#99—1228no—
413#99—1228no—
414#99—1228no—
415#95—21no—
416#95—21no—
417#95—1222no—
418#95—1222no—
419#95—1221no—
420#95—1221no—
421C24—612no—
422C24C2×C4×D6112yes—
423C24C23×D6111yescharacteristic
424#41—2448no—
425#41—2448no—
426#41—2448no—
427#41—2448no—
428C2×Q8—1224no—
429C2×Q8—1228no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—81no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—1228no—
440C2×D4—1224no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1224no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—4881no—
447C2×D4—4882no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—81no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—614no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—1224no—
461C22×C4—1228no—
462C22×C4—1224no—
463C22×C4—1228no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—4881no—
467C22×C4C2×C4×D6112yes—
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—81no—
477C4⋊C4—614no—
478C4⋊C4—614no—
479C4⋊C4—614no—
480C4⋊C4—614no—
481C4⋊C4—1224no—
482C4⋊C4—1228no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—4882no—
486C4⋊C4—4881no—
487C4⋊C4—2444no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—24no—
492C22⋊C4—614no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1224no—
499C22⋊C4D4×D6114yes—
500C42—81no—
501C42—1228no—
502C42—4881no—
503C42—4882no—
504C42—2444no—
505C23—22no—
506C23—24no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—1224no—
512C23—1222no—
513C23—2442no—
514C23—2442no—
515C23C22×C4×D6111yescharacteristic
516C23C2×D4×D6111yescharacteristic
517C23C2×D4×D6111yescharacteristic
518C23C2×D4×D6112yes—
519C23#41×D6112yes—
520Q8—2448no—
521D4—44no—
522D4—42no—
523D4—42no—
524D4—81no—
525D4—81no—
526D4—4881no—
527D4—4881no—
528D4—4884no—
529D4—2448no—
530D4—2444no—
531D4—2442no—
532D4—2442no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—614no—
548C2×C4—618no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—614no—
552C2×C4—614no—
553C2×C4—614no—
554C2×C4—1222no—
555C2×C4—1224no—
556C2×C4—1224no—
557C2×C4—2448no—
558C2×C4—2448no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4C22⋊C4×D6111yescharacteristic
568C2×C4C22⋊C4×D6111yescharacteristic
569C2×C4C22⋊C4×D6111yescharacteristic
570C2×C4C22⋊C4×D6111yescharacteristic
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—84no—
583C22—4884no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22C2×C22⋊C4×D6111yescharacteristic
588C22C4×D4×D6112yes—
589C22#121×D6112yes—
590C22#124×D6111yescharacteristic
591C22#127×D6111yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—42no—
598C4—82no—
599C4—618no—
600C4—1224no—
601C4—4882no—
602C4—2444no—
603C4—2442no—
604C2—22no—
605C2—21no—
606C2—22no—
607C2—21no—
608C2—82no—
609C2—4882no—
610C2#390×D6111yescharacteristic
611C2#2145427711yescharacteristic
612C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1964C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840339C2
cosocleC24×C61C24
hypercenter#1964D61
hypocenterC61#21214
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. C2×#16839734C2semidirect
3. #21448436C2semidirect
4. #21448174C2semidirect
5. #21446656C2semidirect
6. #21446655C2semidirect
7. #21446654C2semidirect
8. #21446653C2semidirect
9. C61×#1964C2semidirect
15. C2×#14519755C4semidirect
16. C2×#14519754C4semidirect
18. C2×#14519750C4semidirect
19. C2×#14519749C4semidirect
69. #14519733D4semidirect
306. #1964D61semidirect
346. C61#21214semidirect

Maximal subgroups

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

subgroupindex
C2×#168403392
C2×#168397342
#214484362
#214481742
#214466562
#214466552
#214466542
#214466532
C61×#19642
#2121461