Finite Group Databasebeta

← #29689121

Subgroups — #29689121

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840333C211yesmaximal, characteristic
2#21448411C211yesmaximal, characteristic
3#21446252C212yesmaximal, complemented
4#21446242C212yesmaximal, complemented
5#21446129C212yesmaximal, complemented
6#21446124C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445755C212yesmaximal, complemented
9C2×C61×#408C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11#16840333—24no—
12#16840333C418yescomplemented
13C2×#14519775—21no—
14C2×#14519764—21no—
15C2×#14519758—21no—
16C2×#14519758—21no—
17C2×#14519756C2212yes—
18C2×#14519756C2212yes—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26#16839669—21no—
27#16839665—21no—
28#16839642—21no—
29#16839642—21no—
30#16839622C2212yes—
31#16839622C2212yes—
32#16839525C2212yes—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36C22×#41×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C61×#408—24no—
40C2×C22⋊C8×C61C2212yes—
41C2×C22⋊C8×C61C2212yes—
42C22×Dic122—21no—
43C22×Dic122—22no—
44C22×Dic122—41no—
45C22×Dic122—41no—
46C22×Dic122—41no—
47C22×Dic122—41no—
48C22×Dic122C2×C414yescomplemented
49C22×Dic122C2312yes—
50C23×Dic61—21no—
51C23×Dic61—21no—
52#14519775—28no—
53#14519764—28no—
54#14519758—28no—
55#14519758—28no—
56#14519756—24no—
57#14519756—24no—
58#14519756C2×C418yes—
59#14519756C2×C418yescomplemented
60#14519736—22no—
61#14519736—41no—
62#14519736—41no—
63#14519736C2×C414yescomplemented
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—41no—
66C2×C4⋊Dic61—41no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80#14519702—42no—
81#14519702—42no—
82#14519702—42no—
83#14519702—42no—
84#14519684—22no—
85#14519684—22no—
86C2×#41×C61—41no—
87C2×#41×C61—41no—
88C2×#41×C61—41no—
89C2×#41×C61—41no—
90C2×#41×C61C2×C414yes—
91C2×#41×C61D414yes—
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61D411yescharacteristic
94C22×D4×C61D411yescharacteristic
95C22×D4×C61D411yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C2×M16×C61—41no—
101C2×M16×C61—41no—
102C22×C8×C61—22no—
103C22×C8×C61—41no—
104C22×C8×C61—41no—
105C22×C8×C61C2312yes—
106C22×C8×C61C2312yes—
107C22⋊C8×C61—24no—
108C22⋊C8×C61—24no—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111C24×C61C2×D411yescharacteristic
112#41×C61—44no—
113#41×C61—44no—
114#41×C61—44no—
115#41×C61—44no—
116C2×Q8×C61—41no—
117C2×Q8×C61—41no—
118C2×Q8×C61C22⋊C414yes—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C2×D4×C61—41no—
125C2×D4×C61C22⋊C414yes—
126C2×D4×C61C22⋊C414yes—
127C2×D4×C61C22⋊C414yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—82no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—24no—
164C2×Dic122—22no—
165C2×Dic122—44no—
166C2×Dic122—44no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—44no—
170C2×Dic122—44no—
171C2×Dic122C22×C414yes—
172M16×C61—44no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—44no—
180C22⋊Dic61—24no—
181C22⋊Dic61—28no—
182C22⋊Dic61—28no—
183C22⋊Dic61—42no—
184C22⋊Dic61—82no—
185C22⋊Dic61—82no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—42no—
192C4⋊Dic61C22×C414yes—
193#11758196—28no—
194#11758196—28no—
195C4×Dic61—24no—
196C4×Dic61—42no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C2×C22⋊C412yes—
211C23×C61C2×C22⋊C412yes—
212C23×C61C22×D411yescharacteristic
213Q8×C61—44no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—48no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—84no—
258Dic122—24no—
259Dic122—24no—
260Dic122—48no—
261Dic122—42no—
262Dic122—42no—
263C8×C61—42no—
264C8×C61—42no—
265#21771—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61#53411yescharacteristic
283C22×C61#53411yescharacteristic
284C22×C61#57712yes—
285C22×C61#57812yes—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61#51811yescharacteristic
290C4×C61#51811yescharacteristic
291C4×C61#53411yescharacteristic
292C4×C61#53411yescharacteristic
293Dic61—44no—
294Dic61—44no—
295Dic61—82no—
296Dic61—82no—
297C2×#536—611no—
298C2×#408D6111yescharacteristic
299#2311—611no—
300#1894—612no—
301#1892—612no—
302#1873—612no—
303#1872—612no—
304#1825—612no—
305#1787—612no—
306C2×C61—42no—
307C2×C61—42no—
308C2×C61—42no—
309C2×C61#230811yescharacteristic
310C2×C61#289812yes—
311C2×C61#320011yescharacteristic
312C2×C61#320112yes—
313C2×C61#320211yescharacteristic
314C22×#41C2×D6111yescharacteristic
315C23×Q8—611no—
316C22×M16C2×D6111yescharacteristic
317C23×C8C2×D6111yescharacteristic
318#536—618no—
319#536—1224no—
320C2×C4⋊Q8—1221no—
321C2×#124—1221no—
322C2×#123—1221no—
323C2×#123—1221no—
324C2×C22⋊Q8—612no—
325C2×C22⋊Q8—612no—
326C2×#117—611no—
327C2×#117—611no—
328C22×C4⋊C4—611no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#103—612no—
333#408—24no—
334C2×C22⋊C8C2×D6112yes—
335C2×C22⋊C8C2×D6112yes—
336#391—1221no—
337#390—1221no—
338#386—1221no—
339#386—1221no—
340#380—612no—
341#380—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345#340—612no—
346C61#2177111yescharacteristic, complemented, Sylow
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C4×D6114yes—
352C2×#41C22⋊D6114yes—
353C22×Q8—614no—
354C22×Q8—612no—
355C22×Q8—1221no—
356C22×Q8—1222no—
357C22×Q8—2441no—
358C22×Q8—2441no—
359C22×Q8—2441no—
360C22×Q8—2441no—
361C22×Q8D24411yescharacteristic
362C22×D4D24411yescharacteristic
363C22×D4D24411yescharacteristic
364C22×D4D24411yescharacteristic
365C23×C4—1221no—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368C23×C4C22×D6111yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×M16—22no—
371C2×M16—41no—
372C2×M16—41no—
373C22×C8—22no—
374C22×C8—41no—
375C22×C8—41no—
376C22×C8C22×D6112yes—
377C22×C8C22×D6112yes—
378C4⋊Q8—1228no—
379#124—1228no—
380#123—1228no—
381#123—1228no—
382C22⋊Q8—618no—
383C22⋊Q8—618no—
384C22⋊Q8—1224no—
385C22⋊Q8—1224no—
386#117—614no—
387#117—1222no—
388#117—2441no—
389#117—2441no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—2441no—
397C2×C4⋊C4—2441no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406#103—2442no—
407#103—2442no—
408#103—2442no—
409#103—2442no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412#95—1222no—
413#95—1222no—
414C24C2×D24411yescharacteristic
415C24C2×D24411yescharacteristic
416C24C2×D24411yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—614no—
424C2×Q8—1222no—
425C2×Q8—1224no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—2444no—
431C2×Q8—2444no—
432C2×Q8—2442no—
433C2×Q8—2442no—
434C2×Q8—2444no—
435C2×Q8—2444no—
436C2×Q8#1175819814yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4#1175819814yes—
444C2×D4#1175819814yes—
445C2×D4#1175819814yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—4882no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4C2×C4×D6112yes—
468C22×C4C2×C4×D6112yes—
469C22×C4C2×C4×D6112yes—
470C22×C4C2×D24411yescharacteristic
471C22×C4C2×D24411yescharacteristic
472C22×C4C2×D24411yescharacteristic
473C22×C4C2×C22⋊D6112yes—
474C22×C4C2×C22⋊D6112yes—
475C22×C4C2×C22⋊D6112yes—
476C22×C4C23×D6111yescharacteristic
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C4⋊C4—614no—
486C4⋊C4—1222no—
487C4⋊C4—1228no—
488C4⋊C4—1228no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—2442no—
494C22⋊C4—1224no—
495C22⋊C4—1228no—
496C22⋊C4—1228no—
497C22⋊C4—4882no—
498C22⋊C4—4882no—
499C22⋊C4—2442no—
500C42—1224no—
501C42—2442no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C2×#1175819812yes—
515C23C2×#1175819812yes—
516C23C2×#1175819812yes—
517C23C22×D24411yescharacteristic
518Q8—44no—
519Q8—1224no—
520Q8—1224no—
521Q8—2448no—
522Q8—2442no—
523Q8—2442no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—4884no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2448no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4C2×#1175819812yes—
560C2×C4C2×#1175819812yes—
561C2×C4C2×#1175819812yes—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4#1451982211yescharacteristic
565C2×C4#1451982211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982211yescharacteristic
568C8—42no—
569C8—42no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×#1175819811yescharacteristic
586C22#1684032411yescharacteristic
587C22#1684032411yescharacteristic
588C22#1684057412yes—
589C22#1684057912yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—4882no—
594C4—4882no—
595C4—2444no—
596C4—2444no—
597C4#1684023511yescharacteristic
598C4#1684023511yescharacteristic
599C4#1684032111yescharacteristic
600C4#1684032111yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#2144839411yescharacteristic
605C2#2145435312yes—
606C2#2145669211yescharacteristic
607C2#2145670312yes—
608C2#2145671411yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#408D61
hypocenterC61#21771
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
3. #21446252C2semidirect
4. #21446242C2semidirect
5. #21446129C2semidirect
6. #21446124C2semidirect
7. #21445915C2semidirect
8. #21445755C2semidirect
12. #16840333C4semidirect
48. C22×Dic122C2×C4semidirect
59. #14519756C2×C4semidirect
63. #14519736C2×C4semidirect
346. C61#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
#214484112
#214462522
#214462422
#214461292
#214461242
#214459152
#214457552
C2×C61×#4082
#2177161