Finite Group Databasebeta

← #29707359

Subgroups — #29707359

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839761C211yesmaximal, characteristic
2C2×#16839756C211yesmaximal, characteristic
3#21448407C211yesmaximal, characteristic
4#21446967C212yesmaximal, complemented
5#21446962C212yesmaximal, complemented
6#21446957C212yesmaximal, complemented
7#21446952C212yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×#16839524C211yesmaximal, characteristic, complemented
10#21445516C211yesmaximal, characteristic, complemented
11C2×C61×#544C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839761—24no—
17#16839756—24no—
18C2×#14519700C2211yescharacteristic
19C2×#14519700C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24#16839668C2212yes—
25#16839664C2212yes—
26#16839639—22no—
27#16839619—21no—
28#16839619—21no—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#544—24no—
36C61×#544—24no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61C2212yes—
39C2×#125×C61—21no—
40C2×#125×C61C2212yes—
41C2×C22⋊Q8×C61—22no—
42C2×#117×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2211yescharacteristic
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2312yes—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69C2×#11758198—22no—
70C2×#11758198—22no—
71#14519700—22no—
72#14519700—22no—
73#14519700—24no—
74#14519697—22no—
75#14519697—22no—
76#14519697—24no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61—22no—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C4⋊Q8×C61—28no—
83C4⋊Q8×C61—24no—
84C4⋊Q8×C61—24no—
85#125×C61—28no—
86#125×C61—24no—
87#125×C61—24no—
88C22⋊Q8×C61—216no—
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61D414yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D414yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61D412yes—
106C2×C42×C61—21no—
107C2×C42×C61C2312yes—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C2×Q8×C61—28no—
111C2×Q8×C61—82no—
112C2×Q8×C61—82no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—81no—
118C22×C4×C61—81no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×Q812yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—42no—
135C22×Dic61—81no—
136C22×Dic61—81no—
137#11758222—28no—
138#11758222—28no—
139#11758222—44no—
140#11758222—44no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C2×Q812yes—
147C2×C4×D61—24no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C2×Q812yes—
156C4⋊C4×C61—28no—
157C4⋊C4×C61—28no—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D414yes—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173#11758198—82no—
174#11758198—82no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—42no—
178#11758194—44no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—22no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—44no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#12012yes—
194C23×C61C22⋊Q812yes—
195C23×C61#12412yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204Q8×C61—88no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—84no—
221C2×C4×C61#12012yes—
222C2×C4×C61C22⋊Q814yes—
223C2×C4×C61C22⋊Q814yes—
224C2×C4×C61C22⋊Q812yes—
225C2×C4×C61#12414yes—
226C2×C4×C61#12414yes—
227C2×C4×C61#12412yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—48no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—84no—
242D244—22no—
243D244—22no—
244D244—41no—
245D244—41no—
246C4×D61—48no—
247Dic122—22no—
248Dic122—22no—
249Dic122—41no—
250Dic122—41no—
251#92418—41no—
252#92418—41no—
253#92418—42no—
254#23458—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61#39314yes—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61C2×#12411yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57811yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—42no—
277C4×C61—44no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#39314yes—
282C4×C61C2×C22⋊Q811yescharacteristic
283C4×C61C2×C22⋊Q811yescharacteristic
284C4×C61C2×#12411yescharacteristic
285C4×C61C2×#12411yescharacteristic
286Dic61—42no—
287Dic61—82no—
288C2×#544D6111yescharacteristic, complemented
289C2×#417—611no—
290C2×#416—611no—
291#2311—611no—
292#2026—612no—
293#2025—612no—
294#2024—612no—
295#2023—612no—
296#1736—611no—
297C2×#340—611no—
298#1729—611no—
299C2×C61—24no—
300C2×C61C2×#39311yescharacteristic
301C2×C61#301211yescharacteristic
302C2×C61#301311yescharacteristic
303C2×C61#308311yescharacteristic
304C2×C61#308411yescharacteristic
305C2×C61#313611yescharacteristic
306C2×C61#313711yescharacteristic
307D61—82no—
308C22×#41—611no—
309C22×M16—611no—
310C23×C8—611no—
311#544—24no—
312#544—24no—
313C2×C4⋊Q8—21no—
314C2×C4⋊Q8C2×D6112yes—
315C2×#125—21no—
316C2×#125C2×D6112yes—
317C2×C22⋊Q8—22no—
318C2×#117C2×D6111yescharacteristic
319C2×#117C2×D6111yescharacteristic
320C22×C4⋊C4—611no—
321C22×C4⋊C4C2×D6111yescharacteristic
322#417—1224no—
323#416—1224no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—612no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—612no—
330#391—612no—
331#390—612no—
332#386—1222no—
333#380—1221no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#340—612no—
338#340—612no—
339#336—612no—
340#336—612no—
341C61#2345811yescharacteristic, complemented, Sylow
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—22no—
349C22×Q8—611no—
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—1221no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—612no—
362C22×C8—1222no—
363C22×C8—1222no—
364C22×C8—1222no—
365C4⋊Q8—28no—
366C4⋊Q8—24no—
367C4⋊Q8—24no—
368#125—28no—
369#125—24no—
370#125—24no—
371C22⋊Q8—216no—
372#117—22no—
373#117—22no—
374#117—22no—
375#117C22⋊D6114yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C4⋊C4C22⋊D6111yescharacteristic
387C2×C4⋊C4C22⋊D6114yes—
388C2×C4⋊C4C22×D6112yes—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4C22⋊D6112yes—
395C2×C42—21no—
396C2×C42C22×D6112yes—
397#103—1222no—
398#103—1222no—
399#103—1224no—
400#102—1222no—
401#102—1222no—
402#102—1224no—
403#95—1222no—
404#95—1222no—
405C24—1221no—
406C24C2×C22⋊D6111yescharacteristic
407#41—1228no—
408#41—1228no—
409#41—2444no—
410#41—2444no—
411C2×Q8—28no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—612no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×D4—612no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—2442no—
428C2×D4—2442no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—1221no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4Q8×D6112yes—
444C22×C4#1175822612yes—
445C22×C4#1175822612yes—
446C22×C4#1175822612yes—
447C22×C4C2×C22⋊D6112yes—
448C22×C4C2×C22⋊D6112yes—
449C22×C4C2×C22⋊D6112yes—
450C22×C4C2×C22⋊D6112yes—
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C23×D6111yescharacteristic
453M16—2444no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—2444no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—2442no—
476C4⋊C4C2×C22⋊D6114yes—
477C4⋊C4C2×C22⋊D6112yes—
478C4⋊C4C2×C22⋊D6112yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C42—22no—
484C42—22no—
485C42—24no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23#1451974612yes—
498C23#1451975312yes—
499C23#1451976612yes—
500C23C22×C22⋊D6111yescharacteristic
501Q8—88no—
502Q8—1222no—
503Q8—1222no—
504Q8—2441no—
505Q8—2441no—
506D4—1222no—
507D4—1222no—
508D4—2448no—
509D4—2441no—
510D4—2441no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—1222no—
528C2×C4—4884no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2448no—
535C2×C4#1451974612yes—
536C2×C4#1451975314yes—
537C2×C4#1451975314yes—
538C2×C4#1451975312yes—
539C2×C4#1451976614yes—
540C2×C4#1451976614yes—
541C2×C4#1451976612yes—
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×Q8×D6111yescharacteristic
545C2×C4C2×#1175822611yescharacteristic
546C2×C4C2×#1175822611yescharacteristic
547C2×C4C2×#1175822611yescharacteristic
548C8—2441no—
549C8—2441no—
550C8—2442no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22#1683967314yes—
565C22C2×#1451974611yescharacteristic
566C22C2×#1451975311yescharacteristic
567C22C2×#1451976611yescharacteristic
568C22#1684056511yescharacteristic
569C22#1684057011yescharacteristic
570C22#1684057311yescharacteristic
571C22#1684057711yescharacteristic
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—4882no—
578C4—2442no—
579C4#1683967314yes—
580C4C2×#1451975311yescharacteristic
581C4C2×#1451975311yescharacteristic
582C4C2×#1451976611yescharacteristic
583C4C2×#1451976611yescharacteristic
584C2—24no—
585C2—4882no—
586C2C2×#1683967311yescharacteristic
587C2#2145512311yescharacteristic
588C2#2145513411yescharacteristic
589C2#2145566611yescharacteristic
590C2#2145567711yescharacteristic
591C2#2145617811yescharacteristic
592C2#2145618911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#544D61
hypocenterC61#23458
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #21446967C2semidirect
5. #21446962C2semidirect
6. #21446957C2semidirect
7. #21446952C2semidirect
8. #21445540C2semidirect
9. C2×#16839524C2semidirect
10. #21445516C2semidirect
11. C2×C61×#544C2semidirect
50. C2×#11758222Q8semidirect
288. C2×#544D61semidirect
341. C61#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#168397612
C2×#168397562
#214484072
#214469672
#214469622
#214469572
#214469522
#214455402
C2×#168395242
#214455162
C2×C61×#5442
#2345861