Finite Group Databasebeta

← #29721607

Subgroups — #29721607

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839970C211yesmaximal, characteristic
2C2×#16839970C211yesmaximal, characteristic
3C2×Q16×Dic61C211yesmaximal, characteristic
4#21450805C211yesmaximal, characteristic
5Q8×#11758196C211yesmaximal, characteristic
6#21447213C211yesmaximal, characteristic
7#21447212C211yesmaximal, characteristic
8#21447063C211yesmaximal, characteristic
9#21447062C211yesmaximal, characteristic
10#21446619C211yesmaximal, characteristic
11#21446534C211yesmaximal, characteristic
12#21446162C211yesmaximal, characteristic
13#21446160C211yesmaximal, characteristic
14#21445822C211yesmaximal, characteristic
15C2×C61×#462C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22#16839970—24no—
23#16839970—24no—
24Q16×Dic61—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839695—21no—
38#16839695—21no—
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839660—21no—
42#16839660C2211yescharacteristic
43#16839658—21no—
44#16839658C2211yescharacteristic
45#16839657—21no—
46#16839657—21no—
47#16839631—21no—
48#16839631C2211yescharacteristic
49C4×#11758196—21no—
50C4×#11758196C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×Q16×C61C2211yescharacteristic
56C22×Q16×C61C2211yescharacteristic
57C2×C4⋊Q8×C61C2211yescharacteristic
58C2×C4×Q8×C61C2211yescharacteristic
59C61×#462—24no—
60C2×C4⋊C8×C61C2211yescharacteristic
61C2×#103×C61C2211yescharacteristic
62C2×#103×C61C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519775—22no—
70#14519775—22no—
71#14519775—22no—
72#14519775—41no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78Q8×Dic61D414yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196D411yescharacteristic
94C2×#11758196D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519711—22no—
101#14519706—22no—
102#14519706—22no—
103#14519702—22no—
104#14519702—22no—
105#14519701—22no—
106#14519701—22no—
107#14519701—22no—
108#14519701—22no—
109#14519701—41no—
110#14519700—41no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123C2×Q16×C61—22no—
124C2×Q16×C61—22no—
125C2×Q16×C61—22no—
126C2×Q16×C61—22no—
127C2×Q16×C61—22no—
128C2×Q16×C61D414yes—
129C22×C8×C61C2311yescharacteristic
130C22×C8×C61C2311yescharacteristic
131C4⋊Q8×C61—22no—
132C4⋊Q8×C61—41no—
133C4×Q8×C61—22no—
134C4×Q8×C61—41no—
135C2×C4⋊C4×C61—21no—
136C2×C4⋊C4×C61—21no—
137C2×C4⋊C4×C61—21no—
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C42×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C4⋊C8×C61—22no—
142#103×C61—22no—
143#103×C61—22no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61C2×D412yes—
159C2×Q8×C61C2×D412yes—
160C2×Q8×C61C2×D412yes—
161C2×Q8×C61#4112yes—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183Q16×C61—28no—
184Q16×C61—44no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61C2×D412yes—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61—42no—
195C4⋊C4×C61—42no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—42no—
199#11758202—44no—
200C4⋊Dic61—21no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61—42no—
208C4⋊Dic61C2×D412yes—
209#11758196—22no—
210#11758196—42no—
211#11758196—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—81no—
216#11758196—81no—
217#11758196—81no—
218#11758196Q1614yescomplemented
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—24no—
222C4×Dic61—42no—
223C4×Dic61—42no—
224C4×Dic61C2×D412yes—
225C4×Dic61#4112yes—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C23×C61C22×D411yescharacteristic
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—41no—
244Q8×C61#12114yes—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C2×Q1612yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61#12112yes—
289C2×Dic61#12112yes—
290C2×Dic61C2×Q1612yes—
291C2×Dic61C2×Q1612yes—
292C2×Dic61#13712yes—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—22no—
298C8×C61—41no—
299#92418—41no—
300#92418—41no—
301#24732—611nomaximal, Sylow
302C22×C61C2×#12111yescharacteristic
303C22×C61D4211yescharacteristic
304C22×C61#54811yescharacteristic
305C22×C61C22×Q1611yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57811yescharacteristic
309C4×C61—24no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C2×#12111yescharacteristic
316C4×C61#53711yescharacteristic
317C4×C61#54711yescharacteristic
318C4×C61#55311yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—41no—
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326Dic61#46214yescomplemented
327C2×#462—611no—
328C2×#462—611no—
329C2×#462D6111yescharacteristic
330C2×C4×Q16—611no—
331#2546—611no—
332Q8×C4⋊C4—611no—
333#2068—611no—
334#2068—611no—
335#2043—611no—
336#2043—611no—
337#1956—611no—
338#1939—611no—
339#1878—611no—
340#1878—611no—
341#1803—611no—
342C2×C61#241611yescharacteristic
343C2×C61C2×#46211yescharacteristic
344C2×C61#304611yescharacteristic
345C2×C61#327811yescharacteristic
346C2×C61D4×Q1611yescharacteristic
347C2×C61#336311yescharacteristic
348C2×C61#337411yescharacteristic
349C22×Q16—611no—
350C22×Q16C2×D6111yescharacteristic
351C22×Q16C2×D6111yescharacteristic
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—611no—
354C2×C4⋊Q8C2×D6111yescharacteristic
355C2×C4×Q8—611no—
356C2×C4×Q8—611no—
357C2×C4×Q8—611no—
358C2×C4×Q8C2×D6111yescharacteristic
359#462—24no—
360#462—1224no—
361#462—1224no—
362C4×Q16—1224no—
363C2×#107—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103C2×D6111yescharacteristic
376C2×#103C2×D6111yescharacteristic
377C2×C4×C8—611no—
378#398—1221no—
379#398—1221no—
380#391—611no—
381#391—611no—
382#389—611no—
383#389—611no—
384#389—1221no—
385#389—1221no—
386#389—1221no—
387#389—1221no—
388#384—611no—
389#384—1221no—
390C4×C4⋊C4—611no—
391C4×C4⋊C4—1221no—
392#340—611no—
393#340—611no—
394#336—611no—
395#336—611no—
396C61#2473211yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8C22×D6111yescharacteristic
399C22×Q8C22×D6111yescharacteristic
400C22×Q8C22×D6111yescharacteristic
401C2×Q16—22no—
402C2×Q16—22no—
403C2×Q16—22no—
404C2×Q16—22no—
405C2×Q16—22no—
406C2×Q16—1222no—
407C2×Q16—1222no—
408C2×Q16—1222no—
409C2×Q16C22⋊D6114yes—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C4⋊Q8—22no—
415C4⋊Q8—41no—
416C4⋊Q8—1222no—
417C4⋊Q8—1222no—
418C4⋊Q8—1222no—
419C4⋊Q8—2441no—
420C4×Q8—22no—
421C4×Q8—41no—
422C4×Q8—614no—
423C4×Q8—1222no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×Q8—1222no—
427C4×Q8—2441no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C42—21no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455#107—1222no—
456C4⋊C8—22no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459#103—22no—
460#103—22no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467#103—2441no—
468#103—2441no—
469C4×C8—1222no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8D4×D6112yes—
499C2×Q8#1175822412yes—
500C2×Q8C2×C22⋊D6112yes—
501C2×Q8C2×C22⋊D6112yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517Q16—28no—
518Q16—44no—
519Q16—2444no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8C2×C22⋊D6112yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—614no—
538C4⋊C4—612no—
539C4⋊C4—1222no—
540C4⋊C4—1224no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—4881no—
548C4⋊C4—4881no—
549C4⋊C4—2442no—
550C4⋊C4—2442no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C42—21no—
557C42—21no—
558C42—42no—
559C42—612no—
560C42—612no—
561C42—1224no—
562C42—1221no—
563C42—1221no—
564C42—2442no—
565C42—2442no—
566C23C2×D4×D6111yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—2441no—
579Q8—2441no—
580Q8—2441no—
581Q8—2441no—
582Q8#1451974914yes—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4Q16×D6112yes—
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C22×C22⋊D6111yescharacteristic
627C2×C4#41×D6111yescharacteristic
628C2×C4C2×#1175822411yescharacteristic
629C2×C4#1451982211yescharacteristic
630C2×C4#1451982411yescharacteristic
631C8—22no—
632C8—41no—
633C8—2441no—
634C8—2441no—
635C22#121×D6111yescharacteristic
636C22D4×C22⋊D6111yescharacteristic
637C22#1684041811yescharacteristic
638C22C2×Q16×D6111yescharacteristic
639C22#137×D6111yescharacteristic
640C22#1684057111yescharacteristic
641C22#1684057511yescharacteristic
642C4—24no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—81no—
647C4—81no—
648C4—614no—
649C4—1222no—
650C4—1222no—
651C4—4881no—
652C4—4881no—
653C4—4881no—
654C4—2442no—
655C4—2441no—
656C4C2×#1451974911yescharacteristic
657C4#1684033911yescharacteristic
658C4#1684041111yescharacteristic
659C4#1684045511yescharacteristic
660C2#2144925611yescharacteristic
661C2#462×D6111yescharacteristic
662C2#2145536611yescharacteristic
663C2#2145747911yescharacteristic
664C2Q16×C22⋊D6111yescharacteristic
665C2#2145849211yescharacteristic
666C2#2145861711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×#462D61
hypocenterC61#24732
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
218. #11758196Q16semidirect
326. Dic61#462semidirect
396. C61#24732semidirect

Maximal subgroups

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

subgroupindex
C2×#168399702
C2×Q16×Dic612
#214508052
Q8×#117581962
#214472132
#214472122
#214470632
#214470622
#214466192
#214465342
#214461622
#214461602
#214458222
C2×C61×#4622
#2473261