Finite Group Databasebeta

← #29698381

Subgroups — #29698381

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839865C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic
3C2×#16839847C211yesmaximal, characteristic
4Q8×#11758196C211yesmaximal, characteristic, complemented
5#21448728C211yesmaximal, characteristic
6#21446554C211yesmaximal, characteristic, complemented
7#21446551C211yesmaximal, characteristic, complemented
8#21446077C211yesmaximal, characteristic
9#21446075C211yesmaximal, characteristic, complemented
10#21446069C211yesmaximal, characteristic
11#21445984C211yesmaximal, characteristic, complemented
12#21445832C211yesmaximal, characteristic, complemented
13#21445818C211yesmaximal, characteristic
14#21445692C211yesmaximal, characteristic, complemented
15C61×#1865C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22#16839865C418yescomplemented
23#16839860—24no—
24#16839847—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660—21no—
39#16839660C2211yescharacteristic
40#16839658—21no—
41#16839658C2211yescharacteristic
42#16839636—21no—
43#16839633—21no—
44#16839631—21no—
45#16839631C2211yescharacteristic
46#16839627C2211yescharacteristic
47#16839619—21no—
48C4×#11758196—21no—
49C4×#11758196C2211yescharacteristic
50#95×D61—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C2×#11758201C2×C414yes—
67C2×#11758201D414yes—
68C2×#11758201Q814yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74Q8×Dic61C2×C414yescomplemented
75C4×D244—22no—
76C4×D244—41no—
77#14519739—22no—
78#14519739—22no—
79#14519739—41no—
80#14519739C2×C414yes—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519711—41no—
101#14519710—22no—
102#14519710C2×C414yescomplemented
103#14519708—22no—
104#14519706—22no—
105#14519701—22no—
106#14519699—22no—
107#14519699C2×C414yes—
108#14519697—22no—
109#14519688—22no—
110#14519687C2×C414yescomplemented
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×Q8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×Q8×C61—22no—
123C4×Q8×C61—41no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—41no—
132#103×C61—22no—
133#103×C61C2×C414yescomplemented
134#95×C61—21no—
135C23×D61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C22×C412yes—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×Q812yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×D244C2×D412yes—
161C2×D244C2×Q812yes—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C22×C412yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61C22×C412yes—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188#11758201C4⋊C4116yescomplemented
189#11758198—24no—
190#11758198—42no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—24no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61—81no—
200C4⋊Dic61C22×C412yes—
201#11758196—22no—
202#11758196—41no—
203#11758196—42no—
204#11758196—81no—
205#11758196—81no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61C22×C412yes—
212C4×Dic61#4112yes—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C22×C412yes—
217C2×#92418C2×D412yes—
218C2×#92418C2×Q812yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61C2×C4⋊C414yes—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C4×D412yes—
274C2×Dic61C4×D412yes—
275C2×Dic61#13712yes—
276D244—22no—
277D244—22no—
278D244—22no—
279D244C2×C4⋊C414yes—
280C4×D61—84no—
281C4×D61—84no—
282C4×D61—84no—
283C8×C61—22no—
284C8×C61—41no—
285#92418—41no—
286#92418C2×C4⋊C414yes—
287#22678—611nomaximal, Sylow
288C22×C61C2×C4×D411yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61D4×Q811yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—24no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—24no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C22×C4⋊C411yescharacteristic
309C4×C61#52011yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55411yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319C2×#440—611no—
320C2×#440—611no—
321C2×C4×SD16—611no—
322Q8×C4⋊C4—611no—
323#2349—611no—
324#1943—611no—
325#1942—611no—
326#1865D6111yescharacteristic, complemented
327#1864—611no—
328#1863—611no—
329#1862—611no—
330#1847—611no—
331#1805—611no—
332#1802—611no—
333#1771—611no—
334C2×C61D4×C4⋊C411yescharacteristic
335C2×C61C4×#13711yescharacteristic
336C2×C61#294611yescharacteristic
337C2×C61#326611yescharacteristic
338C2×C61#326811yescharacteristic
339C2×C61#338511yescharacteristic
340C2×C61#338811yescharacteristic
341D61—44no—
342C22×SD16—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349#440—618no—
350#440—1224no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#107—611no—
354C2×#107—611no—
355C2×#106—611no—
356C2×#106C2×D6111yescharacteristic
357C2×C4⋊C8—611no—
358C2×#103—611no—
359C2×#103C2×D6111yescharacteristic
360C2×#103C2×D6111yescharacteristic
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#97—611no—
365C2×#97—611no—
366C2×C4×C8—611no—
367#389—611no—
368#389—611no—
369#389—611no—
370#389—1221no—
371#389—1221no—
372#389C2×D6111yescharacteristic
373#385—1221no—
374#385—1221no—
375#384—611no—
376#384—1221no—
377#382—611no—
378#380—1221no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381C2×#95—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340C2×D6111yescharacteristic
386#336—611no—
387#336C2×D6111yescharacteristic
388C61#2267811yescharacteristic, complemented, Sylow
389C22×Q8C22×D6111yescharacteristic
390C22×D4—611no—
391C23×C4—1221no—
392C23×C4—1221no—
393C23×C4—1221no—
394C2×SD16—614no—
395C2×SD16—614no—
396C2×SD16—614no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4×Q8—22no—
402C4×Q8—41no—
403C4×Q8—614no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C4×D4—614no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—2441no—
412C4×D4—2441no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#107—614no—
440#107—1222no—
441#107—2441no—
442#106—41no—
443#106—1222no—
444C4⋊C8—1222no—
445#103—22no—
446#103—1222no—
447#103C4×D6114yescomplemented
448#102—614no—
449#102—1222no—
450#102—1222no—
451#97—614no—
452#97—1222no—
453C4×C8—1222no—
454#95—21no—
455#95—611no—
456#95—611no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—1221no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D6112yes—
467C2×Q8D4×D6112yes—
468C2×Q8Q8×D6112yes—
469C2×D4—612no—
470C2×D4—612no—
471C2×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—6116no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—612no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8C2×C4×D6112yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—612no—
519C4⋊C4—1222no—
520C4⋊C4—1224no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—2441no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4C2×C4×D6112yes—
534C22⋊C4—1224no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—612no—
541C42—612no—
542C42—1224no—
543C42—1221no—
544C42—1221no—
545C42—2442no—
546C23—1222no—
547C23—1222no—
548C23—1222no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23C2×D4×D6111yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D6114yes—
558D4—614no—
559D4—1222no—
560D4—1222no—
561D4—1222no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1222no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—4884no—
594C2×C4—4884no—
595C2×C4—4884no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4#1451982211yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—22no—
611C8—41no—
612C8—614no—
613C8—2441no—
614C22—1224no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22C4×D4×D6111yescharacteristic
622C22#1684039411yescharacteristic
623C22#1684042511yescharacteristic
624C22#137×D6111yescharacteristic
625C22#1684056111yescharacteristic
626C22#1684056411yescharacteristic
627C22#1684056811yescharacteristic
628C4—24no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—1222no—
635C4—1222no—
636C4—1222no—
637C4—4881no—
638C4—4881no—
639C4—2442no—
640C4—2441no—
641C4C2×C4⋊C4×D6111yescharacteristic
642C4#1684024111yescharacteristic
643C4#1684041011yescharacteristic
644C4#1684046611yescharacteristic
645C2—2444no—
646C2#2144871711yescharacteristic
647C2#2145463511yescharacteristic
648C2#2145464611yescharacteristic
649C2#2145733211yescharacteristic
650C2#2145736211yescharacteristic
651C2#2145873211yescharacteristic
652C2#2145877111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1865D61
hypocenterC61#22678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
4. Q8×#11758196C2semidirect
6. #21446554C2semidirect
7. #21446551C2semidirect
9. #21446075C2semidirect
11. #21445984C2semidirect
12. #21445832C2semidirect
14. #21445692C2semidirect
15. C61×#1865C2semidirect
22. #16839865C4semidirect
74. Q8×Dic61C2×C4semidirect
102. #14519710C2×C4semidirect
110. #14519687C2×C4semidirect
133. #103×C61C2×C4semidirect
188. #11758201C4⋊C4semidirect
326. #1865D61semidirect
388. C61#22678semidirect
447. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398652
C2×#168398602
C2×#168398472
Q8×#117581962
#214487282
#214465542
#214465512
#214460772
#214460752
#214460692
#214459842
#214458322
#214458182
#214456922
C61×#18652
#2267861