Finite Group Databasebeta

← #28605315

Subgroups — #28605315

Order 15104.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838111C211yesmaximal, characteristic
2C2×#16837530C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic118C211yesmaximal, characteristic, complemented
4#21425065C211yesmaximal, characteristic, complemented
5#21425063C211yesmaximal, characteristic, complemented
6#21425055C211yesmaximal, characteristic, complemented
7#21425053C211yesmaximal, characteristic, complemented
8#21425052C211yesmaximal, characteristic, complemented
9#21425042C211yesmaximal, characteristic, complemented
10#21424096C211yesmaximal, characteristic
11#21424094C211yesmaximal, characteristic
12#21424094C211yesmaximal, characteristic
13#21423923C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#1762C211yesmaximal, characteristic
16#16838111—24no—
17#16838111C418yescomplemented
18C2×#14273726C2211yescharacteristic
19C2×#14273719C2211yescharacteristic
20C2×#14273718C2211yescharacteristic
21C2×#14273706—21no—
22C2×#14273704—21no—
23C2×#14273704—21no—
24C2×C4×Dic118—21no—
25C2×C4×Dic118C2211yescharacteristic
26C2×D4×Dic59C2211yescharacteristic
27C2×D4×Dic59C2211yescharacteristic
28C22×Dic59⋊C4—21no—
29#16837530—24no—
30C2×C4⋊(C59⋊C8)C2211yescharacteristic
31C2×Dic59⋊C8C2211yescharacteristic
32C2×Dic59⋊C8C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×#14273648C2211yescharacteristic
35C2×#14273637C2211yescharacteristic
36C2×C8×Dic59C2211yescharacteristic
37#16837300—21no—
38#16837299—21no—
39#16837299—21no—
40#16837272C2211yescharacteristic
41#16837271C2211yescharacteristic
42#16837271C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C2×C4×D4×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×#97×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118—21no—
63C22×Dic118D411yescharacteristic
64C23×Dic59—21no—
65C23×Dic59—21no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273726—22no—
69#14273726C2×C414yes—
70#14273719—22no—
71#14273719C2×C414yes—
72#14273718—22no—
73#14273718C2×C414yes—
74#14273706—28no—
75#14273704—28no—
76#14273704—28no—
77C4×Dic118—44no—
78C4×Dic118C2×C414yes—
79C4×Dic118D414yes—
80D4×Dic59—22no—
81D4×Dic59—22no—
82D4×Dic59C2×C414yes—
83D4×Dic59C2×C414yes—
84C2×C4⋊Dic59—21no—
85C2×C4⋊Dic59—21no—
86C2×C4⋊Dic59—21no—
87C2×C4⋊Dic59D411yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2311yescharacteristic
90C2×C4⋊Dic59C2311yescharacteristic
91C2×C4⋊Dic59C2311yescharacteristic
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×Dic59⋊C4—28no—
97C2×Dic59⋊C4—21no—
98C2×Dic59⋊C4—21no—
99C2×Dic59⋊C4—21no—
100C2×Dic59⋊C4—21no—
101C2×Dic59⋊C4D411yescharacteristic
102C2×Dic59⋊C4D411yescharacteristic
103C2×C22⋊Dic59—21no—
104C2×C22⋊Dic59—21no—
105C2×C4×Dic59—21no—
106C2×C4×Dic59—21no—
107C2×C4×Dic59C2311yescharacteristic
108C2×C4×Dic59C2311yescharacteristic
109C4⋊(C59⋊C8)—22no—
110Dic59⋊C8—22no—
111Dic59⋊C8—22no—
112#14273648—22no—
113#14273648—22no—
114#14273637—22no—
115C8×Dic59—22no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C22×D4×C59C2311yescharacteristic
119C23×C4×C59—21no—
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C4×D4×C59—22no—
123C4×D4×C59C2×C414yes—
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C22⋊C4×C59—21no—
128C2×C42×C59—21no—
129C2×C42×C59C2311yescharacteristic
130#102×C59—22no—
131#102×C59Q814yescomplemented
132#97×C59—22no—
133#95×C59—21no—
134C24×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59C22×C412yes—
139C2×D4×C59C2×Q812yes—
140C2×D4×C59#4112yes—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—24no—
144C22×C4×C59—21no—
145C22×C4×C59—81no—
146C22×C4×C59—81no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—24no—
153C22×Dic59—24no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—42no—
157C22×Dic59—42no—
158C22×Dic59—81no—
159C22×Dic59—81no—
160C22×Dic59C2×D411yescharacteristic
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C22×Dic59C2×D411yescharacteristic
164C2×Dic118—22no—
165C2×Dic118—22no—
166C2×Dic118—24no—
167C2×Dic118—81no—
168C2×Dic118—81no—
169C2×Dic118C22⋊C414yes—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59C2×Q812yes—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59C22×C412yes—
181C4⋊C4×C59C2×Q812yes—
182C22⋊C4×C59—24no—
183C42×C59—42no—
184C42×C59C22×C412yes—
185C42×C59C2×D412yes—
186C22⋊Dic59—24no—
187C22⋊Dic59—24no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—24no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—81no—
197C4⋊Dic59C22⋊C414yes—
198C4⋊Dic59C22×C412yes—
199C4⋊Dic59C22×C412yes—
200Dic59⋊C4—24no—
201Dic59⋊C4—24no—
202Dic59⋊C4—24no—
203Dic59⋊C4—24no—
204Dic59⋊C4—42no—
205Dic59⋊C4—42no—
206Dic59⋊C4—81no—
207Dic59⋊C4—81no—
208Dic59⋊C4C22⋊C414yes—
209Dic59⋊C4C22⋊C414yes—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C4×Dic59C22×C412yes—
213C4×Dic59C22×C412yes—
214C4×Dic59C2×D412yes—
215C4×Dic59C2×D412yes—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C23×C59—22no—
223C23×C59—22no—
224C23×C59—22no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59C22×D411yescharacteristic
230D4×C59—22no—
231D4×C59—22no—
232D4×C59—22no—
233D4×C59C4×Q814yescomplemented
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—84no—
248C2×C4×C59C2×C22⋊C412yes—
249C2×C4×C59C2×C22⋊C412yes—
250C2×C4×C59C23×C411yescharacteristic
251C2×C4×C59C22×D411yescharacteristic
252C2×C4×C59C22×Q811yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C592−1+411yescharacteristic
255C2×C4×C592−1+411yescharacteristic
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—48no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—84no—
276C2×Dic59C2×C22⋊C412yes—
277C2×Dic59C2×C22⋊C412yes—
278C2×Dic59C2×C22⋊C412yes—
279C2×Dic59C2×C22⋊C412yes—
280C2×Dic59AGL(1,ℤ/8)12yes—
281C2×Dic59#13712yes—
282Dic118—28no—
283Dic118—84no—
284C8×C59—22no—
285C8×C59—41no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#22522—591nomaximal, Sylow
289C22×C59—24no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59C22×C22⋊C411yescharacteristic
297C22×C59#54811yescharacteristic
298C22×C59D4×Q811yescharacteristic
299C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C59C2×#13711yescharacteristic
301C22×C59#57611yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—42no—
306C4×C59—42no—
307C4×C59—81no—
308C4×C59C2×C4×Q811yescharacteristic
309C4×C59#51911yescharacteristic
310C4×C59#54811yescharacteristic
311C4×C59D4×Q811yescharacteristic
312Dic59—22no—
313Dic59—22no—
314Dic59—22no—
315Dic59—42no—
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319Dic59#41714yes—
320C2×#554—591no—
321C2×#446—591no—
322Q8×C4⋊C4—591no—
323#1992—591no—
324#1992—591no—
325#1990—591no—
326#1989—591no—
327#1989—591no—
328#1987—591no—
329#1802—591no—
330#1802—591no—
331#1802—591no—
332#1762—591no—
333#1762D5911yescharacteristic
334C4×#102—591no—
335C2×C59—44no—
336C2×C59Q8×C22⋊C411yescharacteristic
337C2×C59C2×#41711yescharacteristic
338C2×C59#289711yescharacteristic
339C2×C59#336611yescharacteristic
340C2×C59#337211yescharacteristic
341C2×C59#338411yescharacteristic
342C2×C59#338511yescharacteristic
343#554—598no—
344#554—1184no—
345C2×C4⋊Q8—591no—
346C2×#125—591no—
347C2×#125—591no—
348C2×C22⋊Q8—1181no—
349C2×C22⋊Q8—1181no—
350C2×C22⋊Q8—1181no—
351C2×C4×Q8—591no—
352C2×C4×Q8—1181no—
353C2×C4×D4—591no—
354C2×C4×D4—591no—
355C2×C4×D4C2×D5911yescharacteristic
356C22×C4⋊C4—1181no—
357#446—1184no—
358C2×C4⋊C8—591no—
359C2×C4⋊C8—591no—
360C2×C4⋊C8—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×#102C2×D5911yescharacteristic
364C2×#102C2×D5911yescharacteristic
365C2×#97—591no—
366C2×#97C2×D5911yescharacteristic
367C2×C4×C8—591no—
368#389—1181no—
369#389—1181no—
370#389—1181no—
371#384—591no—
372#384—591no—
373#384—591no—
374C4×C4⋊C4—591no—
375C4×C4⋊C4—591no—
376C4×C4⋊C4C2×D5911yescharacteristic
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340C2×D5911yescharacteristic
388#340C2×D5911yescharacteristic
389C59#2252211yescharacteristic, complemented, Sylow
390C22×Q8—591no—
391C22×Q8—1181no—
392C22×D4C22×D5911yescharacteristic
393C23×C4—21no—
394C23×C4—1181no—
395C23×C4—1181no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8C22×D5911yescharacteristic
399C22×C8C22×D5911yescharacteristic
400C4⋊Q8—594no—
401C4⋊Q8—1182no—
402#125—594no—
403#125—594no—
404#125—1182no—
405#125—1182no—
406C22⋊Q8—1188no—
407C22⋊Q8—1188no—
408C22⋊Q8—1188no—
409C4×Q8—594no—
410C4×Q8—594no—
411C4×Q8—2364no—
412C4×D4—22no—
413C4×D4—594no—
414C4×D4—594no—
415C4×D4—1182no—
416C4×D4—1182no—
417C4×D4C4×D5914yes—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1188no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4C22×D5911yescharacteristic
438C2×C4⋊C4C22×D5911yescharacteristic
439C2×C4⋊C4C22×D5911yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4—1181no—
443C2×C42—21no—
444C2×C42—591no—
445C2×C42—591no—
446C2×C42—1181no—
447C2×C42—1181no—
448C2×C42C22×D5911yescharacteristic
449C4⋊C8—1182no—
450C4⋊C8—1182no—
451C4⋊C8—1182no—
452#102—22no—
453#102—1182no—
454#102—1182no—
455#102Dic11814yescomplemented
456#97—22no—
457#97—1182no—
458C4×C8—1182no—
459#95—21no—
460#95—1181no—
461#95—1181no—
462C24—21no—
463C2×Q8—594no—
464C2×Q8—1182no—
465C2×Q8—1182no—
466C2×Q8—1184no—
467C2×Q8—4721no—
468C2×Q8—4721no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic11812yes—
473C2×D4C2×C4×D5912yes—
474C2×D4#1174568512yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—591no—
482C22×C4—591no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—1184no—
486C22×C4—1184no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—4721no—
492C22×C4—4721no—
493C22×C4—2362no—
494C22×C4—2362no—
495C22×C4D4×D5911yescharacteristic
496C22×C4D4×D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8C2×Dic11812yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—594no—
515C4⋊C4—594no—
516C4⋊C4—594no—
517C4⋊C4—592no—
518C4⋊C4—592no—
519C4⋊C4—1184no—
520C4⋊C4—1184no—
521C4⋊C4—1184no—
522C4⋊C4—1184no—
523C4⋊C4—1184no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—4721no—
531C4⋊C4—4721no—
532C4⋊C4—4721no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C4⋊C4C2×Dic11812yes—
537C4⋊C4C2×C4×D5912yes—
538C22⋊C4—24no—
539C22⋊C4—1184no—
540C22⋊C4—1184no—
541C42—42no—
542C42—592no—
543C42—592no—
544C42—592no—
545C42—592no—
546C42—2362no—
547C42—2362no—
548C42C2×C4×D5912yes—
549C42D4×D5912yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D5911yescharacteristic
558Q8—1188no—
559Q8—4724no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4×Dic11814yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—592no—
579C2×C4—592no—
580C2×C4—592no—
581C2×C4—592no—
582C2×C4—592no—
583C2×C4—592no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1182no—
591C2×C4—1182no—
592C2×C4—4724no—
593C2×C4—2368no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4C22⋊C4×D5912yes—
605C2×C4C22⋊C4×D5912yes—
606C2×C4C22×C4×D5911yescharacteristic
607C2×C4C2×D4×D5911yescharacteristic
608C2×C4C22×Dic11811yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4#1427377411yescharacteristic
611C2×C4#1427377411yescharacteristic
612C8—22no—
613C8—41no—
614C8—2361no—
615C8—2361no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C22⋊C4×D5911yescharacteristic
624C22#1683806111yescharacteristic
625C22D4×Dic11811yescharacteristic
626C22AGL(1,ℤ/8)×D5911yescharacteristic
627C22#137×D5911yescharacteristic
628C22#1683820611yescharacteristic
629C22#1683821211yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—594no—
636C4—1182no—
637C4—1182no—
638C4—1182no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2362no—
643C4C2×C4×Dic11811yescharacteristic
644C4#1683787911yescharacteristic
645C4#1683806111yescharacteristic
646C4D4×Dic11811yescharacteristic
647C2—44no—
648C2C22⋊C4×Dic11811yescharacteristic
649C2#417×D5911yescharacteristic
650C2#2143262111yescharacteristic
651C2#2143680411yescharacteristic
652C2#2143687811yescharacteristic
653C2#2143699611yescharacteristic
654C2#2143700711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1762C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1762D59
hypocenterC59#22522
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16837530C2semidirect
3. C4⋊C4×Dic118C2semidirect
4. #21425065C2semidirect
5. #21425063C2semidirect
6. #21425055C2semidirect
7. #21425053C2semidirect
8. #21425052C2semidirect
9. #21425042C2semidirect
17. #16838111C4semidirect
131. #102×C59Q8semidirect
233. D4×C59C4×Q8semidirect
389. C59#22522semidirect
455. #102Dic118semidirect
563. D4C4×Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168381112
C2×#168375302
C4⋊C4×Dic1182
#214250652
#214250632
#214250552
#214250532
#214250522
#214250422
#214240962
#214240942
#214239232
#102×Dic592
C59×#17622
#2252259