Finite Group Databasebeta

← #29697689

Subgroups — #29697689

Order 15616.

838 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×#106×D61C211yesmaximal, characteristic
2#21448695C211yesmaximal, characteristic
3#21448687C211yesmaximal, characteristic
4#21446563C211yesmaximal, characteristic, complemented
5#21446407C211yesmaximal, characteristic, complemented
6#21446401C211yesmaximal, characteristic, complemented
7#21446400C211yesmaximal, characteristic, complemented
8#21446394C211yesmaximal, characteristic, complemented
9#21445908C211yesmaximal, characteristic
10#21445904C211yesmaximal, characteristic
11#21445845C211yesmaximal, characteristic
12#21445718C211yesmaximal, characteristic, complemented
13#21445716C211yesmaximal, characteristic, complemented
14#21445596C211yesmaximal, characteristic
15C61×#1944C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519735—21no—
18C2×C4⋊C4×D61C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24#106×D61C418yescomplemented
25C2×#14519689C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839649—21no—
30#16839649—21no—
31#16839643—21no—
32#16839643—21no—
33#16839624—21no—
34#16839619—21no—
35#16839619C2211yescharacteristic
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×#11758198C2211yescharacteristic
42#16839609—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C2×C8×D61—22no—
66C2×C8×D61—22no—
67C2×C8×D61C2×C414yes—
68#14519735—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—22no—
78C4⋊C4×D61—22no—
79C4⋊C4×D61C2×C414yes—
80C4⋊C4×D61C2×C414yes—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198D411yescharacteristic
84C2×#11758198D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519711—22no—
91#14519709C2×C414yes—
92#14519708C2×C414yes—
93#14519708C2×C414yes—
94#14519689—22no—
95#14519687—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#107×C61—22no—
116#106×C61C2×C414yes—
117#97×C61—22no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—24no—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—81no—
144C2×C4×D61—81no—
145C2×C4×D61C22×C412yes—
146C2×C4×D61#4112yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61C22×C412yes—
151C2×C8×C61#4112yes—
152C2×C8×C61#4112yes—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—81no—
161C4⋊C4×C61C22×C412yes—
162C4⋊C4×C61C22×C412yes—
163C42×C61—21no—
164C42×C61—21no—
165#11758198—22no—
166#11758198—22no—
167#11758198—42no—
168#11758198—42no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—42no—
176C4⋊Dic61—81no—
177C4⋊Dic61C22×C412yes—
178C4⋊Dic61C22×C412yes—
179#11758196—24no—
180#11758196—24no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—42no—
184C4×Dic61—42no—
185C4×Dic61—81no—
186C4×Dic61—81no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418C22×C412yes—
193C8×D61—28no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61C4×D412yes—
200C22×D61AGL(1,ℤ/8)12yes—
201C22×D61#13712yes—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C23×C411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61C4×D412yes—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—48no—
252C4×D61—84no—
253C8×C61—41no—
254C8×C61#11714yescomplemented
255#92418—22no—
256#92418—41no—
257#22615—611nomaximal, Sylow
258C22×C61C2×C4×D411yescharacteristic
259C22×C61#54711yescharacteristic
260C22×C61#54711yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C22×C61#57711yescharacteristic
264C22×C61#57711yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61#44014yes—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C2×#11711yescharacteristic
279C4×C61#51911yescharacteristic
280C4×C61#55211yescharacteristic
281C4×C61#55211yescharacteristic
282Dic61—22no—
283Dic61—41no—
284Dic61—41no—
285Dic61—42no—
286Dic61—42no—
287Dic61—81no—
288Dic61—81no—
289C22×#106—611no—
290#2345—611no—
291#2344—611no—
292#1944—611no—
293#1944D6111yescharacteristic, complemented
294#1920—611no—
295#1920—611no—
296#1919—611no—
297#1919—611no—
298#1824—611no—
299#1823—611no—
300#1808—611no—
301#1776—611no—
302#1776—611no—
303#1752—611no—
304C2×C61#234411yescharacteristic
305C2×C61C2×#44011yescharacteristic
306C2×C61#294911yescharacteristic
307C2×C61#334411yescharacteristic
308C2×C61#334411yescharacteristic
309C2×C61#334511yescharacteristic
310C2×C61#334511yescharacteristic
311D61—24no—
312C23×C8—611no—
313C2×#117—1221no—
314C22×C4⋊C4—611no—
315C22×C4⋊C4—611no—
316C2×#107—611no—
317C2×#107C2×D6111yescharacteristic
318C2×#106—611no—
319C2×#106—611no—
320C2×#106—611no—
321C2×#106—618no—
322C2×#106C2×D6111yescharacteristic
323C2×C22⋊C8—611no—
324C2×#97—611no—
325C2×#97C2×D6111yescharacteristic
326#389—611no—
327#389—611no—
328#389C2×D6111yescharacteristic
329#389C2×D6111yescharacteristic
330#387—1221no—
331#387—1221no—
332#387—1221no—
333#387—1221no—
334#381—1221no—
335#380—611no—
336#380—611no—
337#380—1221no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—611no—
342C4×C22⋊C4—611no—
343C4×C22⋊C4—1221no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340C2×D6111yescharacteristic
353#340C2×D6111yescharacteristic
354#336—611no—
355#336—611no—
356C61#2261511yescharacteristic, complemented, Sylow
357C23×C4—611no—
358C23×C4—1221no—
359C23×C4—1221no—
360C22×C8—611no—
361C22×C8—614no—
362C22×C8—611no—
363C22×C8—1222no—
364C22×C8—1222no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367#117—2444no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—614no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—611no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—22no—
397#107—1222no—
398#106—614no—
399#106—614no—
400#106—614no—
401#106C4×D6114yes—
402C22⋊C8—1222no—
403#97—22no—
404#97—1222no—
405#95—21no—
406#95—21no—
407#95—611no—
408#95—611no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417C24—611no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1224no—
428C22×C4—1224no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4D4×D6111yescharacteristic
442C22×C4D4×D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—612no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1228no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8C2×C4×D6112yes—
455C2×C8#1175822412yes—
456C2×C8#1175822612yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4C2×C4×D6112yes—
478C4⋊C4C2×C4×D6112yes—
479C22⋊C4—1222no—
480C22⋊C4—1222no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C42—21no—
484C42—21no—
485C42—1221no—
486C42—1221no—
487C42—4881no—
488C42—4881no—
489C42—2442no—
490C42—2442no—
491C23—612no—
492C23—612no—
493C23—612no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23C2×D4×D6111yescharacteristic
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—612no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—4884no—
533C2×C4—2448no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4#41×D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4C2×#1175822611yescharacteristic
548C2×C4#1451982411yescharacteristic
549C2×C4#1451982511yescharacteristic
550C8—41no—
551C8—1222no—
552C8—2441no—
553C8#1451973514yescomplemented
554C22—614no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22C4×D4×D6111yescharacteristic
562C22#1684041211yescharacteristic
563C22#1684041511yescharacteristic
564C22AGL(1,ℤ/8)×D6111yescharacteristic
565C22#137×D6111yescharacteristic
566C22#1684057111yescharacteristic
567C22#1684057211yescharacteristic
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—1222no—
575C4—4881no—
576C4—4881no—
577C4—2442no—
578C4—2442no—
579C4—2441no—
580C4—2441no—
581C4C2×#1451973511yescharacteristic
582C4#1684023811yescharacteristic
583C4#1684044611yescharacteristic
584C4#1684045011yescharacteristic
585C2—1224no—
586C2#2144868711yescharacteristic
587C2#440×D6111yescharacteristic
588C2#2145467511yescharacteristic
589C2#2145826911yescharacteristic
590C2#2145827311yescharacteristic
591C2#2145828411yescharacteristic
592C2#2145828811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1944C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1944D61
hypocenterC61#22615
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
4. #21446563C2semidirect
5. #21446407C2semidirect
6. #21446401C2semidirect
7. #21446400C2semidirect
8. #21446394C2semidirect
12. #21445718C2semidirect
13. #21445716C2semidirect
15. C61×#1944C2semidirect
24. #106×D61C4semidirect
254. C8×C61#117semidirect
293. #1944D61semidirect
356. C61#22615semidirect
553. C8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D612
#214486952
#214486872
#214465632
#214464072
#214464012
#214464002
#214463942
#214459082
#214459042
#214458452
#214457182
#214457162
#214455962
C61×#19442
#2261561