Finite Group Databasebeta

← #29708073

Subgroups — #29708073

Order 15616.

1202 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840315C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic
3C2×#16839658C211yesmaximal, characteristic
4#21447159C212yesmaximal, complemented
5#21447155C212yesmaximal, complemented
6#21447150C212yesmaximal, complemented
7#21447145C212yesmaximal, complemented
8#21445540C211yesmaximal, characteristic
9#21445539C211yesmaximal, characteristic
10#21445512C211yesmaximal, characteristic
11C2×C61×#428C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840315—24no—
14#16840315—24no—
15C2×#14519776C2211yescharacteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519754—21no—
20C2×#14519754—21no—
21C2×C4×Dic122—21no—
22C2×C4×Dic122—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4⋊Dic61C2211yescharacteristic
25C22×#11758196—21no—
26C22×C4×Dic61C2211yescharacteristic
27#16839802—24no—
28C2×#14519710C2212yes—
29C2×#14519708C2212yes—
30#16839658—21no—
31#16839658—21no—
32#16839658C2212yes—
33#16839658C2212yes—
34C2×#14519684—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#428—24no—
45C2×#107×C61C2212yes—
46C2×#106×C61C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194Q814yescomplemented
55C22×#92418C2312yes—
56#14519776—22no—
57#14519776—22no—
58#14519776—22no—
59#14519776—22no—
60#14519768—22no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519754—28no—
65#14519754—28no—
66C4×Dic122—28no—
67C4×Dic122—28no—
68#14519736—22no—
69#14519736D414yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61Q814yescomplemented
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—22no—
83C2×#11758196—22no—
84C2×#11758196—44no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519710—24no—
94#14519708—24no—
95#14519684—22no—
96#14519684—22no—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C2×M16×C61—22no—
101C2×M16×C61Q814yescomplemented
102C22×C8×C61C2312yes—
103#117×C61—22no—
104#117×C61D414yes—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—21no—
113#107×C61—24no—
114#106×C61—24no—
115C24×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×Q812yes—
126C22×C4×C61C2×Q812yes—
127C22×C4×C61C2×Q812yes—
128C22×C4×C61#4112yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×Dic122—24no—
147C2×Dic122—24no—
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152M16×C61—28no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61C2×Q814yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C42×C61—24no—
165C22⋊Dic61—24no—
166C4⋊Dic61—24no—
167C4⋊Dic61—24no—
168C4⋊Dic61—24no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—22no—
172C4⋊Dic61C2×D412yes—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×Q814yes—
175#11758196—24no—
176#11758196—24no—
177#11758196—24no—
178#11758196—24no—
179#11758196—44no—
180#11758196—44no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—24no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188#11758194—28no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418C2×Q814yes—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61#12012yes—
197C23×C61C22⋊Q812yes—
198C23×C61C4⋊Q812yes—
199C23×C61C22×D411yescharacteristic
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—84no—
210C2×C4×C61#12012yes—
211C2×C4×C61C22⋊Q814yes—
212C2×C4×C61C22⋊Q814yes—
213C2×C4×C61C22⋊Q812yes—
214C2×C4×C61C4⋊Q814yes—
215C2×C4×C61C4⋊Q814yes—
216C2×C4×C61C4⋊Q812yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×Q811yescharacteristic
220C2×C4×C61C22×Q811yescharacteristic
221C2×C4×C61C22×Q811yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—44no—
237C2×Dic61—44no—
238C2×Dic61—48no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61AGL(1,ℤ/8)12yes—
246C2×Dic61#13712yes—
247Dic122—48no—
248Dic122—48no—
249C8×C61—24no—
250#92418—24no—
251#23554—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61#39514yes—
259C22×C61C2×#12011yescharacteristic
260C22×C61C2×C22⋊Q811yescharacteristic
261C22×C61C2×C4⋊Q811yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C22×C61#57511yescharacteristic
265C22×C61#57511yescharacteristic
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—82no—
269C4×C61#39514yes—
270C4×C61C2×C22⋊Q811yescharacteristic
271C4×C61C2×C22⋊Q811yescharacteristic
272C4×C61C2×C4⋊Q811yescharacteristic
273C4×C61C2×C4⋊Q811yescharacteristic
274Dic61—22no—
275Dic61—22no—
276Dic61—44no—
277Dic61—42no—
278Dic61—41no—
279Dic61—42no—
280Dic61—41no—
281C2×#533—611no—
282C2×#428—611no—
283C2×#428D6111yescharacteristic
284C2×#389—611no—
285#2059—612no—
286#2059—612no—
287#2058—612no—
288#2057—612no—
289#1736—611no—
290#1736—611no—
291#1728—611no—
292C2×C61—24no—
293C2×C61C2×#39511yescharacteristic
294C2×C61#300611yescharacteristic
295C2×C61#300711yescharacteristic
296C2×C61#307711yescharacteristic
297C2×C61#307811yescharacteristic
298C2×C61#316511yescharacteristic
299C2×C61#316611yescharacteristic
300C22×M16—611no—
301C22×M16C2×D6111yescharacteristic
302#533—1224no—
303#533—1224no—
304C2×C4⋊Q8—611no—
305C2×C4⋊Q8—611no—
306C2×#125—611no—
307C2×#125—611no—
308C2×C22⋊Q8—1221no—
309C2×C22⋊Q8—1221no—
310C2×C4×Q8—1221no—
311C2×C4×Q8—1221no—
312C2×#117—611no—
313C2×#117C2×D6111yescharacteristic
314C22×C4⋊C4—611no—
315C22×C4⋊C4—1221no—
316C22×C4⋊C4C2×D6111yescharacteristic
317C22×C42—611no—
318#428—24no—
319#428—1224no—
320C2×#107—612no—
321C2×#107C2×D6112yes—
322C2×#106—612no—
323C2×#106C2×D6112yes—
324#389—612no—
325#389—612no—
326#389—1221no—
327#389—1221no—
328C2×#95—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340—612no—
333#336—612no—
334#336—612no—
335C61#2355411yescharacteristic, complemented, Sylow
336C22×Q8—1221no—
337C22×Q8—1221no—
338C23×C4—21no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—1221no—
342C23×C4C22×D6111yescharacteristic
343C2×M16—22no—
344C2×M16—22no—
345C2×M16—614no—
346C2×M16—1222no—
347C2×M16—1222no—
348C2×M16Dic12214yescomplemented
349C22×C8—612no—
350C22×C8C22×D6112yes—
351C4⋊Q8—1222no—
352C4⋊Q8—1222no—
353C4⋊Q8—1222no—
354C4⋊Q8—1222no—
355#125—1222no—
356#125—1222no—
357#125—1222no—
358#125—1222no—
359C22⋊Q8—1228no—
360C22⋊Q8—1228no—
361C4×Q8—1228no—
362C4×Q8—1228no—
363#117—22no—
364#117—614no—
365#117—1222no—
366#117C22⋊D6114yes—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—2444no—
385C2×C4⋊C4C22×D6112yes—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—1221no—
396C2×C42—1222no—
397C2×C42—1222no—
398#107—24no—
399#107—1224no—
400#106—24no—
401#106—1224no—
402#95—1222no—
403#95—1222no—
404C24D4×D6111yescharacteristic
405C2×Q8—1224no—
406C2×Q8—1224no—
407C2×Q8—2442no—
408C2×Q8—2442no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1224no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4C2×Dic12212yes—
434C22×C4#1175822212yes—
435C22×C4D4×D6112yes—
436C22×C4D4×D6111yescharacteristic
437C22×C4Q8×D6112yes—
438C22×C4Q8×D6112yes—
439C22×C4C2×C22⋊D6112yes—
440C22×C4C23×D6111yescharacteristic
441M16—28no—
442M16—1228no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—614no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8C2×Dic12214yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—612no—
455C4⋊C4—612no—
456C4⋊C4—614no—
457C4⋊C4—1224no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1222no—
466C4⋊C4—1224no—
467C4⋊C4—2444no—
468C4⋊C4—2444no—
469C4⋊C4C2×C22⋊D6112yes—
470C4⋊C4C2×C22⋊D6112yes—
471C22⋊C4—24no—
472C22⋊C4—1224no—
473C42—24no—
474C42—1221no—
475C42—1221no—
476C42—1221no—
477C42—1221no—
478C42—1222no—
479C42—1222no—
480C42—1224no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#1451974512yes—
486C23#1451975512yes—
487C23#1451977612yes—
488C23C2×D4×D6111yescharacteristic
489Q8—2448no—
490Q8—2448no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—84no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1224no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—2444no—
517C2×C4—2444no—
518C2×C4—2448no—
519C2×C4—2442no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4#1451974512yes—
526C2×C4#1451975512yes—
527C2×C4#1451975514yes—
528C2×C4#1451975614yes—
529C2×C4#1451977514yes—
530C2×C4#1451977612yes—
531C2×C4#1451977614yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×Q8×D6111yescharacteristic
535C2×C4C2×Q8×D6111yescharacteristic
536C2×C4C22×Dic12211yescharacteristic
537C2×C4C2×#1175822211yescharacteristic
538C8—24no—
539C8—1224no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22#1683968214yes—
547C22C2×#1451974511yescharacteristic
548C22C2×#1451975511yescharacteristic
549C22C2×#1451977611yescharacteristic
550C22AGL(1,ℤ/8)×D6111yescharacteristic
551C22#137×D6111yescharacteristic
552C22#1684055811yescharacteristic
553C22#1684056111yescharacteristic
554C4—42no—
555C4—42no—
556C4—82no—
557C4—1222no—
558C4—1222no—
559C4—2441no—
560C4—2442no—
561C4—2441no—
562C4—2442no—
563C4—2444no—
564C4#1683968214yes—
565C4C2×#1451975511yescharacteristic
566C4C2×#1451975611yescharacteristic
567C4C2×#1451977511yescharacteristic
568C4C2×#1451977611yescharacteristic
569C2—24no—
570C2C2×#1683968211yescharacteristic
571C2#2145506711yescharacteristic
572C2#2145507511yescharacteristic
573C2#2145561511yescharacteristic
574C2#2145562311yescharacteristic
575C2#2145636311yescharacteristic
576C2#2145636911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#428D61
hypocenterC61#23554
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #21447159C2semidirect
5. #21447155C2semidirect
6. #21447150C2semidirect
7. #21447145C2semidirect
54. C2×#11758194Q8semidirect
74. C2×C4⋊Dic61Q8semidirect
101. C2×M16×C61Q8semidirect
335. C61#23554semidirect
348. C2×M16Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168403152
C2×#168398022
C2×#168396582
#214471592
#214471552
#214471502
#214471452
#214455402
#214455392
#214455122
C2×C61×#4282
#2355461