Finite Group Databasebeta

← #29680838

Subgroups — #29680838

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#519×D61C211yesmaximal, characteristic
2#21454406C212yesmaximal
3C2×#101×D61C211yesmaximal, characteristic
4#21448402C211yesmaximal, characteristic
5#21446036C211yesmaximal, characteristic, complemented
6#21446035C212yesmaximal, complemented
7#21446034C211yesmaximal, characteristic, complemented
8#21446033C211yesmaximal, characteristic, complemented
9#21446032C212yesmaximal, complemented
10#21445527C212yesmaximal
11C61×#1857C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840240C2211yescharacteristic
15#16840239—22no—
16#16840239C2212yes—
17#16840238—22no—
18#16840238—22no—
19C4×Q8×D61—22no—
20C4×Q8×D61—22no—
21C2×#14519735C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23#117×D61—22no—
24#117×D61C2212yes—
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#16839799—22no—
31#16839797—22no—
32C2×#14519696C2211yescharacteristic
33C2×#14519695C2212yes—
34#101×D61C418yescomplemented
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839666—22no—
39#16839624—22no—
40#16839523C2212yes—
41#16839522C2212yes—
42#16839521C2212yes—
43C61×#519C2211yescharacteristic
44C61×#428C2212yes—
45C2×#101×C61C2211yescharacteristic
46C61×#391C2211yescharacteristic
47C61×#339C2212yes—
48C22×Dic122D412yes—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61C2×C414yes—
54C2×Q8×D61Q814yes—
55C22×C4×D61D412yes—
56C22×C4×D61C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758189—22no—
59M16×D61—24no—
60M16×D61C2×C418yes—
61C2×C8×D61—22no—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64C4×Dic122—24no—
65C4×Dic122—24no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68#14519742—24no—
69#14519742—24no—
70#14519742—22no—
71#14519742—22no—
72#14519736—22no—
73#14519736C2312yes—
74#14519735—22no—
75#14519735—22no—
76#14519735—22no—
77#14519735—22no—
78#14519735D412yes—
79#14519735D412yes—
80#14519734—24no—
81#14519734—22no—
82#14519734—24no—
83#14519733—24no—
84#14519733—22no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61D412yes—
90C4⋊C4×D61D412yes—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C22⋊C4×D61—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C42×D61—24no—
97C42×D61—22no—
98#14519711—22no—
99#14519710—22no—
100#14519710—22no—
101#14519709—22no—
102#14519708—22no—
103#14519708—22no—
104#14519696C2×C414yescomplemented
105#14519695C2×C418yescomplemented
106#14519684—22no—
107#14519684—22no—
108#14519684—22no—
109C22×Q8×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#107×C61—22no—
118#106×C61—22no—
119#101×C61C2×C414yescomplemented
120#95×C61—22no—
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61C22×C412yes—
125C2×Q8×C61C2×Q812yes—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D411yescharacteristic
134Q8×D61—28no—
135Q8×D61—28no—
136Q8×D61—28no—
137Q8×D61—28no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61C4⋊C418yes—
144C2×C4×D61C22×C412yes—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×Q812yes—
148C2×C4×D61#4114yes—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—24no—
152C2×Dic122—24no—
153C2×Dic122C4⋊C418yes—
154C2×Dic122C22×C412yes—
155C2×Dic122C2×Q812yes—
156M16×C61—22no—
157M16×C61C22×C414yes—
158C2×C8×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—22no—
165C42×C61—24no—
166C42×C61—22no—
167C42×C61—22no—
168C22⋊Dic61—22no—
169#11758198—22no—
170#11758198—22no—
171#11758198—41no—
172#11758198—41no—
173#11758198—42no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—22no—
182#11758196—22no—
183#11758196—42no—
184C4×Dic61—24no—
185C4×Dic61—24no—
186C4×Dic61—24no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—42no—
192#11758194—22no—
193#11758194C22×C414yes—
194C2×#92418—22no—
195#11758189—28no—
196C8×D61—28no—
197C23×C61C22×D411yescharacteristic
198C22×D61—41no—
199C22×D61—41no—
200C22×D61C2×C4⋊C412yes—
201C22×D61#12011yescharacteristic
202C22×D61#12011yescharacteristic
203C22×D61#12312yes—
204Q8×C61—24no—
205Q8×C61—24no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61C2×C4⋊C414yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D412yes—
218C2×C4×C61C22×Q811yescharacteristic
219C2×C4×C61C2×#4112yes—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C2×C4⋊C414yes—
230C2×Dic61C2×C4⋊C412yes—
231C2×Dic61#12012yes—
232C2×Dic61#12312yes—
233C4×D61—28no—
234C4×D61—28no—
235C4×D61—44no—
236C4×D61—44no—
237C4×D61C4×D414yes—
238C4×D61C4×D414yes—
239C4×D61C22⋊Q814yes—
240C4×D61C22⋊Q814yes—
241Dic122—24no—
242Dic122—24no—
243Dic122—216no—
244C8×C61—24no—
245#92418—24no—
246#20857—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61C22×C4⋊C411yescharacteristic
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×#12311yescharacteristic
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61#38512yes—
259C2×D61#38512yes—
260C4×C61—24no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C2×C4×D412yes—
266C4×C61C2×C22⋊Q812yes—
267Dic61—24no—
268Dic61—42no—
269Dic61—42no—
270Dic61—81no—
271Dic61—81no—
272Dic61#38514yes—
273C2×#519—611no—
274#2911—612no—
275C22×#101—611no—
276#2310—611no—
277#1857—612no—
278#1857—611no—
279#1857—611no—
280#1857—612no—
281#1857—611no—
282#1857D6111yescharacteristic, complemented
283#1733—612no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61C2×#38511yescharacteristic
287C2×C61#302011yescharacteristic
288C2×C61#302011yescharacteristic
289D61—22no—
290D61—22no—
291C23×Q8—611no—
292C22×M16—612no—
293#519—611no—
294#519—612no—
295#519—1222no—
296#519—1222no—
297#519—1222no—
298#519C2×D6111yescharacteristic
299C2×C4×Q8—1222no—
300C2×C4×Q8—1222no—
301C2×#117—611no—
302C2×#117—612no—
303C2×#117—611no—
304C2×#117—1222no—
305#428—612no—
306#428—612no—
307#428—612no—
308#428—1222no—
309#428—1222no—
310#428C2×D6112yes—
311#427—1222no—
312#427—1222no—
313C2×#101—611no—
314C2×#101—612no—
315C2×#101—618no—
316C2×#101C2×D6111yescharacteristic
317#391—611no—
318#391—611no—
319#391—611no—
320#391—1222no—
321#391C2×D6111yescharacteristic
322#381—1222no—
323#339—612no—
324#339—612no—
325#339—612no—
326#339C2×D6112yes—
327C61#2085711yescharacteristic, complemented, Sylow
328C22×Q8—614no—
329C22×Q8—614no—
330C22×Q8—612no—
331C22×Q8—611no—
332C22×Q8—1222no—
333C22×Q8—1222no—
334C22×Q8—1224no—
335C22×Q8C22×D6111yescharacteristic
336C23×C4—611no—
337C23×C4—612no—
338C2×M16—612no—
339C2×M16—618no—
340C2×M16—1222no—
341C2×M16—1224no—
342C2×M16C22×D6112yes—
343C22×C8—1222no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—1222no—
347C4×Q8—1222no—
348C4×Q8—1224no—
349C4×Q8—1224no—
350C4×Q8—1224no—
351C4×Q8—1224no—
352C4×Q8—1222no—
353C4×Q8—1222no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356#117—22no—
357#117—612no—
358#117—612no—
359#117—612no—
360#117—1222no—
361#117—1224no—
362#117—1224no—
363#117—1222no—
364#117—1224no—
365#117—1222no—
366#117—1222no—
367#117—1222no—
368#117—1222no—
369#117—1222no—
370#117C22×D6112yes—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C42—611no—
382C2×C42—1222no—
383C2×C42—1224no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#107—22no—
387#107—1222no—
388#107—1222no—
389#107—1222no—
390#106—22no—
391#106—1222no—
392#106—1222no—
393#106—1222no—
394#101—614no—
395#101—618no—
396#101C4×D6114yescomplemented
397#95—22no—
398#95—1222no—
399#95—1222no—
400#95—1222no—
401C24—611no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—612no—
405C2×Q8—612no—
406C2×Q8—618no—
407C2×Q8—1224no—
408C2×Q8—1224no—
409C2×Q8—1228no—
410C2×Q8—1228no—
411C2×Q8—1228no—
412C2×Q8—1228no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8C2×C4×D6112yes—
416C2×Q8Q8×D6112yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—611no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—612no—
426C22×C4—618no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4D4×D6112yes—
435C22×C4C23×D6111yescharacteristic
436M16—22no—
437M16—614no—
438M16—1228no—
439M16—1222no—
440M16C2×C4×D6114yes—
441C2×C8—22no—
442C2×C8—1228no—
443C2×C8—1222no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—612no—
448C4⋊C4—612no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—2442no—
457C4⋊C4D4×D6112yes—
458C4⋊C4D4×D6112yes—
459C22⋊C4—22no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—2441no—
464C22⋊C4—2441no—
465C22⋊C4—2442no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—1222no—
470C42—1222no—
471C42—1224no—
472C42—1224no—
473C42—1224no—
474C42—1222no—
475C42—1222no—
476C42—2442no—
477C23—611no—
478C23—611no—
479C23—612no—
480C23—612no—
481C23—2441no—
482C23—2441no—
483C23C2×D4×D6111yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—12216no—
487Q8—1224no—
488Q8—1224no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—614no—
502C2×C4—614no—
503C2×C4—614no—
504C2×C4—614no—
505C2×C4—614no—
506C2×C4—1228no—
507C2×C4—1228no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—2444no—
513C2×C4—2444no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C4⋊C4×D6114yes—
520C2×C4C22×C4×D6111yescharacteristic
521C2×C4C2×D4×D6112yes—
522C2×C4C2×Q8×D6111yescharacteristic
523C2×C4#41×D6112yes—
524C8—24no—
525C8—1224no—
526C22—21no—
527C22—21no—
528C22—612no—
529C22—612no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1221no—
534C22—2442no—
535C22—2442no—
536C22C2×C4⋊C4×D6111yescharacteristic
537C22#120×D6111yescharacteristic
538C22#123×D6111yescharacteristic
539C4—24no—
540C4—42no—
541C4—42no—
542C4—81no—
543C4—81no—
544C4—614no—
545C4—1224no—
546C4—4881no—
547C4—4881no—
548C4—2442no—
549C4—2442no—
550C4C4×D4×D6112yes—
551C4C22⋊Q8×D6112yes—
552C2—21no—
553C2—21no—
554C2—1222no—
555C2—1222no—
556C2#385×D6111yescharacteristic
557C2#2145518011yescharacteristic
558C2#2145518411yescharacteristic
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1857C2
socleC22×C61C2×#120
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1857D61
hypocenterC61#20857
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
5. #21446036C2semidirect
6. #21446035C2semidirect
7. #21446034C2semidirect
8. #21446033C2semidirect
9. #21446032C2semidirect
11. C61×#1857C2semidirect
34. #101×D61C4semidirect
104. #14519696C2×C4semidirect
105. #14519695C2×C4semidirect
119. #101×C61C2×C4semidirect
282. #1857D61semidirect
327. C61#20857semidirect
396. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#519×D612
#214544062
C2×#101×D612
#214484022
#214460362
#214460352
#214460342
#214460332
#214460322
#214455272
C61×#18572
#2085761