Finite Group Databasebeta

← #29712866

Subgroups — #29712866

Order 15616.

1286 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2#21448411C211yesmaximal, characteristic, complemented
3#21447033C214yesmaximal, complemented
4#21446268C212yesmaximal, complemented
5#21446029C212yesmaximal, complemented
6#21445773C212yesmaximal, complemented
7#21445747C211yesmaximal, characteristic, complemented
8C61×#3485C211yesmaximal, characteristic
9C2×#14519736C2211yescharacteristic, complemented
10C22×C4⋊Dic61C2211yescharacteristic, complemented
11#16840095—22no—
12#16840090—22no—
13#16840078—22no—
14#16840073—22no—
15#16839805C2214yescomplemented
16C2×#14519704C2214yes—
17#16839763C2214yes—
18C2×#14519696C2212yescomplemented
19C2×#14519694C2212yes—
20#16839726C2212yescomplemented
21#16839669C2212yescomplemented
22#16839665C2212yes—
23#16839642—22no—
24#16839622—22no—
25#16839521C2214yescomplemented
26C2×H(5,𝔽2)×C61C2211yescharacteristic
27C22×#41×C61C2211yescharacteristic
28C61×#547—22no—
29C61×#547—22no—
30C61×#546—22no—
31C61×#546—22no—
32C61×#536C2212yes—
33C61×#534—22no—
34C61×#525C2212yes—
35C2×#123×C61—22no—
36C61×#518C2211yescharacteristic
37C23×Dic61—21no—
38C2×#11758194C2314yes—
39#14519736—42no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2312yes—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—24no—
47C2×C4×Dic61C2312yes—
48#14519710—24no—
49#14519708—24no—
50#14519704—44no—
51#14519701—24no—
52#14519698—24no—
53#14519696—42no—
54#14519694—24no—
55C22⋊#92418—22no—
56C22⋊#92418—22no—
57#14519684—24no—
58H(5,𝔽2)×C61—24no—
59H(5,𝔽2)×C61—42no—
60C2×#41×C61—22no—
61C2×#41×C61—24no—
62C2×#41×C61—22no—
63C2×#41×C61—22no—
64C2×#41×C61—42no—
65C2×#41×C61C2312yes—
66C22×Q8×C61C2311yescharacteristic
67C22×D4×C61—22no—
68C22×D4×C61—22no—
69C22×D4×C61D412yes—
70C22×D4×C61C2311yescharacteristic
71C23×C4×C61D412yes—
72C23×C4×C61C2311yescharacteristic
73C4⋊Q8×C61—22no—
74#124×C61—22no—
75#123×C61—24no—
76#123×C61—24no—
77#123×C61—24no—
78#123×C61—24no—
79#123×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61D412yes—
83C22⋊Q8×C61D412yes—
84#121×C61—24no—
85#121×C61—24no—
86#121×C61D412yes—
87#121×C61D412yes—
88#120×C61—24no—
89#120×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94#117×C61C2312yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C24×C61—22no—
101C24×C61C2×D411yescharacteristic
102C24×C61C2×D412yes—
103#41×C61—24no—
104#41×C61—24no—
105#41×C61—48no—
106#41×C61—48no—
107#41×C61—48no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D412yes—
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—42no—
121C2×D4×C61—48no—
122C2×D4×C61—44no—
123C2×D4×C61—44no—
124C2×D4×C61C2×D412yes—
125C2×D4×C61#4114yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—42no—
144C22×Dic61—82no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C42×C61—22no—
160C22⋊Dic61—42no—
161C22⋊Dic61—84no—
162C4⋊Dic61—24no—
163C4⋊Dic61—42no—
164C4⋊Dic61—41no—
165C4⋊Dic61—41no—
166C4×Dic61—24no—
167#11758194—24no—
168#11758194—24no—
169C2×#92418—24no—
170C23×C61—22no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—44no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61#12012yes—
183C23×C61#12314yes—
184C23×C61C22×D411yescharacteristic
185Q8×C61—24no—
186Q8×C61—44no—
187D4×C61—28no—
188D4×C61—24no—
189D4×C61—48no—
190D4×C61—48no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—82no—
194D4×C61—82no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—82no—
210C2×C4×C61—82no—
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C22×D412yes—
214C2×C4×C61C2×#4112yes—
215C2×C4×C61H(5,𝔽2)12yes—
216C2×Dic61—42no—
217C2×Dic61—44no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—84no—
222#92418—44no—
223#24025—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61—164no—
240C22×C61C2×#12011yescharacteristic
241C22×C61C2×#12311yescharacteristic
242C22×C61#53411yescharacteristic
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61#54614yes—
250Dic61—82no—
251Dic61—82no—
252#3485D6111yescharacteristic
253#3092—612no—
254#2311—611no—
255#2038—614no—
256#1897—612no—
257#1856—612no—
258#1792—612no—
259#1785—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—42no—
263C2×C61—44no—
264C2×C61—82no—
265C2×C61#240511yescharacteristic
266C2×C61#302111yescharacteristic
267C2×C61#302211yescharacteristic
268C2×H(5,𝔽2)C2×D6111yescharacteristic
269C22×#41C2×D6111yescharacteristic
270#547—22no—
271#547—22no—
272#546—22no—
273#546—22no—
274#536C2×D6112yes—
275#534—22no—
276#525C2×D6112yes—
277C2×#123—22no—
278#518C2×D6111yescharacteristic
279C2×#117—611no—
280C22×C4⋊C4—611no—
281#484—1222no—
282#483—1222no—
283#481—1222no—
284#480—1222no—
285#428—614no—
286C2×#104—614no—
287#417—614no—
288C2×#101—612no—
289C2×#100—612no—
290#407—612no—
291#391—612no—
292#390—612no—
293#386—1222no—
294#380—1222no—
295#339—614no—
296C61#2402511yescharacteristic, complemented, Sylow
297H(5,𝔽2)—24no—
298H(5,𝔽2)—42no—
299C2×#41—22no—
300C2×#41—24no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—42no—
304C2×#41C22×D6112yes—
305C22×Q8C22×D6111yescharacteristic
306C22×D4—22no—
307C22×D4—22no—
308C22×D4C22⋊D6112yes—
309C22×D4C22×D6111yescharacteristic
310C23×C4—1221no—
311C23×C4C22⋊D6112yes—
312C23×C4C22×D6111yescharacteristic
313C2×M16—614no—
314C4⋊Q8—22no—
315#124—22no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—22no—
320#123—24no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8C22⋊D6112yes—
324C22⋊Q8C22⋊D6112yes—
325#121—24no—
326#121—24no—
327#121C22⋊D6112yes—
328#121C22⋊D6112yes—
329#120—24no—
330#120—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335#117—2442no—
336#117C22×D6112yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1224no—
348C2×C22⋊C4—1221no—
349C2×C42—612no—
350#107—1224no—
351#106—1224no—
352#104—2444no—
353#103—1224no—
354#102—1224no—
355#101—2442no—
356#100—1224no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359#95—1224no—
360C24—22no—
361C24C2×C22⋊D6111yescharacteristic
362C24C2×C22⋊D6112yes—
363#41—24no—
364#41—24no—
365#41—48no—
366#41—48no—
367#41—48no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C2×C22⋊D6112yes—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4#1175822414yes—
385C2×D4C2×C22⋊D6112yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—4882no—
399C22×C4—2442no—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6112yes—
402C22×C4C2×C22⋊D6112yes—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C23×D6111yescharacteristic
405M16—1224no—
406M16—1224no—
407C2×C8—1224no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—1224no—
413C4⋊C4—2442no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4C2×C22⋊D6112yes—
417C4⋊C4C2×C22⋊D6112yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—4884no—
427C22⋊C4—2442no—
428C42—22no—
429C42—1224no—
430C23—22no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—82no—
441C23—82no—
442C23#1451974612yes—
443C23#1451975814yes—
444C23C22×C22⋊D6111yescharacteristic
445Q8—24no—
446Q8—44no—
447D4—28no—
448D4—24no—
449D4—48no—
450D4—48no—
451D4—42no—
452D4—42no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—4884no—
472C2×C4—2442no—
473C2×C4—2444no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451974611yescharacteristic
478C2×C4#1451974611yescharacteristic
479C2×C4C22×C22⋊D6112yes—
480C2×C4C2×#1175822412yes—
481C2×C4#1451982212yes—
482C8—2444no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—24no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—164no—
499C22C2×#1451974611yescharacteristic
500C22C2×#1451975811yescharacteristic
501C22#1684032111yescharacteristic
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—4882no—
509C4—4882no—
510C4#1684040614yes—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—44no—
515C2—82no—
516C2#2144914211yescharacteristic
517C2#2145519111yescharacteristic
518C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3485C2
socleC22×C61C2×#120
hypercenter#3485D61
hypocenterC61#24025
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #21455739C2semidirect
2. #21448411C2semidirect
3. #21447033C2semidirect
4. #21446268C2semidirect
5. #21446029C2semidirect
6. #21445773C2semidirect
7. #21445747C2semidirect
9. C2×#14519736C22semidirect
10. C22×C4⋊Dic61C22semidirect
15. #16839805C22semidirect
18. C2×#14519696C22semidirect
20. #16839726C22semidirect
21. #16839669C22semidirect
25. #16839521C22semidirect
296. C61#24025semidirect

Maximal subgroups

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

subgroupindex
#214557392
#214484112
#214470332
#214462682
#214460292
#214457732
#214457472
C61×#34852
#2402561