Finite Group Databasebeta

← #29712858

Subgroups — #29712858

Order 15616.

1274 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2#21448411C211yesmaximal, characteristic, complemented
3#21447026C214yesmaximal, complemented
4#21446268C212yesmaximal, complemented
5#21446034C212yesmaximal, complemented
6#21445778C212yesmaximal, complemented
7#21445747C211yesmaximal, characteristic, complemented
8C61×#3487C211yesmaximal, characteristic
9C2×#14519736C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16840095—22no—
12#16840090—22no—
13#16840078—22no—
14#16840073—22no—
15#16839805C2214yes—
16C2×#14519704C2214yes—
17#16839763C2214yes—
18C2×#14519696C2212yes—
19C2×#14519694C2212yes—
20#16839726C2212yes—
21#16839669C2212yes—
22#16839665C2212yes—
23#16839642—22no—
24#16839622—22no—
25#16839521C2214yes—
26C2×2−1+4×C61C2211yescharacteristic
27C22×#41×C61C2211yescharacteristic
28C61×#550—22no—
29C61×#550—22no—
30C61×#548—22no—
31C61×#548—22no—
32C61×#535—22no—
33C61×#535C2212yes—
34C61×#525—22no—
35C61×#525C2212yes—
36C61×#519C2211yescharacteristic
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—24no—
54#14519694—42no—
55C22⋊#92418—22no—
56C22⋊#92418—22no—
57#14519684—24no—
582−1+4×C61—24no—
592−1+4×C61—42no—
60C2×#41×C61—24no—
61C2×#41×C61—22no—
62C2×#41×C61—22no—
63C2×#41×C61—22no—
64C2×#41×C61—22no—
65C2×#41×C61—42no—
66C2×#41×C61C2312yes—
67C22×Q8×C61—22no—
68C22×Q8×C61C2311yescharacteristic
69C22×D4×C61D412yes—
70C22×D4×C61C2311yescharacteristic
71C23×C4×C61D412yes—
72C23×C4×C61C2311yescharacteristic
73#127×C61—24no—
74#127×C61—22no—
75#124×C61—24no—
76#124×C61—22no—
77#123×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61D412yes—
81C22⋊Q8×C61D412yes—
82#121×C61—24no—
83#121×C61—24no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—22no—
88#121×C61D412yes—
89#121×C61D412yes—
90#120×C61—22no—
91#120×C61—24no—
92C4×Q8×C61—22no—
93C4×Q8×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—24no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61C2312yes—
100C24×C61C2×D412yes—
101C24×C61C2×D411yescharacteristic
102#41×C61—24no—
103#41×C61—24no—
104#41×C61—48no—
105#41×C61—48no—
106#41×C61—44no—
107#41×C61—48no—
108#41×C61—44no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—48no—
113C2×Q8×C61C2×D412yes—
114C2×Q8×C61#4114yes—
115C2×D4×C61—22no—
116C2×D4×C61—24no—
117C2×D4×C61—22no—
118C2×D4×C61—24no—
119C2×D4×C61—24no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—42no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
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—22no—
146C4⋊C4×C61—22no—
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—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C42×C61—24no—
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—21no—
171C23×C61—21no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61#12012yes—
181C23×C61C22×D411yescharacteristic
182Q8×C61—28no—
183Q8×C61—24no—
184Q8×C61—44no—
185D4×C61—24no—
186D4×C61—48no—
187D4×C61—44no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—44no—
191D4×C61—82no—
192D4×C61—82no—
193D4×C61—84no—
194D4×C61—84no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—44no—
205C2×C4×C61—44no—
206C2×C4×C61—48no—
207C2×C4×C61—44no—
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12314yes—
211C2×C4×C61C22×D412yes—
212C2×C4×C61C2×#4112yes—
213C2×C4×C61H(5,𝔽2)12yes—
214C2×Dic61—42no—
215C2×Dic61—44no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—84no—
220#92418—44no—
221#24024—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—44no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—82no—
237C22×C61C2×#12011yescharacteristic
238C22×C61C2×#12311yescharacteristic
239C22×C61#53411yescharacteristic
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—24no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61#54614yes—
247Dic61—82no—
248Dic61—82no—
249#3487D6111yescharacteristic
250#3092—612no—
251#2311—611no—
252#2037—614no—
253#1897—612no—
254#1857—612no—
255#1793—612no—
256#1785—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—42no—
260C2×C61—82no—
261C2×C61—82no—
262C2×C61#240511yescharacteristic
263C2×C61#302111yescharacteristic
264C2×C61#302211yescharacteristic
265C2×2−1+4C2×D6111yescharacteristic
266C22×#41C2×D6111yescharacteristic
267#550—22no—
268#550—22no—
269#548—22no—
270#548—22no—
271#535—22no—
272#535C2×D6112yes—
273#525—22no—
274#525C2×D6112yes—
275#519C2×D6111yescharacteristic
276C2×#117—611no—
277C22×C4⋊C4—611no—
278#484—1222no—
279#483—1222no—
280#481—1222no—
281#480—1222no—
282#428—614no—
283C2×#104—614no—
284#417—614no—
285C2×#101—612no—
286C2×#100—612no—
287#407—612no—
288#391—612no—
289#390—612no—
290#386—1222no—
291#380—1222no—
292#339—614no—
293C61#2402411yescharacteristic, complemented, Sylow
2942−1+4—24no—
2952−1+4—42no—
296C2×#41—24no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—42no—
302C2×#41C22×D6112yes—
303C22×Q8—22no—
304C22×Q8C22×D6111yescharacteristic
305C22×D4C22⋊D6112yes—
306C22×D4C22×D6111yescharacteristic
307C23×C4—1221no—
308C23×C4C22⋊D6112yes—
309C23×C4C22×D6111yescharacteristic
310C2×M16—614no—
311#127—24no—
312#127—22no—
313#124—24no—
314#124—22no—
315#123—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8C22⋊D6112yes—
319C22⋊Q8C22⋊D6112yes—
320#121—22no—
321#121—22no—
322#121—24no—
323#121—24no—
324#121—22no—
325#121—22no—
326#121C22⋊D6112yes—
327#121C22⋊D6112yes—
328#120—24no—
329#120—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—22no—
335C4×D4—22no—
336#117—22no—
337#117—2442no—
338#117C22×D6112yes—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C22⋊C4—1224no—
345C2×C22⋊C4—1221no—
346C2×C42—612no—
347#107—1224no—
348#106—1224no—
349#104—2444no—
350#103—1224no—
351#102—1224no—
352#101—1224no—
353#100—2442no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356#95—1224no—
357C24C2×C22⋊D6112yes—
358C24C2×C22⋊D6111yescharacteristic
359#41—24no—
360#41—24no—
361#41—48no—
362#41—48no—
363#41—44no—
364#41—48no—
365#41—44no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—48no—
370C2×Q8#1175822414yes—
371C2×Q8C2×C22⋊D6112yes—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—42no—
384C2×D4C2×C22⋊D6112yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—42no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—4882no—
396C22×C4—2442no—
397C22×C4C2×C22⋊D6112yes—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6112yes—
401C22×C4C23×D6111yescharacteristic
402M16—1224no—
403M16—1224no—
404C2×C8—1224no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—1224no—
410C4⋊C4—2442no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—4884no—
423C22⋊C4—2442no—
424C42—24no—
425C42—22no—
426C42—1224no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23#1451974612yes—
438C23C22×C22⋊D6111yescharacteristic
439Q8—28no—
440Q8—24no—
441Q8—44no—
442D4—24no—
443D4—48no—
444D4—44no—
445D4—42no—
446D4—42no—
447D4—44no—
448D4—82no—
449D4—82no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—4884no—
466C2×C4—2442no—
467C2×C4—2444no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4#1451974611yescharacteristic
472C2×C4#1451974611yescharacteristic
473C2×C4#1451975814yes—
474C2×C4C22×C22⋊D6112yes—
475C2×C4C2×#1175822412yes—
476C2×C4#1451982212yes—
477C8—2444no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22C2×#1451974611yescharacteristic
494C22C2×#1451975811yescharacteristic
495C22#1684032111yescharacteristic
496C4—21no—
497C4—21no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—4882no—
503C4—4882no—
504C4#1684040614yes—
505C2—21no—
506C2—21no—
507C2—42no—
508C2—82no—
509C2—82no—
510C2#2144914211yescharacteristic
511C2#2145519111yescharacteristic
512C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3487C2
socleC22×C61C2×#120
hypercenter#3487D61
hypocenterC61#24024
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3487

Complements

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

NH ≅ G/Ntype
1. #21455739C2semidirect
2. #21448411C2semidirect
3. #21447026C2semidirect
4. #21446268C2semidirect
5. #21446034C2semidirect
6. #21445778C2semidirect
7. #21445747C2semidirect
293. C61#24024semidirect

Maximal subgroups

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

subgroupindex
#214557392
#214484112
#214470262
#214462682
#214460342
#214457782
#214457472
C61×#34872
#2402461