Finite Group Databasebeta

← #29571986

Subgroups — #29571986

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445834C211yesmaximal, characteristic, complemented
2#21444468C211yesmaximal, characteristic, complemented
3#21444379C211yesmaximal, characteristic, complemented
4#21444379C211yesmaximal, characteristic, complemented
5#21444218C211yesmaximal, characteristic, complemented
6#21444214C211yesmaximal, characteristic, complemented
7C61×#3497C211yesmaximal, characteristic
8#16839815C2211yescharacteristic
9#16839815C2211yescharacteristic
10C2×C4⋊#92418C2211yescharacteristic
11C2×#14519701—21no—
12#16839628C2211yescharacteristic
13#16839524—21no—
14#16839508—21no—
15#16839506—21no—
16#16839496—22no—
17#16839496—22no—
18Q8⋊#92418—22no—
19D4⋊#92418—22no—
20C2×2−1+4×C61—21no—
21C61×#550—21no—
22C61×#550C412yescomplemented
23D4×Q8×C61—21no—
24D4×Q8×C61C412yescomplemented
25C61×#548—22no—
26C61×#548—22no—
27C61×#537—22no—
28C61×#537—21no—
29C61×#532—22no—
30C61×#532C2211yescharacteristic
31C61×#532C2211yescharacteristic
32C2×C4×Q8×C61C2211yescharacteristic
33C22×#92418—21no—
34C2×C4⋊Dic61—21no—
35C4⋊#92418—21no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39#14519701—42no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C4×#92418—21no—
43C4×#92418—21no—
44#14519684—21no—
45#14519684—21no—
462−1+4×C61—42no—
472−1+4×C61—42no—
482−1+4×C61—42no—
492−1+4×C61—42no—
50C2×#41×C61—24no—
51C2×#41×C61—22no—
52C2×#41×C61—22no—
53C2×#41×C61—21no—
54C2×#41×C61—21no—
55C22×Q8×C61—22no—
56C22×Q8×C61—22no—
57C22×Q8×C61D411yescharacteristic
58C4⋊Q8×C61—22no—
59C4⋊Q8×C61C2×C411yescharacteristic
60C4⋊Q8×C61D411yescharacteristic
61#127×C61—22no—
62#127×C61C2×C411yescharacteristic
63#127×C61D411yescharacteristic
64#124×C61—24no—
65#124×C61—21no—
66#124×C61—21no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—24no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—24no—
71#121×C61—24no—
72#121×C61—22no—
73#121×C61—24no—
74#121×C61—24no—
75#121×C61—21no—
76#121×C61—21no—
77#121×C61—22no—
78#121×C61D412yes—
79#121×C61D412yes—
80C4×Q8×C61—21no—
81C4×Q8×C61—21no—
82C4×Q8×C61C2×C412yes—
83C4×Q8×C61D412yes—
84C4×D4×C61—24no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61C2×C412yes—
90C4×D4×C61D412yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C42×C61—21no—
94C2×C42×C61C2311yescharacteristic
95#41×C61—48no—
96#41×C61—48no—
97#41×C61—44no—
98#41×C61—44no—
99#41×C61—44no—
100#41×C61—44no—
101#41×C61—41no—
102#41×C61—41no—
103#41×C61—41no—
104#41×C61—41no—
105#41×C61—41no—
106#41×C61—41no—
107#41×C61—41no—
108#41×C61—41no—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—44no—
116C2×Q8×C61—44no—
117C2×Q8×C61—44no—
118C2×Q8×C61—44no—
119C2×Q8×C61C22⋊C412yes—
120C2×Q8×C61D812yes—
121C2×Q8×C61SD1612yes—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—24no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C22⋊C412yes—
131C2×D4×C61C22⋊C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—24no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×Dic61—41no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C22⋊C412yes—
149C4⋊C4×C61C22⋊C412yes—
150C4⋊C4×C61C22⋊C412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C22×C411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C4⋊Dic61—41no—
164C4⋊Dic61—41no—
165C2×#92418—41no—
166C2×#92418—41no—
167C2×#92418—41no—
168C2×#92418—41no—
169C2×#92418—41no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61#12011yescharacteristic
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—48no—
177Q8×C61—41no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61#10214yes—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—48no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—81no—
195D4×C61—84no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—48no—
205C2×C4×C61—48no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—81no—
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C2×D811yescharacteristic
218C2×C4×C61C2×SD1611yescharacteristic
219C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
220C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
221C2×Dic61—81no—
222C2×Dic61—81no—
223C2×Dic61—81no—
224#92418—81no—
225#92418—81no—
226#92418—81no—
227#8429—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61#37911yescharacteristic
237C22×C61#44711yescharacteristic
238C22×C61#45011yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—24no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61C2×#10211yescharacteristic
248C4×C61#41811yescharacteristic
249C4×C61#44711yescharacteristic
250C4×C61#45011yescharacteristic
251Dic61—161no—
252#3497D6111yescharacteristic
253#1805—611no—
254#1540—611no—
255#1517—611no—
256#1517—611no—
257#1480—611no—
258#1479—611no—
259C2×C61—21no—
260C2×C61—21no—
261C2×C61—82no—
262C2×C61—82no—
263C2×C61#178611yescharacteristic
264C2×C61#179411yescharacteristic
265C2×C61#179611yescharacteristic
266C2×2−1+4—21no—
267#550—21no—
268#550Dic6112yescomplemented
269D4×Q8—21no—
270D4×Q8Dic6112yescomplemented
271#548—22no—
272#548—22no—
273#537—22no—
274#537—21no—
275#532—22no—
276#532C2×D6111yescharacteristic
277#532C2×D6111yescharacteristic
278C2×C4×Q8C2×D6111yescharacteristic
279#431—611no—
280#431—611no—
281C2×C4⋊C8—611no—
282C2×#103—1221no—
283#382—611no—
284#340—1221no—
285#332—1221no—
286#331—1221no—
287#327—1222no—
288#327—1222no—
289Q8⋊C8—1222no—
290D4⋊C8—1222no—
291C61#842911yescharacteristic, complemented, Sylow
2922−1+4—42no—
2932−1+4—42no—
2942−1+4—42no—
2952−1+4—42no—
296C2×#41—24no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—21no—
300C2×#41—21no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8C22⋊D6111yescharacteristic
304C22×C8—1221no—
305C4⋊Q8—22no—
306C4⋊Q8C2×Dic6111yescharacteristic
307C4⋊Q8C22⋊D6111yescharacteristic
308#127—22no—
309#127C2×Dic6111yescharacteristic
310#127C22⋊D6111yescharacteristic
311#124—24no—
312#124—21no—
313#124—21no—
314C22⋊Q8—22no—
315C22⋊Q8—24no—
316C22⋊Q8—22no—
317C22⋊Q8—24no—
318#121—24no—
319#121—22no—
320#121—24no—
321#121—24no—
322#121—21no—
323#121—21no—
324#121—22no—
325#121C22⋊D6112yes—
326#121C22⋊D6112yes—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8C2×Dic6112yes—
330C4×Q8C22⋊D6112yes—
331C4×D4—24no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4C2×Dic6112yes—
337C4×D4C22⋊D6112yes—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C42—21no—
342C2×C42C22×D6111yescharacteristic
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347#103—2442no—
348C22⋊C8—1221no—
349C22⋊C8—1221no—
350C4×C8—1221no—
351C4×C8—1221no—
352#95—1221no—
353#95—1221no—
354#41—48no—
355#41—48no—
356#41—44no—
357#41—44no—
358#41—44no—
359#41—44no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8#1175819912yes—
379C2×Q8#1175820012yes—
380C2×Q8C22⋊Dic6112yes—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4C22⋊Dic6112yes—
390C2×D4C22⋊Dic6112yes—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—2441no—
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C2×C8—2441no—
402C2×C8—2441no—
403C2×C8—2441no—
404C2×C8—2441no—
405C2×C8—2441no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4C22⋊Dic6112yes—
415C4⋊C4C22⋊Dic6112yes—
416C4⋊C4C22⋊Dic6112yes—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C42—22no—
424C42—22no—
425C42C22×Dic6111yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
429C23—22no—
430C23—22no—
431C23#1451974611yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—22no—
435Q8—48no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8#1451969814yes—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—48no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—81no—
454D4—84no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—81no—
470C2×C4—4881no—
471C2×C4—4881no—
472C2×C4—4881no—
473C2×C4C2×C22⋊Dic6111yescharacteristic
474C2×C4C2×C22⋊Dic6111yescharacteristic
475C2×C4C2×C22⋊Dic6111yescharacteristic
476C2×C4#1451974611yescharacteristic
477C2×C4#1451974611yescharacteristic
478C2×C4#1451974611yescharacteristic
479C2×C4C2×#1175819911yescharacteristic
480C2×C4C2×#1175820011yescharacteristic
481C2×C4#1451980311yescharacteristic
482C2×C4#1451980311yescharacteristic
483C8—4881no—
484C8—4881no—
485C8—4881no—
486C22—21no—
487C22—21no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22#1683961811yescharacteristic
495C22#1683989611yescharacteristic
496C22#1683991311yescharacteristic
497C4—24no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—9761no—
506C4C2×#1451969811yescharacteristic
507C4#1683976711yescharacteristic
508C4#1683989611yescharacteristic
509C4#1683991311yescharacteristic
510C2—21no—
511C2—21no—
512C2—82no—
513C2—82no—
514C2#2144575011yescharacteristic
515C2#2144578311yescharacteristic
516C2#2144579411yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3497D61
hypocenterC61#8429
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21445834C2semidirect
2. #21444468C2semidirect
3. #21444379C2semidirect
4. #21444379C2semidirect
5. #21444218C2semidirect
6. #21444214C2semidirect
22. C61×#550C4semidirect
24. D4×Q8×C61C4semidirect
268. #550Dic61semidirect
270. D4×Q8Dic61semidirect
291. C61#8429semidirect

Maximal subgroups

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

subgroupindex
#214458342
#214444682
#214443792
#214442182
#214442142
C61×#34972
#842961