Finite Group Databasebeta

← #29658535

Subgroups — #29658535

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21455287C211yesmaximal, characteristic, complemented
4#21445032C214yesmaximal, complemented
5#21444889C214yesmaximal, complemented
6#21444499C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444213C211yesmaximal, characteristic, complemented
9C61×#3100C211yesmaximal, characteristic
10C2×#14519807—21no—
11#16840469—22no—
12#16840469—22no—
13#16840469—22no—
14#16840469C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519756—21no—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21C22×C4⋊Dic61—21no—
22#16840134C2212yescomplemented
23#16840114C2212yescomplemented
24#16839981—22no—
25#16839969C2212yescomplemented
26#16839967—22no—
27#16839959C2212yescomplemented
28#16839900—22no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839763—21no—
32#16839510C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#532C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#480—21no—
38C61×#474C2212yes—
39C61×#459C2212yes—
40C61×#447—21no—
41C4×D8×C61—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122D411yescharacteristic
44C22×Dic122D412yes—
45C23×Dic61—22no—
46#14519807—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758200—24no—
50C2×#11758194—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777D412yes—
54#14519777C2312yes—
55#14519768—24no—
56#14519768—24no—
57#14519756—24no—
58#14519756—24no—
59#14519754—28no—
60#14519754—28no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×Dic122D412yes—
64C4×Dic122C2312yes—
65D4×Dic61—24no—
66D4×Dic61—24no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—24no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×#11758196—22no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—22no—
76C2×C4×Dic61—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80#14519701—22no—
81#14519700—24no—
82#14519698—22no—
83C22⋊#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×D8×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90#127×C61C2311yescharacteristic
91#124×C61—21no—
92#121×C61—21no—
93#121×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#102×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108C4×C8×C61—21no—
109C24×C61—21no—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C2×D4×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—28no—
133C22×Dic61—84no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—22no—
138C2×Dic122—82no—
139C2×Dic122C2×D412yes—
140C2×Dic122C2×D414yes—
141D8×C61—44no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2411yescharacteristic
155C22⋊Dic61—28no—
156#11758202—44no—
157#11758200—44no—
158#11758200—84no—
159C4⋊Dic61—28no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C2×D412yes—
166#11758196—28no—
167#11758196—22no—
168#11758196—22no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—41no—
179C23×C61#12011yescharacteristic
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—24no—
183D4×C61—22no—
184D4×C61—42no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—42no—
188D4×C61—82no—
189D4×C61#13714yescomplemented
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—44no—
197C2×C4×C61—41no—
198C2×C4×C61—82no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—24no—
210C2×Dic61—44no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—88no—
214Dic122—22no—
215Dic122—22no—
216Dic122—42no—
217Dic122—44no—
218Dic122—82no—
219C8×C61—42no—
220#92418—42no—
221#92418—42no—
222#18752—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—24no—
226C22×C61—48no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61#44912yes—
233C22×C61C2×#12011yescharacteristic
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—81no—
239C4×C61#44912yes—
240C4×C61C2×#13511yescharacteristic
241C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C4×C61C2×#13712yes—
243Dic61—42no—
244Dic61—44no—
245Dic61—42no—
246C2×#554—611no—
247#3206—611no—
248#3100D6111yescharacteristic
249#3037—611no—
250#1644—614no—
251#1622—614no—
252#1548—611no—
253#1486—611no—
254#1479—611no—
255C2×C61—22no—
256C2×C61—44no—
257C2×C61—161no—
258C2×C61C2×#44911yescharacteristic
259C2×C61#300411yescharacteristic
260C2×C61#301911yescharacteristic
261C2×#137—1221no—
262#554—614no—
263#554—1222no—
264#554—1222no—
265#554—1222no—
266#532C2×D6111yescharacteristic
267C2×C4⋊Q8—611no—
268C2×#125—1221no—
269C2×C22⋊Q8—1222no—
270C2×C22⋊Q8—1221no—
271C2×C4×Q8—611no—
272C2×C4×D4—1221no—
273C2×C4×D4C2×D6111yescharacteristic
274C22×C4⋊C4—1221no—
275#492—612no—
276#487—612no—
277#480—21no—
278#474C2×D6112yes—
279#464—1222no—
280#462—612no—
281#461—1222no—
282#460—612no—
283#459C2×D6112yes—
284#448—1222no—
285#447—21no—
286C4×D8—22no—
287#431—611no—
288#431C2×D6111yescharacteristic
289#423—611no—
290#417—1221no—
291#333—614no—
292Q8⋊C8—614no—
293D4⋊C8—614no—
294C61#1875211yescharacteristic, complemented, Sylow
295C2×#41—21no—
296C22×Q8—611no—
297C22×Q8—612no—
298C22×D4C22⋊D6111yescharacteristic
299C23×C4—21no—
300C23×C4—1222no—
301#137—2444no—
302C2×Q16—1222no—
303C2×SD16—1224no—
304C2×SD16—1222no—
305C2×D8—22no—
306C2×M16—1221no—
307C4⋊Q8—612no—
308C4⋊Q8—612no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8C22×D6111yescharacteristic
312#127C22×D6111yescharacteristic
313#125—1224no—
314#125—1224no—
315#124—21no—
316C22⋊Q8—1228no—
317C22⋊Q8—1228no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320#121—21no—
321#121—21no—
322C4×Q8—612no—
323C4×Q8—612no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1224no—
331C4×D4—1224no—
332C4×D4C22⋊D6112yes—
333C4×D4C22×D6112yes—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1224no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—1222no—
345C2×C42—1221no—
346C2×C42C22×D6111yescharacteristic
347#107—22no—
348C4⋊C8—612no—
349C4⋊C8—1221no—
350C4⋊C8—1221no—
351C4⋊C8C22×D6112yes—
352#103—1224no—
353#103—1222no—
354#102—22no—
355#102—22no—
356#102—1222no—
357C22⋊C8—21no—
358C22⋊C8—1221no—
359C4×C8—21no—
360C4×C8—612no—
361C24—21no—
362#41—42no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—612no—
366C2×Q8—614no—
367C2×Q8—1222no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—4882no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4C2×C22⋊D6112yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—1228no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—4884no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393Q16—2444no—
394SD16—4884no—
395SD16—2444no—
396D8—44no—
397M16—2442no—
398C2×C8—22no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—612no—
405C4⋊C4—1228no—
406C4⋊C4—1228no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—1228no—
420C42—1222no—
421C42—1222no—
422C42D4×D6111yescharacteristic
423C42D4×D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C23×D6111yescharacteristic
426C23—24no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23#1451974511yescharacteristic
432Q8—41no—
433Q8—41no—
434Q8—1222no—
435Q8—1222no—
436Q8—4882no—
437Q8—2444no—
438Q8—2442no—
439D4—24no—
440D4—22no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—82no—
446D4#1451980714yescomplemented
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1224no—
459C2×C4—4888no—
460C2×C4—2442no—
461C2×C4—2444no—
462C2×C4—2442no—
463C2×C4#1451974511yescharacteristic
464C2×C4#1451974511yescharacteristic
465C2×C4#1451974511yescharacteristic
466C2×C4#1451979612yes—
467C2×C4#1451980112yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C8—42no—
472C8—2442no—
473C8—2442no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22#1683990412yes—
484C22C2×#1451974511yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—81no—
490C4—2442no—
491C4—2444no—
492C4—2442no—
493C4#1683990412yes—
494C4C2×#1451979611yescharacteristic
495C4C2×#1451980111yescharacteristic
496C4C2×#1451980712yes—
497C2—22no—
498C2—44no—
499C2—161no—
500C2C2×#1683990411yescharacteristic
501C2#2145505211yescharacteristic
502C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3100C2
socleC22×C61C2×#120
hypercenter#3100D61
hypocenterC61#18752
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. #21456741C2semidirect
3. #21455287C2semidirect
4. #21445032C2semidirect
5. #21444889C2semidirect
6. #21444499C2semidirect
7. #21444244C2semidirect
8. #21444213C2semidirect
22. #16840134C22semidirect
23. #16840114C22semidirect
25. #16839969C22semidirect
27. #16839959C22semidirect
32. #16839510C22semidirect
33. #16839492C22semidirect
189. D4×C61#137semidirect
294. C61#18752semidirect
446. D4#14519807semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
#214567412
#214552872
#214450322
#214448892
#214444992
#214442442
#214442132
C61×#31002
#1875261