Finite Group Databasebeta

← #29731506

Subgroups — #29731506

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456314C211yesmaximal, characteristic, complemented
2C2×#16840037C211yesmaximal, characteristic, complemented
3#21455512C212yesmaximal, complemented
4C2×#16839879C211yesmaximal, characteristic, complemented
5#21447034C212yesmaximal, complemented
6#21446866C212yesmaximal, complemented
7#21446678C211yesmaximal, characteristic, complemented
8#21446602C211yesmaximal, characteristic, complemented
9#21446332C212yesmaximal, complemented
10#21445654C211yesmaximal, characteristic, complemented
11C61×#3061C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519807C2212yes—
14C22×Dic244C2211yescharacteristic
15C2×#14519784C2212yes—
16#16840333C2212yes—
17#16840153—22no—
18#16840145—21no—
19#16840140—21no—
20#16840037—24no—
21#16840018—22no—
22#16840006—22no—
23#16840001—22no—
24#16839989—22no—
25#16839879—24no—
26C2×#14519713C2211yescharacteristic
27#16839802C2212yes—
28C2×#14519704C2211yescharacteristic
29C2×#14519704C2212yes—
30#16839758C2211yescharacteristic
31C2×#14519700C2212yes—
32C2×#14519695C2212yes—
33#16839717C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839667C2212yes—
36#16839531C2212yes—
37#16839526C2211yescharacteristic
38#16839523C2211yescharacteristic
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C61×#537C2212yes—
42C61×#468—22no—
43C61×#467—21no—
44C61×#466—21no—
45C2×#107×C61C2211yescharacteristic
46C61×#417C2212yes—
47C22×Dic122C2312yes—
48#14519807—24no—
49#14519807—24no—
50#14519807—42no—
51#14519807—42no—
52C2×Dic244—22no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—22no—
56C2×#11758202—22no—
57C2×#11758202—22no—
58C2×#11758200—22no—
59C2×#11758200—22no—
60#14519784—24no—
61#14519784—42no—
62C2×#11758194—22no—
63C2×#11758194C2312yes—
64C2×#11758194C2311yescharacteristic
65C22×#92418—22no—
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519764—22no—
69#14519758—42no—
70#14519756—22no—
71#14519756—22no—
72#14519736C2311yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74#14519713D414yescomplemented
75#14519710—22no—
76#14519708—22no—
77#14519704—24no—
78#14519704—22no—
79#14519701D411yescharacteristic
80#14519701D411yescharacteristic
81#14519700—22no—
82#14519700—22no—
83#14519698D411yescharacteristic
84#14519698D411yescharacteristic
85#14519695—24no—
86#14519687—21no—
87C8×Dic61—22no—
88C8×Dic61—21no—
89#14519684—42no—
90C2×#41×C61C2311yescharacteristic
91C2×#41×C61C2312yes—
92#135×C61—22no—
93#135×C61—22no—
94C2×Q16×C61D411yescharacteristic
95C2×SD16×C61D412yes—
96C2×D8×C61D411yescharacteristic
97#131×C61—22no—
98#131×C61—22no—
99C2×M16×C61—21no—
100C2×M16×C61C2311yescharacteristic
101C22×C8×C61—21no—
102C22×C8×C61C2311yescharacteristic
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105#121×C61—22no—
106#121×C61—22no—
107#121×C61—42no—
108C2×C4⋊C4×C61C2312yes—
109#107×C61—21no—
110#107×C61—21no—
111#103×C61—22no—
112#102×C61—22no—
113#41×C61—21no—
114#41×C61—22no—
115#41×C61—21no—
116#41×C61—22no—
117#41×C61—21no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—21no—
121C2×Q8×C61C2×D411yescharacteristic
122C2×Q8×C61C2×D412yes—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—42no—
126C2×D4×C61C2×D411yescharacteristic
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140Q16×C61—22no—
141SD16×C61—24no—
142D8×C61—22no—
143M16×C61—21no—
144M16×C61—21no—
145M16×C61—42no—
146C2×C8×C61—21no—
147C2×C8×C61—42no—
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61#4112yes—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C22⋊Dic61—21no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160#11758202—44no—
161#11758202—44no—
162#11758200—44no—
163#11758200—44no—
164C4⋊Dic61C2×D411yescharacteristic
165C4⋊Dic61C2×D411yescharacteristic
166#11758196—42no—
167#11758196—42no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171#11758194—22no—
172#11758194—21no—
173#11758194—44no—
174#11758194C2×D414yes—
175#11758194#4112yes—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418—44no—
181C2×#92418C2×D411yescharacteristic
182C2×#92418C2×D411yescharacteristic
183Dic244—42no—
184Dic244—42no—
185C23×C61—22no—
186C23×C61—21no—
187C23×C61H(5,𝔽2)11yescharacteristic
188Q8×C61—22no—
189Q8×C61—21no—
190Q8×C61—21no—
191Q8×C61—22no—
192D4×C61—22no—
193D4×C61—21no—
194D4×C61—21no—
195D4×C61—22no—
196D4×C61—41no—
197D4×C61—42no—
198D4×C61—41no—
199D4×C61—42no—
200D4×C61—162no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—82no—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223Dic122—42no—
224Dic122—42no—
225Dic122—42no—
226Dic122—42no—
227C8×C61—41no—
228C8×C61#12112yes—
229C8×C61#12112yes—
230#92418—21no—
231#92418—21no—
232#92418—24no—
233#92418—22no—
234#25442—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—41no—
238C22×C61—42no—
239C22×C61—41no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—41no—
243C22×C61#54011yescharacteristic
244C22×C61D4211yescharacteristic
245C22×C61#55011yescharacteristic
246C4×C61—42no—
247C4×C61—41no—
248C4×C61—82no—
249C4×C61C2×#12111yescharacteristic
250C4×C61#53211yescharacteristic
251C4×C61D4211yescharacteristic
252C4×C61#54611yescharacteristic
253Dic61—42no—
254Dic61—41no—
255Dic61—41no—
256Dic61—82no—
257#3156—611no—
258C2×#473—611no—
259#3063—612no—
260#3061D6111yescharacteristic
261C2×#443—611no—
262#2038—612no—
263#2007—612no—
264#1969—611no—
265#1952—611no—
266#1908—612no—
267#1764—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—81no—
271C2×C61—82no—
272C2×C61#242811yescharacteristic
273C2×C61#329011yescharacteristic
274C2×C61#329311yescharacteristic
275C2×#137—612no—
276C2×#137—612no—
277C2×#135C2×D6111yescharacteristic
278C22×Q16—611no—
279C2×#131—612no—
280C2×#131C2×D6111yescharacteristic
281#537C2×D6112yes—
282#536—612no—
283#497—1222no—
284#495—1221no—
285#494—1221no—
286#473—1224no—
287#470—1222no—
288#468—22no—
289#468—1222no—
290#467—21no—
291#467—1222no—
292#466—21no—
293#465—1222no—
294#443—1224no—
295C2×#108—611no—
296#428—612no—
297C2×#107C2×D6111yescharacteristic
298C2×#104—612no—
299C2×#104—611no—
300#417C2×D6112yes—
301#416—611no—
302C2×#103—612no—
303C2×#101—612no—
304#405—611no—
305C2×C4×C8—611no—
306#391—612no—
307#343—612no—
308#341—611no—
309#339—611no—
310C61#2544211yescharacteristic, complemented, Sylow
311C2×#41C22×D6111yescharacteristic
312C2×#41C22×D6112yes—
313C22×Q8—612no—
314#137—1224no—
315#137—1224no—
316#137—2442no—
317#137—2442no—
318#135—22no—
319#135—22no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×Q16—1222no—
323C2×Q16—1222no—
324C2×Q16—1221no—
325C2×Q16—1221no—
326C2×Q16C22⋊D6111yescharacteristic
327C2×SD16—1222no—
328C2×SD16—1222no—
329C2×SD16C22⋊D6112yes—
330C2×D8C22⋊D6111yescharacteristic
331#131—22no—
332#131—22no—
333#131—1224no—
334#131—2442no—
335C2×M16—21no—
336C2×M16—612no—
337C2×M16—611no—
338C2×M16—1222no—
339C2×M16C22×D6111yescharacteristic
340C22×C8—21no—
341C22×C8—611no—
342C22×C8—1222no—
343C22×C8C22×D6111yescharacteristic
344C4⋊Q8—1222no—
345#124—1222no—
346#123—2442no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—1222no—
350C22⋊Q8—1222no—
351#121—22no—
352#121—22no—
353#121—42no—
354#117—611no—
355C2×C4⋊C4C22×D6112yes—
356C2×C42—611no—
357#108—614no—
358#107—21no—
359#107—21no—
360#107—1222no—
361#106—1222no—
362#104—1222no—
363#104—1224no—
364#103—22no—
365#103—611no—
366#103—611no—
367#103—1222no—
368#103—1222no—
369#102—22no—
370#102—611no—
371#102—611no—
372#101—1224no—
373#97—1221no—
374C4×C8—1222no—
375C4×C8—1221no—
376#95—2442no—
377#41—22no—
378#41—21no—
379#41—21no—
380#41—22no—
381#41—21no—
382#41—22no—
383#41—22no—
384#41—21no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8D4×D6111yescharacteristic
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4D4×D6111yescharacteristic
397C2×D4C2×C22⋊D6112yes—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4C23×D6111yescharacteristic
404Q16—22no—
405Q16—2444no—
406Q16—2444no—
407Q16—2442no—
408Q16—2442no—
409SD16—24no—
410SD16—2444no—
411SD16—2444no—
412D8—22no—
413M16—21no—
414M16—21no—
415M16—42no—
416M16—614no—
417M16—612no—
418M16—1222no—
419M16—1221no—
420M16—2444no—
421C2×C8—21no—
422C2×C8—42no—
423C2×C8—611no—
424C2×C8—611no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—2444no—
430C2×C8D4×D6112yes—
431C2×C8#1175822412yes—
432C2×C8C2×C22⋊D6111yescharacteristic
433C2×C8C2×C22⋊D6111yescharacteristic
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—611no—
438C4⋊C4—611no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—1221no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C42—1221no—
447C42—1221no—
448C42—1221no—
449C23—21no—
450C23—22no—
451C23#1451982211yescharacteristic
452Q8—22no—
453Q8—21no—
454Q8—21no—
455Q8—22no—
456Q8—2442no—
457Q8—2442no—
458Q8—2442no—
459Q8—2442no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—22no—
464D4—41no—
465D4—42no—
466D4—41no—
467D4—42no—
468D4—162no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—1222no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4#41×D6111yescharacteristic
489C2×C4C2×#1175822411yescharacteristic
490C2×C4#1451982311yescharacteristic
491C8—41no—
492C8—1221no—
493C8—1221no—
494C8—1224no—
495C8—1222no—
496C8#1451974912yes—
497C8#1451974912yes—
498C22—21no—
499C22—21no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—42no—
506C22#1684036511yescharacteristic
507C22D4×C22⋊D6111yescharacteristic
508C22#1684043011yescharacteristic
509C4—41no—
510C4—42no—
511C4—82no—
512C4—4882no—
513C4—2442no—
514C4—2441no—
515C4—2441no—
516C4C2×#1451974911yescharacteristic
517C4#1684030711yescharacteristic
518C4#1684039611yescharacteristic
519C4#1684040011yescharacteristic
520C2—21no—
521C2—21no—
522C2—81no—
523C2—82no—
524C2#2144940911yescharacteristic
525C2#2145760111yescharacteristic
526C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3061C2
socleC22×C61D42
hypercenter#3061D61
hypocenterC61#25442
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. #21456314C2semidirect
2. C2×#16840037C2semidirect
3. #21455512C2semidirect
4. C2×#16839879C2semidirect
5. #21447034C2semidirect
6. #21446866C2semidirect
7. #21446678C2semidirect
8. #21446602C2semidirect
9. #21446332C2semidirect
10. #21445654C2semidirect
74. #14519713D4semidirect
310. C61#25442semidirect

Maximal subgroups

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

subgroupindex
#214563142
C2×#168400372
#214555122
C2×#168398792
#214470342
#214468662
#214466782
#214466022
#214463322
#214456542
C61×#30612
#2544261