Finite Group Databasebeta

← #29686956

Subgroups — #29686956

Order 15616.

868 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#21455539C211yesmaximal, characteristic
2#21454718C212yesmaximal, complemented
3C2×#16839883C211yesmaximal, characteristic, complemented
4#21454245C211yesmaximal, characteristic, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446607C211yesmaximal, characteristic, complemented
7#21446486C212yesmaximal, complemented
8#21446318C212yesmaximal, complemented
9#21445705C211yesmaximal, characteristic, complemented
10#21445618C212yesmaximal, complemented
11C61×#1814C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519785C2212yes—
15C2×#14519784C2211yescharacteristic
16#41×Dic61C2211yescharacteristic
17#16840037C412yescomplemented
18#16840032C412yescomplemented
19#16840025C414yescomplemented
20#16839883—21no—
21#16839883—21no—
22#16839883—21no—
23#16839883—21no—
24#16839881—24no—
25#16839876—24no—
26#16839810C2212yes—
27C2×#14519713C2211yescharacteristic
28#16839780C2212yes—
29C2×#14519703C2212yes—
30#16839744C2212yes—
31#16839735C2212yes—
32#16839720—21no—
33#16839720C2211yescharacteristic
34#16839718C2212yes—
35#16839717—21no—
36#16839717C2211yescharacteristic
37M16×Dic61—21no—
38M16×Dic61C2211yescharacteristic
39C2×#14519687—21no—
40C2×#14519687C2211yescharacteristic
41#16839533C2212yes—
42#16839521C2212yes—
43C2×#131×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C61×#427C2211yescharacteristic
46C61×#421C2212yes—
47C61×#344C2212yes—
48C2×#11758222C2312yes—
49#14519808—22no—
50#14519807C2×C414yes—
51#14519803—22no—
52#14519802C2×C414yes—
53#14519799—21no—
54#14519799—21no—
55#14519799—21no—
56#14519799—21no—
57#14519797—24no—
58C2×Dic244C2×C411yescharacteristic, complemented
59C2×#11758190C2×C412yescomplemented
60C2×D488C2×C411yescharacteristic, complemented
61#14519785—22no—
62#14519785—22no—
63#14519785—22no—
64#14519785—22no—
65#14519784—22no—
66#14519784D414yes—
67C2×#11758194—21no—
68C2×#11758194C2312yes—
69C2×#11758194C2311yescharacteristic
70C2×#11758189—22no—
71C22×#92418—21no—
72C22×#92418C2311yescharacteristic
73C2×C8×D61—22no—
74Q8×Dic61—21no—
75D4×Dic61—21no—
76D4×Dic61—22no—
77#14519736—21no—
78#14519736C2311yescharacteristic
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2311yescharacteristic
81#14519713C2×C412yescomplemented
82#14519713C2×C412yescomplemented
83#14519712—24no—
84#14519705—24no—
85#14519703—22no—
86#14519703—22no—
87#14519695C2×C414yescomplemented
88#14519693C2×C414yescomplemented
89#14519691—24no—
90#14519688—22no—
91#14519687—21no—
92#14519687—21no—
93#14519687—21no—
94#14519687D411yescharacteristic
95#14519687D411yescharacteristic
96#14519687D411yescharacteristic
97#14519687D411yescharacteristic
98C4×#92418—22no—
99C8×Dic61—21no—
100C8×Dic61—21no—
101C8×Dic61—21no—
102C2×#41×C61C2311yescharacteristic
103#131×C61—21no—
104#131×C61—21no—
105#131×C61C2×C412yes—
106#131×C61C2×C412yes—
107C2×M16×C61—21no—
108C2×M16×C61C2312yes—
109C2×M16×C61C2311yescharacteristic
110C22×C8×C61—21no—
111C22×C8×C61C2311yescharacteristic
112#117×C61C2312yes—
113#108×C61—21no—
114#108×C61—21no—
115#107×C61—21no—
116#106×C61—21no—
117#104×C61—24no—
118#41×C61—21no—
119#41×C61—21no—
120#41×C61C22×C412yes—
121#41×C61C2×D412yes—
122C2×Q8×C61#4111yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61#4111yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134C2×D244C22×C412yes—
135C2×C4×D61—22no—
136C2×Dic122C22×C412yes—
137M16×C61—21no—
138M16×C61—22no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61—22no—
142M16×C61C22×C412yes—
143C2×C8×C61—21no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61C22×C411yescharacteristic
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61#4111yescharacteristic
152C2×C8×C61#4111yescharacteristic
153C4⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C42×C61—22no—
156C22⋊Dic61—22no—
157C22⋊Dic61#4112yes—
158#11758202—24no—
159#11758201—24no—
160#11758200—24no—
161#11758199—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4×Dic61—22no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—22no—
170#11758194—21no—
171#11758194—24no—
172#11758194C22×C414yes—
173#11758194C2×D412yes—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—24no—
177C2×#92418C2×D411yescharacteristic
178C2×#92418C2×D412yes—
179C2×#92418C2×D411yescharacteristic
180C2×#92418#4112yes—
181Dic244—21no—
182Dic244—21no—
183D488—21no—
184D488—21no—
185#11758190—22no—
186#11758190—22no—
187#11758189—42no—
188#11758189—42no—
189C8×D61—42no—
190C8×D61—42no—
191C23×C61—21no—
192C23×C61C2×#4111yescharacteristic
193C22×D61—22no—
194Q8×C61—21no—
195Q8×C61C4×D412yes—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—21no—
199D4×C61C4×D412yes—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—42no—
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C22⋊D61—42no—
211C22⋊D61—42no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—81no—
217C2×Dic61—82no—
218D244—22no—
219D244—22no—
220C4×D61—42no—
221C4×D61—42no—
222Dic122—22no—
223Dic122—22no—
224C8×C61—22no—
225C8×C61—42no—
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61C4×D411yescharacteristic
230#92418—24no—
231#92418—22no—
232#92418#12112yes—
233#92418#12112yes—
234#21559—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61C4×#4111yescharacteristic
239C22×C61#52511yescharacteristic
240C22×C61#53211yescharacteristic
241C2×D61—42no—
242C2×D61—82no—
243C4×C61—22no—
244C4×C61—82no—
245C4×C61C2×C4×D411yescharacteristic
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×#12111yescharacteristic
248C4×C61#52911yescharacteristic
249Dic61—42no—
250Dic61—41no—
251Dic61—41no—
252Dic61—42no—
253#3067—611no—
254#2955—612no—
255C2×#444—611no—
256#2875—611no—
257#1976—611no—
258#1953—611no—
259#1932—612no—
260#1906—612no—
261#1814D6111yescharacteristic, complemented
262#1773—611no—
263#1757—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—42no—
268C2×C61C4×#12111yescharacteristic
269D61—82no—
270#575—612no—
271C2×#135—611no—
272C2×#131—611no—
273C2×#131—612no—
274C2×#131C2×D6111yescharacteristic
275C4×#41—611no—
276#473—612no—
277#472—612no—
278#471—614no—
279#444—1221no—
280#444—1221no—
281#444—1221no—
282#444—1221no—
283#444—1224no—
284#443—1224no—
285#430—612no—
286C2×#108—611no—
287C2×#108C2×D6111yescharacteristic
288#427C2×D6111yescharacteristic
289#421—612no—
290#421C2×D6112yes—
291C2×#104—612no—
292#413—612no—
293#410—612no—
294#405—611no—
295#405—611no—
296#405—612no—
297#405—1221no—
298#405—1221no—
299C4×M16—611no—
300C4×M16—1221no—
301C2×#97—611no—
302C2×#97—1221no—
303#344—612no—
304#344C2×D6112yes—
305#339—612no—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41—612no—
308C2×#41C22×D6111yescharacteristic
309#137—614no—
310#137—1222no—
311AGL(1,ℤ/8)—614no—
312AGL(1,ℤ/8)—1222no—
313#135—1224no—
314#135—1221no—
315#135—1221no—
316#135—1221no—
317#135—1221no—
318C2×Q16—611no—
319C2×SD16—612no—
320C2×D8—611no—
321#131—21no—
322#131—21no—
323#131—614no—
324#131—1222no—
325#131—1222no—
326#131—1222no—
327#131—1222no—
328#131—1222no—
329#131C4×D6112yes—
330#131C4×D6112yes—
331C2×M16—21no—
332C2×M16—611no—
333C2×M16—612no—
334C2×M16—1222no—
335C2×M16—1221no—
336C2×M16C22×D6112yes—
337C2×M16C22×D6111yescharacteristic
338C22×C8—21no—
339C22×C8—611no—
340C22×C8—1222no—
341C22×C8—1221no—
342C22×C8C22×D6111yescharacteristic
343C4×Q8—1221no—
344C4×D4—1222no—
345C4×D4—1221no—
346#117—611no—
347#117—1221no—
348#117C22×D6112yes—
349C2×C42—611no—
350C2×C42—1221no—
351#108—21no—
352#108—21no—
353#108—612no—
354#108—612no—
355#108—1224no—
356#107—21no—
357#106—21no—
358#104—24no—
359#104—1224no—
360#104—1222no—
361#104—1222no—
362#101—614no—
363#100—614no—
364#99—1224no—
365#97—611no—
366#97—611no—
367#97—611no—
368#97—611no—
369#97—1222no—
370#97—1221no—
371#97—1221no—
372#97—1221no—
373C4×C8—1222no—
374C4×C8—1221no—
375C4×C8—1221no—
376C4×C8—1221no—
377#41—21no—
378#41—21no—
379#41—1222no—
380#41—1222no—
381#41—1222no—
382#41—1222no—
383#41C2×C4×D6112yes—
384#41D4×D6112yes—
385C2×Q8—612no—
386C2×Q8#1175822611yescharacteristic
387C2×D4—21no—
388C2×D4—612no—
389C2×D4—1222no—
390C2×D4#1175822611yescharacteristic
391C22×C4—21no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4C23×D6111yescharacteristic
396Q16—1224no—
397Q16—1221no—
398Q16—1221no—
399SD16—1224no—
400SD16—1224no—
401SD16—1222no—
402SD16—1222no—
403D8—1224no—
404D8—1221no—
405D8—1221no—
406M16—21no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—614no—
412M16—612no—
413M16—1224no—
414M16—1221no—
415M16—1222no—
416M16—2442no—
417M16—2442no—
418M16C2×C4×D6112yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—611no—
424C2×C8—612no—
425C2×C8—611no—
426C2×C8—612no—
427C2×C8—1224no—
428C2×C8—1221no—
429C2×C8—1222no—
430C2×C8—2442no—
431C2×C8—2442no—
432C2×C8C2×C4×D6111yescharacteristic
433C2×C8C2×C4×D6111yescharacteristic
434C2×C8D4×D6111yescharacteristic
435C2×C8D4×D6111yescharacteristic
436C2×C8#1175822411yescharacteristic
437C2×C8#1175822411yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C22⋊C4—22no—
442C22⋊C4—612no—
443C22⋊C4—1222no—
444C42—22no—
445C42—611no—
446C42—611no—
447C42—1222no—
448C42—1221no—
449C42—1221no—
450C23—21no—
451C23—1222no—
452C23#41×D6111yescharacteristic
453Q8—21no—
454Q8—1222no—
455Q8—1222no—
456Q8#1451973912yes—
457D4—22no—
458D4—22no—
459D4—21no—
460D4—1222no—
461D4—1222no—
462D4—2442no—
463D4—2442no—
464D4#1451973912yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—1222no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1222no—
473C2×C4—4881no—
474C2×C4—4882no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822411yescharacteristic
482C2×C4C2×#1175822611yescharacteristic
483C8—22no—
484C8—42no—
485C8—612no—
486C8—612no—
487C8—1224no—
488C8—1222no—
489C8#1451973711yescharacteristic
490C8#1451973711yescharacteristic
491C8#1451973711yescharacteristic
492C8#1451973711yescharacteristic
493C22—22no—
494C22—41no—
495C22—42no—
496C22—4882no—
497C22—2442no—
498C22#1684023011yescharacteristic
499C22#1684027511yescharacteristic
500C22#1684030711yescharacteristic
501C4—22no—
502C4—82no—
503C4—2441no—
504C4—2442no—
505C4—2441no—
506C4—2442no—
507C4C2×#1451973711yescharacteristic
508C4C2×#1451973911yescharacteristic
509C4#121×D6111yescharacteristic
510C4#1684029811yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—4882no—
516C2#2144832511yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1814C2
socleC2×C61C4×#121
prime radical#21455539C2
cosocleC22×C4×C61C24
hypercenter#1814D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #21454718C2semidirect
3. C2×#16839883C2semidirect
4. #21454245C2semidirect
5. #21446715C2semidirect
6. #21446607C2semidirect
7. #21446486C2semidirect
8. #21446318C2semidirect
9. #21445705C2semidirect
10. #21445618C2semidirect
11. C61×#1814C2semidirect
17. #16840037C4semidirect
18. #16840032C4semidirect
19. #16840025C4semidirect
58. C2×Dic244C2×C4semidirect
59. C2×#11758190C2×C4semidirect
60. C2×D488C2×C4semidirect
81. #14519713C2×C4semidirect
82. #14519713C2×C4semidirect
87. #14519695C2×C4semidirect
88. #14519693C2×C4semidirect
261. #1814D61semidirect
306. C61#21559semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214547182
C2×#168398832
#214542452
#214467152
#214466072
#214464862
#214463182
#214457052
#214456182
C61×#18142
#2155961