Finite Group Databasebeta

← #29686966

Subgroups — #29686966

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#21455535C211yesmaximal, characteristic
2#21454718C212yesmaximal, complemented
3C2×#16839884C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic, complemented
5#21446609C211yesmaximal, characteristic, complemented
6#21446491C212yesmaximal, complemented
7#21446317C212yesmaximal, complemented
8#21445870C211yesmaximal, characteristic, complemented
9#21445705C211yesmaximal, characteristic, complemented
10#21445621C212yesmaximal, complemented
11C61×#1976C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519785C2211yescharacteristic
14C2×#14519784C2212yes—
15C2×#14519783C2211yescharacteristic
16#16840230C2211yescharacteristic
17#16840034C412yescomplemented
18#16840029C412yescomplemented
19#16840022C414yescomplemented
20#16839884—21no—
21#16839884—21no—
22#16839884—21no—
23#16839884—21no—
24#16839881—24no—
25#16839876—24no—
26#16839810C2212yes—
27C2×#14519713C2211yescharacteristic
28C2×#14519713C2211yescharacteristic
29#16839780C2212yes—
30C2×#14519704C2212yes—
31#16839746C2212yes—
32#16839719—21no—
33#16839718C2212yes—
34#16839718C2211yescharacteristic
35#16839716—21no—
36M16×Dic61C2211yescharacteristic
37C4×#11758189—21no—
38C2×#14519687C2211yescharacteristic
39#97×D61—21no—
40#16839532C2212yes—
41#16839532C2212yes—
42#16839521C2212yes—
43C2×#135×C61C2211yescharacteristic
44C61×#427C2211yescharacteristic
45C61×#421C2212yes—
46C61×#410C2212yes—
47C61×#405C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808C2×C414yes—
50#14519807—22no—
51#14519803C2×C414yes—
52#14519802—22no—
53#14519797—24no—
54#14519785—22no—
55#14519785D414yes—
56#14519784—22no—
57#14519784—22no—
58#14519784—22no—
59#14519784—22no—
60#14519783—21no—
61#14519783—21no—
62#14519783C2×C412yes—
63#14519783C2×C412yes—
64C2×#11758194—22no—
65C2×#11758194C2312yes—
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2312yes—
68C2×#11758189—21no—
69C2×#11758189—21no—
70C22×#92418—22no—
71C22×#92418C2311yescharacteristic
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74#14519742—21no—
75#14519739—21no—
76#14519737—22no—
77#14519734—21no—
78C2×C4×Dic61C2311yescharacteristic
79C42×D61—21no—
80#14519713—21no—
81#14519713—21no—
82#14519713C2×C412yescomplemented
83#14519713C2×C412yescomplemented
84#14519712—24no—
85#14519705—24no—
86#14519704—22no—
87#14519704—22no—
88#14519696C2×C414yescomplemented
89#14519694C2×C414yescomplemented
90#14519688—21no—
91#14519688—22no—
92#14519687—21no—
93#14519687D411yescharacteristic
94#14519687D411yescharacteristic
95#14519687D411yescharacteristic
96#14519687D411yescharacteristic
97C4×#92418—21no—
98C4×#92418—22no—
99C8×Dic61—21no—
100C2×#41×C61C2312yes—
101#135×C61—21no—
102#135×C61—21no—
103#135×C61—21no—
104#135×C61—21no—
105C2×Q16×C61C2×C411yescharacteristic, complemented
106C2×SD16×C61C2×C412yescomplemented
107C2×D8×C61C2×C411yescharacteristic, complemented
108C2×M16×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#117×C61C2312yes—
111#117×C61C2311yescharacteristic
112#107×C61—21no—
113#106×C61—21no—
114#104×C61—24no—
115#99×C61—24no—
116#97×C61—21no—
117C4×C8×C61—21no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61C22×C412yes—
123C2×D4×C61—22no—
124C2×D4×C61C22×C412yes—
125C22×C4×C61—22no—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—21no—
128C22×Dic61—21no—
129#11758222—21no—
130#11758222—21no—
131#11758222C22×C412yes—
132#11758222C2×D412yes—
133C2×D244#4111yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×Dic122#4111yescharacteristic
137Q16×C61—21no—
138Q16×C61—21no—
139SD16×C61—22no—
140SD16×C61—22no—
141D8×C61—21no—
142D8×C61—21no—
143M16×C61—21no—
144M16×C61C2×D412yes—
145C2×C8×C61—21no—
146C2×C8×C61C22×C411yescharacteristic
147C2×C8×C61C22×C411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61#4111yescharacteristic
151C2×C8×C61#4111yescharacteristic
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61#4112yes—
156C42×C61—21no—
157C42×C61—22no—
158#11758202—24no—
159#11758201—24no—
160#11758200—24no—
161#11758199—24no—
162#11758198—22no—
163#11758196—22no—
164C4×Dic61—22no—
165C4×Dic61—21no—
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61C2×D411yescharacteristic
168#11758194—21no—
169#11758194—22no—
170#11758194—22no—
171#11758194—22no—
172#11758194—42no—
173#11758194—42no—
174#11758194C22×C414yes—
175#11758194C22×C412yes—
176C2×#92418—21no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—42no—
180C2×#92418—42no—
181C2×#92418C2×D411yescharacteristic
182C2×#92418C2×D412yes—
183C2×#92418C2×D411yescharacteristic
184C2×#92418#4112yes—
185#11758189—22no—
186#11758189—22no—
187#11758189—24no—
188C8×D61—22no—
189C8×D61—22no—
190C8×D61—24no—
191C23×C61—22no—
192C23×C61C2×#4111yescharacteristic
193C22×D61—21no—
194Q8×C61—22no—
195Q8×C61—22no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—42no—
199D4×C61—42no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C22⋊D61—22no—
212C22⋊D61—22no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—81no—
217D244—21no—
218D244C4×D412yes—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—82no—
222Dic122—21no—
223Dic122C4×D412yes—
224C8×C61—22no—
225C8×C61C4×D411yescharacteristic
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229#92418—22no—
230#92418—24no—
231#92418—42no—
232#92418#12112yes—
233#92418#12112yes—
234#21559—611nomaximal, Sylow
235C22×C61—41no—
236C22×C61—42no—
237C22×C61—82no—
238C22×C61C4×#4111yescharacteristic
239C22×C61#52511yescharacteristic
240C22×C61#53211yescharacteristic
241C2×D61—22no—
242C2×D61—42no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—82no—
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×#12111yescharacteristic
249C4×C61#52911yescharacteristic
250Dic61—22no—
251Dic61—41no—
252Dic61—41no—
253#3067—611no—
254#2955—612no—
255C2×#444—611no—
256#2875—611no—
257#1976D6111yescharacteristic, complemented
258#1953—611no—
259#1932—612no—
260#1906—612no—
261#1814—611no—
262#1773—611no—
263#1757—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—82no—
268C2×C61C4×#12111yescharacteristic
269D61—42no—
270#575—612no—
271C2×#135C2×D6111yescharacteristic
272C2×#131—612no—
273C2×#131—611no—
274C2×#131—611no—
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×#108—611no—
288#427C2×D6111yescharacteristic
289#421—612no—
290#421C2×D6112yes—
291C2×#104—612no—
292#413—612no—
293#410C2×D6112yes—
294#405—611no—
295#405—612no—
296#405—1221no—
297#405—1221no—
298#405C2×D6111yescharacteristic
299C4×M16—611no—
300C4×M16—1221no—
301C2×#97—611no—
302C2×#97—1221no—
303#344—612no—
304#344—612no—
305#339—612no—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41—611no—
308C2×#41C22×D6112yes—
309#137—614no—
310#137—1222no—
311AGL(1,ℤ/8)—614no—
312AGL(1,ℤ/8)—1222no—
313#135—21no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—1224no—
318C2×Q16C4×D6111yescharacteristic, complemented
319C2×SD16C4×D6112yescomplemented
320C2×D8C4×D6111yescharacteristic, complemented
321#131—614no—
322#131—612no—
323#131—612no—
324#131—1222no—
325#131—1222no—
326#131—1222no—
327#131—1222no—
328#131—1222no—
329#131—1221no—
330#131—1221no—
331C2×M16—611no—
332C2×M16—612no—
333C2×M16—612no—
334C2×M16—1221no—
335C2×M16—1222no—
336C2×M16—1221no—
337C2×M16C22×D6111yescharacteristic
338C22×C8—611no—
339C22×C8—1221no—
340C22×C8—1222no—
341C22×C8—1221no—
342C22×C8C22×D6111yescharacteristic
343C4×Q8—1221no—
344C4×D4—1221no—
345C4×D4—1222no—
346#117—1221no—
347#117C22×D6112yes—
348#117C22×D6111yescharacteristic
349C2×C42—611no—
350C2×C42—1221no—
351#108—612no—
352#108—612no—
353#108—1224no—
354#108—1221no—
355#108—1221no—
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—24no—
365#97—21no—
366#97—611no—
367#97—611no—
368#97—611no—
369#97—611no—
370#97—1221no—
371#97—1222no—
372#97—1221no—
373C4×C8—21no—
374C4×C8—1221no—
375C4×C8—1222no—
376C4×C8—1221no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—612no—
382#41—612no—
383#41—1221no—
384#41—1221no—
385C2×Q8—611no—
386C2×Q8C2×C4×D6112yes—
387C2×D4—22no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4C2×C4×D6112yes—
391C22×C4—22no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4C23×D6111yescharacteristic
396Q16—21no—
397Q16—21no—
398Q16—1224no—
399SD16—22no—
400SD16—22no—
401SD16—1224no—
402SD16—1224no—
403D8—21no—
404D8—21no—
405D8—1224no—
406M16—21no—
407M16—612no—
408M16—614no—
409M16—1224no—
410M16—1222no—
411M16—1221no—
412M16—1222no—
413M16—1222no—
414M16—1222no—
415M16—1222no—
416M16—2442no—
417M16—2442no—
418M16D4×D6112yes—
419C2×C8—21no—
420C2×C8—611no—
421C2×C8—612no—
422C2×C8—611no—
423C2×C8—612no—
424C2×C8—1224no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1222no—
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—21no—
439C4⋊C4—22no—
440C4⋊C4—1222no—
441C22⋊C4—22no—
442C22⋊C4—1222no—
443C22⋊C4#1175822412yes—
444C42—21no—
445C42—22no—
446C42—611no—
447C42—611no—
448C42—1222no—
449C42—1221no—
450C23—22no—
451C23—1221no—
452C23C2×#1175822411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—612no—
456Q8—1221no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—612no—
462D4—1222no—
463D4—1222no—
464D4—1221no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—1221no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—4882no—
476C2×C4—4881no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4#41×D6111yescharacteristic
482C2×C4C2×#1175822411yescharacteristic
483C8—22no—
484C8—612no—
485C8—612no—
486C8—1224no—
487C8—1222no—
488C8—2442no—
489C8#1451973711yescharacteristic
490C8#1451973711yescharacteristic
491C8#1451973711yescharacteristic
492C8#1451973711yescharacteristic
493C22—41no—
494C22—42no—
495C22—82no—
496C22—1222no—
497C22—2442no—
498C22C4×#1175822411yescharacteristic
499C22#1684027211yescharacteristic
500C22#1684031011yescharacteristic
501C4—42no—
502C4—42no—
503C4—82no—
504C4—1222no—
505C4—2441no—
506C4—2441no—
507C4C4×D4×D6111yescharacteristic
508C4C2×#1451973711yescharacteristic
509C4#121×D6111yescharacteristic
510C4#1684029111yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—2442no—
516C2#2144833511yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1976C2
socleC2×C61C4×#121
prime radical#21455535C2
cosocleC22×C4×C61C24
hypercenter#1976D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #21454718C2semidirect
3. C2×#16839884C2semidirect
4. #21454246C2semidirect
5. #21446609C2semidirect
6. #21446491C2semidirect
7. #21446317C2semidirect
8. #21445870C2semidirect
9. #21445705C2semidirect
10. #21445621C2semidirect
11. C61×#1976C2semidirect
17. #16840034C4semidirect
18. #16840029C4semidirect
19. #16840022C4semidirect
82. #14519713C2×C4semidirect
83. #14519713C2×C4semidirect
88. #14519696C2×C4semidirect
89. #14519694C2×C4semidirect
105. C2×Q16×C61C2×C4semidirect
106. C2×SD16×C61C2×C4semidirect
107. C2×D8×C61C2×C4semidirect
257. #1976D61semidirect
306. C61#21559semidirect
318. C2×Q16C4×D61semidirect
319. C2×SD16C4×D61semidirect
320. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555352
#214547182
C2×#168398842
#214542462
#214466092
#214464912
#214463172
#214458702
#214457052
#214456212
C61×#19762
#2155961