Finite Group Databasebeta

← #29686963

Subgroups — #29686963

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#21454720C212yesmaximal, complemented
3C2×#16839885C211yesmaximal, characteristic, complemented
4#21446717C211yesmaximal, characteristic, complemented
5#21446609C211yesmaximal, characteristic, complemented
6#21446488C212yesmaximal, complemented
7#21446320C212yesmaximal, complemented
8#21445873C211yesmaximal, characteristic, complemented
9#21445703C211yesmaximal, characteristic, complemented
10#21445620C212yesmaximal, complemented
11C61×#2875C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519783C2212yes—
15#16840037C412yescomplemented
16#16840032C412yescomplemented
17#16840025C414yescomplemented
18#16839885—21no—
19#16839885—21no—
20#16839885—21no—
21#16839885—21no—
22#16839882—24no—
23#16839877—24no—
24#16839810C2212yes—
25C2×#14519713C2211yescharacteristic
26C2×#14519713C2211yescharacteristic
27#16839800C2211yescharacteristic
28#16839779C2212yes—
29#16839778C2212yes—
30C2×#14519705C2212yes—
31#16839744C2212yes—
32#16839735C2212yes—
33#16839717C2212yes—
34#16839534C2212yes—
35#16839532C2212yes—
36#16839522C2212yes—
37C2×#131×C61C2211yescharacteristic
38C2×#131×C61C2211yescharacteristic
39C4×#41×C61C2211yescharacteristic
40C61×#405—21no—
41C61×#405—21no—
42C61×#405C2211yescharacteristic
43C61×#405C2211yescharacteristic
44C4×M16×C61—21no—
45C4×M16×C61C2211yescharacteristic
46C2×#97×C61—21no—
47C2×#97×C61C2211yescharacteristic
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#14519783—22no—
62#14519783—22no—
63#14519783—22no—
64#14519783—22no—
65C2×#11758194C2312yes—
66C2×#11758194C2312yes—
67C2×#11758189—22no—
68C2×C8×D61—22no—
69#14519736C2312yes—
70#14519713—21no—
71#14519713—21no—
72#14519713C2×C412yescomplemented
73#14519713C2×C412yescomplemented
74#14519712—24no—
75#14519711—21no—
76#14519709—21no—
77#14519705—22no—
78#14519705—22no—
79#14519704—24no—
80#14519703—24no—
81#14519695C2×C414yescomplemented
82#14519693C2×C414yescomplemented
83#14519691—24no—
84#14519687—22no—
85C8×Dic61—22no—
86C2×#41×C61C2311yescharacteristic
87#131×C61—21no—
88#131×C61—21no—
89#131×C61—22no—
90#131×C61C2×C412yes—
91#131×C61C2×C412yes—
92#131×C61D414yes—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61C2311yescharacteristic
96C2×M16×C61C2311yescharacteristic
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99C22×C8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C4×Q8×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—22no—
104#117×C61—21no—
105#117×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108#97×C61—21no—
109#97×C61—21no—
110#97×C61—21no—
111#97×C61D411yescharacteristic
112#97×C61D411yescharacteristic
113#97×C61D411yescharacteristic
114#97×C61D411yescharacteristic
115C4×C8×C61—21no—
116C4×C8×C61—21no—
117C4×C8×C61—21no—
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×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—22no—
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—21no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61—24no—
142M16×C61C22×C412yes—
143M16×C61C2×D412yes—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—24no—
149C2×C8×C61C22×C411yescharacteristic
150C2×C8×C61C22×C411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D411yescharacteristic
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61#4111yescharacteristic
157C2×C8×C61#4111yescharacteristic
158C2×C8×C61#4112yes—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61#4112yes—
163C42×C61—22no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C22⋊Dic61—22no—
169#11758202—24no—
170#11758201—24no—
171#11758200—24no—
172#11758199—24no—
173C4⋊Dic61—22no—
174C4×Dic61—22no—
175#11758194—22no—
176#11758194—22no—
177#11758194—22no—
178#11758194C22×C414yes—
179C2×#92418—22no—
180C2×#92418—22no—
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—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—81no—
207C2×C4×C61—82no—
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C22⋊D61—42no—
215C22⋊D61—42no—
216C2×Dic61—22no—
217C2×Dic61—42no—
218D244—22no—
219D244—22no—
220C4×D61—42no—
221C4×D61—42no—
222Dic122—22no—
223Dic122—22no—
224C8×C61—22no—
225C8×C61—22no—
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61C4×D411yescharacteristic
230C8×C61#12112yes—
231C8×C61#12112yes—
232#92418—24no—
233#92418—42no—
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—42no—
245C4×C61—41no—
246C4×C61—41no—
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×C4×D411yescharacteristic
249C4×C61C2×#12111yescharacteristic
250C4×C61#52911yescharacteristic
251Dic61—42no—
252Dic61—82no—
253#3067—611no—
254#2955—612no—
255C2×#444—611no—
256#2875D6111yescharacteristic, complemented
257#1976—611no—
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—42no—
268C2×C61C4×#12111yescharacteristic
269D61—82no—
270#575—612no—
271C2×#135—611no—
272C2×#131—612no—
273C2×#131C2×D6111yescharacteristic
274C2×#131C2×D6111yescharacteristic
275C4×#41C2×D6111yescharacteristic
276#473—612no—
277#472—612no—
278#471—614no—
279#444—1224no—
280#444—1221no—
281#444—1221no—
282#444—1221no—
283#444—1221no—
284#443—1224no—
285#430—612no—
286C2×#108—611no—
287C2×#108—611no—
288#427—611no—
289#421—612no—
290#421—612no—
291C2×#104—612no—
292#413—612no—
293#410—612no—
294#405—21no—
295#405—21no—
296#405—612no—
297#405C2×D6111yescharacteristic
298#405C2×D6111yescharacteristic
299C4×M16—21no—
300C4×M16C2×D6111yescharacteristic
301C2×#97—21no—
302C2×#97C2×D6111yescharacteristic
303#344—612no—
304#344—612no—
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—22no—
324#131—1222no—
325#131—1222no—
326#131—1222no—
327#131—1222no—
328#131C4×D6112yes—
329#131C4×D6112yes—
330#131D24414yes—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—612no—
334C2×M16—612no—
335C2×M16—1222no—
336C2×M16C22×D6111yescharacteristic
337C2×M16C22×D6111yescharacteristic
338C22×C8—21no—
339C22×C8—21no—
340C22×C8—1222no—
341C22×C8C22×D6111yescharacteristic
342C22×C8C22×D6111yescharacteristic
343C4×Q8—21no—
344C4×D4—21no—
345C4×D4—22no—
346#117—21no—
347#117—612no—
348#117C22×D6111yescharacteristic
349C2×C42—21no—
350C2×C42C22×D6111yescharacteristic
351#108—612no—
352#108—612no—
353#108—1224no—
354#108—1221no—
355#108—1221no—
356#107—1221no—
357#106—1221no—
358#104—1224no—
359#104—1224no—
360#104—1222no—
361#104—1222no—
362#101—614no—
363#100—614no—
364#99—1224no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—1222no—
369#97C22⋊D6111yescharacteristic
370#97C22⋊D6111yescharacteristic
371#97C22⋊D6111yescharacteristic
372#97C22⋊D6111yescharacteristic
373C4×C8—21no—
374C4×C8—21no—
375C4×C8—21no—
376C4×C8—1222no—
377#41—21no—
378#41—21no—
379#41—1222no—
380#41—1222no—
381#41—1222no—
382#41—1222no—
383#41C2×C4×D6112yes—
384#41C2×D24412yes—
385C2×Q8—612no—
386C2×Q8#1175822211yescharacteristic
387C2×D4—21no—
388C2×D4—612no—
389C2×D4—1222no—
390C2×D4#1175822211yescharacteristic
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
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—21no—
408M16—22no—
409M16—22no—
410M16—24no—
411M16—614no—
412M16—1222no—
413M16—1222no—
414M16—1222no—
415M16—2442no—
416M16—2442no—
417M16C2×C4×D6112yes—
418M16C2×D24412yes—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—2442no—
427C2×C8—2442no—
428C2×C8C2×C4×D6111yescharacteristic
429C2×C8C2×C4×D6111yescharacteristic
430C2×C8C2×D24412yes—
431C2×C8#1175822211yescharacteristic
432C2×C8#1175822211yescharacteristic
433C2×C8#1175822212yes—
434C2×C8C2×C22⋊D6111yescharacteristic
435C2×C8C2×C22⋊D6111yescharacteristic
436C2×C8C2×C22⋊D6111yescharacteristic
437C2×C8C2×C22⋊D6111yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—1222no—
441C22⋊C4—22no—
442C22⋊C4—1222no—
443C22⋊C4#1175822212yes—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—1222no—
448C42C2×C22⋊D6111yescharacteristic
449C42C2×C22⋊D6111yescharacteristic
450C23—21no—
451C23—1222no—
452C23C2×#1175822211yescharacteristic
453Q8—21no—
454Q8—1222no—
455Q8—1222no—
456Q8C4×D24412yes—
457D4—22no—
458D4—22no—
459D4—21no—
460D4—1222no—
461D4—1222no—
462D4—2442no—
463D4—2442no—
464D4C4×D24412yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—1222no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C22×C22⋊D6111yescharacteristic
479C2×C4C22×D24411yescharacteristic
480C2×C4C2×#1175822211yescharacteristic
481C2×C4C2×#1175822211yescharacteristic
482C2×C4C2×#1175822211yescharacteristic
483C8—22no—
484C8—22no—
485C8—1224no—
486C8—2442no—
487C8C4×C22⋊D6111yescharacteristic
488C8C4×C22⋊D6111yescharacteristic
489C8C4×C22⋊D6111yescharacteristic
490C8C4×C22⋊D6111yescharacteristic
491C8#1451975112yes—
492C8#1451975112yes—
493C22—22no—
494C22—41no—
495C22—42no—
496C22—4882no—
497C22—2442no—
498C22C4×#1175822211yescharacteristic
499C22#1684027311yescharacteristic
500C22#1684031211yescharacteristic
501C4—22no—
502C4—42no—
503C4—41no—
504C4—41no—
505C4—4882no—
506C4—2442no—
507C4C2×C4×C22⋊D6111yescharacteristic
508C4C2×C4×D24411yescharacteristic
509C4C2×#1451975111yescharacteristic
510C4#1684029611yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—4882no—
516C2C4×#1451975111yescharacteristic
subgroupG/H
centerC4C2×#14519751
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2875C2
socleC2×C61C4×#121
prime radical#21455539C2
cosocleC22×C4×C61C24
hypercenter#2875D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21454720C2semidirect
3. C2×#16839885C2semidirect
4. #21446717C2semidirect
5. #21446609C2semidirect
6. #21446488C2semidirect
7. #21446320C2semidirect
8. #21445873C2semidirect
9. #21445703C2semidirect
10. #21445620C2semidirect
11. C61×#2875C2semidirect
15. #16840037C4semidirect
16. #16840032C4semidirect
17. #16840025C4semidirect
58. C2×Dic244C2×C4semidirect
59. C2×#11758190C2×C4semidirect
60. C2×D488C2×C4semidirect
72. #14519713C2×C4semidirect
73. #14519713C2×C4semidirect
81. #14519695C2×C4semidirect
82. #14519693C2×C4semidirect
256. #2875D61semidirect
306. C61#21559semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214547202
C2×#168398852
#214467172
#214466092
#214464882
#214463202
#214458732
#214457032
#214456202
C61×#28752
#2155961