Finite Group Databasebeta

← #29686958

Subgroups — #29686958

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#21455540C211yesmaximal, characteristic
2#21454721C212yesmaximal, complemented
3C2×#16839883C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic, complemented
5#21446717C211yesmaximal, characteristic, complemented
6#21446487C212yesmaximal, complemented
7#21446321C212yesmaximal, complemented
8#21445874C211yesmaximal, characteristic, complemented
9#21445706C211yesmaximal, characteristic, complemented
10#21445618C212yesmaximal, complemented
11C61×#1953C211yesmaximal, characteristic, complemented
12#16840562C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519785C2212yes—
15C2×#14519785C2211yescharacteristic
16C2×#14519783C2211yescharacteristic
17#16840230C2211yescharacteristic
18#16840038C412yescomplemented
19#16840033C412yescomplemented
20#16840026C414yescomplemented
21#16839883—24no—
22#16839883—21no—
23#16839883—21no—
24#16839883—21no—
25#16839883—21no—
26#16839878—24no—
27#16839800C2211yescharacteristic
28#16839778C2212yes—
29#16839778C2212yes—
30C2×#14519703C2212yes—
31#16839744C2212yes—
32#16839735C2212yes—
33#16839720C2212yes—
34#16839719—21no—
35#16839718C2211yescharacteristic
36#16839716—21no—
37M16×Dic61C2211yescharacteristic
38C4×#11758189—21no—
39C2×#14519687C2211yescharacteristic
40#97×D61—21no—
41#16839533C2212yes—
42#16839523C2212yes—
43C61×#430C2212yes—
44C2×#108×C61C2211yescharacteristic
45C2×#108×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#344C2212yes—
48C2×#11758222C2312yes—
49C2×#11758222C2311yescharacteristic
50#14519810—22no—
51#14519810C2×C414yes—
52#14519805—22no—
53#14519805C2×C414yes—
54#14519799—24no—
55#14519799—21no—
56#14519799—21no—
57#14519799—21no—
58#14519799—21no—
59C2×Dic244C2×C411yescharacteristic, complemented
60C2×#11758190C2×C412yescomplemented
61C2×D488C2×C411yescharacteristic, complemented
62#14519785—22no—
63#14519785—22no—
64#14519785—22no—
65#14519785—22no—
66#14519785—22no—
67#14519785D414yes—
68#14519783—21no—
69#14519783—21no—
70#14519783C2×C412yes—
71#14519783C2×C412yes—
72C2×#11758194C2311yescharacteristic
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—22no—
76C22×#92418C2311yescharacteristic
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61—22no—
80#14519742—21no—
81#14519739—21no—
82#14519737—22no—
83#14519736C2312yes—
84#14519734—21no—
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—21no—
87#14519711—21no—
88#14519709—21no—
89#14519703—24no—
90#14519703—24no—
91#14519703—22no—
92#14519703—22no—
93#14519695C2×C414yescomplemented
94#14519693C2×C414yescomplemented
95#14519691—24no—
96#14519688—21no—
97#14519687—21no—
98#14519687—22no—
99#14519687D411yescharacteristic
100#14519687D411yescharacteristic
101#14519687D411yescharacteristic
102#14519687D411yescharacteristic
103C4×#92418—21no—
104C8×Dic61—21no—
105C8×Dic61—22no—
106C2×M16×C61C2312yes—
107C2×M16×C61C2312yes—
108C2×M16×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#117×C61C2311yescharacteristic
111#108×C61—24no—
112#108×C61—21no—
113#108×C61—21no—
114#108×C61C2×C412yescomplemented
115#108×C61C2×C412yescomplemented
116#97×C61—21no—
117C4×C8×C61—21no—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—21no—
120C2×C22⋊D61—22no—
121C22×Dic61—21no—
122#11758222—21no—
123#11758222—21no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222C22×C412yes—
129#11758222C2×D412yes—
130C2×D244C22×C412yes—
131C2×D244#4111yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×Dic122C22×C412yes—
136C2×Dic122#4111yescharacteristic
137M16×C61—22no—
138M16×C61—21no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61C22×C414yes—
142M16×C61C2×D412yes—
143C2×C8×C61—21no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
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—
153C22⋊C4×C61#4112yes—
154C42×C61—21no—
155C22⋊Dic61—22no—
156#11758198—22no—
157C4⋊Dic61—22no—
158#11758196—22no—
159C4×Dic61—22no—
160C4×Dic61—21no—
161C4×Dic61—22no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164#11758194—21no—
165#11758194C22×C412yes—
166C2×#92418—21no—
167C2×#92418C2×D412yes—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170C2×#92418#4112yes—
171Dic244—24no—
172Dic244—21no—
173Dic244—21no—
174D488—24no—
175D488—21no—
176D488—21no—
177#11758190—24no—
178#11758190—24no—
179#11758190—22no—
180#11758190—22no—
181#11758189—22no—
182#11758189—22no—
183#11758189—24no—
184#11758189—42no—
185#11758189—42no—
186C8×D61—22no—
187C8×D61—22no—
188C8×D61—24no—
189C8×D61—42no—
190C8×D61—42no—
191C23×C61C2×#4111yescharacteristic
192C22×D61—21no—
193C22×D61—22no—
194C2×C4×C61—22no—
195C2×C4×C61C23×C411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—22no—
202C22⋊D61—22no—
203C22⋊D61—42no—
204C22⋊D61—42no—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—42no—
210C2×Dic61—81no—
211D244—22no—
212D244—22no—
213D244—21no—
214D244C4×D412yes—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61—82no—
220Dic122—21no—
221Dic122—22no—
222Dic122—22no—
223Dic122C4×D412yes—
224C8×C61—24no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61C4×D411yescharacteristic
230C8×C61C4×D411yescharacteristic
231#92418—22no—
232#92418#12112yes—
233#92418#12112yes—
234#21559—611nomaximal, Sylow
235C22×C61—41no—
236C22×C61C4×#4111yescharacteristic
237C22×C61#52511yescharacteristic
238C22×C61#53211yescharacteristic
239C2×D61—22no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—82no—
243C4×C61—42no—
244C4×C61C2×C4×D411yescharacteristic
245C4×C61C2×C4×D411yescharacteristic
246C4×C61C2×#12111yescharacteristic
247C4×C61#52911yescharacteristic
248Dic61—22no—
249Dic61—41no—
250Dic61—41no—
251Dic61—42no—
252Dic61—82no—
253#3067—611no—
254#2955—612no—
255C2×#444—611no—
256#2875—611no—
257#1976—611no—
258#1953D6111yescharacteristic, complemented
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×C61C4×#12111yescharacteristic
268D61—42no—
269D61—82no—
270#575—612no—
271C2×#135—611no—
272C2×#131—611no—
273C2×#131—612no—
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#430C2×D6112yes—
286C2×#108C2×D6111yescharacteristic
287C2×#108C2×D6111yescharacteristic
288#427—611no—
289#421—612no—
290#421—612no—
291C2×#104—612no—
292#413—612no—
293#410—612no—
294#405—612no—
295#405—611no—
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#344C2×D6112yes—
305#339—612no—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41—611no—
308C2×#41—612no—
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—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—1221no—
333C2×M16—1222no—
334C2×M16—1221no—
335C2×M16C22×D6112yes—
336C2×M16C22×D6112yes—
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—612no—
347#117—1221no—
348#117C22×D6111yescharacteristic
349C2×C42—611no—
350C2×C42—1221no—
351#108—24no—
352#108—21no—
353#108—21no—
354#108C4×D6112yescomplemented
355#108C4×D6112yescomplemented
356#107—1221no—
357#106—1221no—
358#104—1222no—
359#104—1222no—
360#104—1224no—
361#104—1224no—
362#101—614no—
363#100—614no—
364#99—1224no—
365#97—21no—
366#97—611no—
367#97—611no—
368#97—611no—
369#97—611no—
370#97—1221no—
371#97—1221no—
372#97—1222no—
373C4×C8—21no—
374C4×C8—1221no—
375C4×C8—1221no—
376C4×C8—1222no—
377#41—612no—
378#41—612no—
379#41—1221no—
380#41—1221no—
381#41—1222no—
382#41—1222no—
383#41—1222no—
384#41—1222no—
385C2×Q8—612no—
386C2×Q8—611no—
387C2×D4—612no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—1222no—
391C22×C4—1222no—
392C22×C4—1221no—
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—612no—
411M16—1224no—
412M16—1221no—
413M16—1222no—
414M16—1222no—
415M16—2442no—
416M16—2442no—
417M16C2×C4×D6114yes—
418M16D4×D6112yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—611no—
423C2×C8—612no—
424C2×C8—611no—
425C2×C8—612no—
426C2×C8—1224no—
427C2×C8—1221no—
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#1175822611yescharacteristic
437C2×C8#1175822611yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C22⋊C4—1222no—
442C22⋊C4—1222no—
443C22⋊C4#1175822612yes—
444C42—21no—
445C42—611no—
446C42—611no—
447C42—1222no—
448C42—1221no—
449C42—1222no—
450C23—1221no—
451C23—1222no—
452C23C2×#1175822611yescharacteristic
453Q8—612no—
454Q8—1221no—
455Q8—1222no—
456Q8—1222no—
457D4—612no—
458D4—1222no—
459D4—1222no—
460D4—1222no—
461D4—1222no—
462D4—1221no—
463D4—2442no—
464D4—2442no—
465C2×C4—22no—
466C2×C4—1221no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—4882no—
473C2×C4—4881no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4#41×D6111yescharacteristic
482C2×C4C2×#1175822611yescharacteristic
483C8—24no—
484C8—22no—
485C8—42no—
486C8—612no—
487C8—612no—
488C8—1222no—
489C8#1451973911yescharacteristic
490C8#1451973911yescharacteristic
491C8#1451973911yescharacteristic
492C8#1451973911yescharacteristic
493C22—41no—
494C22—1222no—
495C22—4882no—
496C22—2442no—
497C22—2442no—
498C22C4×#1175822611yescharacteristic
499C22#1684027511yescharacteristic
500C22#1684031311yescharacteristic
501C4—42no—
502C4—1222no—
503C4—4882no—
504C4—2441no—
505C4—2441no—
506C4—2442no—
507C4C4×D4×D6111yescharacteristic
508C4C2×#1451973911yescharacteristic
509C4#121×D6111yescharacteristic
510C4#1684029711yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—4882no—
515C2—2442no—
516C2#2144832711yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1953C2
socleC2×C61C4×#121
prime radical#21455540C2
cosocleC22×C4×C61C24
hypercenter#1953D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #21454721C2semidirect
3. C2×#16839883C2semidirect
4. #21454246C2semidirect
5. #21446717C2semidirect
6. #21446487C2semidirect
7. #21446321C2semidirect
8. #21445874C2semidirect
9. #21445706C2semidirect
10. #21445618C2semidirect
11. C61×#1953C2semidirect
18. #16840038C4semidirect
19. #16840033C4semidirect
20. #16840026C4semidirect
59. C2×Dic244C2×C4semidirect
60. C2×#11758190C2×C4semidirect
61. C2×D488C2×C4semidirect
93. #14519695C2×C4semidirect
94. #14519693C2×C4semidirect
114. #108×C61C2×C4semidirect
115. #108×C61C2×C4semidirect
258. #1953D61semidirect
306. C61#21559semidirect
354. #108C4×D61semidirect
355. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555402
#214547212
C2×#168398832
#214542462
#214467172
#214464872
#214463212
#214458742
#214457062
#214456182
C61×#19532
#2155961