Finite Group Databasebeta

← #29686955

Subgroups — #29686955

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455535C211yesmaximal, characteristic
2#21454718C212yesmaximal, complemented
3C2×#16839879C211yesmaximal, characteristic, complemented
4#21454241C211yesmaximal, characteristic, complemented
5#21446609C211yesmaximal, characteristic, complemented
6#21446491C212yesmaximal, complemented
7#21446317C212yesmaximal, complemented
8#21445870C211yesmaximal, characteristic, complemented
9#21445700C211yesmaximal, 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#16839881—24no—
21#16839879—21no—
22#16839879—21no—
23#16839879—21no—
24#16839879—21no—
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—
36#16839714—21no—
37M16×Dic61C2211yescharacteristic
38C2×C8×Dic61C2211yescharacteristic
39C4×C8×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—21no—
55#14519785—21no—
56#14519785D412yes—
57#14519785D412yes—
58#14519784—22no—
59#14519784—22no—
60#14519784—22no—
61#14519784—22no—
62#14519783—21no—
63#14519783—21no—
64#14519783C2×C412yes—
65#14519783C2×C412yes—
66C2×#11758194—22no—
67C2×#11758194C2312yes—
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2312yes—
70C2×#11758189—21no—
71C2×#11758189—21no—
72C22×#92418—22no—
73C22×#92418C2311yescharacteristic
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76#14519742—21no—
77#14519739—21no—
78#14519737—22no—
79#14519734—21no—
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—21no—
82#14519713—21no—
83#14519713—21no—
84#14519713C2×C412yescomplemented
85#14519713C2×C412yescomplemented
86#14519712—24no—
87#14519705—24no—
88#14519704—22no—
89#14519704—22no—
90#14519696C2×C414yescomplemented
91#14519694C2×C414yescomplemented
92#14519688—21no—
93#14519688—22no—
94#14519687—21no—
95C4×#92418—21no—
96C4×#92418—22no—
97C8×Dic61—21no—
98C8×Dic61D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C8×Dic61D411yescharacteristic
101C8×Dic61D411yescharacteristic
102C2×#41×C61C2312yes—
103#135×C61—21no—
104#135×C61—21no—
105#135×C61—21no—
106#135×C61—21no—
107C2×Q16×C61C2×C411yescharacteristic, complemented
108C2×SD16×C61C2×C412yescomplemented
109C2×D8×C61C2×C411yescharacteristic, complemented
110C2×M16×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61C2312yes—
113#117×C61C2311yescharacteristic
114#107×C61—21no—
115#106×C61—21no—
116#104×C61—24no—
117#99×C61—24no—
118#97×C61—21no—
119C4×C8×C61—21no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124C2×Q8×C61C22×C412yes—
125C2×D4×C61—22no—
126C2×D4×C61C22×C412yes—
127C22×C4×C61—22no—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—21no—
130C22×Dic61—21no—
131#11758222—21no—
132#11758222—21no—
133#11758222C22×C412yes—
134#11758222C2×D412yes—
135C2×D244#4111yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×Dic122#4111yescharacteristic
139Q16×C61—21no—
140Q16×C61—21no—
141SD16×C61—22no—
142SD16×C61—22no—
143D8×C61—21no—
144D8×C61—21no—
145M16×C61—21no—
146M16×C61C2×D412yes—
147C2×C8×C61—21no—
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C22×C411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61#4111yescharacteristic
153C2×C8×C61#4111yescharacteristic
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61#4112yes—
158C42×C61—21no—
159C42×C61—22no—
160#11758202—24no—
161#11758201—24no—
162#11758200—24no—
163#11758199—24no—
164#11758198—22no—
165#11758196—22no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170#11758194—21no—
171#11758194—22no—
172#11758194—22no—
173#11758194—22no—
174#11758194—42no—
175#11758194—42no—
176#11758194C22×C414yes—
177#11758194C22×C412yes—
178C2×#92418—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—42no—
182C2×#92418—42no—
183C2×#92418C2×D411yescharacteristic
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186C2×#92418C2×D411yescharacteristic
187C2×#92418#4111yescharacteristic
188C2×#92418#4111yescharacteristic
189#11758189—22no—
190#11758189—22no—
191#11758189—22no—
192#11758189—22no—
193C8×D61—22no—
194C8×D61—22no—
195C8×D61—22no—
196C8×D61—22no—
197C23×C61—22no—
198C23×C61C2×#4111yescharacteristic
199C22×D61—21no—
200Q8×C61—22no—
201Q8×C61—22no—
202D4×C61—22no—
203D4×C61—22no—
204D4×C61—42no—
205D4×C61—42no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61C23×C411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C22⋊D61—22no—
218C22⋊D61—22no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—42no—
223D244—21no—
224D244C4×D412yes—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—44no—
228Dic122—21no—
229Dic122C4×D412yes—
230C8×C61—22no—
231C8×C61C4×D411yescharacteristic
232C8×C61C4×D411yescharacteristic
233C8×C61C4×D411yescharacteristic
234C8×C61C4×D411yescharacteristic
235#92418—22no—
236#92418—24no—
237#92418—42no—
238#92418#12111yescharacteristic
239#92418#12111yescharacteristic
240#92418#12111yescharacteristic
241#92418#12111yescharacteristic
242#21558—611nomaximal, Sylow
243C22×C61—41no—
244C22×C61—42no—
245C22×C61—82no—
246C22×C61C4×#4111yescharacteristic
247C22×C61#52511yescharacteristic
248C22×C61#53211yescharacteristic
249C2×D61—22no—
250C2×D61—42no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61C2×C4×D411yescharacteristic
255C4×C61C2×C4×D411yescharacteristic
256C4×C61C2×#12111yescharacteristic
257C4×C61#52911yescharacteristic
258Dic61—22no—
259Dic61—22no—
260Dic61—41no—
261#3067—611no—
262#2955—612no—
263C2×#443—611no—
264C4×#131—611no—
265#1976D6111yescharacteristic, complemented
266#1953—611no—
267#1932—612no—
268#1906—612no—
269#1814—611no—
270#1772—611no—
271#1757—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61—82no—
276C2×C61C4×#12111yescharacteristic
277D61—42no—
278#575—612no—
279C2×#135C2×D6111yescharacteristic
280C2×#131—612no—
281C2×#131—611no—
282C2×#131—611no—
283C4×#41—611no—
284#473—612no—
285#472—612no—
286#471—614no—
287#444—1224no—
288#443—1221no—
289#443—1221no—
290#443—1221no—
291#443—1221no—
292#443—1224no—
293#430—612no—
294C2×#108—611no—
295C2×#108—611no—
296#427C2×D6111yescharacteristic
297#421—612no—
298#421C2×D6112yes—
299C2×#104—612no—
300#413—612no—
301#410C2×D6112yes—
302#405—611no—
303#405—612no—
304#405—1221no—
305#405—1221no—
306#405C2×D6111yescharacteristic
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—611no—
310C2×C4×C8—1221no—
311#344—612no—
312#344—612no—
313#339—612no—
314C61#2155811yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41C22×D6112yes—
317#137—614no—
318#137—1222no—
319AGL(1,ℤ/8)—614no—
320AGL(1,ℤ/8)—1222no—
321#135—21no—
322#135—21no—
323#135—21no—
324#135—21no—
325#135—1224no—
326C2×Q16C4×D6111yescharacteristic, complemented
327C2×SD16C4×D6112yescomplemented
328C2×D8C4×D6111yescharacteristic, complemented
329#131—612no—
330#131—612no—
331#131—612no—
332#131—612no—
333#131—1221no—
334#131—1221no—
335#131—1222no—
336#131—1222no—
337#131—1222no—
338#131—1222no—
339#131—1221no—
340#131—1221no—
341C2×M16—611no—
342C2×M16—612no—
343C2×M16—612no—
344C2×M16—1221no—
345C2×M16—1222no—
346C2×M16—1221no—
347C2×M16C22×D6111yescharacteristic
348C22×C8—611no—
349C22×C8—1221no—
350C22×C8—1222no—
351C22×C8—1221no—
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1221no—
355C4×D4—1222no—
356#117—1221no—
357#117C22×D6112yes—
358#117C22×D6111yescharacteristic
359C2×C42—611no—
360C2×C42—1221no—
361#108—612no—
362#108—612no—
363#108—1224no—
364#108—1221no—
365#108—1221no—
366#107—21no—
367#106—21no—
368#104—24no—
369#104—1224no—
370#104—1222no—
371#104—1222no—
372#101—614no—
373#100—614no—
374#99—24no—
375#97—21no—
376#97—1221no—
377#97—1222no—
378#97—1221no—
379C4×C8—21no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—611no—
384C4×C8—1221no—
385C4×C8—1222no—
386C4×C8—1221no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—612no—
392#41—612no—
393#41—1221no—
394#41—1221no—
395C2×Q8—611no—
396C2×Q8C2×C4×D6112yes—
397C2×D4—22no—
398C2×D4—611no—
399C2×D4—1221no—
400C2×D4C2×C4×D6112yes—
401C22×C4—22no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4C23×D6111yescharacteristic
406Q16—21no—
407Q16—21no—
408Q16—1224no—
409SD16—22no—
410SD16—22no—
411SD16—1224no—
412SD16—1224no—
413D8—21no—
414D8—21no—
415D8—1224no—
416M16—21no—
417M16—612no—
418M16—614no—
419M16—1222no—
420M16—1222no—
421M16—1222no—
422M16—1221no—
423M16—1222no—
424M16—1222no—
425M16—1222no—
426M16—1222no—
427M16—2442no—
428M16—2442no—
429M16D4×D6112yes—
430C2×C8—21no—
431C2×C8—611no—
432C2×C8—611no—
433C2×C8—611no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—2442no—
445C2×C8—2442no—
446C2×C8C2×C4×D6111yescharacteristic
447C2×C8C2×C4×D6111yescharacteristic
448C2×C8D4×D6111yescharacteristic
449C2×C8D4×D6111yescharacteristic
450C2×C8#1175822411yescharacteristic
451C2×C8#1175822411yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—1222no—
455C22⋊C4—22no—
456C22⋊C4—1222no—
457C22⋊C4#1175822412yes—
458C42—21no—
459C42—22no—
460C42—611no—
461C42—611no—
462C42—1222no—
463C42—1221no—
464C23—22no—
465C23—1221no—
466C23C2×#1175822411yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—612no—
470Q8—1221no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—42no—
475D4—612no—
476D4—1222no—
477D4—1222no—
478D4—1221no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1221no—
489C2×C4—2444no—
490C2×C4—2442no—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C2×C4C2×#1175822411yescharacteristic
497C8—22no—
498C8—611no—
499C8—611no—
500C8—611no—
501C8—611no—
502C8—1224no—
503C8—1222no—
504C8—2442no—
505C8#1451973711yescharacteristic
506C8#1451973711yescharacteristic
507C8#1451973711yescharacteristic
508C8#1451973711yescharacteristic
509C22—41no—
510C22—42no—
511C22—82no—
512C22—1222no—
513C22—2442no—
514C22C4×#1175822411yescharacteristic
515C22#1684027211yescharacteristic
516C22#1684031011yescharacteristic
517C4—42no—
518C4—42no—
519C4—82no—
520C4—1222no—
521C4—1222no—
522C4—2441no—
523C4C4×D4×D6111yescharacteristic
524C4C2×#1451973711yescharacteristic
525C4#121×D6111yescharacteristic
526C4#1684029111yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—2442no—
532C2#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#21558
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×#16839879C2semidirect
4. #21454241C2semidirect
5. #21446609C2semidirect
6. #21446491C2semidirect
7. #21446317C2semidirect
8. #21445870C2semidirect
9. #21445700C2semidirect
10. #21445621C2semidirect
11. C61×#1976C2semidirect
17. #16840034C4semidirect
18. #16840029C4semidirect
19. #16840022C4semidirect
84. #14519713C2×C4semidirect
85. #14519713C2×C4semidirect
90. #14519696C2×C4semidirect
91. #14519694C2×C4semidirect
107. C2×Q16×C61C2×C4semidirect
108. C2×SD16×C61C2×C4semidirect
109. C2×D8×C61C2×C4semidirect
265. #1976D61semidirect
314. C61#21558semidirect
326. C2×Q16C4×D61semidirect
327. C2×SD16C4×D61semidirect
328. 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×#168398792
#214542412
#214466092
#214464912
#214463172
#214458702
#214457002
#214456212
C61×#19762
#2155861