Finite Group Databasebeta

← #29686954

Subgroups — #29686954

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#21455540C211yesmaximal, characteristic
2#21454723C212yesmaximal, complemented
3C2×#16839880C211yesmaximal, characteristic, complemented
4C4×#14519783C211yesmaximal, characteristic, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446610C211yesmaximal, characteristic, complemented
7#21446489C212yesmaximal, complemented
8#21446319C212yesmaximal, complemented
9#21445870C211yesmaximal, characteristic, complemented
10#21445620C212yesmaximal, complemented
11C61×#1772C211yesmaximal, characteristic, complemented
12#16840562C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519783C2212yes—
15C2×#14519783C2211yescharacteristic
16C2×#14519783C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840038C412yescomplemented
19#16840033C412yescomplemented
20#16840026C414yescomplemented
21#16839885—24no—
22#16839880—24no—
23#16839880—21no—
24#16839880—21no—
25#16839880—21no—
26#16839880—21no—
27#16839811C2212yes—
28C2×#14519713C2211yescharacteristic
29#16839780C2212yes—
30#16839780C2212yes—
31C2×#14519705C2212yes—
32#16839744C2212yes—
33#16839735C2212yes—
34#16839717C2211yescharacteristic
35#16839717C2211yescharacteristic
36#16839716—21no—
37#16839716—21no—
38C4×#11758194C2211yescharacteristic
39C4×#11758189—21no—
40C4×C8×D61—21no—
41#16839534C2212yes—
42#16839532C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#427C2211yescharacteristic
45C61×#405C2212yes—
46C2×C4×C8×C61C2211yescharacteristic
47C61×#339C2212yes—
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#14519783—21no—
63#14519783—21no—
64#14519783—21no—
65#14519783—21no—
66#14519783—22no—
67#14519783—22no—
68#14519783—22no—
69#14519783—22no—
70#14519783C2×C412yes—
71#14519783C2×C412yes—
72#14519783D412yes—
73#14519783D412yes—
74C2×#11758194C2312yes—
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758189—21no—
78C2×#11758189—21no—
79C2×#11758189—22no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—22no—
83C4×Dic122—21no—
84C4×D244—21no—
85C4×C22⋊D61—22no—
86#14519736C2311yescharacteristic
87#14519734—21no—
88C42×D61—21no—
89#14519713—21no—
90#14519713—21no—
91#14519713—24no—
92#14519705—24no—
93#14519705—22no—
94#14519705—22no—
95#14519705—24no—
96#14519695C2×C414yescomplemented
97#14519693C2×C414yescomplemented
98#14519691—24no—
99#14519688—21no—
100#14519687—21no—
101#14519687—21no—
102C4×#92418—21no—
103C8×Dic61—21no—
104C8×Dic61—21no—
105C2×M16×C61C2312yes—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61C2312yes—
109C2×C42×C61C2311yescharacteristic
110#108×C61C2×C412yescomplemented
111#108×C61C2×C412yescomplemented
112#107×C61—21no—
113#106×C61—21no—
114#97×C61—22no—
115C4×C8×C61—22no—
116C4×C8×C61D411yescharacteristic
117C4×C8×C61D411yescharacteristic
118C4×C8×C61D411yescharacteristic
119C4×C8×C61D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—21no—
123C2×C22⋊D61—22no—
124#11758222—21no—
125#11758222—21no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222C22×C412yes—
131#11758222C2×D412yes—
132C2×D244C22×C412yes—
133C2×D244#4111yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×Dic122C22×C412yes—
138C2×Dic122#4111yescharacteristic
139M16×C61—22no—
140M16×C61C22×C414yes—
141C2×C8×C61—22no—
142C2×C8×C61C22×C411yescharacteristic
143C2×C8×C61C22×C411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61#4111yescharacteristic
151C2×C8×C61#4111yescharacteristic
152C2×C8×C61#4111yescharacteristic
153C2×C8×C61#4111yescharacteristic
154C4⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—21no—
157C42×C61—22no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C22⋊Dic61#4112yes—
161#11758198—22no—
162C4⋊Dic61—21no—
163#11758196—22no—
164C4×Dic61—22no—
165C4×Dic61—21no—
166#11758194—21no—
167#11758194—21no—
168#11758194—22no—
169#11758194—22no—
170#11758194C22×C412yes—
171#11758194C2×D412yes—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175Dic244—24no—
176Dic244—21no—
177Dic244—21no—
178D488—24no—
179D488—21no—
180D488—21no—
181#11758190—24no—
182#11758190—24no—
183#11758190—22no—
184#11758190—22no—
185#11758189—22no—
186#11758189—22no—
187#11758189—22no—
188#11758189—22no—
189#11758189—42no—
190#11758189—42no—
191C8×D61—22no—
192C8×D61—22no—
193C8×D61—22no—
194C8×D61—22no—
195C8×D61—42no—
196C8×D61—42no—
197C23×C61C2×#4111yescharacteristic
198C22×D61—21no—
199C22×D61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—42no—
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—22no—
211C22⋊D61—22no—
212C22⋊D61—42no—
213C22⋊D61—42no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217D244—22no—
218D244—22no—
219D244—21no—
220D244C4×D412yes—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—44no—
224C4×D61—42no—
225C4×D61—42no—
226Dic122—21no—
227Dic122—22no—
228Dic122—22no—
229Dic122C4×D412yes—
230C8×C61—24no—
231C8×C61C4×D411yescharacteristic
232C8×C61C4×D411yescharacteristic
233C8×C61C4×D411yescharacteristic
234C8×C61C4×D411yescharacteristic
235C8×C61#12111yescharacteristic
236C8×C61#12111yescharacteristic
237C8×C61#12111yescharacteristic
238C8×C61#12111yescharacteristic
239#92418—22no—
240#92418—22no—
241#92418—42no—
242#21558—611nomaximal, Sylow
243C22×C61—41no—
244C22×C61C4×#4111yescharacteristic
245C22×C61#52511yescharacteristic
246C22×C61#53211yescharacteristic
247C2×D61—22no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—82no—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—82no—
254C4×C61C2×C4×D411yescharacteristic
255C4×C61C2×C4×D411yescharacteristic
256C4×C61C2×#12111yescharacteristic
257C4×C61#52911yescharacteristic
258Dic61—22no—
259Dic61—42no—
260Dic61—42no—
261#3067—611no—
262#2955—612no—
263C2×#443—611no—
264C4×#131—611no—
265#1976—611no—
266#1953—611no—
267#1932—612no—
268#1906—612no—
269#1814—611no—
270#1772D6111yescharacteristic, complemented
271#1757—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61C4×#12111yescharacteristic
276D61—42no—
277D61—82no—
278#575—612no—
279C2×#135—611no—
280C2×#131—611no—
281C2×#131—611no—
282C2×#131—612no—
283C4×#41—611no—
284#473—612no—
285#472—612no—
286#471—614no—
287#444—1224no—
288#443—1224no—
289#443—1221no—
290#443—1221no—
291#443—1221no—
292#443—1221no—
293#430—612no—
294C2×#108—611no—
295C2×#108C2×D6111yescharacteristic
296#427C2×D6111yescharacteristic
297#421—612no—
298#421—612no—
299C2×#104—612no—
300#413—612no—
301#410—612no—
302#405—611no—
303#405—611no—
304#405—1221no—
305#405—1221no—
306#405C2×D6112yes—
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—1221no—
310C2×C4×C8C2×D6111yescharacteristic
311#344—612no—
312#344—612no—
313#339C2×D6112yes—
314C61#2155811yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41—612no—
317#137—614no—
318#137—1222no—
319AGL(1,ℤ/8)—614no—
320AGL(1,ℤ/8)—1222no—
321#135—1224no—
322#135—1221no—
323#135—1221no—
324#135—1221no—
325#135—1221no—
326C2×Q16—611no—
327C2×SD16—612no—
328C2×D8—611no—
329#131—612no—
330#131—612no—
331#131—612no—
332#131—612no—
333#131—1221no—
334#131—1221no—
335#131—1221no—
336#131—1221no—
337#131—1222no—
338#131—1222no—
339#131—1222no—
340#131—1222no—
341C2×M16—611no—
342C2×M16—611no—
343C2×M16—612no—
344C2×M16—1221no—
345C2×M16—1221no—
346C2×M16—1222no—
347C2×M16C22×D6112yes—
348C22×C8—1221no—
349C22×C8—1221no—
350C22×C8—1222no—
351C22×C8C22×D6111yescharacteristic
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1221no—
355C4×D4—1222no—
356#117—611no—
357#117—1221no—
358#117C22×D6112yes—
359C2×C42—1221no—
360C2×C42C22×D6111yescharacteristic
361#108—1224no—
362#108—1221no—
363#108—1221no—
364#108C4×D6112yescomplemented
365#108C4×D6112yescomplemented
366#107—21no—
367#106—21no—
368#104—1224no—
369#104—1224no—
370#104—1222no—
371#104—1222no—
372#101—614no—
373#100—614no—
374#99—1224no—
375#97—22no—
376#97—1221no—
377#97—1221no—
378#97—1221no—
379C4×C8—22no—
380C4×C8—1221no—
381C4×C8—1221no—
382C4×C8—1221no—
383C4×C8D24411yescharacteristic
384C4×C8D24411yescharacteristic
385C4×C8D24411yescharacteristic
386C4×C8D24411yescharacteristic
387#41—612no—
388#41—612no—
389#41—1221no—
390#41—1221no—
391#41—1222no—
392#41—1222no—
393#41—1222no—
394#41—1222no—
395C2×Q8—612no—
396C2×Q8—611no—
397C2×D4—612no—
398C2×D4—611no—
399C2×D4—1221no—
400C2×D4—1222no—
401C22×C4—21no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4C23×D6111yescharacteristic
406Q16—1224no—
407Q16—1221no—
408Q16—1221no—
409SD16—1224no—
410SD16—1224no—
411SD16—1222no—
412SD16—1222no—
413D8—1224no—
414D8—1221no—
415D8—1221no—
416M16—22no—
417M16—612no—
418M16—612no—
419M16—1221no—
420M16—1221no—
421M16—1222no—
422M16—1222no—
423M16—1222no—
424M16—1222no—
425M16—1222no—
426M16—1222no—
427M16—2442no—
428M16—2442no—
429M16C2×C4×D6114yes—
430C2×C8—22no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—2442no—
439C2×C8—2442no—
440C2×C8C2×C4×D6111yescharacteristic
441C2×C8C2×C4×D6111yescharacteristic
442C2×C8C2×D24411yescharacteristic
443C2×C8C2×D24411yescharacteristic
444C2×C8C2×D24411yescharacteristic
445C2×C8C2×D24411yescharacteristic
446C2×C8#1175822211yescharacteristic
447C2×C8#1175822211yescharacteristic
448C2×C8D4×D6111yescharacteristic
449C2×C8D4×D6111yescharacteristic
450C2×C8#1175822611yescharacteristic
451C2×C8#1175822611yescharacteristic
452C4⋊C4—22no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C22⋊C4—22no—
456C22⋊C4—612no—
457C22⋊C4—1222no—
458C42—21no—
459C42—22no—
460C42—1222no—
461C42—1221no—
462C42C2×D24411yescharacteristic
463C42C2×D24411yescharacteristic
464C23—1221no—
465C23—1222no—
466C23#41×D6111yescharacteristic
467Q8—612no—
468Q8—1221no—
469Q8—1222no—
470Q8—1222no—
471D4—612no—
472D4—1222no—
473D4—1222no—
474D4—1222no—
475D4—1222no—
476D4—1221no—
477D4—2442no—
478D4—2442no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—2444no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C22×D24411yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822211yescharacteristic
496C2×C4C2×#1175822611yescharacteristic
497C8—24no—
498C8—1222no—
499C8—1222no—
500C8—2442no—
501C8C4×D24411yescharacteristic
502C8C4×D24411yescharacteristic
503C8C4×D24411yescharacteristic
504C8C4×D24411yescharacteristic
505C8#1451975211yescharacteristic
506C8#1451975211yescharacteristic
507C8#1451975211yescharacteristic
508C8#1451975211yescharacteristic
509C22—41no—
510C22—1222no—
511C22—4882no—
512C22—2442no—
513C22—2442no—
514C22C4×#1175822611yescharacteristic
515C22#1684027411yescharacteristic
516C22#1684030911yescharacteristic
517C4—22no—
518C4—41no—
519C4—82no—
520C4—1222no—
521C4—2442no—
522C4—2442no—
523C4C4×D4×D6111yescharacteristic
524C4C2×C4×D24411yescharacteristic
525C4C2×#1451975211yescharacteristic
526C4#1684029911yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—4882no—
531C2—2442no—
532C2C4×#1451975211yescharacteristic
subgroupG/H
centerC8#14519752
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1772C2
socleC2×C61C4×#121
prime radical#21455540C2
cosocleC22×C4×C61C24
hypercenter#1772D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. #21454723C2semidirect
3. C2×#16839880C2semidirect
4. C4×#14519783C2semidirect
5. #21446715C2semidirect
6. #21446610C2semidirect
7. #21446489C2semidirect
8. #21446319C2semidirect
9. #21445870C2semidirect
10. #21445620C2semidirect
11. C61×#1772C2semidirect
18. #16840038C4semidirect
19. #16840033C4semidirect
20. #16840026C4semidirect
59. C2×Dic244C2×C4semidirect
60. C2×#11758190C2×C4semidirect
61. C2×D488C2×C4semidirect
96. #14519695C2×C4semidirect
97. #14519693C2×C4semidirect
110. #108×C61C2×C4semidirect
111. #108×C61C2×C4semidirect
270. #1772D61semidirect
314. C61#21558semidirect
364. #108C4×D61semidirect
365. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555402
#214547232
C2×#168398802
C4×#145197832
#214467152
#214466102
#214464892
#214463192
#214458702
#214456202
C61×#17722
#2155861