Finite Group Databasebeta

← #29686946

Subgroups — #29686946

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#21454720C212yesmaximal, complemented
3C4×#14519783C211yesmaximal, characteristic, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446607C211yesmaximal, characteristic, complemented
6#21446485C212yesmaximal, complemented
7#21446323C212yesmaximal, complemented
8#21445874C211yesmaximal, characteristic, complemented
9#21445698C211yesmaximal, characteristic, complemented
10#21445619C212yesmaximal, complemented
11C2×C61×#443C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519783C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840034C412yescomplemented
17#16840029C412yescomplemented
18#16840022C414yescomplemented
19#16839882—24no—
20#16839877—24no—
21#16839810C2212yes—
22C2×#14519713C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839779C2212yes—
25#16839778C2212yes—
26#16839746C2212yes—
27#16839736C2212yes—
28#16839717C2211yescharacteristic
29#16839717C2212yes—
30#16839717C2211yescharacteristic
31#16839716—21no—
32#16839716—21no—
33C4×#11758194C2211yescharacteristic
34C4×#11758189—21no—
35C4×C8×D61—21no—
36#16839533C2212yes—
37#16839533C2212yes—
38#16839522C2212yes—
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2212yes—
41C61×#443—21no—
42C61×#443—21no—
43C61×#443—21no—
44C61×#443—21no—
45C2×#108×C61C2211yescharacteristic
46C2×#104×C61C2212yes—
47C2×C4×C8×C61C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808C2×C414yes—
50#14519807—22no—
51#14519803C2×C414yes—
52#14519802—22no—
53#14519797—24no—
54#14519783—21no—
55#14519783—21no—
56#14519783—21no—
57#14519783—21no—
58#14519783C2×C412yes—
59#14519783C2×C412yes—
60#14519783D412yes—
61#14519783D412yes—
62C2×#11758194C2311yescharacteristic
63C2×#11758194C2312yes—
64C2×#11758194C2311yescharacteristic
65C2×#11758189—21no—
66C2×#11758189—21no—
67C2×C8×D61—21no—
68C2×C8×D61—21no—
69C4×Dic122—21no—
70C4×D244—21no—
71C4×C22⋊D61—22no—
72#14519736C2312yes—
73#14519736C2311yescharacteristic
74#14519734—21no—
75C42×D61—21no—
76#14519713C2×C412yescomplemented
77#14519713C2×C412yescomplemented
78#14519712—24no—
79#14519711—21no—
80#14519709—21no—
81#14519704—24no—
82#14519703—24no—
83#14519696C2×C414yescomplemented
84#14519694C2×C414yescomplemented
85#14519692—24no—
86#14519688—21no—
87#14519687—21no—
88#14519687—21no—
89#14519687—22no—
90C4×#92418—21no—
91C8×Dic61—21no—
92C8×Dic61—21no—
93C8×Dic61—22no—
94C2×#41×C61C2312yes—
95#135×C61—21no—
96#135×C61—21no—
97#135×C61—21no—
98#135×C61—21no—
99C2×Q16×C61C2×C411yescharacteristic, complemented
100C2×SD16×C61C2×C412yescomplemented
101C2×D8×C61C2×C411yescharacteristic, complemented
102#131×C61—22no—
103#131×C61—22no—
104#131×C61—22no—
105#131×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61C2312yes—
108C22×C8×C61—22no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#108×C61—21no—
113#108×C61—21no—
114#104×C61—22no—
115#104×C61—22no—
116C4×C8×C61D411yescharacteristic
117C4×C8×C61D411yescharacteristic
118C4×C8×C61D411yescharacteristic
119C4×C8×C61D411yescharacteristic
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—21no—
128C22×C4×C61—22no—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—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—22no—
146M16×C61—22no—
147M16×C61—42no—
148M16×C61—42no—
149C2×C8×C61—22no—
150C2×C8×C61—42no—
151C2×C8×C61—42no—
152C2×C8×C61C22×C411yescharacteristic
153C2×C8×C61C22×C411yescharacteristic
154C2×C8×C61C2×D411yescharacteristic
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61#4111yescharacteristic
161C2×C8×C61#4111yescharacteristic
162C2×C8×C61#4111yescharacteristic
163C2×C8×C61#4111yescharacteristic
164C42×C61—21no—
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C22⋊Dic61—22no—
168C22⋊Dic61#4112yes—
169#11758202—24no—
170#11758201—24no—
171#11758200—24no—
172#11758199—24no—
173#11758198—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176#11758196—22no—
177C4×Dic61—22no—
178C4×Dic61—21no—
179C4×Dic61—22no—
180#11758194—21no—
181#11758194—21no—
182#11758194—22no—
183#11758194C22×C414yes—
184#11758194C22×C412yes—
185#11758194C2×D412yes—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—22no—
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—21no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61C23×C411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C22⋊D61—22no—
219C22⋊D61—22no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223D244—21no—
224D244C4×D412yes—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—44no—
228Dic122—21no—
229Dic122C4×D412yes—
230C8×C61—42no—
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—24no—
241#92418—22no—
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—22no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61C2×C4×D411yescharacteristic
255C4×C61C2×C4×D411yescharacteristic
256C4×C61C2×#12111yescharacteristic
257C4×C61#52911yescharacteristic
258Dic61—22no—
259Dic61—42no—
260Dic61—82no—
261#3067—611no—
262#2955—612no—
263C2×#443D6111yescharacteristic, complemented
264C4×#131—611no—
265#1976—611no—
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—611no—
281C2×#131—611no—
282C2×#131C2×D6112yes—
283C4×#41—611no—
284#473—612no—
285#472—612no—
286#471—614no—
287#444—1224no—
288#443—21no—
289#443—21no—
290#443—21no—
291#443—21no—
292#443—1224no—
293#430—612no—
294C2×#108—611no—
295C2×#108C2×D6111yescharacteristic
296#427—611no—
297#421—612no—
298#421—612no—
299C2×#104C2×D6112yes—
300#413—612no—
301#410—612no—
302#405—611no—
303#405—611no—
304#405—612no—
305#405—1221no—
306#405—1221no—
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—1221no—
310C2×C4×C8C2×D6111yescharacteristic
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—22no—
330#131—22no—
331#131—22no—
332#131—22no—
333#131—612no—
334#131—612no—
335#131—612no—
336#131—612no—
337#131—1221no—
338#131—1221no—
339#131—1221no—
340#131—1221no—
341C2×M16—22no—
342C2×M16—611no—
343C2×M16—611no—
344C2×M16—612no—
345C2×M16—1221no—
346C2×M16—1221no—
347C2×M16C22×D6112yes—
348C22×C8—22no—
349C22×C8—1221no—
350C22×C8—1221no—
351C22×C8C22×D6111yescharacteristic
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1221no—
355C4×D4—1222no—
356#117—611no—
357#117—612no—
358#117—1221no—
359C2×C42—1221no—
360C2×C42C22×D6111yescharacteristic
361#108—21no—
362#108—21no—
363#108—612no—
364#108—612no—
365#108—1224no—
366#107—1221no—
367#106—1221no—
368#104—22no—
369#104—22no—
370#104—1224no—
371#104—1224no—
372#101—614no—
373#100—614no—
374#99—1224no—
375#97—1222no—
376#97—1221no—
377#97—1221no—
378#97—1221no—
379C4×C8—1222no—
380C4×C8—1221no—
381C4×C8—1221no—
382C4×C8—1221no—
383C4×C8C22⋊D6111yescharacteristic
384C4×C8C22⋊D6111yescharacteristic
385C4×C8C22⋊D6111yescharacteristic
386C4×C8C22⋊D6111yescharacteristic
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—21no—
402C22×C4—22no—
403C22×C4—1222no—
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—22no—
417M16—22no—
418M16—42no—
419M16—42no—
420M16—612no—
421M16—614no—
422M16—612no—
423M16—1222no—
424M16—1221no—
425M16—1221no—
426M16—1222no—
427M16—1222no—
428M16—1222no—
429M16—1222no—
430C2×C8—22no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—1222no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8C2×C4×D6111yescharacteristic
441C2×C8C2×C4×D6111yescharacteristic
442C2×C8#1175822211yescharacteristic
443C2×C8#1175822211yescharacteristic
444C2×C8D4×D6111yescharacteristic
445C2×C8D4×D6111yescharacteristic
446C2×C8#1175822411yescharacteristic
447C2×C8#1175822411yescharacteristic
448C2×C8C2×C22⋊D6111yescharacteristic
449C2×C8C2×C22⋊D6111yescharacteristic
450C2×C8C2×C22⋊D6111yescharacteristic
451C2×C8C2×C22⋊D6111yescharacteristic
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C22⋊C4—612no—
456C22⋊C4—1222no—
457C22⋊C4—1222no—
458C42—21no—
459C42—1222no—
460C42—1221no—
461C42—1222no—
462C42C2×C22⋊D6111yescharacteristic
463C42C2×C22⋊D6111yescharacteristic
464C23—22no—
465C23—1221no—
466C23#41×D6111yescharacteristic
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—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—2444no—
490C2×C4—2442no—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822411yescharacteristic
496C2×C4C2×#1175822211yescharacteristic
497C8—42no—
498C8—1224no—
499C8—1222no—
500C8—1222no—
501C8C4×C22⋊D6111yescharacteristic
502C8C4×C22⋊D6111yescharacteristic
503C8C4×C22⋊D6111yescharacteristic
504C8C4×C22⋊D6111yescharacteristic
505C8#1451974811yescharacteristic
506C8#1451974811yescharacteristic
507C8#1451974811yescharacteristic
508C8#1451974811yescharacteristic
509C22—41no—
510C22—42no—
511C22—82no—
512C22—1222no—
513C22—2442no—
514C22C4×#1175822411yescharacteristic
515C22#1684026911yescharacteristic
516C22#1684030911yescharacteristic
517C4—22no—
518C4—41no—
519C4—42no—
520C4—1222no—
521C4—4882no—
522C4—2442no—
523C4C4×D4×D6111yescharacteristic
524C4C2×C4×C22⋊D6111yescharacteristic
525C4C2×#1451974811yescharacteristic
526C4#1684029411yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—2442no—
532C2C4×#1451974811yescharacteristic
subgroupG/H
centerC8#14519748
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#443C2
socleC2×C61C4×#121
prime radical#21455535C2
cosocleC22×C4×C61C24
hypercenterC2×#443D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. #21454720C2semidirect
3. C4×#14519783C2semidirect
4. #21446718C2semidirect
5. #21446607C2semidirect
6. #21446485C2semidirect
7. #21446323C2semidirect
8. #21445874C2semidirect
9. #21445698C2semidirect
10. #21445619C2semidirect
11. C2×C61×#443C2semidirect
16. #16840034C4semidirect
17. #16840029C4semidirect
18. #16840022C4semidirect
76. #14519713C2×C4semidirect
77. #14519713C2×C4semidirect
83. #14519696C2×C4semidirect
84. #14519694C2×C4semidirect
99. C2×Q16×C61C2×C4semidirect
100. C2×SD16×C61C2×C4semidirect
101. C2×D8×C61C2×C4semidirect
263. C2×#443D61semidirect
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
#214547202
C4×#145197832
#214467182
#214466072
#214464852
#214463232
#214458742
#214456982
#214456192
C2×C61×#4432
#2155861