Finite Group Databasebeta

← #29657441

Subgroups — #29657441

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459604C211yesmaximal, characteristic
2#21456940C211yesmaximal, characteristic, complemented
3C2×#16840165C211yesmaximal, characteristic, complemented
4#21445066C214yesmaximal, complemented
5#21445052C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444269C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3039C211yesmaximal, characteristic, complemented
10C2×#14519824—21no—
11Q8×D244—21no—
12Q8×D244C2212yes—
13#16840416—22no—
14#16840416—22no—
15#16840416—21no—
16#16840416C2212yes—
17#16840338—22no—
18#16840338—21no—
19#16840312—21no—
20#16840312C2211yescharacteristic
21C2×#14519777—21no—
22C2×#14519777C2211yescharacteristic
23#16840165—21no—
24#16840165—22no—
25#16840165C2214yes—
26#16840110—21no—
27#16840075—22no—
28#16840039C2214yes—
29#16839815C2211yescharacteristic
30C2×#14519710—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839509C2214yes—
33Q8⋊#92418C2212yes—
34D4⋊#92418C2212yes—
35#16839489C2214yes—
36C2×#137×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C4×#41×C61C2211yescharacteristic
39C61×#462—21no—
40C61×#462C2212yes—
41C61×#461—21no—
42C61×#461C2212yes—
43C61×#423C2211yescharacteristic
44C61×#419—21no—
45#14519824—44no—
46#14519824—44no—
47C2×#11758222—22no—
48C2×#11758222—21no—
49C2×#11758224—24no—
50C2×#11758224—22no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C2×Q8×D61—22no—
54C22×#92418—21no—
55#14519777—22no—
56#14519777—22no—
57#14519777—21no—
58#14519777—21no—
59#14519777—22no—
60#14519777D411yescharacteristic
61#14519777D412yes—
62#14519777C2312yes—
63#14519774D411yescharacteristic
64#14519765—22no—
65#14519765—21no—
66#14519761—24no—
67#14519761—24no—
68#14519761—24no—
69#14519757—24no—
70#14519757—22no—
71#14519757—22no—
72#14519757—24no—
73#14519751—22no—
74#14519751—21no—
75#14519751—22no—
76#14519751D412yes—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—21no—
80C4×D244C2312yes—
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—22no—
84C2×C4⋊Dic61D412yes—
85#14519710—24no—
86#14519710—28no—
87C4⋊#92418C2312yes—
88#14519697—24no—
89C22⋊#92418—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C4×#92418C2312yes—
93C2×#41×C61D411yescharacteristic
94C22×Q8×C61—21no—
95#137×C61—44no—
96C2×Q16×C61—22no—
97C2×SD16×C61—22no—
98C2×M16×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61—21no—
102C4⋊Q8×C61C2312yes—
103C4×Q8×C61D411yescharacteristic
104C4×Q8×C61C2311yescharacteristic
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61D411yescharacteristic
108C4×D4×C61C2311yescharacteristic
109#117×C61—21no—
110#117×C61D411yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—21no—
116C4⋊C8×C61C2312yes—
117#103×C61—22no—
118#103×C61—22no—
119#41×C61—22no—
120#41×C61D814yes—
121C2×Q8×C61—22no—
122C2×Q8×C61—41no—
123C2×Q8×C61—42no—
124C2×Q8×C61—42no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61C2×D411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—22no—
137C22×Dic61—24no—
138C22×Dic61—22no—
139Q8×D61—44no—
140Q8×D61—44no—
141#11758224—416no—
142#11758224—44no—
143#11758224—44no—
144#11758222—48no—
145#11758222—42no—
146#11758222—42no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C2×D412yes—
150C2×C4×D61—24no—
151C2×C4×D61—22no—
152C2×Dic122—24no—
153C2×Dic122—21no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—48no—
157C2×Dic122—42no—
158C2×Dic122—42no—
159Q16×C61—44no—
160SD16×C61—44no—
161M16×C61—42no—
162C2×C8×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61C2×D411yescharacteristic
170C4⋊C4×C61C2×D411yescharacteristic
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2411yescharacteristic
181#11758198—24no—
182#11758198—24no—
183#11758198—22no—
184#11758198—22no—
185C4⋊Dic61—24no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C2×D412yes—
192C4⋊Dic61C2×D414yes—
193C2×#92418—24no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—21no—
197C23×C61#12011yescharacteristic
198C22×D61—22no—
199Q8×C61—21no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—82no—
203Q8×C61C2×D812yes—
204D4×C61—24no—
205D4×C61—21no—
206D4×C61C2×D812yes—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—82no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61C2×D812yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—48no—
226C22⋊D61—44no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—48no—
230C2×Dic61—44no—
231D244—24no—
232D244—42no—
233D244—42no—
234C4×D61—48no—
235C4×D61—42no—
236C4×D61—42no—
237Dic122—48no—
238Dic122—42no—
239Dic122—42no—
240Dic122—42no—
241Dic122—44no—
242C8×C61—42no—
243#92418—24no—
244#92418—42no—
245#18650—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—22no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61#44712yes—
252C22×C61C2×#12011yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—82no—
262C4×C61#44712yes—
263C4×C61#45612yes—
264C4×C61C22×D811yescharacteristic
265C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
266C4×C61#57812yes—
267Dic61—42no—
268Dic61—44no—
269#3499—611no—
270#3226—611no—
271C2×#500—611no—
272#3039D6111yescharacteristic, complemented
273#1649—614no—
274#1647—614no—
275#1546—611no—
276#1492—611no—
277#1476—611no—
278C2×C61—22no—
279C2×C61—42no—
280C2×C61C2×#44711yescharacteristic
281C2×C61#301111yescharacteristic
282C2×C61C2×#45611yescharacteristic
283D61—82no—
284C2×2−1+4—1221no—
285C2×#137—21no—
286D4×Q8—612no—
287D4×Q8—1221no—
288#547—612no—
289#547—1222no—
290#547—1222no—
291#547—1221no—
292#536—1222no—
293#536—1221no—
294#532—611no—
295#532—1221no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—1221no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299C4×#41C2×D6111yescharacteristic
300#500—614no—
301#500—1222no—
302#500—1221no—
303#486—1221no—
304#480—1222no—
305#474—614no—
306#462—21no—
307#462C2×D6112yes—
308#461—21no—
309#461C2×D6112yes—
310#431—611no—
311C2×#107—1222no—
312#423C2×D6111yescharacteristic
313#419—21no—
314C2×C4×C8—611no—
315#332—614no—
316Q8⋊C8—612no—
317D4⋊C8—612no—
318D4⋊C8—614no—
319C61#1865011yescharacteristic, complemented, Sylow
3202−1+4—2444no—
3212−1+4—2444no—
322C2×#41—1222no—
323C2×#41—1224no—
324C2×#41—1222no—
325C2×#41—1221no—
326C2×#41C22⋊D6111yescharacteristic
327C22×Q8—21no—
328C22×Q8—1222no—
329C22×Q8—1222no—
330C22×Q8—1221no—
331#137—44no—
332C2×Q16—22no—
333C2×SD16—22no—
334C2×M16—21no—
335C22×C8—1221no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—612no—
340C4⋊Q8—611no—
341C4⋊Q8—612no—
342C4⋊Q8—1222no—
343C4⋊Q8—1222no—
344C4⋊Q8—1222no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8C22×D6112yes—
348#127—611no—
349#124—1222no—
350#124—1221no—
351#123—1224no—
352#123—1224no—
353#123—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358#121—612no—
359#121—1222no—
360#121—1222no—
361#121—1221no—
362C4×Q8C22⋊D6111yescharacteristic
363C4×Q8C22×D6111yescharacteristic
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—612no—
367C4×D4—1222no—
368C4×D4—1222no—
369C4×D4—1221no—
370C4×D4C22⋊D6111yescharacteristic
371C4×D4C22×D6111yescharacteristic
372#117—21no—
373#117C22⋊D6111yescharacteristic
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C42—21no—
381C2×C42C22×D6111yescharacteristic
382#107—1228no—
383#107—1224no—
384C4⋊C8—21no—
385C4⋊C8—612no—
386C4⋊C8C22×D6112yes—
387#103—22no—
388#103—22no—
389#102—1224no—
390C22⋊C8—1221no—
391C4×C8—612no—
392C4×C8—1221no—
393C4×C8—1221no—
394#41—22no—
395#41—2448no—
396#41—24416no—
397#41—2444no—
398#41—2444no—
399#41—2442no—
400#41—2442no—
401#41#1175819914yes—
402C2×Q8—22no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—1224no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1221no—
410C2×Q8—2448no—
411C2×Q8—2444no—
412C2×Q8—2444no—
413C2×Q8—2442no—
414C2×Q8—2442no—
415C2×Q8C2×C22⋊D6111yescharacteristic
416C2×D4—21no—
417C2×D4—612no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4C2×C22⋊D6111yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—1224no—
428C22×C4—1224no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4D4×D6111yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434Q16—44no—
435SD16—44no—
436M16—42no—
437C2×C8—22no—
438C2×C8—1224no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—614no—
448C4⋊C4—612no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4C2×C22⋊D6111yescharacteristic
456C4⋊C4C2×C22⋊D6111yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—1222no—
463C22⋊C4—1222no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42D4×D6111yescharacteristic
468C42D4×D6111yescharacteristic
469C42C2×C22⋊D6111yescharacteristic
470C42C23×D6111yescharacteristic
471C23—21no—
472C23—1222no—
473C23#1451974511yescharacteristic
474Q8—21no—
475Q8—42no—
476Q8—42no—
477Q8—82no—
478Q8—2448no—
479Q8—2442no—
480Q8—2442no—
481Q8—2442no—
482Q8—2444no—
483Q8C2×#1175819912yes—
484D4—24no—
485D4—21no—
486D4—1224no—
487D4—2448no—
488D4—2444no—
489D4—2442no—
490D4—2442no—
491D4C2×#1175819912yes—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—1224no—
503C2×C4—1222no—
504C2×C4—2448no—
505C2×C4—2448no—
506C2×C4—2444no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4#1451974511yescharacteristic
510C2×C4#1451974511yescharacteristic
511C2×C4#1451974511yescharacteristic
512C2×C4C2×#1175819912yes—
513C2×C4#1451980212yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C8—42no—
518C8—1224no—
519C8—2442no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—2442no—
526C22—2442no—
527C22—2442no—
528C22#1683989512yes—
529C22C2×#1451974511yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—82no—
536C4—2444no—
537C4—2442no—
538C4#1683989512yes—
539C4#1683994312yes—
540C4C22×#1175819911yescharacteristic
541C4C2×#1451980211yescharacteristic
542C4#1684057812yes—
543C2—22no—
544C2—42no—
545C2—4882no—
546C2C2×#1683989511yescharacteristic
547C2#2145511311yescharacteristic
548C2C2×#1683994311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3039C2
socleC22×C61C2×#120
prime radical#21459604C2
hypercenter#3039D61
hypocenterC61#18650
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
2. #21456940C2semidirect
3. C2×#16840165C2semidirect
4. #21445066C2semidirect
5. #21445052C2semidirect
6. #21444492C2semidirect
7. #21444269C2semidirect
8. #21444201C2semidirect
9. C61×#3039C2semidirect
272. #3039D61semidirect
319. C61#18650semidirect

Maximal subgroups

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

subgroupindex
#214596042
#214569402
C2×#168401652
#214450662
#214450522
#214444922
#214442692
#214442012
C61×#30392
#1865061