Finite Group Databasebeta

← #29689205

Subgroups — #29689205

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459599C211yesmaximal, characteristic
2#21454301C211yesmaximal, characteristic, complemented
3#21454281C211yesmaximal, characteristic, complemented
4#21446353C212yesmaximal, complemented
5#21446341C211yesmaximal, characteristic, complemented
6#21446339C211yesmaximal, characteristic, complemented
7#21446211C212yesmaximal, complemented
8#21446203C211yesmaximal, characteristic, complemented
9#21445922C212yesmaximal, complemented
10#21445793C212yesmaximal, complemented
11C61×#1884C211yesmaximal, characteristic, complemented
12C2×#14519824C2211yescharacteristic
13C2×#14519824C2211yescharacteristic
14#16840529C2211yescharacteristic
15D4×Dic122—22no—
16#16840417—22no—
17#16840412—24no—
18#16840333C2211yescharacteristic
19#16840333C2211yescharacteristic
20#16840332C412yescomplemented
21#16840332C412yescomplemented
22#16840310C412yescomplemented
23C2×#14519777C2211yescharacteristic
24C4⋊Q8×D61C412yescomplemented
25#16840239C2212yes—
26C4×#11758222C2211yescharacteristic
27#16839779C2212yes—
28#16839772C2212yes—
29#16839770C2212yes—
30#16839744—22no—
31#16839744C2212yes—
32C2×#14519695—22no—
33C2×#14519695C2212yes—
34#16839736C2211yescharacteristic
35#16839735C2212yes—
36#16839734—24no—
37#16839522C2212yes—
38#16839519C2212yes—
39#16839518C2212yes—
40C61×#568C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#421C2212yes—
43C61×#410C2211yescharacteristic
44C61×#339C2212yes—
45#14519824—42no—
46#14519824—42no—
47#14519824—42no—
48#14519824—42no—
49C2×#11758222—21no—
50C2×#11758222D412yes—
51C2×#11758222C2311yescharacteristic
52C2×#11758224—22no—
53C2×#11758224C2×C412yes—
54C2×#11758224C2×C412yes—
55C2×#11758224D414yes—
56C22×Dic122—21no—
57C22×Dic122D412yes—
58C22×Dic122C2311yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×Q8×D61C2×C412yes—
61C2×Q8×D61C2×C412yes—
62#14519784—42no—
63C2×#11758194—22no—
64C2×#11758194C2312yes—
65#14519777—22no—
66#14519777C2×C412yes—
67#14519777C2×C412yes—
68#14519776—24no—
69#14519775C2×C412yes—
70#14519775C2×C412yes—
71#14519764C2×C412yes—
72#14519764C2×C412yes—
73#14519759—24no—
74#14519759—24no—
75#14519758—22no—
76#14519758—22no—
77#14519756—22no—
78#14519756—22no—
79#14519754—24no—
80#14519754—24no—
81#14519748—24no—
82C4×Dic122—21no—
83C4×Dic122—42no—
84#14519742—42no—
85C4×D244—21no—
86D4×Dic61—24no—
87C4×C22⋊D61—22no—
88#14519736C2312yes—
89#14519736C2311yescharacteristic
90#14519734C2×C412yes—
91#14519733—24no—
92C2×C4⋊Dic61—22no—
93C4⋊C4×D61—24no—
94C42×D61C2×C412yes—
95#14519704—42no—
96#14519702—42no—
97#14519700—42no—
98#14519695—22no—
99#14519695—24no—
100#14519693—22no—
101#14519692—22no—
102#14519691—24no—
103C2×#41×C61C2311yescharacteristic
104C2×#41×C61C2311yescharacteristic
105#131×C61—42no—
106C2×M16×C61—22no—
107C2×M16×C61C2312yes—
108C4⋊Q8×C61C2×C411yescharacteristic
109#127×C61C2×C411yescharacteristic
110#124×C61C2×C411yescharacteristic
111#124×C61C2×C411yescharacteristic
112#121×C61—22no—
113C4×D4×C61—22no—
114#117×C61C2312yes—
115C2×C42×C61C2311yescharacteristic
116#104×C61—42no—
117#99×C61—22no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61C22×C411yescharacteristic
123C2×Q8×C61C22×C411yescharacteristic
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61C22×C411yescharacteristic
127C2×D4×C61C22×C411yescharacteristic
128C2×D4×C61C2×D412yes—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—22no—
134C2×C22⋊D61C22⋊C414yes—
135C2×C22⋊D61C2×D412yes—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61C22⋊C414yes—
139C22×Dic61C2×D412yes—
140Q8×D61—44no—
141Q8×D61—44no—
142#11758224—44no—
143#11758224—44no—
144#11758224—48no—
145#11758224—48no—
146#11758222—22no—
147#11758222—22no—
148#11758222—42no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152C2×D244C2×D411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22×C412yes—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—42no—
164C2×Dic122—42no—
165C2×Dic122—42no—
166C2×Dic122—42no—
167C2×Dic122C22⋊C414yes—
168C2×Dic122C22×C412yes—
169C2×Dic122C22×C412yes—
170C2×Dic122C2×D411yescharacteristic
171C2×Dic122C2×D412yes—
172M16×C61—22no—
173M16×C61—22no—
174M16×C61—42no—
175C2×C8×C61—22no—
176C2×C8×C61—42no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61—21no—
183C42×C61—22no—
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C22⋊Dic61—24no—
187C22⋊Dic61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—24no—
193C4⋊Dic61—24no—
194C4⋊Dic61—22no—
195#11758196—24no—
196#11758196—24no—
197#11758196—22no—
198#11758196—44no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—44no—
202C4×Dic61C22×C412yes—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—21no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61C2×C22⋊C412yes—
212Q8×C61—22no—
213Q8×C61—22no—
214D4×C61—24no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—44no—
218D4×C61—42no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61H(5,𝔽2)12yes—
232C2×C4×C61H(5,𝔽2)12yes—
233C22⋊D61—24no—
234C22⋊D61—44no—
235C22⋊D61—44no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—44no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C414yes—
245D244—21no—
246D244—21no—
247C4×D61—24no—
248C4×D61—44no—
249C4×D61—44no—
250C4×D61—44no—
251Dic122—24no—
252Dic122—22no—
253Dic122—22no—
254Dic122—21no—
255Dic122—21no—
256Dic122—24no—
257Dic122—44no—
258Dic122—44no—
259Dic122—41no—
260Dic122—41no—
261C8×C61—42no—
262#92418—42no—
263#21783—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61C22×C22⋊C411yescharacteristic
270C22×C61#53412yes—
271C2×D61—22no—
272C2×D61—42no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—82no—
278C4×C61#51812yes—
279C4×C61#53411yescharacteristic
280C4×C61#53411yescharacteristic
281Dic61—24no—
282Dic61—22no—
283Dic61—42no—
284Dic61—42no—
285Dic61—82no—
286#3499—611no—
287#2889—611no—
288#2884—611no—
289#1912—612no—
290#1909—611no—
291#1909—611no—
292#1886—612no—
293#1884—611no—
294#1884D6111yescharacteristic, complemented
295#1827—612no—
296#1796—612no—
297C2×C61—21no—
298C2×C61—22no—
299C2×C61—42no—
300C2×C61—81no—
301C2×C61#230811yescharacteristic
302D61—42no—
303C2×2−1+4—611no—
304C2×2−1+4—611no—
305#568—611no—
306#568C2×D6111yescharacteristic
307D4×Q8—1222no—
308#547—1222no—
309#547—1224no—
310#536—611no—
311#536—611no—
312#536—612no—
313#536—612no—
314#532—612no—
315#532C2×D6111yescharacteristic
316C2×C4⋊Q8—612no—
317C2×C4⋊Q8—611no—
318#519—612no—
319C4×#41—611no—
320#421—612no—
321#421C2×D6112yes—
322#419—612no—
323#419—612no—
324#413—612no—
325#413—1222no—
326C2×#101—612no—
327C2×#101—1222no—
328#410—611no—
329#410—612no—
330#410—1224no—
331#410C2×D6111yescharacteristic
332#339—612no—
333#339C2×D6112yes—
334H(3,ℤ/4)—612no—
335H(3,ℤ/4)—612no—
336C61#2178311yescharacteristic, complemented, Sylow
3372−1+4—2442no—
3382−1+4—2442no—
3392−1+4—2442no—
3402−1+4—2442no—
341C2×#41—612no—
342C2×#41—614no—
343C2×#41—612no—
344C2×#41—611no—
345C2×#41—612no—
346C2×#41—1221no—
347C2×#41—1222no—
348C2×#41C22×D6111yescharacteristic
349C2×#41C22×D6111yescharacteristic
350C22×Q8—612no—
351C22×Q8—612no—
352C22×Q8—611no—
353C22×Q8—612no—
354C22×Q8—611no—
355C22×Q8—1221no—
356#131—42no—
357#131—2442no—
358C2×M16—22no—
359C2×M16—612no—
360C2×M16—1222no—
361C2×M16C22×D6112yes—
362C4⋊Q8—612no—
363C4⋊Q8—612no—
364C4⋊Q8—612no—
365C4⋊Q8—612no—
366C4⋊Q8—1222no—
367C4⋊Q8—1224no—
368C4⋊Q8C4×D6111yescharacteristic
369#127C4×D6111yescharacteristic
370#124—612no—
371#124—612no—
372#124C4×D6111yescharacteristic
373#124C4×D6111yescharacteristic
374#123—1224no—
375#123—1224no—
376#123—1222no—
377#123—1222no—
378C22⋊Q8—1224no—
379C22⋊Q8—1224no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382#121—22no—
383#121—1224no—
384C4×Q8—1221no—
385C4×Q8—2442no—
386C4×Q8—2442no—
387C4×D4—22no—
388C4×D4—1221no—
389C4×D4—1222no—
390C4×D4—1224no—
391#117—611no—
392#117—612no—
393#117—612no—
394#117—1224no—
395#117C22×D6112yes—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1222no—
398C2×C42—612no—
399C2×C42C22×D6111yescharacteristic
400#104—42no—
401#104—2442no—
402#103—2442no—
403#103—2442no—
404#101—1224no—
405#101—1222no—
406#100—1222no—
407#99—22no—
408#99—1224no—
409#99—1222no—
410#41—22no—
411#41—22no—
412#41—22no—
413#41—22no—
414#41—1222no—
415#41—1222no—
416#41—2442no—
417#41—2442no—
418#41—2442no—
419#41—2442no—
420#41—2448no—
421#41—2444no—
422#41—2444no—
423#41—2448no—
424C2×Q8—612no—
425C2×Q8—611no—
426C2×Q8—612no—
427C2×Q8—614no—
428C2×Q8—612no—
429C2×Q8—1222no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8—2442no—
439C2×Q8—2444no—
440C2×Q8—2444no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×Q8C2×C4×D6111yescharacteristic
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—612no—
446C2×D4—614no—
447C2×D4—611no—
448C2×D4—1222no—
449C2×D4C2×C4×D6111yescharacteristic
450C2×D4C2×C4×D6111yescharacteristic
451C2×D4D4×D6112yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—612no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—614no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4D4×D6112yes—
463C22×C4C23×D6111yescharacteristic
464M16—22no—
465M16—22no—
466M16—42no—
467M16—1222no—
468M16—1222no—
469M16—2442no—
470C2×C8—22no—
471C2×C8—42no—
472C2×C8—1222no—
473C2×C8—2442no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—2444no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—1222no—
488C22⋊C4—1224no—
489C22⋊C4—1221no—
490C22⋊C4—1224no—
491C22⋊C4—1222no—
492C42—21no—
493C42—22no—
494C42—612no—
495C42—1222no—
496C42—1222no—
497C42—2444no—
498C42C2×C4×D6111yescharacteristic
499C42C2×C4×D6111yescharacteristic
500C23—21no—
501C23—612no—
502C23C22⋊C4×D6112yes—
503C23C2×D4×D6111yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—1224no—
507Q8—1222no—
508Q8—1221no—
509Q8—1222no—
510Q8—1221no—
511Q8—1224no—
512Q8—2444no—
513Q8—2444no—
514Q8—2441no—
515Q8—2441no—
516D4—24no—
517D4—22no—
518D4—22no—
519D4—44no—
520D4—42no—
521D4—1224no—
522D4—1221no—
523D4—1221no—
524D4—2444no—
525D4—2444no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—612no—
536C2×C4—614no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4C22⋊C4×D6112yes—
549C2×C4C22×C4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4#1451982212yes—
552C2×C4#1451982312yes—
553C8—42no—
554C8—2442no—
555C22—22no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—1222no—
561C22—2442no—
562C22C2×C22⋊C4×D6111yescharacteristic
563C22#1684032012yes—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—82no—
569C4—1222no—
570C4—1224no—
571C4—4882no—
572C4—2442no—
573C4—2442no—
574C4#1684023412yes—
575C4#1684032211yescharacteristic
576C4#1684032311yescharacteristic
577C2—21no—
578C2—22no—
579C2—42no—
580C2—81no—
581C2—2442no—
582C2#2144839311yescharacteristic
subgroupG/H
centerC2#21448393
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1884C2
socleC2×C61#2308
prime radical#21459599C2
cosocleC22×C4×C61C24
hypercenter#1884D61
hypocenterC61#21783
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1884

Complements

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

NH ≅ G/Ntype
2. #21454301C2semidirect
3. #21454281C2semidirect
4. #21446353C2semidirect
5. #21446341C2semidirect
6. #21446339C2semidirect
7. #21446211C2semidirect
8. #21446203C2semidirect
9. #21445922C2semidirect
10. #21445793C2semidirect
11. C61×#1884C2semidirect
20. #16840332C4semidirect
21. #16840332C4semidirect
22. #16840310C4semidirect
24. C4⋊Q8×D61C4semidirect
294. #1884D61semidirect
336. C61#21783semidirect

Maximal subgroups

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

subgroupindex
#214595992
#214543012
#214542812
#214463532
#214463412
#214463392
#214462112
#214462032
#214459222
#214457932
C61×#18842
#2178361