Finite Group Databasebeta

← #29688175

Subgroups — #29688175

Order 15616.

722 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#21456139C211yesmaximal, characteristic
2#21454757C211yesmaximal, characteristic, complemented
3#21454535C211yesmaximal, characteristic
4#21446798C211yesmaximal, characteristic, complemented
5#21446649C211yesmaximal, characteristic
6#21446532C211yesmaximal, characteristic, complemented
7#21446528C211yesmaximal, characteristic, complemented
8#21446524C211yesmaximal, characteristic, complemented
9#21446520C211yesmaximal, characteristic, complemented
10#21445955C211yesmaximal, characteristic, complemented
11#21445663C211yesmaximal, characteristic, complemented
12#21445658C211yesmaximal, characteristic, complemented
13#21445654C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1765C211yesmaximal, characteristic, complemented
16C2×#14519783C2211yescharacteristic
17#16840315C2211yescharacteristic
18#16840308C2211yescharacteristic
19#16840118C412yescomplemented
20#16840115C412yescomplemented
21#16840111C412yescomplemented
22#16840107C412yescomplemented
23#16839892—22no—
24#16839890—22no—
25#16839833—22no—
26#16839831—22no—
27#16839790C2211yescharacteristic
28#16839789C2211yescharacteristic
29#16839785C2211yescharacteristic
30#16839785C2211yescharacteristic
31C2×#14519705C2211yescharacteristic
32C2×#14519704C2211yescharacteristic
33C2×#14519703C2211yescharacteristic
34#16839759C2211yescharacteristic
35#16839758C2211yescharacteristic
36#16839756C2211yescharacteristic
37#16839730C2211yescharacteristic
38#16839729C2211yescharacteristic
39#16839717C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×C4×#92418C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45#16839608C2211yescharacteristic
46#16839527C2211yescharacteristic
47#16839526C2211yescharacteristic
48#16839526C2211yescharacteristic
49#16839523C2211yescharacteristic
50#16839522C2211yescharacteristic
51#16839521C2211yescharacteristic
52C2×#131×C61C2211yescharacteristic
53C2×#104×C61C2211yescharacteristic
54C61×#416C2211yescharacteristic
55C61×#405C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C61×#341C2211yescharacteristic
58C61×#339C2211yescharacteristic
59C2×#11758222C2311yescharacteristic
60#14519783—22no—
61#14519783—22no—
62C2×#11758194C2311yescharacteristic
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C8×D61—21no—
68#14519776C2×C412yes—
69#14519775C2×C411yescharacteristic
70#14519773C2×C411yescharacteristic
71#14519768C2×C412yes—
72#14519766C2×C411yescharacteristic
73#14519764C2×C411yescharacteristic
74#14519754—22no—
75#14519749—22no—
76C4×Dic122—22no—
77C4×C22⋊D61—22no—
78#14519736C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C4⋊#92418—22no—
82#14519706—22no—
83#14519706—22no—
84#14519706—22no—
85#14519705—22no—
86#14519704—22no—
87#14519703—22no—
88#14519702C2×C412yes—
89#14519701C2×C412yes—
90#14519700C2×C412yes—
91#14519699C2×C412yes—
92#14519698C2×C412yes—
93#14519697C2×C412yes—
94C22⋊#92418—22no—
95#14519689—22no—
96#14519687—21no—
97#14519687—22no—
98#14519687D412yes—
99#14519687D412yes—
100C4×#92418—22no—
101C8×Dic61—21no—
102C8×Dic61—22no—
103C8×Dic61C2×C412yes—
104C8×Dic61C2×C412yes—
105#14519684—22no—
106C2×#41×C61C2311yescharacteristic
107#131×C61—22no—
108#131×C61—22no—
109C2×M16×C61—21no—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C22×C8×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#117×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#104×C61—22no—
118#103×C61C2×C412yes—
119#102×C61C2×C412yes—
120#97×C61—21no—
121#97×C61—22no—
122C4×C8×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126#41×C61—21no—
127C2×Q8×C61C22×C411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61C22×C411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—21no—
134C2×C22⋊D61—22no—
135C22×Dic61—21no—
136C22×Dic61#4112yes—
137#11758222—21no—
138#11758222—21no—
139#11758222—21no—
140#11758222—21no—
141C2×D244C22×C411yescharacteristic
142C2×C4×D61—21no—
143C2×Dic122—22no—
144C2×Dic122C22×C411yescharacteristic
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—42no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—42no—
155C2×C8×C61C22×C412yes—
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61#4112yes—
159C2×C8×C61#4112yes—
160C4⋊C4×C61C22×C412yes—
161C22⋊C4×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61#4112yes—
165C22⋊Dic61—21no—
166C22⋊Dic61—22no—
167#11758198—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C22×C412yes—
170#11758196—22no—
171#11758196—22no—
172#11758196—22no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C4×Dic61C22×C411yescharacteristic
176C4×Dic61C22×C411yescharacteristic
177C4×Dic61C2×D412yes—
178C4×Dic61#4112yes—
179#11758194—21no—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418C22×C412yes—
190C2×#92418C2×D412yes—
191#11758189—42no—
192C8×D61—42no—
193C23×C61—21no—
194C23×C61C2×#4111yescharacteristic
195C22×D61—21no—
196Q8×C61—21no—
197Q8×C61—21no—
198D4×C61—21no—
199D4×C61—21no—
200D4×C61—41no—
201D4×C61—41no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C22⋊D61—41no—
215C22⋊D61—41no—
216C22⋊D61—82no—
217C2×Dic61—22no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61C4×D412yes—
223C2×Dic61C4×D412yes—
224D244—21no—
225D244—21no—
226C4×D61—41no—
227C4×D61—41no—
228Dic122—21no—
229Dic122—21no—
230Dic122—82no—
231C8×C61—22no—
232C8×C61—41no—
233C8×C61—41no—
234C8×C61#12412yes—
235C8×C61#12412yes—
236#92418—22no—
237#92418—22no—
238#92418—41no—
239#92418—41no—
240#21659—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—42no—
243C22×C61—41no—
244C22×C61C2×C4×D411yescharacteristic
245C22×C61#52911yescharacteristic
246C22×C61#52911yescharacteristic
247C2×D61—42no—
248C2×D61—41no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—41no—
252C4×C61C4×#4111yescharacteristic
253C4×C61C4×#4111yescharacteristic
254C4×C61C2×#12411yescharacteristic
255C4×C61#53211yescharacteristic
256Dic61—22no—
257Dic61—22no—
258Dic61—42no—
259Dic61—42no—
260Dic61—41no—
261#3131—611no—
262#2961—611no—
263#2932—611no—
264#1995—611no—
265#1963—611no—
266#1938—611no—
267#1938—611no—
268#1937—611no—
269#1937—611no—
270#1837—611no—
271#1765—611no—
272#1765D6111yescharacteristic, complemented
273#1764—611no—
274#1764—611no—
275#1726—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—81no—
279C2×C61C4×#12411yescharacteristic
280C2×C61#295712yes—
281D61—81no—
282C2×#131—611no—
283C2×#131C2×D6111yescharacteristic
284#533—611no—
285#532—611no—
286#488—612no—
287#488—612no—
288#487—612no—
289#486—612no—
290#446—1222no—
291#446—1222no—
292#435—1222no—
293#435—1222no—
294#424—611no—
295#424—611no—
296#423—611no—
297#423—611no—
298C2×#104—611no—
299C2×#104—611no—
300C2×#104—611no—
301C2×#104C2×D6111yescharacteristic
302#416—611no—
303#416—611no—
304#416—611no—
305#416C2×D6111yescharacteristic
306#408—611no—
307#408—611no—
308#405—611no—
309#405C2×D6111yescharacteristic
310C2×#97—611no—
311C2×#97—611no—
312C2×#97C2×D6111yescharacteristic
313C2×C4×C8—611no—
314C2×C4×C8—611no—
315C2×C4×C8—611no—
316#376—611no—
317#341—611no—
318#341—611no—
319#341—611no—
320#341C2×D6111yescharacteristic
321#339—611no—
322#339—611no—
323#339—611no—
324#339C2×D6111yescharacteristic
325C61#2165911yescharacteristic, complemented, Sylow
326C2×#41—611no—
327C2×#41C22×D6111yescharacteristic
328#131—22no—
329#131—22no—
330#131—1222no—
331#131—1222no—
332C2×M16—21no—
333C2×M16—611no—
334C2×M16—611no—
335C2×M16—1221no—
336C2×M16C22×D6111yescharacteristic
337C2×M16C22×D6111yescharacteristic
338C22×C8—21no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—1221no—
342C22×C8C22×D6111yescharacteristic
343C22×C8C22×D6111yescharacteristic
344C4⋊Q8—612no—
345C4⋊Q8—611no—
346#127—611no—
347#125—612no—
348#124—611no—
349#124—611no—
350C22⋊Q8—1222no—
351#121—1222no—
352C4×Q8—1222no—
353C4×D4—1222no—
354#117—611no—
355#117C22×D6111yescharacteristic
356C2×C42—611no—
357C2×C42—611no—
358C2×C42C22×D6111yescharacteristic
359C4⋊C8—1222no—
360C4⋊C8—1222no—
361C4⋊C8—1222no—
362C4⋊C8—1222no—
363#104—22no—
364#104—1222no—
365#104—1222no—
366#104—1222no—
367#103—612no—
368#103—612no—
369#103—612no—
370#103C4×D6112yes—
371#102—612no—
372#102—612no—
373#102—612no—
374#102C4×D6112yes—
375C22⋊C8—1222no—
376C22⋊C8—1222no—
377#97—21no—
378#97—22no—
379#97—612no—
380#97—612no—
381#97—1222no—
382#97—1221no—
383C4×C8—21no—
384C4×C8—612no—
385C4×C8—612no—
386C4×C8—1222no—
387C4×C8—1221no—
388C4×C8—1222no—
389#95—1222no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1221no—
395#41—1221no—
396#41—1221no—
397#41—1221no—
398C2×Q8—611no—
399C2×Q8—1222no—
400C2×Q8C2×C4×D6111yescharacteristic
401C2×D4—21no—
402C2×D4—611no—
403C2×D4—1222no—
404C2×D4—1221no—
405C2×D4C2×C4×D6111yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—612no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4C23×D6111yescharacteristic
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—42no—
417M16—1221no—
418M16—1221no—
419M16—1221no—
420M16—1221no—
421M16—2442no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—612no—
428C2×C8—612no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—2442no—
436C2×C8C2×C4×D6112yes—
437C2×C8#1175822212yes—
438C2×C8D4×D6111yescharacteristic
439C2×C8D4×D6111yescharacteristic
440C2×C8#1175822412yes—
441C4⋊C4—612no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4C2×C4×D6112yes—
447C22⋊C4—21no—
448C22⋊C4—1222no—
449C22⋊C4—1222no—
450C22⋊C4—1221no—
451C42—21no—
452C42—21no—
453C42—611no—
454C42—611no—
455C42—612no—
456C42—612no—
457C42—1221no—
458C42—1221no—
459C42#1175822212yes—
460C23—21no—
461C23—1221no—
462C23#41×D6111yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—1221no—
466Q8—1221no—
467Q8—4882no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—1221no—
473D4—1221no—
474D4—4882no—
475D4—2441no—
476D4—2441no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—612no—
484C2×C4—612no—
485C2×C4—1222no—
486C2×C4—1221no—
487C2×C4—1222no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4C22×C4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822411yescharacteristic
496C2×C4C2×#1175822211yescharacteristic
497C2×C4C2×#1175822211yescharacteristic
498C8—22no—
499C8—41no—
500C8—41no—
501C8—1222no—
502C8—1222no—
503C8—2441no—
504C8—2441no—
505C8#1451976312yes—
506C8#1451976312yes—
507C22—22no—
508C22—42no—
509C22—41no—
510C22—2442no—
511C22—2441no—
512C22C4×D4×D6111yescharacteristic
513C22#1684029011yescharacteristic
514C22#1684029211yescharacteristic
515C4—42no—
516C4—42no—
517C4—41no—
518C4—1222no—
519C4—1222no—
520C4—2442no—
521C4—2442no—
522C4—2441no—
523C4C4×#1175822411yescharacteristic
524C4C4×#1175822211yescharacteristic
525C4C2×#1451976311yescharacteristic
526C4#1684030911yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—4881no—
531C2C4×#1451976311yescharacteristic
532C2#2145473312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1765C2
socleC22×C61C2×C4×D4
prime radical#16840308C22
cosocleC22×C4×C61C24
hypercenter#1765D61
hypocenterC61#21659
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #21454757C2semidirect
4. #21446798C2semidirect
6. #21446532C2semidirect
7. #21446528C2semidirect
8. #21446524C2semidirect
9. #21446520C2semidirect
10. #21445955C2semidirect
11. #21445663C2semidirect
12. #21445658C2semidirect
13. #21445654C2semidirect
14. #21445507C2semidirect
15. C61×#1765C2semidirect
19. #16840118C4semidirect
20. #16840115C4semidirect
21. #16840111C4semidirect
22. #16840107C4semidirect
272. #1765D61semidirect
325. C61#21659semidirect

Maximal subgroups

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

subgroupindex
#214561392
#214547572
#214545352
#214467982
#214466492
#214465322
#214465282
#214465242
#214465202
#214459552
#214456632
#214456582
#214456542
#214455072
C61×#17652
#2165961