Finite Group Databasebeta

← #29683235

Subgroups — #29683235

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519799C211yesmaximal, characteristic
2#21454464C211yesmaximal, characteristic, complemented
3C4×#14519785C211yesmaximal, characteristic, complemented
4#21446466C212yesmaximal, complemented
5#21446417C212yesmaximal, complemented
6#21446147C212yesmaximal
7#21446091C212yesmaximal, complemented
8C4×#14519703C211yesmaximal, characteristic, complemented
9C4×#14519703C211yesmaximal, characteristic, complemented
10#21445564C211yesmaximal, characteristic, complemented
11C4×#108×C61C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519785C2212yes—
14C4×#11758222C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16C4×Dic244C411yescharacteristic, complemented
17C4×Dic244C411yescharacteristic, complemented
18C4×#11758190C411yescharacteristic, complemented
19C4×#11758190C411yescharacteristic, complemented
20C4×#11758190C411yescharacteristic, complemented
21C4×#11758190C411yescharacteristic, complemented
22C4×D488C411yescharacteristic, complemented
23C4×D488C411yescharacteristic, complemented
24#16839824—22no—
25#16839813—21no—
26#16839813—21no—
27#16839800C2211yescharacteristic
28#16839790C2212yes—
29C2×#14519703C2212yes—
30C2×#14519703C2212yes—
31#16839759C2211yescharacteristic
32#16839759C2211yescharacteristic
33#16839730C2212yes—
34#16839720C2212yes—
35#16839719—22no—
36C4×#11758189—21no—
37C2×C4×#92418C2211yescharacteristic
38C4×C8×D61—21no—
39C42×Dic61C2211yescharacteristic
40#16839523C2212yes—
41#16839523C2212yes—
42#16839516C2212yes—
43#16839516C2212yes—
44C2×#108×C61C2212yes—
45C4×M16×C61C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C2×C4×C8×C61C2211yescharacteristic
48C61×#341C2212yes—
49C2×#11758222C2311yescharacteristic
50C2×#11758222C2311yescharacteristic
51#14519799—24no—
52C2×Dic244C2×C411yescharacteristic
53C2×#11758190C2×C411yescharacteristic
54C2×#11758190C2×C411yescharacteristic
55C2×D488C2×C411yescharacteristic
56#14519785—24no—
57#14519785—24no—
58C2×#11758189—22no—
59C22×#92418C2312yes—
60C2×C8×D61—22no—
61C4×Dic122C2×C411yescharacteristic
62C4×Dic122C2×C411yescharacteristic
63C4×Dic122C2×C411yescharacteristic
64C4×Dic122C2×C411yescharacteristic
65C4×D244C2×C411yescharacteristic
66C4×D244C2×C411yescharacteristic
67C4×D244C2×C411yescharacteristic
68C4×D244C2×C411yescharacteristic
69C4×C22⋊D61—21no—
70C4×C22⋊D61—21no—
71C4×C22⋊D61—21no—
72C4×C22⋊D61—21no—
73#14519736C2311yescharacteristic
74#14519736C2311yescharacteristic
75#14519734—21no—
76#14519734—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80C42×D61—21no—
81C42×D61—21no—
82#14519711C2×C411yescharacteristic
83#14519711C2×C411yescharacteristic
84#14519709C2×C411yescharacteristic
85#14519709C2×C411yescharacteristic
86#14519706—22no—
87#14519706—22no—
88#14519703—24no—
89#14519703—24no—
90#14519702C2×C411yescharacteristic
91#14519702C2×C411yescharacteristic
92#14519702C2×C411yescharacteristic
93#14519702C2×C411yescharacteristic
94#14519699C2×C411yescharacteristic
95#14519699C2×C411yescharacteristic
96#14519699C2×C411yescharacteristic
97#14519699C2×C411yescharacteristic
98#14519689—22no—
99#14519689—22no—
100#14519687—22no—
101C4×#92418D411yescharacteristic
102C4×#92418D411yescharacteristic
103C4×#92418D412yes—
104C8×Dic61—22no—
105C2×M16×C61C2312yes—
106C2×M16×C61C2312yes—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110#108×C61—24no—
111#97×C61—21no—
112#97×C61—21no—
113C4×C8×C61—21no—
114C4×C8×C61—21no—
115C4×C8×C61C2×C411yescharacteristic
116C4×C8×C61C2×C411yescharacteristic
117C4×C8×C61C2×C411yescharacteristic
118C4×C8×C61C2×C411yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—21no—
123C2×C22⋊D61—21no—
124C22×Dic61—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222#4114yes—
129C2×D244C22×C411yescharacteristic
130C2×D244C22×C411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×Dic122C22×C411yescharacteristic
138C2×Dic122C22×C411yescharacteristic
139M16×C61—22no—
140M16×C61—22no—
141M16×C61—22no—
142M16×C61#4114yes—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61C22×C411yescharacteristic
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C22×C411yescharacteristic
150C2×C8×C61C22×C411yescharacteristic
151C42×C61C22×C411yescharacteristic
152C42×C61C22×C411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C22⋊Dic61—21no—
156C22⋊Dic61—21no—
157#11758198—21no—
158#11758198—21no—
159#11758198—21no—
160#11758198—21no—
161C4⋊Dic61C22×C411yescharacteristic
162C4⋊Dic61C22×C411yescharacteristic
163C4⋊Dic61C22×C411yescharacteristic
164C4⋊Dic61C22×C411yescharacteristic
165#11758196—21no—
166#11758196—21no—
167#11758196—21no—
168#11758196—21no—
169C4×Dic61—24no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418C2×D412yes—
182C2×#92418C2×D412yes—
183Dic244—22no—
184D488—22no—
185#11758190—22no—
186#11758190—22no—
187#11758189—28no—
188C8×D61—28no—
189C23×C61C2×#4111yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61C4×D411yescharacteristic
195C2×C4×C61C4×D411yescharacteristic
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61C4×D411yescharacteristic
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4112yes—
201C2×C4×C61C2×#4112yes—
202C22⋊D61—24no—
203C22⋊D61—42no—
204C2×Dic61—21no—
205C2×Dic61—24no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213D244—21no—
214D244—21no—
215D244—21no—
216D244#11712yes—
217C4×D61—24no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—42no—
221C4×D61—42no—
222C4×D61—42no—
223Dic122—21no—
224Dic122—21no—
225Dic122—21no—
226Dic122#11712yes—
227C8×C61—21no—
228C8×C61—21no—
229C8×C61—21no—
230C8×C61—21no—
231C8×C61—24no—
232C8×C61—42no—
233#92418—22no—
234#92418—22no—
235#21137—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61C2×C4×D411yescharacteristic
239C22×C61#52912yes—
240C2×D61—22no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—41no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61#44312yes—
249C4×C61#44412yes—
250C4×C61C2×#11711yescharacteristic
251C4×C61C4×#4111yescharacteristic
252C4×C61#52512yes—
253Dic61—22no—
254Dic61—44no—
255Dic61—41no—
256Dic61—41no—
257Dic61—41no—
258Dic61—41no—
259Dic61—41no—
260C4×#135—611no—
261#2922—611no—
262C4×#131—611no—
263#1929—612no—
264#1922—612no—
265#1876—612no—
266#1866—612no—
267C4×#108D6111yescharacteristic, complemented
268C4×#104—611no—
269C4×#104—611no—
270#1744—611no—
271C2×C61—22no—
272C2×C61#229011yescharacteristic
273C2×C61C2×#44311yescharacteristic
274C2×C61C2×#44411yescharacteristic
275D61—42no—
276D61—81no—
277C2×#135—611no—
278C2×#131—612no—
279C4×#41—611no—
280C4×#41—611no—
281C4×Q16—611no—
282C4×Q16—611no—
283C4×SD16—611no—
284C4×SD16—611no—
285C4×SD16—611no—
286C4×SD16—611no—
287C4×D8—611no—
288C4×D8—611no—
289#433—1222no—
290#431—1221no—
291#431—1221no—
292C2×#108C2×D6112yes—
293#427—611no—
294#424—612no—
295C2×#104—612no—
296C2×#104—612no—
297#416—611no—
298#416—611no—
299#408—612no—
300#405—612no—
301#405—1222no—
302C4×M16—1221no—
303C4×M16C2×D6111yescharacteristic
304C4×M16C2×D6111yescharacteristic
305C2×C4×C8—611no—
306C2×C4×C8—1221no—
307C2×C4×C8C2×D6111yescharacteristic
308C43—611no—
309#341C2×D6112yes—
310#339—612no—
311#339—612no—
312#336—612no—
313#336—612no—
314C61#2113711yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41—611no—
317#135—1224no—
318C2×Q16—611no—
319C2×SD16—611no—
320C2×SD16—611no—
321C2×D8—611no—
322#131—1224no—
323#131—1224no—
324C2×M16—1222no—
325C2×M16C22×D6112yes—
326C2×M16C22×D6112yes—
327C22×C8—612no—
328C22×C8—1222no—
329C22×C8C22×D6111yescharacteristic
330C22×C8C22×D6111yescharacteristic
331C4×Q8—611no—
332C4×Q8—611no—
333C4×Q8—611no—
334C4×Q8—611no—
335C4×D4—611no—
336C4×D4—611no—
337C4×D4—611no—
338C4×D4—611no—
339C4×D4—1221no—
340C4×D4—1221no—
341C4×D4—1221no—
342C4×D4—1221no—
343#117—611no—
344#117—611no—
345#117—1221no—
346#117—1221no—
347C2×C42—612no—
348C2×C42—1221no—
349C2×C42—1221no—
350C2×C42—1221no—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353#108—24no—
354#107—611no—
355#107—611no—
356#106—611no—
357#106—611no—
358C4⋊C8—1222no—
359C4⋊C8—1222no—
360#104—1224no—
361#104—1224no—
362#103—611no—
363#103—611no—
364#103—611no—
365#103—611no—
366#102—611no—
367#102—611no—
368#102—611no—
369#102—611no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372#97—21no—
373#97—21no—
374#97—1222no—
375C4×C8—21no—
376C4×C8—21no—
377C4×C8—612no—
378C4×C8—611no—
379C4×C8—611no—
380C4×C8—1222no—
381C4×C8C4×D6111yescharacteristic
382C4×C8C4×D6111yescharacteristic
383C4×C8C4×D6111yescharacteristic
384C4×C8C4×D6111yescharacteristic
385#41—614no—
386#41—1222no—
387#41—1222no—
388#41—1222no—
389C2×Q8—611no—
390C2×Q8—611no—
391C2×D4—611no—
392C2×D4—611no—
393C2×D4—1221no—
394C2×D4—1221no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4#1175822611yescharacteristic
403C22×C4#1175822611yescharacteristic
404C22×C4C23×D6111yescharacteristic
405Q16—1222no—
406SD16—1222no—
407SD16—1222no—
408D8—1222no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—1228no—
413M16#1175822214yes—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1228no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8C2×C4×D6111yescharacteristic
424C2×C8C2×C4×D6111yescharacteristic
425C2×C8C2×C4×D6111yescharacteristic
426C2×C8C2×C4×D6111yescharacteristic
427C4⋊C4—611no—
428C4⋊C4—611no—
429C4⋊C4—611no—
430C4⋊C4—611no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C22⋊C4—1221no—
436C22⋊C4—1221no—
437C22⋊C4—1221no—
438C22⋊C4—1221no—
439C22⋊C4—1221no—
440C22⋊C4—1221no—
441C42—1224no—
442C42—1221no—
443C42—1221no—
444C42—1221no—
445C42—1221no—
446C42—1221no—
447C42—1221no—
448C42—1222no—
449C42—1222no—
450C42—2442no—
451C42C2×C4×D6111yescharacteristic
452C42C2×C4×D6111yescharacteristic
453C42D4×D6111yescharacteristic
454C42D4×D6111yescharacteristic
455C23—1221no—
456C23—1221no—
457C23C2×#1175822611yescharacteristic
458Q8—612no—
459Q8—1221no—
460Q8—1221no—
461Q8—1221no—
462D4—612no—
463D4—1224no—
464D4—1221no—
465D4—1221no—
466D4—1221no—
467D4—2442no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—1221no—
471C2×C4—1224no—
472C2×C4—1224no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4#1451973911yescharacteristic
486C2×C4#1451973911yescharacteristic
487C2×C4#1451973911yescharacteristic
488C2×C4#1451973911yescharacteristic
489C2×C4C22×C4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4C2×#1175822212yes—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—24no—
498C8—42no—
499C8—1222no—
500C8—1222no—
501C22—21no—
502C22—21no—
503C22—1222no—
504C22—2441no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22—2441no—
509C22C2×#1451973911yescharacteristic
510C22#1684029912yes—
511C4—21no—
512C4—21no—
513C4—1222no—
514C4—2444no—
515C4—2441no—
516C4—2441no—
517C4—2441no—
518C4—2441no—
519C4—2441no—
520C4#1683987812yes—
521C4#1683988312yes—
522C4#117×D6111yescharacteristic
523C4C4×#1175822211yescharacteristic
524C4#1684027112yes—
525C2—22no—
526C2—4881no—
527C2—2442no—
528C2#2144826011yescharacteristic
529C2C2×#1683987811yescharacteristic
530C2C2×#1683988311yescharacteristic
subgroupG/H
centerC42D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#108×C61C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenterC4×#108D61
hypocenterC61#21137
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#108

Complements

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

NH ≅ G/Ntype
2. #21454464C2semidirect
3. C4×#14519785C2semidirect
4. #21446466C2semidirect
5. #21446417C2semidirect
7. #21446091C2semidirect
8. C4×#14519703C2semidirect
9. C4×#14519703C2semidirect
10. #21445564C2semidirect
11. C4×#108×C61C2semidirect
16. C4×Dic244C4semidirect
17. C4×Dic244C4semidirect
18. C4×#11758190C4semidirect
19. C4×#11758190C4semidirect
20. C4×#11758190C4semidirect
21. C4×#11758190C4semidirect
22. C4×D488C4semidirect
23. C4×D488C4semidirect
267. C4×#108D61semidirect
314. C61#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214544642
C4×#145197852
#214464662
#214464172
#214461472
#214460912
C4×#145197032
#214455642
C4×#108×C612
#2113761