Finite Group Databasebeta

← #29683240

Subgroups — #29683240

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×#14519797C211yesmaximal, characteristic
2#21454463C211yesmaximal, characteristic, complemented
3C4×#14519785C211yesmaximal, characteristic, complemented
4#21446470C212yesmaximal, complemented
5#21446421C212yesmaximal, complemented
6#21446143C212yesmaximal
7#21446095C212yesmaximal, complemented
8C4×#14519712C211yesmaximal, characteristic, complemented
9C4×#14519705C211yesmaximal, characteristic, complemented
10#21445565C211yesmaximal, characteristic, complemented
11C4×#104×C61C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519785C2212yes—
14C4×#11758222C2211yescharacteristic
15C4×#11758202C411yescharacteristic, complemented
16C4×#11758202C411yescharacteristic, complemented
17C4×#11758201C411yescharacteristic, complemented
18C4×#11758201C411yescharacteristic, complemented
19C4×#11758200C411yescharacteristic, complemented
20C4×#11758200C411yescharacteristic, complemented
21C4×#11758199C411yescharacteristic, complemented
22C4×#11758199C411yescharacteristic, complemented
23#16839823—22no—
24#16839815—21no—
25#16839815—21no—
26C2×#14519712C2212yes—
27#16839798C2211yescharacteristic
28#16839789C2212yes—
29C2×#14519705C2212yes—
30#16839758C2211yescharacteristic
31#16839756C2211yescharacteristic
32#16839729C2212yes—
33#16839720C2212yes—
34#16839719—22no—
35C4×#11758194C2211yescharacteristic
36C4×#11758189—21no—
37C2×C4×#92418C2211yescharacteristic
38C2×C4×#92418C2211yescharacteristic
39C4×C8×D61—21no—
40#16839526C2212yes—
41#16839522C2212yes—
42#16839517C2212yes—
43#16839517C2212yes—
44C4×#41×C61C2211yescharacteristic
45C2×#104×C61C2212yes—
46C4×M16×C61C2211yescharacteristic
47C43×C61C2211yescharacteristic
48C61×#339C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519797—24no—
51C2×#11758202C2×C411yescharacteristic
52C2×#11758200C2×C411yescharacteristic
53C2×#11758201C2×C411yescharacteristic
54C2×#11758199C2×C411yescharacteristic
55#14519785—24no—
56#14519785—24no—
57C2×#11758194C2312yes—
58C2×#11758189—22no—
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2312yes—
62C2×C8×D61—22no—
63C4×Dic122C2×C411yescharacteristic
64C4×Dic122C2×C411yescharacteristic
65C4×D244C2×C411yescharacteristic
66C4×D244C2×C411yescharacteristic
67C4×C22⋊D61—21no—
68C4×C22⋊D61—21no—
69#14519736C2311yescharacteristic
70#14519734—21no—
71C42×D61—21no—
72#14519712—24no—
73#14519710C2×C411yescharacteristic
74#14519710C2×C411yescharacteristic
75#14519708C2×C411yescharacteristic
76#14519708C2×C411yescharacteristic
77C4⋊#92418—22no—
78C4⋊#92418—22no—
79#14519705—24no—
80#14519701C2×C411yescharacteristic
81#14519701C2×C411yescharacteristic
82#14519700C2×C411yescharacteristic
83#14519700C2×C411yescharacteristic
84#14519698C2×C411yescharacteristic
85#14519698C2×C411yescharacteristic
86#14519697C2×C411yescharacteristic
87#14519697C2×C411yescharacteristic
88C22⋊#92418—22no—
89C22⋊#92418—22no—
90#14519688—21no—
91#14519687—22no—
92C4×#92418—21no—
93C4×#92418C2×C411yescharacteristic
94C4×#92418C2×C411yescharacteristic
95C4×#92418C2×C411yescharacteristic
96C4×#92418C2×C411yescharacteristic
97C4×#92418D411yescharacteristic
98C4×#92418D411yescharacteristic
99C4×#92418D412yes—
100C8×Dic61—22no—
101C2×#41×C61C2311yescharacteristic
102C2×M16×C61C2312yes—
103C4×Q8×C61C2×C411yescharacteristic
104C4×Q8×C61C2×C411yescharacteristic
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61C2×C411yescharacteristic
108C4×D4×C61C2×C411yescharacteristic
109#117×C61—21no—
110#117×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2312yes—
115C2×C42×C61C2311yescharacteristic
116#104×C61—24no—
117#97×C61—21no—
118C4×C8×C61—21no—
119#41×C61—22no—
120#41×C61—22no—
121C2×Q8×C61C22×C411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61C22×C411yescharacteristic
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—21no—
132#11758222—22no—
133#11758222#4114yes—
134C2×D244C22×C411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×Dic122C22×C411yescharacteristic
139M16×C61—22no—
140M16×C61#4114yes—
141C2×C8×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61C22×C411yescharacteristic
145C4⋊C4×C61C22×C411yescharacteristic
146C22⋊C4×C61—21no—
147C22⋊C4×C61—21no—
148C22⋊C4×C61—21no—
149C42×C61—24no—
150C42×C61—21no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61—42no—
156C42×C61C22×C411yescharacteristic
157C42×C61C22×C411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C22⋊Dic61—21no—
161#11758202—22no—
162#11758201—22no—
163#11758200—22no—
164#11758199—22no—
165#11758198—21no—
166#11758198—21no—
167C4⋊Dic61C22×C411yescharacteristic
168C4⋊Dic61C22×C411yescharacteristic
169#11758196—21no—
170#11758196—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174#11758194—22no—
175#11758194—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418C22×C411yescharacteristic
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184C2×#92418C22×C411yescharacteristic
185C2×#92418C2×D412yes—
186C2×#92418C2×D412yes—
187#11758189—28no—
188C8×D61—28no—
189C23×C61—21no—
190C23×C61C2×#4111yescharacteristic
191C22×D61—21no—
192Q8×C61—21no—
193Q8×C61—21no—
194D4×C61—21no—
195D4×C61—21no—
196D4×C61—42no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61C4×D411yescharacteristic
209C2×C4×C61C4×D411yescharacteristic
210C2×C4×C61C4×D411yescharacteristic
211C2×C4×C61C4×D411yescharacteristic
212C2×C4×C61C23×C411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C61C2×#4112yes—
216C22⋊D61—24no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220D244—21no—
221D244#11712yes—
222C4×D61—24no—
223C4×D61—42no—
224C4×D61—42no—
225Dic122—21no—
226Dic122#11712yes—
227C8×C61—24no—
228#92418—21no—
229#92418—21no—
230#92418—21no—
231#92418—21no—
232#92418—22no—
233#92418—22no—
234#92418—42no—
235#21137—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61C2×C4×D411yescharacteristic
242C22×C61#52912yes—
243C2×D61—22no—
244C2×D61—41no—
245C2×D61—41no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—44no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61#44312yes—
253C4×C61#44412yes—
254C4×C61C2×#11711yescharacteristic
255C4×C61C4×#4111yescharacteristic
256C4×C61#52512yes—
257Dic61—22no—
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×#108—611no—
268C4×#104—611no—
269C4×#104D6111yescharacteristic, complemented
270#1744—611no—
271C2×C61—22no—
272C2×C61—81no—
273C2×C61#229011yescharacteristic
274C2×C61C2×#44311yescharacteristic
275C2×C61C2×#44411yescharacteristic
276D61—42no—
277C2×#135—611no—
278C2×#131—612no—
279C4×#41—611no—
280C4×#41C2×D6111yescharacteristic
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×#108—612no—
293#427—611no—
294#424—612no—
295C2×#104—612no—
296C2×#104C2×D6112yes—
297#416—611no—
298#416—611no—
299#408—612no—
300#405—612no—
301#405—1222no—
302C4×M16—611no—
303C4×M16—1221no—
304C4×M16C2×D6111yescharacteristic
305C2×C4×C8—611no—
306C2×C4×C8—611no—
307C2×C4×C8—1221no—
308C43C2×D6111yescharacteristic
309#341—612no—
310#339—612no—
311#339C2×D6112yes—
312#336—612no—
313#336—612no—
314C61#2113711yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41C22×D6111yescharacteristic
317#135—1224no—
318C2×Q16—611no—
319C2×SD16—611no—
320C2×SD16—611no—
321C2×D8—611no—
322#131—1224no—
323#131—1224no—
324C2×M16—612no—
325C2×M16—1222no—
326C2×M16C22×D6112yes—
327C22×C8—612no—
328C22×C8—611no—
329C22×C8—611no—
330C22×C8—1222no—
331C4×Q8—611no—
332C4×Q8—611no—
333C4×Q8C4×D6111yescharacteristic
334C4×Q8C4×D6111yescharacteristic
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—611no—
338C4×D4—611no—
339C4×D4—1221no—
340C4×D4—1221no—
341C4×D4C4×D6111yescharacteristic
342C4×D4C4×D6111yescharacteristic
343#117—21no—
344#117—611no—
345#117—1221no—
346#117C22×D6111yescharacteristic
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—1221no—
351C2×C42C22×D6112yes—
352C2×C42C22×D6111yescharacteristic
353#108—1224no—
354#107—611no—
355#107—611no—
356#106—611no—
357#106—611no—
358C4⋊C8—1222no—
359C4⋊C8—1222no—
360#104—24no—
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—1222no—
374#97—1221no—
375C4×C8—21no—
376C4×C8—611no—
377C4×C8—611no—
378C4×C8—611no—
379C4×C8—611no—
380C4×C8—612no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—1222no—
384C4×C8—1221no—
385#41—22no—
386#41—22no—
387#41—614no—
388#41—1222no—
389C2×Q8—611no—
390C2×Q8C2×C4×D6111yescharacteristic
391C2×D4—21no—
392C2×D4—611no—
393C2×D4—1221no—
394C2×D4C2×C4×D6111yescharacteristic
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4#1175822411yescharacteristic
403C22×C4#1175822411yescharacteristic
404C22×C4C23×D6111yescharacteristic
405Q16—1222no—
406SD16—1222no—
407SD16—1222no—
408D8—1222no—
409M16—22no—
410M16—1228no—
411M16—1222no—
412M16—1222no—
413M16#1175822214yes—
414C2×C8—22no—
415C2×C8—611no—
416C2×C8—611no—
417C2×C8—611no—
418C2×C8—611no—
419C2×C8—612no—
420C2×C8—612no—
421C2×C8—1228no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1222no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—611no—
430C4⋊C4—611no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4C2×C4×D6111yescharacteristic
434C4⋊C4C2×C4×D6111yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—1221no—
439C22⋊C4—1221no—
440C22⋊C4—1221no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—1221no—
449C42—1221no—
450C42—1221no—
451C42C2×C4×D6111yescharacteristic
452C42C2×C4×D6111yescharacteristic
453C42D4×D6111yescharacteristic
454C42D4×D6111yescharacteristic
455C23—21no—
456C23—1221no—
457C23C2×#1175822411yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—612no—
461Q8—1221no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—612no—
466D4—1224no—
467D4—1221no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—1221no—
480C2×C4—1224no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4#1451973711yescharacteristic
486C2×C4#1451973711yescharacteristic
487C2×C4#1451973711yescharacteristic
488C2×C4#1451973711yescharacteristic
489C2×C4C22×C4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4#41×D6112yes—
492C2×C4C2×#1175822212yes—
493C8—24no—
494C8—1221no—
495C8—1221no—
496C8—1221no—
497C8—1221no—
498C8—1222no—
499C8—1222no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—1222no—
507C22—2441no—
508C22—2441no—
509C22C2×#1451973711yescharacteristic
510C22#1684029412yes—
511C4—21no—
512C4—21no—
513C4—44no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—1222no—
518C4—2441no—
519C4—2441no—
520C4#1683987612yes—
521C4#1683988112yes—
522C4#117×D6111yescharacteristic
523C4C4×#1175822211yescharacteristic
524C4#1684027112yes—
525C2—22no—
526C2—81no—
527C2—2442no—
528C2#2144826511yescharacteristic
529C2C2×#1683987611yescharacteristic
530C2C2×#1683988111yescharacteristic
subgroupG/H
centerC42D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#104×C61C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenterC4×#104D61
hypocenterC61#21137
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. #21454463C2semidirect
3. C4×#14519785C2semidirect
4. #21446470C2semidirect
5. #21446421C2semidirect
7. #21446095C2semidirect
8. C4×#14519712C2semidirect
9. C4×#14519705C2semidirect
10. #21445565C2semidirect
11. C4×#104×C61C2semidirect
15. C4×#11758202C4semidirect
16. C4×#11758202C4semidirect
17. C4×#11758201C4semidirect
18. C4×#11758201C4semidirect
19. C4×#11758200C4semidirect
20. C4×#11758200C4semidirect
21. C4×#11758199C4semidirect
22. C4×#11758199C4semidirect
269. C4×#104D61semidirect
314. C61#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#145197972
#214544632
C4×#145197852
#214464702
#214464212
#214461432
#214460952
C4×#145197122
C4×#145197052
#214455652
C4×#104×C612
#2113761