Finite Group Databasebeta

← #29690527

Subgroups — #29690527

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839814C211yesmaximal, characteristic
2#21448223C211yesmaximal, characteristic
3#384×D61C211yesmaximal, characteristic
4#21446583C211yesmaximal, characteristic, complemented
5#21446582C211yesmaximal, characteristic, complemented
6#21446580C211yesmaximal, characteristic, complemented
7#21446013C212yesmaximal
8#21446010C212yesmaximal
9#21445974C212yesmaximal, complemented
10#21445945C212yesmaximal, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C2×#14519734—21no—
13C2×C4⋊C4×D61—21no—
14C2×C4⋊C4×D61—21no—
15C2×C4⋊C4×D61C2212yes—
16C2×C42×D61C2211yescharacteristic
17#16839814C418yescomplemented
18C2×#14519709C2212yes—
19C2×#14519708C2212yes—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519689C2212yes—
23C2×C8×Dic61C2212yes—
24#16839632C2211yescharacteristic
25#16839631—21no—
26#16839631—21no—
27#16839631C2212yes—
28#16839631C2211yescharacteristic
29#16839631C2211yescharacteristic
30#16839626—21no—
31#16839621—21no—
32#16839619C2211yescharacteristic
33#16839619C2211yescharacteristic
34C4×#11758196—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#106×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×C4×D61—22no—
45C22×C4×D61D411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49#14519734—44no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C4⋊C4×D61—42no—
63C4⋊C4×D61—42no—
64C4⋊C4×D61—44no—
65C4⋊C4×D61—44no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61C2311yescharacteristic
72C42×D61C2×C414yes—
73C42×D61Q814yes—
74#14519709—42no—
75#14519708—42no—
76#14519708—42no—
77C4⋊#92418C2×C414yescomplemented
78#14519689C2×C418yes—
79C8×Dic61C2×C418yescomplemented
80#14519684—21no—
81#14519684—21no—
82#14519684—21no—
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#106×C61—42no—
92C4⋊C8×C61C2×C414yescomplemented
93#95×C61—21no—
94C23×D61C2×D411yescharacteristic
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—44no—
114C2×C4×D61—44no—
115C2×C4×D61—44no—
116C2×C4×D61C4⋊C414yes—
117C2×C4×D61C4⋊C414yes—
118C2×C4×D61SD1614yes—
119C2×C4×D61SD1614yes—
120C2×C4×D61C22×C412yes—
121C2×C4×D61C2×Q812yes—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61C22×C414yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—42no—
133C42×C61—42no—
134C42×C61C22×C412yes—
135C42×C61C2×Q812yes—
136#11758198—42no—
137#11758198—42no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—42no—
145C4⋊Dic61—42no—
146#11758196—44no—
147#11758196—44no—
148#11758196—42no—
149#11758196—42no—
150#11758196—44no—
151#11758196—42no—
152#11758196—42no—
153C4×Dic61—42no—
154C4×Dic61C4⋊C414yes—
155C4×Dic61C4⋊C414yes—
156C4×Dic61C22×C412yes—
157C4×Dic61C2×Q812yes—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418C22×C414yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61C2×C4⋊C412yes—
167C22×D61C2×SD1612yes—
168C22×D61C2×SD1612yes—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C2×C4⋊C412yes—
181C2×C4×C61C2×C4⋊C412yes—
182C2×C4×C61C2×SD1612yes—
183C2×C4×C61C2×SD1612yes—
184C2×C4×C61C23×C411yescharacteristic
185C2×C4×C61C22×Q811yescharacteristic
186C2×C4×C61H(5,𝔽2)11yescharacteristic
187C2×C4×C61H(5,𝔽2)11yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61C2×C4⋊C412yes—
202C2×Dic61C2×C4⋊C412yes—
203C2×Dic61C2×C4⋊C412yes—
204C2×Dic61C2×SD1612yes—
205C2×Dic61C2×SD1612yes—
206C2×Dic61AGL(1,ℤ/8)12yes—
207C2×Dic61#13712yes—
208C4×D61—22no—
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—28no—
213C4×D61—88no—
214C4×D61#106116yescomplemented
215C8×C61—24no—
216#92418—24no—
217#21981—611nomaximal, Sylow
218C22×C61C22×C4⋊C411yescharacteristic
219C22×C61#53511yescharacteristic
220C22×C61#53611yescharacteristic
221C22×C61C22×SD1611yescharacteristic
222C22×C61C22×SD1611yescharacteristic
223C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
224C22×C61C2×#13711yescharacteristic
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61C2×#10614yes—
232C4×C61—22no—
233C4×C61—81no—
234C4×C61—82no—
235C4×C61—81no—
236C4×C61C2×#10614yes—
237C4×C61#52011yescharacteristic
238C4×C61#52011yescharacteristic
239C4×C61#54411yescharacteristic
240C4×C61#54411yescharacteristic
241Dic61—22no—
242Dic61—81no—
243Dic61—82no—
244Dic61—81no—
245Dic61C2×#10614yes—
246Dic61#42814yes—
247C2×#431—611no—
248#2285—611no—
249C2×#384—611no—
250#1948—611no—
251#1948—611no—
252#1948—611no—
253#1948D6111yescharacteristic, complemented
254#1853—612no—
255#1853—612no—
256#1844—612no—
257#1834—612no—
258C2×C61#231711yescharacteristic
259C2×C61C22×#10611yescharacteristic
260C2×C61C2×#42811yescharacteristic
261C2×C61#320611yescharacteristic
262C2×C61#320811yescharacteristic
263C2×C61#322711yescharacteristic
264C2×C61#322911yescharacteristic
265D61—24no—
266C2×#117—1221no—
267C22×C4⋊C4—612no—
268C22×C4⋊C4—1221no—
269C22×C4⋊C4—1221no—
270C22×C42—611no—
271#431—618no—
272C2×#106—612no—
273C2×#106—612no—
274C2×#106—612no—
275C2×#106C2×D6112yes—
276C2×C4⋊C8—611no—
277C2×C4⋊C8C2×D6111yescharacteristic
278C2×C22⋊C8—612no—
279C2×C4×C8—612no—
280#384—611no—
281#384—612no—
282#384—611no—
283#384—611no—
284#384—1221no—
285#384—1221no—
286#384C2×D6111yescharacteristic
287#382—1221no—
288#380—611no—
289#380—611no—
290#380—1221no—
291C4×C4⋊C4—611no—
292C4×C4⋊C4—1221no—
293C4×C4⋊C4C2×D6111yescharacteristic
294#340—612no—
295#340—612no—
296#340—612no—
297#340C2×D6112yes—
298C61#2198111yescharacteristic, complemented, Sylow
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—1222no—
303C22×C8—612no—
304C22×C8C22×D6112yes—
305#117—2444no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—2442no—
321C2×C4⋊C4—2442no—
322C2×C4⋊C4—2444no—
323C2×C4⋊C4—2444no—
324C2×C4⋊C4C22×D6112yes—
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C4⋊C4C22×D6111yescharacteristic
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C42—21no—
330C2×C42—611no—
331C2×C42—611no—
332C2×C42—611no—
333C2×C42—614no—
334C2×C42—614no—
335C2×C42—1221no—
336C2×C42C22×D6111yescharacteristic
337#106—42no—
338#106—2442no—
339#106—2442no—
340#106—2442no—
341C4⋊C8—614no—
342C4⋊C8C4×D6114yescomplemented
343C22⋊C8—618no—
344C4×C8—618no—
345#95—21no—
346#95—1221no—
347#95—1221no—
348#95—1221no—
349C24—611no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—614no—
356C22×C4—614no—
357C22×C4—614no—
358C22×C4—611no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—614no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—2444no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4C23×D6111yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—614no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8C2×C4×D6114yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—1221no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—2442no—
398C4⋊C4—2444no—
399C4⋊C4—2444no—
400C4⋊C4—2442no—
401C4⋊C4—2442no—
402C4⋊C4—2442no—
403C4⋊C4—2442no—
404C4⋊C4—2444no—
405C4⋊C4—2442no—
406C22⋊C4—2442no—
407C22⋊C4—2442no—
408C42—42no—
409C42—612no—
410C42—612no—
411C42—614no—
412C42—614no—
413C42—2442no—
414C42C2×C4×D6112yes—
415C42Q8×D6112yes—
416C23—612no—
417C23—612no—
418C23—612no—
419C23—1221no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23C2×D4×D6111yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—612no—
436C2×C4—612no—
437C2×C4—612no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—6116no—
442C2×C4—612no—
443C2×C4—1228no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—4888no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4C4⋊C4×D6112yes—
461C2×C4C4⋊C4×D6112yes—
462C2×C4SD16×D6112yes—
463C2×C4SD16×D6112yes—
464C2×C4C22×C4×D6111yescharacteristic
465C2×C4C2×Q8×D6111yescharacteristic
466C2×C4#1451982211yescharacteristic
467C2×C4#1451982311yescharacteristic
468C2×C4#1451982411yescharacteristic
469C2×C4#1451982511yescharacteristic
470C8—24no—
471C8—1224no—
472C22—614no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22—1222no—
477C22—1222no—
478C22—1222no—
479C22C2×C4⋊C4×D6111yescharacteristic
480C22#1684032811yescharacteristic
481C22#1684033511yescharacteristic
482C22C2×SD16×D6111yescharacteristic
483C22C2×SD16×D6111yescharacteristic
484C22AGL(1,ℤ/8)×D6111yescharacteristic
485C22#137×D6111yescharacteristic
486C4—22no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4—614no—
491C4—614no—
492C4—1222no—
493C4—4881no—
494C4—4881no—
495C4—4882no—
496C4#106×D6114yes—
497C4#1684024111yescharacteristic
498C4#1684024411yescharacteristic
499C4#1684038911yescharacteristic
500C4#1684039211yescharacteristic
501C2—1224no—
502C2#2144844111yescharacteristic
503C2C2×#106×D6111yescharacteristic
504C2#428×D6111yescharacteristic
505C2#2145674311yescharacteristic
506C2#2145675911yescharacteristic
507C2#2145694611yescharacteristic
508C2#2145696211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1948D61
hypocenterC61#21981
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #21446583C2semidirect
5. #21446582C2semidirect
6. #21446580C2semidirect
9. #21445974C2semidirect
10. #21445945C2semidirect
11. C61×#1948C2semidirect
17. #16839814C4semidirect
77. C4⋊#92418C2×C4semidirect
79. C8×Dic61C2×C4semidirect
92. C4⋊C8×C61C2×C4semidirect
214. C4×D61#106semidirect
253. #1948D61semidirect
298. C61#21981semidirect
342. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
#214482232
#384×D612
#214465832
#214465822
#214465802
#214460132
#214460102
#214459742
#214459452
C61×#19482
#2198161