Finite Group Databasebeta

← #29686269

Subgroups — #29686269

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839968C211yesmaximal, characteristic
2C2×#16839861C212yesmaximal
3#21448386C211yesmaximal, characteristic, complemented
4C4×#14519739C211yesmaximal, characteristic
5#21446742C211yesmaximal, characteristic, complemented
6#21446459C212yesmaximal, complemented
7#21446154C212yesmaximal
8#21445816C211yesmaximal, characteristic
9#21445674C211yesmaximal, characteristic, complemented
10#21445648C212yesmaximal, complemented
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519742C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×#14519739C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×C42×D61—21no—
19#16839968C418yescomplemented
20#16839861—28no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519702C2212yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25C2×#14519697C2212yes—
26C2×#14519687C2212yes—
27#16839667C2212yes—
28#16839658—21no—
29#16839658—21no—
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61C2211yescharacteristic
33C4⋊C4×Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839609—21no—
36#16839609—21no—
37C42×Dic61C2211yescharacteristic
38#16839524C2212yes—
39#16839516C2212yes—
40C2×#107×C61C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122C2312yes—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C2×#11758190—24no—
51C2×#11758190—24no—
52C2×#11758190C2×C418yes—
53C22×#92418C2312yes—
54#14519777—22no—
55#14519777C2×C414yescomplemented
56#14519742—24no—
57#14519742—24no—
58C4×D244—22no—
59C4×D244C2×C414yes—
60#14519739—22no—
61#14519739—22no—
62#14519739—22no—
63#14519739D414yes—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—22no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—44no—
81C4⋊#92418—22no—
82#14519702C2×C418yescomplemented
83#14519700—24no—
84#14519700—24no—
85#14519697—24no—
86#14519687—24no—
87#14519684—21no—
88#14519684—21no—
89#14519684—22no—
90#14519684—22no—
91C22×C8×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100C4⋊C8×C61C2×C414yescomplemented
101#95×C61—21no—
102C23×D61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61C2×D411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244C22×C412yes—
119C2×D244C2×D412yes—
120C2×D244#4112yes—
121C2×C4×D61—24no—
122C2×C4×D61—24no—
123C2×C4×D61—24no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122C22×C414yes—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61C22×C414yes—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61#4112yes—
147C42×C61—42no—
148C42×C61C22×C412yes—
149C42×C61#4112yes—
150#11758198—24no—
151#11758198—24no—
152#11758198—24no—
153C4⋊Dic61—24no—
154C4⋊Dic61—24no—
155C4⋊Dic61—21no—
156C4⋊Dic61C22×C412yes—
157#11758196—44no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—22no—
163C4×Dic61—41no—
164C4×Dic61—42no—
165C4×Dic61—42no—
166C4×Dic61—44no—
167C4×Dic61C2×D412yes—
168C4×Dic61#4112yes—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172#11758190—216no—
173C23×C61C22×D411yescharacteristic
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61C4×D412yes—
193C2×C4×C61C4×D412yes—
194C2×C4×C61AGL(1,ℤ/8)12yes—
195C2×C4×C61#13712yes—
196C2×C4×C61C23×C411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—24no—
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61#12112yes—
218C2×Dic61#12112yes—
219C2×Dic61#13712yes—
220D244—22no—
221D244—22no—
222D244—22no—
223D244C4×D414yes—
224C4×D61—48no—
225C4×D61—48no—
226C4×D61—48no—
227Dic122—24no—
228Dic122—24no—
229Dic122—42no—
230Dic122—42no—
231C8×C61—24no—
232#92418—42no—
233#21504—611nomaximal, Sylow
234C22×C61C2×C4×D411yescharacteristic
235C22×C61C2×#12111yescharacteristic
236C22×C61#52511yescharacteristic
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C61C2×#13711yescharacteristic
239C22×C61C2×#13711yescharacteristic
240C22×C61#57511yescharacteristic
241C2×D61—24no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C4×C61—22no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—82no—
252C4×C61#44014yescomplemented
253C4×C61C2×C4×D411yescharacteristic
254C4×C61C4×#4111yescharacteristic
255C4×C61#52911yescharacteristic
256C4×C61#53211yescharacteristic
257Dic61—22no—
258Dic61—22no—
259Dic61—22no—
260Dic61—41no—
261Dic61—42no—
262Dic61—44no—
263C2×#461—611no—
264C2×#440—612no—
265C4×C4⋊Q8—611no—
266C42×D4—611no—
267#1981—611no—
268#1947D6111yescharacteristic, complemented
269#1928—612no—
270#1877—612no—
271#1802—611no—
272#1767—611no—
273#1763—612no—
274C2×C61C4×#12111yescharacteristic
275C2×C61C2×#44011yescharacteristic
276C2×C61C4×#13711yescharacteristic
277C2×C61#303611yescharacteristic
278C2×C61#303911yescharacteristic
279C2×C61#311011yescharacteristic
280C2×C61#311611yescharacteristic
281D61—44no—
282C22×SD16—612no—
283C2×C4⋊Q8—611no—
284C2×C4×Q8—612no—
285C2×C4×D4—611no—
286C2×C4×D4—611no—
287C2×C4×D4—611no—
288C22×C42—1221no—
289#461—618no—
290#440—1228no—
291C2×#107C2×D6112yes—
292C2×C4⋊C8—611no—
293C2×C4⋊C8C2×D6111yescharacteristic
294C2×#103—612no—
295C2×#103—612no—
296C2×#103—612no—
297C2×#102—612no—
298C2×#97—612no—
299#391—612no—
300#389—1221no—
301#389—1221no—
302#384C2×D6111yescharacteristic
303C4×C4⋊C4—611no—
304C4×C4⋊C4—611no—
305C4×C4⋊C4—1221no—
306C4×C4⋊C4—1221no—
307C4×C4⋊C4C2×D6111yescharacteristic
308C4×C22⋊C4—1221no—
309C4×C22⋊C4—1221no—
310C4×C22⋊C4—1221no—
311C43—611no—
312#340—612no—
313#340C2×D6112yes—
314#336—612no—
315C61#2150411yescharacteristic, complemented, Sylow
316C22×Q8—612no—
317C22×D4—611no—
318C23×C4—1221no—
319C23×C4—1221no—
320C23×C4—1221no—
321C2×SD16—618no—
322C2×SD16—1224no—
323C2×SD16—1224no—
324C22×C8—612no—
325C22×C8C22×D6112yes—
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328C4×Q8—1224no—
329C4×Q8—1224no—
330C4×D4—614no—
331C4×D4—614no—
332C4×D4—1222no—
333C4×D4—1222no—
334C4×D4—1222no—
335C4×D4—1222no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4C22×D6112yes—
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C42—21no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—1222no—
356C2×C42—1221no—
357C2×C42—1221no—
358C2×C42—2444no—
359C2×C42C22×D6111yescharacteristic
360#107—24no—
361C4⋊C8—1222no—
362C4⋊C8C4×D6114yescomplemented
363#103—618no—
364#103—1224no—
365#103—1224no—
366#102—1224no—
367#97—1224no—
368#95—21no—
369#95—1221no—
370#95—1221no—
371#95—1222no—
372#95—1222no—
373C24—1221no—
374C2×Q8—614no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×D4—612no—
381C2×D4—612no—
382C2×D4—612no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—1224no—
394C22×C4—1224no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—1224no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407SD16—12216no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8C2×C4×D6114yes—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—612no—
421C4⋊C4—1221no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—2444no—
425C4⋊C4D4×D6112yes—
426C4⋊C4#1175822612yes—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C42—42no—
431C42—612no—
432C42—612no—
433C42—1222no—
434C42—1222no—
435C42—1221no—
436C42—1221no—
437C42—1222no—
438C42—2441no—
439C42—2444no—
440C42—2442no—
441C42—2442no—
442C42C2×C4×D6112yes—
443C42#1175822612yes—
444C23—1222no—
445C23—1222no—
446C23—1222no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23C2×D4×D6111yescharacteristic
452Q8—1224no—
453Q8—1224no—
454Q8—2442no—
455Q8—2442no—
456D4—614no—
457D4—1222no—
458D4—1222no—
459D4—1222no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1224no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—2448no—
486C2×C4—2448no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2448no—
492C2×C4#1451973912yes—
493C2×C4#1451973912yes—
494C2×C4#1451980112yes—
495C2×C4#1451980912yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#41×D6111yescharacteristic
499C2×C4#41×D6111yescharacteristic
500C2×C4C2×#1175822611yescharacteristic
501C2×C4C2×#1175822611yescharacteristic
502C8—24no—
503C8—2442no—
504C22—1224no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22C2×#1451973911yescharacteristic
512C22#121×D6111yescharacteristic
513C22#1684027511yescharacteristic
514C22C2×#1451980111yescharacteristic
515C22#137×D6111yescharacteristic
516C22C2×#1451980911yescharacteristic
517C22#1684055811yescharacteristic
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—1222no—
523C4—1222no—
524C4—1222no—
525C4—2442no—
526C4—2441no—
527C4—2444no—
528C4#1683986114yescomplemented
529C4C4×D4×D6111yescharacteristic
530C4C4×#1175822611yescharacteristic
531C4#1684029711yescharacteristic
532C4#1684031311yescharacteristic
533C2—2444no—
534C2#2144832711yescharacteristic
535C2C2×#1683986111yescharacteristic
536C2#2145463711yescharacteristic
537C2#2145527811yescharacteristic
538C2#2145530311yescharacteristic
539C2#2145592011yescharacteristic
540C2#2145598411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1947D61
hypocenterC61#21504
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. #21448386C2semidirect
5. #21446742C2semidirect
6. #21446459C2semidirect
9. #21445674C2semidirect
10. #21445648C2semidirect
11. C61×#1947C2semidirect
19. #16839968C4semidirect
55. #14519777C2×C4semidirect
82. #14519702C2×C4semidirect
100. C4⋊C8×C61C2×C4semidirect
252. C4×C61#440semidirect
268. #1947D61semidirect
315. C61#21504semidirect
362. C4⋊C8C4×D61semidirect
528. C4#16839861semidirect

Maximal subgroups

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

subgroupindex
C2×#168399682
C2×#168398612
#214483862
C4×#145197392
#214467422
#214464592
#214461542
#214458162
#214456742
#214456482
C61×#19472
#2150461