Finite Group Databasebeta

← #29696287

Subgroups — #29696287

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2C2×#16839827C211yesmaximal, characteristic
3#21448770C211yesmaximal, characteristic
4#21446779C211yesmaximal, characteristic, complemented
5#21446770C211yesmaximal, characteristic, complemented
6#21446239C212yesmaximal, complemented
7#21446128C212yesmaximal, complemented
8#21445913C212yesmaximal, complemented
9C4×#14519701C211yesmaximal, characteristic, complemented
10#21445534C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840485—24no—
14#16840485C418yescomplemented
15C2×#14519777C2211yescharacteristic
16C2×#14519772—22no—
17C2×#14519765—21no—
18C2×#14519753C2212yes—
19C2×C4×Dic122—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×#14519735C2212yes—
23#16839827—24no—
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519689C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839663—21no—
29#16839652—22no—
30#16839619C2212yes—
31C4⋊C4×Dic61—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122—21no—
45C22×D244D411yescharacteristic
46C22×C4×D61C2312yes—
47C22×#92418C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49C2×C8×D61—24no—
50C2×C8×D61—24no—
51C2×C8×D61—42no—
52#14519777—22no—
53#14519777C2×C414yes—
54#14519772—216no—
55#14519765—28no—
56#14519753—24no—
57#14519753C2×C418yes—
58C4×Dic122—28no—
59C4×D244C2×C414yes—
60C4×D244D414yes—
61#14519735—24no—
62#14519735—24no—
63#14519735—42no—
64#14519735C2×C418yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—22no—
79#14519701—22no—
80#14519701—22no—
81#14519689—24no—
82C4×#92418—22no—
83#14519684—21no—
84#14519684—22no—
85C22×Q8×C61C2311yescharacteristic
86C22×C8×C61C2312yes—
87C4×Q8×C61—22no—
88C4×Q8×C61C2×C414yes—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97#103×C61—42no—
98C23×D61C2×D412yes—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61C22×C412yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D412yes—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244C22⋊C414yes—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61C22×C414yes—
123C2×C4×D61C2×D414yes—
124C2×Dic122—24no—
125C2×Dic122—42no—
126C2×Dic122—42no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—81no—
136C4⋊C4×C61C22×C412yes—
137C42×C61—24no—
138C42×C61C22×C412yes—
139C42×C61C2×D412yes—
140#11758198—28no—
141#11758198—28no—
142#11758198—44no—
143#11758198—82no—
144#11758198C22⋊C418yes—
145C4⋊Dic61—24no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C22⋊C414yes—
151C4⋊Dic61C22×C414yes—
152C4⋊Dic61#4112yes—
153#11758196—28no—
154C4×Dic61—28no—
155C4×Dic61—44no—
156C4×Dic61—42no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418#4112yes—
163C8×D61—48no—
164C23×C61C22×D411yescharacteristic
165C22×D61—22no—
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61C2×C22⋊C414yes—
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—41no—
175Q8×C61—41no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61C2×C22⋊C412yes—
188C2×C4×C61C2×C22⋊C412yes—
189C2×C4×C61#13512yes—
190C2×C4×C61#13512yes—
191C2×C4×C61C23×C411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—24no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61C2×C22⋊C414yes—
209D244—28no—
210C4×D61—24no—
211C4×D61—24no—
212C4×D61—24no—
213C4×D61—24no—
214Dic122—48no—
215C8×C61—42no—
216#92418—22no—
217#92418—41no—
218#22490—611nomaximal, Sylow
219C22×C61C22×C22⋊C411yescharacteristic
220C22×C61#54611yescharacteristic
221C22×C61#54711yescharacteristic
222C22×C61C2×#13511yescharacteristic
223C22×C61C2×#13511yescharacteristic
224C22×C61#57711yescharacteristic
225C22×C61#57811yescharacteristic
226C2×D61—24no—
227C2×D61—24no—
228C2×D61—24no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61—42no—
233C4×C61—22no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—82no—
237C4×C61#41614yes—
238C4×C61C4×#4111yescharacteristic
239C4×C61#52011yescharacteristic
240C4×C61#54611yescharacteristic
241C4×C61#54711yescharacteristic
242Dic61—24no—
243Dic61—44no—
244Dic61—42no—
245Dic61—81no—
246C2×#556—611no—
247C2×C8×D4—611no—
248#2354—611no—
249#1990—611no—
250#1988—611no—
251#1892—612no—
252#1873—612no—
253#1825—612no—
254#1803D6111yescharacteristic, complemented
255C4×#103—611no—
256#1735—612no—
257C2×C61#233911yescharacteristic
258C2×C61C2×#41611yescharacteristic
259C2×C61#289811yescharacteristic
260C2×C61#330211yescharacteristic
261C2×C61#330411yescharacteristic
262C2×C61#334011yescharacteristic
263C2×C61#334211yescharacteristic
264D61—44no—
265C23×C8—612no—
266#556—618no—
267#556—1224no—
268C2×C4⋊Q8—611no—
269C2×#126—1222no—
270C2×#124—1221no—
271C2×C22⋊Q8—612no—
272C2×C4×Q8—1221no—
273C2×C4×Q8C2×D6111yescharacteristic
274C2×C4×D4—611no—
275C2×#117—612no—
276C2×#117—612no—
277C8×D4—1224no—
278C2×C4⋊C8C2×D6111yescharacteristic
279C2×#103—611no—
280C2×#103—611no—
281C2×#103C2×D6112yes—
282C2×C22⋊C8—612no—
283C2×C4×C8—611no—
284#390—1221no—
285#388—1222no—
286#384C2×D6111yescharacteristic
287#380—612no—
288C4×C4⋊C4—611no—
289C4×C4⋊C4—1221no—
290#340—612no—
291#340—612no—
292#340—611no—
293#340—612no—
294#340—612no—
295#340—611no—
296#340C2×D6112yes—
297C61#2249011yescharacteristic, complemented, Sylow
298C22×Q8—1221no—
299C22×Q8C22×D6111yescharacteristic
300C22×D4—611no—
301C23×C4—612no—
302C22×C8—611no—
303C22×C8—611no—
304C22×C8—1224no—
305C22×C8—1224no—
306C22×C8—2442no—
307C22×C8C22×D6112yes—
308C4⋊Q8—614no—
309C4⋊Q8—1222no—
310#126—12216no—
311#124—1228no—
312C22⋊Q8—618no—
313C22⋊Q8—1224no—
314C4×Q8—22no—
315C4×Q8—1228no—
316C4×Q8C4×D6114yes—
317C4×D4—614no—
318C4×D4—614no—
319#117—618no—
320#117—1224no—
321#117—1224no—
322#117—2442no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6112yes—
333C2×C4⋊C4C22×D6111yescharacteristic
334C2×C22⋊C4—612no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C42—21no—
340C2×C42—1222no—
341C2×C42—1222no—
342C2×C42—1221no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—22no—
345#103—42no—
346#103—1222no—
347#103—1222no—
348C22⋊C8—1224no—
349C4×C8—1222no—
350#95—1222no—
351#95—1221no—
352C24—612no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—1224no—
359C2×Q8—2442no—
360C2×Q8—2442no—
361C2×Q8C2×C4×D6112yes—
362C2×D4—614no—
363C2×D4—1222no—
364C2×D4—1222no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—612no—
369C22×C4—614no—
370C22×C4—614no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4—1221no—
378C22×C4D4×D6111yescharacteristic
379C22×C4D4×D6111yescharacteristic
380C22×C4C23×D6111yescharacteristic
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—612no—
385C2×C8—1221no—
386C2×C8—1221no—
387C2×C8—1221no—
388C2×C8—1221no—
389C2×C8—1221no—
390C2×C8—2448no—
391C4⋊C4—21no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4—614no—
398C4⋊C4—614no—
399C4⋊C4—612no—
400C4⋊C4—1228no—
401C4⋊C4—1222no—
402C4⋊C4—1221no—
403C4⋊C4—1224no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4C2×C4×D6112yes—
407C22⋊C4—618no—
408C22⋊C4—1228no—
409C22⋊C4—1228no—
410C22⋊C4—4882no—
411C22⋊C4—2444no—
412C42—24no—
413C42—1228no—
414C42—2444no—
415C42—2442no—
416C42C2×C4×D6112yes—
417C42D4×D6112yes—
418C23—614no—
419C23—1222no—
420C23—1222no—
421C23—1222no—
422C23—1222no—
423C23—1222no—
424C23—1222no—
425C23C2×D4×D6111yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430Q8—2448no—
431D4—1228no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—614no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1224no—
447C2×C4—1224no—
448C2×C4—1224no—
449C2×C4—1224no—
450C2×C4—1224no—
451C2×C4—1222no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4C22⋊C4×D6112yes—
460C2×C4C22⋊C4×D6112yes—
461C2×C4#1451979812yes—
462C2×C4#1451980012yes—
463C2×C4C22×C4×D6111yescharacteristic
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4#41×D6111yescharacteristic
466C2×C4#41×D6111yescharacteristic
467C2×C4#1451982211yescharacteristic
468C2×C4#1451982511yescharacteristic
469C8—42no—
470C8—1222no—
471C8—2441no—
472C22—1224no—
473C22—1224no—
474C22—1224no—
475C22—2442no—
476C22—2442no—
477C22—2442no—
478C22—2442no—
479C22C2×C22⋊C4×D6111yescharacteristic
480C22#1684040911yescharacteristic
481C22#1684041411yescharacteristic
482C22C2×#1451979811yescharacteristic
483C22C2×#1451980011yescharacteristic
484C22#1684057211yescharacteristic
485C22#1684057811yescharacteristic
486C4—22no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—1224no—
491C4—4881no—
492C4—2442no—
493C4—2444no—
494C4#1683975714yes—
495C4#1684023011yescharacteristic
496C4#1684024311yescharacteristic
497C4#1684040911yescharacteristic
498C4#1684041411yescharacteristic
499C2—2444no—
500C2#2144864711yescharacteristic
501C2C2×#1683975711yescharacteristic
502C2#2145435011yescharacteristic
503C2#2145773911yescharacteristic
504C2#2145776111yescharacteristic
505C2#2145823511yescharacteristic
506C2#2145824811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22490
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21446779C2semidirect
5. #21446770C2semidirect
6. #21446239C2semidirect
7. #21446128C2semidirect
8. #21445913C2semidirect
9. C4×#14519701C2semidirect
10. #21445534C2semidirect
11. C61×#1803C2semidirect
14. #16840485C4semidirect
254. #1803D61semidirect
297. C61#22490semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168398272
#214487702
#214467792
#214467702
#214462392
#214461282
#214459132
C4×#145197012
#214455342
C61×#18032
#2249061