Finite Group Databasebeta

← #29736681

Subgroups — #29736681

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840129C212yesmaximal, complemented
3#21454168C211yesmaximal, characteristic
4#21446961C211yesmaximal, characteristic, complemented
5#21446960C211yesmaximal, characteristic, complemented
6#21446957C211yesmaximal, characteristic, complemented
7#21446786C212yesmaximal, complemented
8#21446130C212yesmaximal
9#21446128C212yesmaximal
10#21445580C211yesmaximal, characteristic
11C61×#2024C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519775C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519753—21no—
19C2×#14519753C2211yescharacteristic
20C2×Q8×Dic61—21no—
21C2×#14519742—21no—
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840129—28no—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519689C2212yes—
29C2×#14519687C2212yes—
30#16839668C2211yescharacteristic
31#16839659—21no—
32#16839659—21no—
33#16839631C2211yescharacteristic
34#16839621—21no—
35#16839621—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#391C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2312yes—
51#14519777—24no—
52#14519775—24no—
53#14519768—24no—
54#14519768D418yes—
55#14519757—28no—
56#14519757—24no—
57#14519753—28no—
58#14519753—24no—
59Q8×Dic61—28no—
60#14519742—28no—
61#14519735—24no—
62#14519735—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—24no—
82#14519702—24no—
83#14519701—24no—
84#14519700—24no—
85#14519689—24no—
86#14519687—24no—
87#14519684—21no—
88#14519684—21no—
89C22×Q8×C61—21no—
90C22×Q8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97#103×C61—24no—
98#95×C61—21no—
99#95×C61—21no—
100C23×D61C2×D411yescharacteristic
101C2×Q8×C61—24no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—42no—
107C2×Q8×C61—42no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C2×C4×D61—22no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61#4114yes—
130C2×Dic122—24no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—42no—
136C2×Dic122—42no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D414yes—
144C42×C61—22no—
145#11758198—28no—
146#11758198—42no—
147#11758198—42no—
148C4⋊Dic61—24no—
149C4⋊Dic61—22no—
150C4⋊Dic61—28no—
151C4⋊Dic61—22no—
152C4⋊Dic61—42no—
153C4⋊Dic61C2×D414yes—
154#11758196—28no—
155#11758196—42no—
156#11758196—42no—
157#11758196—42no—
158#11758196—42no—
159#11758196—42no—
160C4×Dic61—28no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C4×Dic61C2×D414yes—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61C22×D411yescharacteristic
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61#12412yes—
175Q8×C61—48no—
176Q8×C61—42no—
177Q8×C61—42no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#13712yes—
192C2×C4×C61#13712yes—
193C2×C4×C61C22×D412yes—
194C2×C4×C61C2×#4112yes—
195C2×C4×C612−1+411yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—24no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61#12012yes—
213C2×Dic61#12012yes—
214C2×Dic61#12412yes—
215C4×D61—24no—
216C4×D61—24no—
217C4×D61—44no—
218C4×D61—44no—
219Dic122—48no—
220Dic122—42no—
221Dic122—42no—
222C8×C61—42no—
223#92418—42no—
224#25910—611nomaximal, Sylow
225C22×C61C2×#12011yescharacteristic
226C22×C61C2×#12411yescharacteristic
227C22×C61#53611yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57511yescharacteristic
231C22×C61#57511yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—41no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—44no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61#54812yes—
245C4×C61#54812yes—
246Dic61—22no—
247Dic61—22no—
248Dic61—22no—
249Dic61—44no—
250Dic61—81no—
251Dic61—81no—
252C2×#533—611no—
253C2×#491—612no—
254#2857—611no—
255#2024—611no—
256#2024—611no—
257#2024—611no—
258#2024D6111yescharacteristic, complemented
259#1992—612no—
260#1873—612no—
261#1873—612no—
262#1749—611no—
263C2×C61#243611yescharacteristic
264C2×C61#300711yescharacteristic
265C2×C61#300711yescharacteristic
266C2×C61#313511yescharacteristic
267C2×C61#313911yescharacteristic
268C2×C61#323711yescharacteristic
269C2×C61#323711yescharacteristic
270D61—42no—
271#533—1228no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—611no—
274C2×C4⋊Q8C2×D6111yescharacteristic
275C2×#125—612no—
276C2×C22⋊Q8—611no—
277C2×C22⋊Q8—611no—
278C2×C22⋊Q8—1221no—
279C2×C22⋊Q8—1221no—
280C2×C4×Q8—1221no—
281C2×C4×Q8—1221no—
282C2×#117—612no—
283C22×C42—611no—
284#491—1228no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103—612no—
288C2×#103C2×D6112yes—
289C2×C22⋊C8—612no—
290C2×#97—612no—
291#391—611no—
292#391C2×D6111yescharacteristic
293#389—1221no—
294#389—1221no—
295#384—611no—
296#380—1221no—
297#380—1221no—
298#340—612no—
299#340—612no—
300#340—612no—
301#340C2×D6112yes—
302#336—611no—
303#336C2×D6111yescharacteristic
304C61#2591011yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—611no—
307C22×Q8—1221no—
308C22×Q8C22×D6111yescharacteristic
309C23×C4—611no—
310C23×C4—611no—
311C23×C4—611no—
312C22×C8—612no—
313C22×C8C22×D6112yes—
314C4⋊Q8—24no—
315C4⋊Q8—1224no—
316C4⋊Q8—1224no—
317#125—618no—
318#125—1224no—
319C22⋊Q8—1228no—
320C22⋊Q8—1228no—
321C22⋊Q8—1224no—
322C22⋊Q8—1224no—
323C4×Q8—1228no—
324C4×Q8—1228no—
325#117—1224no—
326#117—1224no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C4⋊C4C22×D6112yes—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C42—611no—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42—1224no—
348C2×C42—1222no—
349C2×C42C22×D6111yescharacteristic
350#103—24no—
351#103—1224no—
352#103—1224no—
353#103—1224no—
354C22⋊C8—1224no—
355#97—1224no—
356#95—21no—
357#95—21no—
358#95—1221no—
359#95—1221no—
360C24—611no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—1224no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—1221no—
372C2×Q8—1221no—
373C2×Q8—2442no—
374C2×Q8—2442no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—614no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4D4×D6111yescharacteristic
395C22×C4D4×D6111yescharacteristic
396C22×C4C23×D6111yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—614no—
407C4⋊C4—1228no—
408C4⋊C4—1228no—
409C4⋊C4—1224no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—2442no—
413C4⋊C4—2442no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C4⋊C4—2442no—
417C4⋊C4—2442no—
418C4⋊C4D4×D6114yes—
419C22⋊C4—1228no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—614no—
424C42—1222no—
425C42—1222no—
426C42—1228no—
427C23—612no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23—1221no—
432C23—1221no—
433C23—1221no—
434C23C2×D4×D6111yescharacteristic
435Q8—48no—
436Q8—42no—
437Q8—42no—
438Q8—2448no—
439Q8—2442no—
440Q8—2442no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—612no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1224no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—2444no—
467C2×C4—2444no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4#1451980612yes—
477C2×C4#1451980912yes—
478C2×C4C2×D4×D6112yes—
479C2×C4#41×D6112yes—
480C2×C4#1451982411yescharacteristic
481C2×C4#1451982511yescharacteristic
482C8—42no—
483C8—2442no—
484C22—1222no—
485C22—1222no—
486C22—1222no—
487C22—2441no—
488C22—2441no—
489C22—2441no—
490C22—2441no—
491C22#120×D6111yescharacteristic
492C22#124×D6111yescharacteristic
493C22#1684033511yescharacteristic
494C22C2×#1451980611yescharacteristic
495C22C2×#1451980911yescharacteristic
496C22#1684055811yescharacteristic
497C22#1684056111yescharacteristic
498C4—22no—
499C4—22no—
500C4—44no—
501C4—81no—
502C4—81no—
503C4—1222no—
504C4—1222no—
505C4—1222no—
506C4—4881no—
507C4—4881no—
508C4—2444no—
509C4#1684042012yes—
510C4#1684042212yes—
511C2—2442no—
512C2#2144947211yescharacteristic
513C2#2145507411yescharacteristic
514C2#2145507811yescharacteristic
515C2#2145616111yescharacteristic
516C2#2145619611yescharacteristic
517C2#2145703111yescharacteristic
518C2#2145703211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2024D61
hypocenterC61#25910
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
2. C2×#16840129C2semidirect
4. #21446961C2semidirect
5. #21446960C2semidirect
6. #21446957C2semidirect
7. #21446786C2semidirect
11. C61×#2024C2semidirect
258. #2024D61semidirect
304. C61#25910semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401292
#214541682
#214469612
#214469602
#214469572
#214467862
#214461302
#214461282
#214455802
C61×#20242
#2591061