Finite Group Databasebeta

← #29689401

Subgroups — #29689401

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840332C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic, complemented
3#21448429C211yesmaximal, characteristic, complemented
4#21446750C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446638C211yesmaximal, characteristic, complemented
7#21446636C211yesmaximal, characteristic, complemented
8#21446621C211yesmaximal, characteristic, complemented
9#21445756C212yesmaximal
10#21445752C212yesmaximal, complemented
11C61×#1958C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14#16840332—24no—
15#16840332C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519759—22no—
20C2×#14519754—22no—
21C2×#14519734C2211yescharacteristic
22#16839821—24no—
23C2×#14519706C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839654—22no—
33#16839649—22no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36C4⋊C4×Dic61C2212yes—
37C2×#124×C61C2211yescharacteristic
38C2×#103×C61C2212yes—
39C2×#102×C61C2212yes—
40C2×C4×C8×C61C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C2×#11758224—22no—
43C2×#11758224—42no—
44C2×#11758224—42no—
45C2×#11758224C2×C414yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yes—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—22no—
59#14519777C2×C414yes—
60#14519775—22no—
61#14519775C2×C414yes—
62#14519764—22no—
63#14519764C2×C414yes—
64#14519759—216no—
65#14519754—216no—
66#14519734C2×C414yes—
67#14519734D414yes—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—22no—
72C2×#11758196D412yes—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77C2×C4×Dic61C2311yescharacteristic
78#14519706—24no—
79#14519702—42no—
80#14519702—42no—
81#14519701—42no—
82#14519700—42no—
83#14519700—42no—
84#14519698—42no—
85#14519689—24no—
86#14519688—22no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61C2311yescharacteristic
90C22×D4×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92#124×C61—22no—
93#124×C61C2×C414yes—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—22no—
96C2×C42×C61C2311yescharacteristic
97#103×C61—42no—
98#102×C61—42no—
99C4×C8×C61—22no—
100#95×C61—22no—
101C24×C61—21no—
102C23×D61C2×D411yescharacteristic
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61C22×C412yes—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61C22×C412yes—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—24no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—24no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123C22×Dic61—42no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D412yes—
126Q8×D61—48no—
127Q8×D61—48no—
128#11758224—48no—
129#11758224—48no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—24no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61C22×C412yes—
136C2×C4×D61C2×D412yes—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—24no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C2×Dic122C22×C412yes—
147C2×Dic122C22×C412yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—28no—
154C42×C61C22×C412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—28no—
157#11758198—82no—
158#11758198C22⋊C418yes—
159C4⋊Dic61—28no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—82no—
163#11758196—28no—
164#11758196C22⋊C418yes—
165C4×Dic61—82no—
166C4×Dic61C22×C412yes—
167C4×Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61—22no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61C22×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61C2×C22⋊C412yes—
182Q8×C61—24no—
183Q8×C61—42no—
184D4×C61—24no—
185D4×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61C2×C22⋊C414yes—
195C2×C4×C61C23×C411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C2×C4×C61H(5,𝔽2)12yes—
199C22⋊D61—48no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—48no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61C2×C22⋊C414yes—
215C2×Dic61C2×C22⋊C412yes—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—48no—
219Dic122—24no—
220Dic122—24no—
221Dic122—48no—
222Dic122—42no—
223Dic122—42no—
224C8×C61—42no—
225#92418—42no—
226#21814—611nomaximal, Sylow
227C22×C61—42no—
228C22×C61—44no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—82no—
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61#53511yescharacteristic
234C22×C61#53511yescharacteristic
235C22×C61#57712yes—
236C22×C61#57812yes—
237C2×D61—22no—
238C2×D61—24no—
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—42no—
242C4×C61—24no—
243C4×C61—42no—
244C4×C61—82no—
245C4×C61#51812yes—
246C4×C61#53512yes—
247Dic61—22no—
248Dic61—24no—
249Dic61—42no—
250Dic61—42no—
251Dic61—82no—
252C2×#536—611no—
253C2×#433—611no—
254#2314—611no—
255#1983—612no—
256#1981—612no—
257#1960—611no—
258#1959—611no—
259#1958D6111yescharacteristic, complemented
260#1956—611no—
261#1788—612no—
262#1787—612no—
263C2×C61—82no—
264C2×C61#230911yescharacteristic
265C2×C61#289812yes—
266C2×C61#322111yescharacteristic
267C2×C61#322311yescharacteristic
268C2×C61#322512yes—
269D61—42no—
270C22×#41—611no—
271C23×Q8—611no—
272#536—618no—
273#536—1224no—
274C2×C4⋊Q8—611no—
275C2×C4⋊Q8—611no—
276C2×#124—611no—
277C2×#124C2×D6111yescharacteristic
278C2×#123—1222no—
279C2×C22⋊Q8—1222no—
280C2×#117—611no—
281#433—1224no—
282C2×C4⋊C8—612no—
283C2×#103—612no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103—612no—
288C2×#103C2×D6112yes—
289C2×#102—612no—
290C2×#102C2×D6112yes—
291C2×C22⋊C8—612no—
292C2×#97—611no—
293C2×C4×C8C2×D6111yescharacteristic
294#388—1222no—
295#387—1222no—
296#384—611no—
297#382C2×D6111yescharacteristic
298#380—612no—
299C4×C4⋊C4—612no—
300C61#2181411yescharacteristic, complemented, Sylow
301C2×#41—614no—
302C2×#41—1222no—
303C2×#41—2442no—
304C2×#41—2442no—
305C22×Q8—614no—
306C22×Q8—611no—
307C22×Q8—611no—
308C22×Q8—1221no—
309C22×Q8—1222no—
310C22×Q8—2442no—
311C22×Q8—2442no—
312C22×Q8C22×D6111yescharacteristic
313C22×D4—1221no—
314C22×D4C22×D6111yescharacteristic
315C23×C4—611no—
316C23×C4—1221no—
317C23×C4—1221no—
318C22×C8—612no—
319C22×C8C22×D6112yes—
320C4⋊Q8—614no—
321C4⋊Q8—614no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324#124—22no—
325#124—614no—
326#124—1222no—
327#124C4×D6114yes—
328#123—12216no—
329C22⋊Q8—12216no—
330#117—614no—
331#117—614no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4C22×D6112yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—612no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C42—611no—
343C2×C42—1222no—
344C2×C42C22×D6111yescharacteristic
345C4⋊C8—1224no—
346#103—42no—
347#103—2442no—
348#103—2442no—
349#103—2442no—
350#103—2442no—
351#103—2442no—
352#102—42no—
353#102—2442no—
354C22⋊C8—1224no—
355#97—1222no—
356C4×C8—22no—
357#95—22no—
358#95—1222no—
359#95—1222no—
360C24—21no—
361C24—611no—
362#41—2448no—
363#41—2448no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—612no—
368C2×Q8—612no—
369C2×Q8—1224no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—2442no—
377C2×Q8—2442no—
378C2×Q8—2448no—
379C2×Q8—2448no—
380C2×Q8C2×C4×D6112yes—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—1224no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4C2×C4×D6112yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—611no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—1224no—
395C22×C4—1224no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4D4×D6112yes—
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—618no—
416C4⋊C4—1228no—
417C4⋊C4—1228no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—4882no—
421C22⋊C4—28no—
422C22⋊C4—618no—
423C22⋊C4—1228no—
424C22⋊C4—4882no—
425C42—612no—
426C42—612no—
427C42—4882no—
428C42C2×C4×D6112yes—
429C42D4×D6112yes—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—612no—
436C23—1222no—
437C23—1222no—
438C23—1221no—
439C23—1221no—
440C23C2×D4×D6111yescharacteristic
441Q8—24no—
442Q8—42no—
443Q8—1224no—
444Q8—1224no—
445Q8—2448no—
446Q8—2442no—
447Q8—2442no—
448D4—24no—
449D4—42no—
450D4—2448no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—614no—
460C2×C4—612no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—2448no—
470C2×C4—2448no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4C22⋊C4×D6114yes—
479C2×C4C22×C4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#1451982212yes—
482C2×C4#1451982312yes—
483C8—42no—
484C8—2442no—
485C22—44no—
486C22—42no—
487C22—81no—
488C22—82no—
489C22—81no—
490C22—1224no—
491C22—1222no—
492C22—2441no—
493C22—2442no—
494C22—2441no—
495C22C2×C22⋊C4×D6111yescharacteristic
496C22#1684032511yescharacteristic
497C22#1684033011yescharacteristic
498C22#1684056912yes—
499C22#1684057612yes—
500C4—24no—
501C4—42no—
502C4—82no—
503C4—1224no—
504C4—1222no—
505C4—4882no—
506C4—2442no—
507C4—2442no—
508C4#1684023412yes—
509C4#1684032812yes—
510C2—82no—
511C2—2442no—
512C2#2144839811yescharacteristic
513C2#2145434812yes—
514C2#2145688311yescharacteristic
515C2#2145690511yescharacteristic
516C2#2145692112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21814
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16839821C2semidirect
3. #21448429C2semidirect
4. #21446750C2semidirect
5. #21446740C2semidirect
6. #21446638C2semidirect
7. #21446636C2semidirect
8. #21446621C2semidirect
10. #21445752C2semidirect
11. C61×#1958C2semidirect
15. #16840332C4semidirect
259. #1958D61semidirect
300. C61#21814semidirect

Maximal subgroups

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

subgroupindex
C2×#168403322
C2×#168398212
#214484292
#214467502
#214467402
#214466382
#214466362
#214466212
#214457562
#214457522
C61×#19582
#2181461