Finite Group Databasebeta

← #29696805

Subgroups — #29696805

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2C2×Q8×#92418C211yesmaximal, characteristic, complemented
3#21446751C212yesmaximal
4#21446742C211yesmaximal, characteristic, complemented
5#21446628C212yesmaximal
6#21446620C211yesmaximal, characteristic, complemented
7#21445831C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic
9C4×#14519700C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic
11C61×#2355C211yesmaximal, characteristic, complemented
12#16840485—24no—
13#16840485C418yescomplemented
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519765C2212yes—
17C2×#14519753—21no—
18C2×C4×Dic122C2212yes—
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21Q8×#92418—24no—
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2212yes—
24C2×#14519700C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C4×#92418C2211yescharacteristic
32C2×C4×#92418C2212yes—
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#391—21no—
38C61×#387—22no—
39C61×#384C2211yescharacteristic
40C61×#382C2212yes—
41C4×C4⋊C4×C61—21no—
42C4×C4⋊C4×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2212yes—
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×#92418C2311yescharacteristic
48C22×#92418C2312yes—
49C22×#92418C2311yescharacteristic
50#14519777—22no—
51#14519777C2×C414yes—
52#14519772—216no—
53#14519765—24no—
54#14519765C2×C418yes—
55#14519753—28no—
56C4×Dic122—24no—
57C4×Dic122C2×C418yes—
58C4×D244—22no—
59C4×D244C2×C414yes—
60#14519735—28no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×#11758198—22no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×C4×Dic61—22no—
69C4⋊#92418—22no—
70C4⋊#92418—24no—
71#14519700—24no—
72#14519700—24no—
73#14519700—42no—
74#14519700—42no—
75#14519697—22no—
76#14519697—22no—
77#14519697—42no—
78C4×#92418—22no—
79C4×#92418—24no—
80C22×Q8×C61D411yescharacteristic
81C4×Q8×C61C2×C414yes—
82C4×Q8×C61D414yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C4⋊C4×C61D412yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C42×C61—21no—
91C2×C42×C61—22no—
92C2×C42×C61C2311yescharacteristic
93C2×C42×C61C2312yes—
94#95×C61—22no—
95#95×C61—21no—
96#95×C61—22no—
97#95×C61—22no—
98C23×D61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61C22⋊C414yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—22no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244C22×C412yes—
118C2×C4×D61—24no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×Dic122—22no—
122C2×Dic122—22no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122C22×C414yes—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61—81no—
134C4⋊C4×C61C22⋊C414yes—
135C4⋊C4×C61C22⋊C418yes—
136C4⋊C4×C61#4112yes—
137C42×C61—42no—
138C42×C61—82no—
139C42×C61C22×C412yes—
140C42×C61C22×C414yes—
141C42×C61C2×D412yes—
142C42×C61C2×D414yes—
143#11758198—28no—
144#11758198—24no—
145#11758198—28no—
146C4⋊Dic61—24no—
147C4⋊Dic61—24no—
148C4⋊Dic61—21no—
149C4⋊Dic61C22×C412yes—
150#11758196—28no—
151C4×Dic61—28no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—22no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—21no—
160C2×#92418#4112yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169Q8×C61—28no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—21no—
174C2×C4×C61—22no—
175C2×C4×C61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61C2×C22⋊C414yes—
186C2×C4×C61C2×C22⋊C414yes—
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—24no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203D244—22no—
204D244—22no—
205D244—41no—
206D244—41no—
207C4×D61—48no—
208Dic122—24no—
209Dic122—24no—
210Dic122—42no—
211Dic122—42no—
212#92418—22no—
213#92418—41no—
214#92418—42no—
215#22541—611nomaximal, Sylow
216C22×C61C22×C22⋊C411yescharacteristic
217C22×C61#54811yescharacteristic
218C22×C61#55011yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61C2×#13511yescharacteristic
221C22×C61#57711yescharacteristic
222C22×C61#57811yescharacteristic
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—24no—
231C4×C61—24no—
232C4×C61—22no—
233C4×C61—42no—
234C4×C61—81no—
235C4×C61—82no—
236C4×C61#41614yes—
237C4×C61C4×#4111yescharacteristic
238C4×C61#52011yescharacteristic
239C4×C61#54811yescharacteristic
240C4×C61#55011yescharacteristic
241Dic61—44no—
242Dic61—42no—
243C2×#556—611no—
244C2×C8×Q8—611no—
245#2355D6111yescharacteristic, complemented
246#1983—612no—
247#1981—611no—
248#1958—612no—
249#1956—611no—
250#1805—612no—
251#1802—611no—
252C4×#103—612no—
253C4×#102—611no—
254C2×C61#234111yescharacteristic
255C2×C61C2×#41611yescharacteristic
256C2×C61#289811yescharacteristic
257C2×C61#336011yescharacteristic
258C2×C61#336411yescharacteristic
259C2×C61#339311yescharacteristic
260C2×C61#339411yescharacteristic
261D61—82no—
262#556—618no—
263#556—1224no—
264C2×C4⋊Q8—611no—
265C2×#126—1222no—
266C2×#124—612no—
267C2×C22⋊Q8—1221no—
268C2×C4×Q8—612no—
269C2×C4×Q8C2×D6111yescharacteristic
270C2×C4×D4—611no—
271C2×#117—1221no—
272C8×Q8—1224no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8—612no—
275C2×#103—612no—
276C2×#103—612no—
277C2×#103—612no—
278C2×#103—612no—
279C2×#102—611no—
280C2×#102—611no—
281C2×#102—612no—
282C2×C4×C8—611no—
283C2×C4×C8—612no—
284#391—21no—
285#387—22no—
286#384C2×D6111yescharacteristic
287#382C2×D6112yes—
288C4×C4⋊C4—21no—
289C4×C4⋊C4C2×D6111yescharacteristic
290C4×C4⋊C4C2×D6112yes—
291#340—612no—
292#340—611no—
293#340—611no—
294C61#2254111yescharacteristic, complemented, Sylow
295C22×Q8—612no—
296C22×Q8C22⋊D6111yescharacteristic
297C22×D4—611no—
298C23×C4—1221no—
299C22×C8—611no—
300C22×C8—611no—
301C22×C8—612no—
302C4⋊Q8—614no—
303C4⋊Q8—1222no—
304#126—12216no—
305#124—618no—
306#124—1224no—
307C22⋊Q8—1228no—
308C4×Q8—618no—
309C4×Q8—1224no—
310C4×Q8C4×D6114yes—
311C4×Q8D24414yes—
312C4×D4—614no—
313C4×D4—1222no—
314#117—1228no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C4⋊C4C22⋊D6112yes—
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C4⋊C4C22×D6112yes—
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1222no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—1222no—
332C2×C42C22×D6111yescharacteristic
333C2×C42C22×D6112yes—
334C4⋊C8—1222no—
335C4⋊C8—1224no—
336#103—1224no—
337#103—1224no—
338#103—2442no—
339#103—2442no—
340#102—1222no—
341#102—1222no—
342#102—2442no—
343C4×C8—1222no—
344C4×C8—1224no—
345#95—22no—
346#95—21no—
347#95—22no—
348#95—22no—
349C24—1221no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—614no—
353C2×Q8—1222no—
354C2×Q8—1222no—
355C2×Q8—1222no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×Q8#1175819814yes—
359C2×D4—612no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—1221no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—1224no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4C2×C22⋊D6112yes—
374C22×C4C2×C22⋊D6112yes—
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378C2×C8—612no—
379C2×C8—1221no—
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—1222no—
383C2×C8—1221no—
384C2×C8—1222no—
385C2×C8—1222no—
386C2×C8—1221no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—81no—
394C4⋊C4—612no—
395C4⋊C4—1221no—
396C4⋊C4—1228no—
397C4⋊C4—1224no—
398C4⋊C4—1224no—
399C4⋊C4#1175819814yes—
400C4⋊C4#1175819818yes—
401C4⋊C4#1175822612yes—
402C22⋊C4—1228no—
403C22⋊C4—1224no—
404C22⋊C4—1228no—
405C42—42no—
406C42—82no—
407C42—1228no—
408C42C2×C4×D6112yes—
409C42C2×C4×D6114yes—
410C42C2×D24412yes—
411C42C2×D24414yes—
412C23—1222no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23—2441no—
419C23C22×C22⋊D6111yescharacteristic
420Q8—28no—
421Q8—1224no—
422Q8—1224no—
423Q8—2442no—
424Q8—2442no—
425D4—1222no—
426D4—1222no—
427D4—2441no—
428D4—2441no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—1222no—
445C2×C4—1224no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2448no—
451C2×C4C2×#1175819814yes—
452C2×C4C2×#1175819812yes—
453C2×C4C2×#1175819814yes—
454C2×C4C2×#1175819812yes—
455C2×C4#1451979712yes—
456C2×C4#1451979712yes—
457C2×C4C22×C4×D6111yescharacteristic
458C2×C4C22×D24411yescharacteristic
459C2×C4#41×D6111yescharacteristic
460C2×C4C2×#1175822611yescharacteristic
461C2×C4#1451982311yescharacteristic
462C2×C4#1451982511yescharacteristic
463C8—1222no—
464C8—2441no—
465C8—2442no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22—2442no—
471C22—2442no—
472C22—2442no—
473C22C22×#1175819811yescharacteristic
474C22#1684042311yescharacteristic
475C22#1684043111yescharacteristic
476C22C2×#1451979711yescharacteristic
477C22C2×#1451979711yescharacteristic
478C22#1684057311yescharacteristic
479C22#1684057711yescharacteristic
480C4—24no—
481C4—22no—
482C4—24no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4—2444no—
487C4—2442no—
488C4#1683975614yes—
489C4C4×#1175822611yescharacteristic
490C4#1684024411yescharacteristic
491C4#1684042411yescharacteristic
492C4#1684043411yescharacteristic
493C2—4882no—
494C2#2144865711yescharacteristic
495C2C2×#1683975611yescharacteristic
496C2#2145434911yescharacteristic
497C2#2145846811yescharacteristic
498C2#2145850411yescharacteristic
499C2#2145880311yescharacteristic
500C2#2145881111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2355C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#2355D61
hypocenterC61#22541
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#92418C2semidirect
4. #21446742C2semidirect
6. #21446620C2semidirect
7. #21445831C2semidirect
9. C4×#14519700C2semidirect
11. C61×#2355C2semidirect
13. #16840485C4semidirect
245. #2355D61semidirect
294. C61#22541semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×Q8×#924182
#214467512
#214467422
#214466282
#214466202
#214458312
#214458162
C4×#145197002
C4×#145196972
C61×#23552
#2254161