Finite Group Databasebeta

← #29733673

Subgroups — #29733673

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840375C211yesmaximal, characteristic, complemented
2C2×#16840157C211yesmaximal, characteristic, complemented
3#21451577C211yesmaximal, characteristic
4#21446828C212yesmaximal, complemented
5#21446823C212yesmaximal
6#21446789C211yesmaximal, characteristic, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446171C212yesmaximal, complemented
9#21446157C212yesmaximal, complemented
10#21445586C211yesmaximal, characteristic, complemented
11C2×C61×#488C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×#11758200C2212yes—
14#16840375—24no—
15#16840375—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519772—22no—
18C2×#14519768—22no—
19C2×#14519757C2212yes—
20C2×#14519753C2211yescharacteristic
21C2×#14519748C2211yescharacteristic
22C2×#14519742C2212yes—
23C2×#14519734C2211yescharacteristic
24#16840157—24no—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839685—22no—
30#16839678—22no—
31#16839669C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839632C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×#124×C61C2211yescharacteristic
39C61×#488—24no—
40C2×#103×C61C2212yes—
41C2×#102×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×Dic122C2312yes—
44C22×C22⋊D61—21no—
45C22×C4×D61C2311yescharacteristic
46C2×#11758202—24no—
47C2×#11758202—24no—
48C2×#11758202—42no—
49C2×#11758200—24no—
50C2×#11758200—24no—
51C2×#11758200—42no—
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519772—216no—
56#14519768—216no—
57#14519757—24no—
58#14519757—24no—
59#14519753—42no—
60#14519748—42no—
61#14519742—24no—
62#14519742—24no—
63#14519734—22no—
64#14519734—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×#11758196D412yes—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—22no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77C2×#11758198D412yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61C2311yescharacteristic
80#14519702—42no—
81#14519702—42no—
82#14519689—24no—
83#14519688—22no—
84#14519684—22no—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88#125×C61—22no—
89#125×C61D414yescomplemented
90#124×C61—22no—
91#124×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—22no—
95C2×C42×C61C2311yescharacteristic
96#103×C61—24no—
97#102×C61—24no—
98C4×C8×C61—22no—
99C24×C61—21no—
100C23×D61C2×D411yescharacteristic
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—42no—
114C2×C22⋊D61—82no—
115C22×Dic61—22no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D412yes—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122C2×D414yes—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61C2×D414yes—
136C22⋊C4×C61—44no—
137C42×C61C2×D412yes—
138C42×C61C2×D412yes—
139C22⋊Dic61—42no—
140C22⋊Dic61—82no—
141#11758202—48no—
142#11758200—48no—
143#11758198—28no—
144#11758198—24no—
145#11758198—42no—
146C4⋊Dic61—28no—
147C4⋊Dic61—28no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—82no—
151#11758196—28no—
152#11758196—24no—
153#11758196—42no—
154C4×Dic61—28no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418#4114yes—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61C22×D411yescharacteristic
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—22no—
170C22×D61—22no—
171Q8×C61—42no—
172Q8×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—24no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12014yes—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C2×#4112yes—
187C2×C4×C61H(5,𝔽2)12yes—
188C22⋊D61—84no—
189C2×Dic61—22no—
190C2×Dic61—24no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C4×D61—42no—
203C4×D61—42no—
204Dic122—24no—
205Dic122—24no—
206Dic122—42no—
207Dic122—42no—
208C8×C61—42no—
209#92418—24no—
210#25612—611nomaximal, Sylow
211C22×C61—82no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—82no—
216C22×C61C2×#12011yescharacteristic
217C22×C61#53211yescharacteristic
218C22×C61#53511yescharacteristic
219C22×C61#57712yes—
220C22×C61#57812yes—
221C2×D61—42no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—42no—
226C4×C61—24no—
227C4×C61—44no—
228C4×C61—81no—
229C4×C61#54612yes—
230C4×C61#55012yes—
231Dic61—44no—
232Dic61—41no—
233Dic61—42no—
234Dic61—81no—
235Dic61—81no—
236C2×#541—611no—
237C2×#497—611no—
238C2×#488D6111yescharacteristic, complemented
239#2620—611no—
240#2001—612no—
241#2000—612no—
242#1992—611no—
243#1956—611no—
244#1880—612no—
245#1878—612no—
246#1750—611no—
247C2×C61—161no—
248C2×C61#243111yescharacteristic
249C2×C61#301312yes—
250C2×C61#317812yes—
251C2×C61#322211yescharacteristic
252C2×C61#322311yescharacteristic
253D61—81no—
254C22×Q16—612no—
255C22×SD16—612no—
256#541—1224no—
257#541—1224no—
258C2×C4⋊Q8—611no—
259C2×#126—1222no—
260C2×#125—1222no—
261C2×#125C2×D6111yescharacteristic
262C2×#124C2×D6111yescharacteristic
263C2×C22⋊Q8—611no—
264C2×C22⋊Q8—612no—
265C2×#121—611no—
266C2×C4×Q8—612no—
267C2×#117—611no—
268#497—1224no—
269#488—24no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#103C2×D6112yes—
273C2×#102C2×D6112yes—
274C2×C22⋊C8—612no—
275C2×#97—611no—
276C2×C4×C8C2×D6111yescharacteristic
277#396—1222no—
278#394—1222no—
279#391—611no—
280#390—611no—
281#384—611no—
282#340—612no—
283#340—612no—
284#336—612no—
285C61#2561211yescharacteristic, complemented, Sylow
286C22×Q8—612no—
287C22×Q8C22×D6111yescharacteristic
288C22×D4—1221no—
289C22×D4C22×D6111yescharacteristic
290C23×C4—611no—
291C2×Q16—1224no—
292C2×Q16—1224no—
293C2×Q16—2442no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C2×SD16—2442no—
297C22×C8—612no—
298C22×C8C22×D6112yes—
299C4⋊Q8—614no—
300C4⋊Q8—1222no—
301#126—12216no—
302#125—22no—
303#125—12216no—
304#125C22⋊D6114yescomplemented
305#124—22no—
306#124—22no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—2442no—
310#121—2442no—
311C4×Q8—1224no—
312C4×Q8—1224no—
313#117—1222no—
314#117—1222no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4C22×D6112yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—612no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1221no—
331C2×C42—611no—
332C2×C42—1222no—
333C2×C42C22×D6111yescharacteristic
334#103—24no—
335#103—2442no—
336#103—2442no—
337#102—24no—
338C22⋊C8—1224no—
339#97—1222no—
340C4×C8—22no—
341#95—1222no—
342C24—21no—
343C24—611no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—614no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—1222no—
352C2×Q8—1222no—
353C2×Q8—1222no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—4882no—
359C2×D4—2442no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—612no—
363C22×C4—611no—
364C22×C4—1222no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4D4×D6112yes—
372C22×C4C23×D6111yescharacteristic
373Q16—2448no—
374SD16—2448no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—614no—
379C2×C8—1222no—
380C2×C8—1222no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—1228no—
384C4⋊C4—1228no—
385C4⋊C4—1224no—
386C4⋊C4—1228no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—4882no—
390C4⋊C4—2442no—
391C4⋊C4C2×C22⋊D6114yes—
392C22⋊C4—44no—
393C22⋊C4—1228no—
394C22⋊C4—1224no—
395C22⋊C4—4882no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C42—1228no—
399C42—1221no—
400C42—1221no—
401C42D4×D6112yes—
402C42C2×C22⋊D6112yes—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—1221no—
409C23—1221no—
410C23—1222no—
411C23—1222no—
412C23—1221no—
413C23C2×D4×D6111yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—1224no—
417Q8—1224no—
418Q8—2442no—
419Q8—2442no—
420D4—42no—
421D4—42no—
422D4—4884no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—1222no—
432C2×C4—1224no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1221no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4#1451974514yes—
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C2×C4#41×D6112yes—
450C2×C4#1451982312yes—
451C8—42no—
452C8—1224no—
453C22—81no—
454C22—82no—
455C22—82no—
456C22—81no—
457C22—81no—
458C22—2441no—
459C22—2442no—
460C22—2442no—
461C22—2441no—
462C22—2441no—
463C22C2×#1451974511yescharacteristic
464C22#1684031111yescharacteristic
465C22#1684033011yescharacteristic
466C22#1684056912yes—
467C22#1684057612yes—
468C4—24no—
469C4—44no—
470C4—81no—
471C4—4881no—
472C4—4881no—
473C4—2442no—
474C4—2441no—
475C4—2444no—
476C4#1684040512yes—
477C4#1684043112yes—
478C2—161no—
479C2—4881no—
480C2#2144944111yescharacteristic
481C2#2145513012yes—
482C2#2145648612yes—
483C2#2145689411yescharacteristic
484C2#2145690511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#488C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#488D61
hypocenterC61#25612
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. C2×#16840375C2semidirect
2. C2×#16840157C2semidirect
4. #21446828C2semidirect
6. #21446789C2semidirect
7. #21446621C2semidirect
8. #21446171C2semidirect
9. #21446157C2semidirect
10. #21445586C2semidirect
11. C2×C61×#488C2semidirect
54. #14519777D4semidirect
89. #125×C61D4semidirect
238. C2×#488D61semidirect
285. C61#25612semidirect
304. #125C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168401572
#214515772
#214468282
#214468232
#214467892
#214466212
#214461712
#214461572
#214455862
C2×C61×#4882
#2561261