Finite Group Databasebeta

← #29733664

Subgroups — #29733664

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×#16840154C211yesmaximal, characteristic, complemented
2C2×#16840119C211yesmaximal, characteristic, complemented
3#21451576C211yesmaximal, characteristic
4#21446827C212yesmaximal, complemented
5#21446820C212yesmaximal
6#21446789C211yesmaximal, characteristic, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal, complemented
9#21446160C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C2×C61×#541C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×#11758201C2212yes—
14C2×#14519765C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17#16840154—24no—
18#16840119—24no—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519700C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2212yes—
24C2×#14519688C2211yescharacteristic
25C2×C4×#92418C2211yescharacteristic
26#16839684—22no—
27#16839678—22no—
28#16839668C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839632C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#541—24no—
35C61×#541—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—22no—
38C2×#125×C61—22no—
39C2×#125×C61C2211yescharacteristic
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61C2212yes—
42C2×#117×C61C2211yescharacteristic
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C2×#11758202—24no—
47C2×#11758202—24no—
48C2×#11758202—42no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519765—22no—
55#14519765—22no—
56#14519756—42no—
57#14519751—42no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66#14519701—42no—
67#14519701—42no—
68#14519700—24no—
69#14519697—24no—
70C22⋊#92418—24no—
71#14519688—22no—
72C4×#92418—22no—
73#14519684—22no—
74C22×Q8×C61C2312yes—
75C23×C4×C61C2311yescharacteristic
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D414yescomplemented
78#126×C61—216no—
79#125×C61—216no—
80#125×C61—22no—
81#125×C61D414yescomplemented
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86#117×C61—22no—
87#117×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D412yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61D412yes—
96C2×C42×C61—22no—
97C2×C42×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99C24×C61C2×D411yescharacteristic
100C23×D61—21no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61C2×D414yes—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—42no—
118C2×C22⋊D61—82no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—22no—
125C2×D244—22no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—28no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61C2×D414yes—
137C22⋊C4×C61—28no—
138C22⋊C4×C61—24no—
139C42×C61—28no—
140C42×C61—21no—
141C42×C61—21no—
142C42×C61C2×D412yes—
143C42×C61C2×D412yes—
144C22⋊Dic61—42no—
145#11758202—48no—
146#11758201—48no—
147#11758198—42no—
148#11758198—44no—
149#11758198—82no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—82no—
153#11758196—42no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418#4114yes—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61C22×D411yescharacteristic
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—42no—
171Q8×C61—24no—
172Q8×C61—24no—
173Q8×C61—42no—
174Q8×C61—42no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—24no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61#12014yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4112yes—
195C2×C4×C61H(5,𝔽2)12yes—
196C22⋊D61—84no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204D244—42no—
205D244—42no—
206Dic122—42no—
207Dic122—42no—
208#92418—24no—
209#92418—42no—
210#25612—611nomaximal, Sylow
211C22×C61—42no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—42no—
216C22×C61C2×#12011yescharacteristic
217C22×C61#53211yescharacteristic
218C22×C61#53511yescharacteristic
219C22×C61#57712yes—
220C22×C61#57812yes—
221C2×D61—82no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—82no—
226C4×C61—24no—
227C4×C61—41no—
228C4×C61—44no—
229C4×C61—42no—
230C4×C61—44no—
231C4×C61#54612yes—
232C4×C61#55012yes—
233Dic61—81no—
234Dic61—81no—
235Dic61—81no—
236C2×#541D6111yescharacteristic, complemented
237C2×#497—611no—
238C2×#488—611no—
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—81no—
248C2×C61#243111yescharacteristic
249C2×C61#301312yes—
250C2×C61#317812yes—
251C2×C61#322211yescharacteristic
252C2×C61#322311yescharacteristic
253D61—161no—
254C22×Q16—612no—
255C22×SD16—612no—
256#541—24no—
257#541—24no—
258C2×C4⋊Q8C2×D6111yescharacteristic
259C2×#126—22no—
260C2×#125—22no—
261C2×#125C2×D6111yescharacteristic
262C2×#124—611no—
263C2×C22⋊Q8—611no—
264C2×C22⋊Q8C2×D6112yes—
265C2×#121—611no—
266C2×C4×Q8C2×D6112yes—
267C2×#117C2×D6111yescharacteristic
268#497—1224no—
269#488—1224no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#103—612no—
273C2×#102—612no—
274C2×C22⋊C8—612no—
275C2×#97—611no—
276C2×C4×C8—611no—
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—611no—
287C22×Q8C22×D6112yes—
288C22×D4—611no—
289C22×D4—1221no—
290C23×C4C22×D6111yescharacteristic
291C2×Q16—1224no—
292C2×Q16—1224no—
293C2×Q16—2442no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C2×SD16—2442no—
297C22×C8—612no—
298C22×C8—612no—
299C4⋊Q8—22no—
300C4⋊Q8C22⋊D6114yescomplemented
301#126—216no—
302#125—216no—
303#125—22no—
304#125C22⋊D6114yescomplemented
305#124—1222no—
306#124—1222no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—2442no—
310#121—2442no—
311C4×Q8—24no—
312C4×Q8—24no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4C22⋊D6112yes—
324C2×C4⋊C4C22×D6112yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4C22⋊D6112yes—
331C2×C42—22no—
332C2×C42C22×D6111yescharacteristic
333C2×C42C22×D6111yescharacteristic
334#103—1224no—
335#103—2442no—
336#103—2442no—
337#102—1224no—
338C22⋊C8—1224no—
339#97—1222no—
340C4×C8—1222no—
341#95—1222no—
342C24—1221no—
343C24C2×C22⋊D6111yescharacteristic
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1222no—
352C2×Q8—1222no—
353C2×Q8C2×C22⋊D6114yes—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—4882no—
359C2×D4—2442no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4C2×C22⋊D6112yes—
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6112yes—
372C22×C4C23×D6111yescharacteristic
373Q16—2448no—
374SD16—2448no—
375C2×C8—614no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—1222no—
379C2×C8—1222no—
380C2×C8—1222no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—4882no—
390C4⋊C4—2442no—
391C4⋊C4C2×C22⋊D6114yes—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4—4882no—
395C22⋊C4—2442no—
396C22⋊C4—2442no—
397C22⋊C4—2444no—
398C42—28no—
399C42—21no—
400C42—21no—
401C42C2×C22⋊D6112yes—
402C42C2×C22⋊D6112yes—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—21no—
408C23—2441no—
409C23—2441no—
410C23—2442no—
411C23—2442no—
412C23—2441no—
413C23C22×C22⋊D6111yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—2442no—
419Q8—2442no—
420D4—4884no—
421D4—2442no—
422D4—2442no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4#1451974614yes—
447C2×C4C22×C22⋊D6111yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C2×C4#41×D6112yes—
450C2×C4#1451982312yes—
451C8—1224no—
452C8—2442no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—4881no—
459C22—4882no—
460C22—4882no—
461C22—4881no—
462C22—4881no—
463C22C2×#1451974611yescharacteristic
464C22#1684030811yescharacteristic
465C22#1684032611yescharacteristic
466C22#1684057012yes—
467C22#1684057712yes—
468C4—24no—
469C4—42no—
470C4—44no—
471C4—41no—
472C4—44no—
473C4—4881no—
474C4—4881no—
475C4—4881no—
476C4#1684040012yes—
477C4#1684043112yes—
478C2—81no—
479C2—9761no—
480C2#2144943411yescharacteristic
481C2#2145513412yes—
482C2#2145648912yes—
483C2#2145688511yescharacteristic
484C2#2145689611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#541C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#541D61
hypocenterC61#25612
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×#16840154C2semidirect
2. C2×#16840119C2semidirect
4. #21446827C2semidirect
6. #21446789C2semidirect
7. #21446622C2semidirect
8. #21446174C2semidirect
9. #21446160C2semidirect
10. #21445585C2semidirect
11. C2×C61×#541C2semidirect
77. C4⋊Q8×C61D4semidirect
81. #125×C61D4semidirect
236. C2×#541D61semidirect
285. C61#25612semidirect
300. C4⋊Q8C22⋊D61semidirect
304. #125C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401542
C2×#168401192
#214515762
#214468272
#214468202
#214467892
#214466222
#214461742
#214461602
#214455852
C2×C61×#5412
#2561261