Finite Group Databasebeta

← #29711981

Subgroups — #29711981

Order 15616.

1158 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840375C211yesmaximal, characteristic
2C2×#16839919C212yesmaximal
3#21448402C211yesmaximal, characteristic
4#21446967C211yesmaximal, characteristic, complemented
5#21446921C211yesmaximal, characteristic, complemented
6#21446918C211yesmaximal, characteristic, complemented
7#21446161C212yesmaximal, complemented
8#21446157C212yesmaximal, complemented
9#21446130C212yesmaximal
10#21445889C211yesmaximal, characteristic
11C61×#2026C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×Dic244C2212yes—
14C22×#11758202C2212yes—
15#16840375—28no—
16C2×#14519777C2211yescharacteristic
17C2×#14519772—22no—
18C2×#14519768—22no—
19C2×#14519757C2212yes—
20C2×#14519742C2212yes—
21C2×#14519735C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23#16839919—28no—
24C2×#14519702C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519689C2212yes—
28#16839669C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839666—22no—
32#16839624—22no—
33#16839524C2212yes—
34#16839517C2211yescharacteristic
35#16839516C2212yes—
36C2×#125×C61C2211yescharacteristic
37C2×#103×C61C2212yes—
38C61×#391C2211yescharacteristic
39C61×#340C2212yes—
40C61×#336C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122C2312yes—
43C22×Dic122C2311yescharacteristic
44C2×Q8×D61—22no—
45C2×Q8×D61—22no—
46C2×Q8×D61—24no—
47C2×Q8×D61—42no—
48C22×C4×D61—21no—
49C22×C4×D61C2311yescharacteristic
50C2×Dic244—24no—
51C2×Dic244—24no—
52C2×Dic244—42no—
53C2×#11758202—24no—
54C2×#11758202—24no—
55C2×#11758202—42no—
56C22×#92418C2312yes—
57#14519777—24no—
58#14519772—216no—
59#14519768—216no—
60#14519757—24no—
61#14519757D418yes—
62#14519742—24no—
63#14519742—24no—
64#14519735—42no—
65#14519734—24no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196D412yes—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×#11758198D412yes—
74C2×C4×Dic61—22no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2311yescharacteristic
77#14519702—42no—
78#14519700—24no—
79#14519700—42no—
80#14519689—24no—
81#14519684—22no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86#125×C61—24no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91#103×C61—24no—
92#95×C61—22no—
93C23×D61C2×D411yescharacteristic
94C2×Q8×C61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61#4112yes—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61C2×D412yes—
107C22×Dic61C2×D411yescharacteristic
108Q8×D61—48no—
109Q8×D61—48no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—22no—
113C2×C4×D61—24no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61C2×D414yes—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122—22no—
122C2×Dic122—22no—
123C2×Dic122—24no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122C2×D414yes—
129C2×Dic122#4112yes—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61C2×D414yes—
137C42×C61—22no—
138#11758202—48no—
139#11758198—28no—
140#11758198—24no—
141#11758198—42no—
142C4⋊Dic61—28no—
143C4⋊Dic61—28no—
144C4⋊Dic61—22no—
145#11758196—28no—
146#11758196—24no—
147C4×Dic61—28no—
148C4×Dic61—22no—
149C4×Dic61—42no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—22no—
153Dic244—48no—
154C23×C61C22×D411yescharacteristic
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—22no—
158C22×D61—22no—
159C22×D61#12012yes—
160Q8×C61—24no—
161Q8×C61—42no—
162C2×C4×C61—24no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61C22×D412yes—
174C2×C4×C61C2×#4111yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×C4×C61H(5,𝔽2)12yes—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—22no—
180C2×Dic61—24no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61#12012yes—
191C4×D61—24no—
192C4×D61—24no—
193C4×D61—48no—
194Dic122—24no—
195Dic122—24no—
196Dic122—24no—
197Dic122—48no—
198Dic122—42no—
199Dic122—42no—
200Dic122—42no—
201C8×C61—42no—
202#92418—42no—
203#23943—611nomaximal, Sylow
204C22×C61C2×#12011yescharacteristic
205C22×C61#52511yescharacteristic
206C22×C61#53511yescharacteristic
207C22×C61#57812yes—
208C22×C61#57812yes—
209C2×D61—22no—
210C2×D61—24no—
211C2×D61—42no—
212C2×D61—41no—
213C2×D61—41no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61—44no—
217C4×C61—81no—
218C4×C61#54612yes—
219C4×C61#54612yes—
220Dic61—22no—
221Dic61—42no—
222Dic61—44no—
223Dic61—42no—
224Dic61—81no—
225C2×#541—611no—
226C2×#451—612no—
227#2310—611no—
228#2026—611no—
229#2026D6111yescharacteristic, complemented
230#2016—611no—
231#2016—611no—
232#1878—612no—
233#1878—612no—
234#1873—612no—
235#1818—611no—
236C2×C61#240411yescharacteristic
237C2×C61#301112yes—
238C2×C61#312712yes—
239C2×C61#322311yescharacteristic
240C2×C61#322311yescharacteristic
241D61—42no—
242C23×Q8—611no—
243C22×Q16—612no—
244C22×Q16—612no—
245#541—1228no—
246C2×C4⋊Q8—611no—
247C2×#126—1222no—
248C2×#125—1222no—
249C2×#125C2×D6111yescharacteristic
250C2×C22⋊Q8—612no—
251C2×C4×Q8—612no—
252C2×#117—611no—
253C2×#117—611no—
254#451—1228no—
255C2×#103—612no—
256C2×#103—612no—
257C2×#103—612no—
258C2×#103C2×D6112yes—
259C2×C22⋊C8—612no—
260#391—611no—
261#391—611no—
262#391—611no—
263#391—1222no—
264#391C2×D6111yescharacteristic
265#381—1222no—
266#340—612no—
267#340C2×D6112yes—
268#336—612no—
269#336—611no—
270#336C2×D6111yescharacteristic
271C61#2394311yescharacteristic, complemented, Sylow
272C22×Q8—612no—
273C22×Q8—611no—
274C22×Q8—1221no—
275C22×Q8—1224no—
276C22×Q8—1222no—
277C22×Q8—1222no—
278C22×Q8—2442no—
279C22×Q8C22×D6111yescharacteristic
280C23×C4—611no—
281C23×C4—1221no—
282C2×Q16—1224no—
283C2×Q16—1224no—
284C2×Q16—1224no—
285C2×Q16—1224no—
286C2×Q16—2442no—
287C2×Q16—2442no—
288C22×C8—612no—
289C22×C8C22×D6112yes—
290C4⋊Q8—1224no—
291#126—12216no—
292#125—24no—
293#125—12216no—
294C22⋊Q8—618no—
295C22⋊Q8—1224no—
296C4×Q8—1224no—
297C4×Q8—1224no—
298#117—1224no—
299#117—2442no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—611no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1222no—
306C2×C4⋊C4C22×D6111yescharacteristic
307C2×C4⋊C4C22×D6112yes—
308C2×C22⋊C4—612no—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1221no—
311C2×C42—611no—
312C2×C42—1222no—
313C2×C42—1221no—
314C2×C42C22×D6111yescharacteristic
315#103—24no—
316#103—1224no—
317#103—2442no—
318#103—2442no—
319C22⋊C8—1224no—
320#95—22no—
321#95—1222no—
322#95—1222no—
323#95—1222no—
324C24—611no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—614no—
329C2×Q8—612no—
330C2×Q8—1224no—
331C2×Q8—1222no—
332C2×Q8—1221no—
333C2×Q8—1222no—
334C2×Q8—1222no—
335C2×Q8—1222no—
336C2×Q8—1222no—
337C2×Q8—1222no—
338C2×Q8—1222no—
339C2×Q8—2442no—
340C2×Q8—2442no—
341C2×Q8—2448no—
342C2×Q8—2448no—
343C2×Q8#1175822612yes—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—611no—
348C22×C4—614no—
349C22×C4—612no—
350C22×C4—1224no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1222no—
354C22×C4—1221no—
355C22×C4—1222no—
356C22×C4—1221no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4D4×D6112yes—
360C22×C4C23×D6111yescharacteristic
361Q16—2448no—
362Q16—2448no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—1222no—
367C2×C8—1222no—
368C2×C8—1222no—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—1224no—
373C4⋊C4—1228no—
374C4⋊C4—1228no—
375C4⋊C4—1228no—
376C4⋊C4—1222no—
377C4⋊C4D4×D6114yes—
378C22⋊C4—1224no—
379C22⋊C4—1228no—
380C22⋊C4—2442no—
381C42—22no—
382C42—1228no—
383C42—1222no—
384C42—2442no—
385C23—612no—
386C23—1221no—
387C23—1222no—
388C23—1222no—
389C23—1221no—
390C23C2×D4×D6111yescharacteristic
391Q8—24no—
392Q8—42no—
393Q8—1224no—
394Q8—1224no—
395Q8—1224no—
396Q8—2448no—
397Q8—2442no—
398Q8—2442no—
399Q8—2442no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—612no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1224no—
415C2×C4—1224no—
416C2×C4—1222no—
417C2×C4—1224no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—2448no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2442no—
428C2×C4C2×D4×D6112yes—
429C2×C4#41×D6111yescharacteristic
430C2×C4C2×#1175822611yescharacteristic
431C2×C4#1451982312yes—
432C8—42no—
433C8—2442no—
434C22—1224no—
435C22—1222no—
436C22—2441no—
437C22—2441no—
438C22—2442no—
439C22#120×D6111yescharacteristic
440C22#1684027511yescharacteristic
441C22#1684033011yescharacteristic
442C22#1684057612yes—
443C22#1684057812yes—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—81no—
448C4—1222no—
449C4—4881no—
450C4—2442no—
451C4—2442no—
452C4—2444no—
453C4#1684040312yes—
454C4#1684040512yes—
455C2—2442no—
456C2#2144913911yescharacteristic
457C2#2145511112yes—
458C2#2145612112yes—
459C2#2145690311yescharacteristic
460C2#2145690511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2026C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2026D61
hypocenterC61#23943
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
4. #21446967C2semidirect
5. #21446921C2semidirect
6. #21446918C2semidirect
7. #21446161C2semidirect
8. #21446157C2semidirect
11. C61×#2026C2semidirect
229. #2026D61semidirect
271. C61#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168399192
#214484022
#214469672
#214469212
#214469182
#214461612
#214461572
#214461302
#214458892
C61×#20262
#2394361