Finite Group Databasebeta

← #29739025

Subgroups — #29739025

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448766C211yesmaximal, characteristic
2#21447343C212yesmaximal
3#21447190C212yesmaximal
4#21446585C212yesmaximal, complemented
5#21446577C211yesmaximal, characteristic, complemented
6#21446538C211yesmaximal, characteristic, complemented
7#21446394C212yesmaximal, complemented
8#21445975C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic
10C4×#14519697C211yesmaximal, characteristic
11C61×#2571C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13C2×#14519735—21no—
14C2×#14519710C2212yes—
15C2×#14519708C2212yes—
16C2×#14519708C2212yes—
17C2×C4⋊#92418C2212yes—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519697C2211yescharacteristic
20C2×#14519697C2211yescharacteristic
21C2×#14519697C2212yes—
22C2×C4×#92418C2211yescharacteristic
23#16839664C2212yes—
24#16839654—22no—
25#16839632C2211yescharacteristic
26#16839619—21no—
27C4×C4⋊Dic61C2212yes—
28#16839524C2211yescharacteristic
29#16839524C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33C61×#400—22no—
34C61×#389—21no—
35C61×#389C2212yes—
36C61×#384C2211yescharacteristic
37C61×#382—21no—
38C61×#382C2212yes—
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C22×D244C2311yescharacteristic
42C22×C4×D61—21no—
43C22×#92418C2312yes—
44C22×#92418C2311yescharacteristic
45C22×#92418C2311yescharacteristic
46C4×D244—22no—
47C4×D244Q814yes—
48#14519735—44no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×#11758198—21no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×C4×Dic61—22no—
57#14519710—42no—
58#14519708—24no—
59#14519708—42no—
60C4⋊#92418—22no—
61C4⋊#92418—24no—
62#14519697—22no—
63#14519697—22no—
64#14519697—42no—
65C4×#92418—22no—
66#14519684—22no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61C2311yescharacteristic
74C2×C4⋊C4×C61C2312yes—
75C2×C42×C61—21no—
76C2×C42×C61C2312yes—
77C2×C42×C61C2311yescharacteristic
78#95×C61—21no—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—22no—
82#95×C61D412yes—
83C23×D61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—22no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244C2×Q812yes—
100C2×C4×D61—24no—
101C2×C4×D61—42no—
102C2×C4×D61—42no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—81no—
113C4⋊C4×C61—82no—
114C42×C61—22no—
115C42×C61—22no—
116C42×C61—42no—
117C42×C61C2×Q812yes—
118C42×C61#4112yes—
119#11758198—24no—
120#11758198—44no—
121#11758198—82no—
122C4⋊Dic61—21no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126C4⋊Dic61—81no—
127C4⋊Dic61C2×Q812yes—
128C4×Dic61—44no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418#4112yes—
137C2×#92418#4112yes—
138C23×C61C22×D411yescharacteristic
139C22×D61—22no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61C22⋊Q812yes—
164C2×C4×C61C22⋊Q812yes—
165C2×C4×C61#13512yes—
166C2×C4×C61AGL(1,ℤ/8)12yes—
167C2×C4×C61C22×Q811yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C2×C4×C61H(5,𝔽2)11yescharacteristic
172C2×C4×C612−1+411yescharacteristic
173C2×Dic61—22no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179D244—22no—
180D244—22no—
181D244—41no—
182D244—41no—
183C4×D61—48no—
184#92418—22no—
185#92418—22no—
186#92418—42no—
187#26091—611nomaximal, Sylow
188C22×C61C2×C22⋊Q811yescharacteristic
189C22×C61#54711yescharacteristic
190C22×C61#55011yescharacteristic
191C22×C61C2×#13511yescharacteristic
192C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
193C22×C61#57711yescharacteristic
194C22×C61#57711yescharacteristic
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C4×C61—22no—
203C4×C61—42no—
204C4×C61—42no—
205C4×C61—81no—
206C4×C61—82no—
207C4×C61—81no—
208C4×C61#47814yes—
209C4×C61#53311yescharacteristic
210C4×C61#54411yescharacteristic
211C4×C61#55211yescharacteristic
212C4×C61#55311yescharacteristic
213Dic61—42no—
214Dic61—82no—
215#2571D6111yescharacteristic, complemented
216#2354—611no—
217#2092—612no—
218#2065—612no—
219#1949—612no—
220#1947—611no—
221#1939—611no—
222#1919—612no—
223#1844—611no—
224#1802—611no—
225C4×#102—611no—
226C2×C61#244111yescharacteristic
227C2×C61C2×#47811yescharacteristic
228C2×C61#308411yescharacteristic
229C2×C61#334011yescharacteristic
230C2×C61#335411yescharacteristic
231C2×C61#339411yescharacteristic
232C2×C61#339711yescharacteristic
233D61—82no—
234C2×C4×D4—611no—
235C2×#117—1221no—
236C2×#107—612no—
237C2×#106—612no—
238C2×#106—612no—
239C2×C4⋊C8—611no—
240C2×C4⋊C8—612no—
241C2×#102—611no—
242C2×#102—611no—
243C2×#102—612no—
244C2×C4×C8—611no—
245#400—22no—
246#390—612no—
247#389—21no—
248#389C2×D6112yes—
249#388—1222no—
250#384—611no—
251#384C2×D6111yescharacteristic
252#382—21no—
253#382C2×D6112yes—
254#380—1221no—
255C4×C4⋊C4—612no—
256C4×C4⋊C4C2×D6111yescharacteristic
257C4×C4⋊C4C2×D6111yescharacteristic
258#340—612no—
259#340—612no—
260#340—611no—
261#340—611no—
262#336—612no—
263C61#2609111yescharacteristic, complemented, Sylow
264C22×D4—611no—
265C23×C4—1221no—
266C22×C8—611no—
267C22×C8—611no—
268C22×C8—612no—
269C4×D4—614no—
270C4×D4—1222no—
271#117—2444no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—612no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4C22⋊D6111yescharacteristic
280C2×C4⋊C4C22⋊D6111yescharacteristic
281C2×C4⋊C4C22×D6111yescharacteristic
282C2×C4⋊C4C22×D6111yescharacteristic
283C2×C4⋊C4C22×D6112yes—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1222no—
286C2×C22⋊C4—1222no—
287C2×C42—21no—
288C2×C42—1222no—
289C2×C42C22×D6112yes—
290C2×C42C22×D6111yescharacteristic
291#107—2442no—
292#106—1224no—
293#106—2442no—
294C4⋊C8—1224no—
295C4⋊C8—1222no—
296#102—1222no—
297#102—1222no—
298#102—2442no—
299C4×C8—1222no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—1222no—
305#95C22⋊D6112yes—
306C24—1221no—
307C2×D4—612no—
308C2×D4—1221no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—1221no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—1221no—
317C22×C4—1224no—
318C22×C4—1222no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4C2×C22⋊D6112yes—
322C22×C4C2×C22⋊D6112yes—
323C22×C4C2×C22⋊D6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325C22×C4C23×D6111yescharacteristic
326C2×C8—612no—
327C2×C8—612no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—1221no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8—1222no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—42no—
344C4⋊C4—81no—
345C4⋊C4—82no—
346C4⋊C4—612no—
347C4⋊C4—1221no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—4881no—
351C4⋊C4—2442no—
352C22⋊C4—1224no—
353C22⋊C4—4882no—
354C22⋊C4—2444no—
355C42—22no—
356C42—22no—
357C42—42no—
358C42—2444no—
359C42Q8×D6112yes—
360C42#1175822612yes—
361C23—1222no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23—2441no—
368C23C22×C22⋊D6111yescharacteristic
369D4—1222no—
370D4—1222no—
371D4—2441no—
372D4—2441no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—1222no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2448no—
397C2×C4#1451975312yes—
398C2×C4#1451975312yes—
399C2×C4#1451979712yes—
400C2×C4#1451980212yes—
401C2×C4C2×Q8×D6111yescharacteristic
402C2×C4#41×D6111yescharacteristic
403C2×C4#41×D6111yescharacteristic
404C2×C4C2×#1175822611yescharacteristic
405C2×C4#1451982311yescharacteristic
406C2×C4#1451982411yescharacteristic
407C8—1222no—
408C8—1222no—
409C8—2442no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22—2442no—
415C22—2442no—
416C22—2442no—
417C22C2×#1451975311yescharacteristic
418C22#1684041111yescharacteristic
419C22#1684043111yescharacteristic
420C22C2×#1451979711yescharacteristic
421C22C2×#1451980211yescharacteristic
422C22#1684057011yescharacteristic
423C22#1684057311yescharacteristic
424C4—22no—
425C4—42no—
426C4—42no—
427C4—82no—
428C4—81no—
429C4—81no—
430C4—4882no—
431C4—2442no—
432C4#1684005914yes—
433C4#1684031811yescharacteristic
434C4#1684039211yescharacteristic
435C4#1684045011yescharacteristic
436C4#1684046511yescharacteristic
437C2—4882no—
438C2#2144952811yescharacteristic
439C2C2×#1684005911yescharacteristic
440C2#2145567711yescharacteristic
441C2#2145822611yescharacteristic
442C2#2145840811yescharacteristic
443C2#2145881111yescharacteristic
444C2#2145883911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2571C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2571D61
hypocenterC61#26091
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
4. #21446585C2semidirect
5. #21446577C2semidirect
6. #21446538C2semidirect
7. #21446394C2semidirect
8. #21445975C2semidirect
11. C61×#2571C2semidirect
215. #2571D61semidirect
263. C61#26091semidirect

Maximal subgroups

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

subgroupindex
#214487662
#214473432
#214471902
#214465852
#214465772
#214465382
#214463942
#214459752
#214458162
C4×#145196972
C61×#25712
#2609161