Finite Group Databasebeta

← #29738103

Subgroups — #29738103

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448766C211yesmaximal, characteristic
2#21447190C212yesmaximal
3#21446962C212yesmaximal
4#21446777C212yesmaximal, complemented
5#21446582C211yesmaximal, characteristic, complemented
6#21446538C211yesmaximal, characteristic, complemented
7#21446394C212yesmaximal, complemented
8#21445946C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic
10C4×#14519697C211yesmaximal, characteristic
11C61×#2552C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13C2×#14519735—21no—
14C2×#14519708C2212yes—
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2212yes—
17C2×#14519697C2211yescharacteristic
18C2×#14519697C2211yescharacteristic
19C2×#14519697C2212yes—
20C2×C4×#92418C2211yescharacteristic
21#16839664C2212yes—
22#16839654—22no—
23#16839632C2211yescharacteristic
24#16839619—21no—
25C4×C4⋊Dic61C2212yes—
26#16839524C2211yescharacteristic
27#16839524C2211yescharacteristic
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33C2×#125×C61C2212yes—
34C61×#398—22no—
35C61×#389—21no—
36C61×#389C2212yes—
37C61×#384—21no—
38C61×#384C2211yescharacteristic
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C22×D244C2311yescharacteristic
42C22×C4×D61—21no—
43C22×#92418C2311yescharacteristic
44C22×#92418C2311yescharacteristic
45C22×#92418C2312yes—
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#14519708—24no—
58C4⋊#92418—22no—
59C4⋊#92418—42no—
60#14519697—22no—
61#14519697—22no—
62#14519697—42no—
63C4×#92418—22no—
64#14519684—22no—
65#125×C61—24no—
66#125×C61—42no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61C2312yes—
75C2×C4⋊C4×C61C2311yescharacteristic
76C2×C4⋊C4×C61C2311yescharacteristic
77C2×C42×C61—21no—
78C2×C42×C61C2312yes—
79C2×C42×C61C2311yescharacteristic
80#95×C61—21no—
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—44no—
110C4⋊C4×C61—44no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—81no—
113C4⋊C4×C61—82no—
114C4⋊C4×C61#4112yes—
115C4⋊C4×C61#4112yes—
116C42×C61—22no—
117C42×C61—22no—
118C42×C61—42no—
119C42×C61C2×Q812yes—
120C42×C61#4112yes—
121#11758198—24no—
122#11758198—44no—
123#11758198—82no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—42no—
128C4⋊Dic61—81no—
129C4⋊Dic61C2×Q812yes—
130C4×Dic61—44no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—22no—
138C2×#92418—21no—
139C2×#92418—21no—
140C23×C61C22×D411yescharacteristic
141C22×D61—22no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61C22⋊Q812yes—
166C2×C4×C61C22⋊Q812yes—
167C2×C4×C61#13512yes—
168C2×C4×C61AGL(1,ℤ/8)12yes—
169C2×C4×C61C22×Q811yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C2×C4×C61H(5,𝔽2)11yescharacteristic
174C2×C4×C612−1+411yescharacteristic
175C2×Dic61—22no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181D244—22no—
182D244—22no—
183D244—41no—
184D244—41no—
185C4×D61—48no—
186#92418—41no—
187#92418—42no—
188#92418—41no—
189#26021—611nomaximal, Sylow
190C22×C61C2×C22⋊Q811yescharacteristic
191C22×C61#54611yescharacteristic
192C22×C61#54811yescharacteristic
193C22×C61C2×#13511yescharacteristic
194C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
195C22×C61#57711yescharacteristic
196C22×C61#57711yescharacteristic
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C4×C61—22no—
205C4×C61—42no—
206C4×C61—42no—
207C4×C61—42no—
208C4×C61—42no—
209C4×C61—82no—
210C4×C61#47814yes—
211C4×C61#53311yescharacteristic
212C4×C61#54411yescharacteristic
213C4×C61#55511yescharacteristic
214C4×C61#55611yescharacteristic
215Dic61—42no—
216Dic61—82no—
217#2552D6111yescharacteristic, complemented
218#2354—611no—
219#2065—612no—
220#2025—612no—
221#1989—612no—
222#1948—611no—
223#1939—611no—
224#1919—612no—
225#1834—611no—
226#1802—611no—
227C4×#102—611no—
228C2×C61#243911yescharacteristic
229C2×C61C2×#47811yescharacteristic
230C2×C61#308411yescharacteristic
231C2×C61#330311yescharacteristic
232C2×C61#333211yescharacteristic
233C2×C61#336011yescharacteristic
234C2×C61#336811yescharacteristic
235D61—82no—
236C2×#125C2×D6112yes—
237C2×C4×D4—611no—
238C2×#117—1221no—
239C2×#106—612no—
240C2×C4⋊C8—611no—
241C2×C4⋊C8—612no—
242C2×#102—611no—
243C2×#102—611no—
244C2×#102—612no—
245C2×C4×C8—611no—
246#398—22no—
247#390—612no—
248#389—21no—
249#389C2×D6112yes—
250#388—1222no—
251#384—21no—
252#384—611no—
253#384C2×D6111yescharacteristic
254#380—1221no—
255C4×C4⋊C4—612no—
256C4×C4⋊C4C2×D6111yescharacteristic
257C4×C4⋊C4C2×D6111yescharacteristic
258#340—612no—
259#340—612no—
260#340—612no—
261#340—611no—
262#340—612no—
263#340—611no—
264#336—612no—
265C61#2602111yescharacteristic, complemented, Sylow
266C22×D4—611no—
267C23×C4—1221no—
268C22×C8—611no—
269C22×C8—612no—
270C22×C8—611no—
271#125—24no—
272#125—42no—
273C4×D4—614no—
274C4×D4—1222no—
275#117—2444no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—612no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C4⋊C4C22⋊D6111yescharacteristic
287C2×C4⋊C4C22×D6112yes—
288C2×C4⋊C4C22×D6111yescharacteristic
289C2×C4⋊C4C22×D6111yescharacteristic
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1222no—
292C2×C22⋊C4—1222no—
293C2×C42—21no—
294C2×C42—1222no—
295C2×C42C22×D6112yes—
296C2×C42C22×D6111yescharacteristic
297#106—1224no—
298C4⋊C8—1222no—
299C4⋊C8—2442no—
300#102—1222no—
301#102—1222no—
302#102—2442no—
303C4×C8—1222no—
304#95—21no—
305#95—22no—
306#95—1222no—
307#95C22⋊D6112yes—
308C24—1221no—
309C2×D4—612no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—1221no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—1221no—
319C22×C4—1224no—
320C22×C4—1222no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4C2×C22⋊D6112yes—
324C22×C4C2×C22⋊D6112yes—
325C22×C4C2×C22⋊D6111yescharacteristic
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C23×D6111yescharacteristic
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—1221no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1221no—
336C2×C8—1221no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4—612no—
349C4⋊C4—1221no—
350C4⋊C4—1222no—
351C4⋊C4—1222no—
352C4⋊C4—4881no—
353C4⋊C4—2442no—
354C4⋊C4#1175822412yes—
355C4⋊C4#1175822612yes—
356C22⋊C4—1224no—
357C22⋊C4—4882no—
358C22⋊C4—2444no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—2444no—
363C42Q8×D6112yes—
364C42#1175822612yes—
365C23—1222no—
366C23—2441no—
367C23—2441no—
368C23—2441no—
369C23—2441no—
370C23—2441no—
371C23—2441no—
372C23C22×C22⋊D6111yescharacteristic
373D4—1222no—
374D4—1222no—
375D4—2441no—
376D4—2441no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—1222no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2448no—
401C2×C4#1451975312yes—
402C2×C4#1451975312yes—
403C2×C4#1451979712yes—
404C2×C4#1451980212yes—
405C2×C4C2×Q8×D6111yescharacteristic
406C2×C4C2×#1175822411yescharacteristic
407C2×C4C2×#1175822611yescharacteristic
408C2×C4C2×#1175822611yescharacteristic
409C2×C4#1451982211yescharacteristic
410C2×C4#1451982511yescharacteristic
411C8—2441no—
412C8—2442no—
413C8—2441no—
414C22—4881no—
415C22—4881no—
416C22—4881no—
417C22—4881no—
418C22—2442no—
419C22—2442no—
420C22—2442no—
421C22C2×#1451975311yescharacteristic
422C22#1684040611yescharacteristic
423C22#1684042311yescharacteristic
424C22C2×#1451979711yescharacteristic
425C22C2×#1451980211yescharacteristic
426C22#1684057011yescharacteristic
427C22#1684057311yescharacteristic
428C4—22no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—4882no—
435C4—2442no—
436C4#1684005914yes—
437C4#1684031711yescharacteristic
438C4#1684038911yescharacteristic
439C4#1684047811yescharacteristic
440C4#1684048611yescharacteristic
441C2—4882no—
442C2#2144950611yescharacteristic
443C2C2×#1684005911yescharacteristic
444C2#2145567711yescharacteristic
445C2#2145774711yescharacteristic
446C2#2145814611yescharacteristic
447C2#2145846811yescharacteristic
448C2#2145854811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2552C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2552D61
hypocenterC61#26021
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2552

Complements

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

NH ≅ G/Ntype
4. #21446777C2semidirect
5. #21446582C2semidirect
6. #21446538C2semidirect
7. #21446394C2semidirect
8. #21445946C2semidirect
11. C61×#2552C2semidirect
217. #2552D61semidirect
265. C61#26021semidirect

Maximal subgroups

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

subgroupindex
#214487662
#214471902
#214469622
#214467772
#214465822
#214465382
#214463942
#214459462
#214458162
C4×#145196972
C61×#25522
#2602161