Finite Group Databasebeta

← #29575951

Subgroups — #29575951

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455279C211yesmaximal, characteristic
2#21448730C211yesmaximal, characteristic, complemented
3#21444379C211yesmaximal, characteristic, complemented
4#21444209C211yesmaximal, characteristic, complemented
5#21443722C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1502C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×C4×D244—21no—
10C2×C4×D244C2211yescharacteristic
11#16839974—22no—
12#16839964C412yescomplemented
13#16839954C412yescomplemented
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839763—21no—
17#16839660—21no—
18#16839633—22no—
19#16839628C2211yescharacteristic
20#16839621—21no—
21#95×D61—21no—
22#16839496—22no—
23#16839489—22no—
24C61×#532C2211yescharacteristic
25C61×#431C2211yescharacteristic
26C4×C4⋊C4×C61C2211yescharacteristic
27C61×#327—21no—
28C61×#327—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×C4×D61—22no—
32#14519802—42no—
33#14519802—42no—
34C2×#11758201—22no—
35C2×#11758199—22no—
36C2×#11758194—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—42no—
40C4×D244—42no—
41C4×D244C2×C412yes—
42C4×D244D412yes—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—22no—
46C2×#11758198—21no—
47C2×#11758198—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418C2×C412yescomplemented
51#14519701—22no—
52#14519698—22no—
53C22⋊#92418—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684—21no—
57#14519684D412yes—
58C2×#41×C61—21no—
59C4⋊Q8×C61C2×C411yescharacteristic, complemented
60#127×C61C2×C411yescharacteristic, complemented
61#124×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C4×C8×C61—21no—
72#95×C61D411yescharacteristic
73#95×C61D411yescharacteristic
74C23×D61—21no—
75#41×C61—41no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79C2×Q8×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—41no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—22no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—21no—
95C2×D244C22⋊C412yes—
96C2×D244D812yes—
97C2×D244SD1612yes—
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—44no—
101C2×C4×D61—44no—
102C2×C4×D61—82no—
103C2×C4×D61—82no—
104C2×C8×C61—41no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—42no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C42×C61—41no—
111C42×C61—41no—
112C42×C61C22×C411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61#4111yescharacteristic
116#11758201—44no—
117#11758199—44no—
118#11758198—22no—
119#11758198—22no—
120#11758198—44no—
121C4⋊Dic61—21no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61C22⋊C412yes—
125#11758196—82no—
126#11758194—41no—
127#11758194—41no—
128C2×#92418—22no—
129C2×#92418—41no—
130C23×C61—41no—
131C23×C61#12011yescharacteristic
132C22×D61—22no—
133C22×D61—22no—
134C22×D61—22no—
135C22×D61—42no—
136C22×D61—42no—
137Q8×C61—41no—
138Q8×C61—41no—
139D4×C61—41no—
140D4×C61—41no—
141D4×C61—81no—
142D4×C61—81no—
143D4×C61—81no—
144D4×C61—81no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12311yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61C2×SD1611yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61#13711yescharacteristic
166C2×Dic61—22no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171D244—22no—
172D244—22no—
173D244—22no—
174D244#10214yes—
175C4×D61—82no—
176C4×D61—82no—
177C4×D61—88no—
178C8×C61—81no—
179#92418—42no—
180#92418—81no—
181#9179—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#38611yescharacteristic
188C22×C61#44711yescharacteristic
189C22×C61#44811yescharacteristic
190C2×D61—24no—
191C2×D61—44no—
192C2×D61—44no—
193C2×D61—42no—
194C2×D61—42no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—42no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61C2×#10211yescharacteristic
203C4×C61#41711yescharacteristic
204C4×C61#48011yescharacteristic
205C4×C61#48111yescharacteristic
206Dic61—42no—
207Dic61—82no—
208#3036—611no—
209#2349—611no—
210#1517—611no—
211#1502D6111yescharacteristic, complemented
212#1478—611no—
213#1303—611no—
214#1286—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—161no—
218C2×C61#187411yescharacteristic
219C2×C61#190411yescharacteristic
220C2×C61#190611yescharacteristic
221D61—44no—
222C2×AGL(1,ℤ/8)—1221no—
223#532C2×D6111yescharacteristic
224C2×C4×D4—611no—
225C2×C4×D4—1221no—
226#463—1222no—
227#461—612no—
228#459—612no—
229#431—611no—
230#431C2×D6111yescharacteristic
231#423—611no—
232#417—1221no—
233#389—1221no—
234#385—1222no—
235#382—611no—
236#380—1221no—
237C4×C4⋊C4C2×D6111yescharacteristic
238C2×#95—1221no—
239#327—21no—
240#327—21no—
241#327—1222no—
242D4⋊C8—1222no—
243C61#917911yescharacteristic, complemented, Sylow
244C2×#41—21no—
245C22×D4—611no—
246C23×C4—1222no—
247C23×C4—1221no—
248AGL(1,ℤ/8)—2442no—
249AGL(1,ℤ/8)—2442no—
250C2×SD16—1222no—
251C2×D8—1222no—
252C2×M16—1221no—
253C4⋊Q8C4×D6111yescharacteristic, complemented
254#127C4×D6111yescharacteristic, complemented
255#124—21no—
256#121—21no—
257#121—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—612no—
261C4×D4—612no—
262C4×D4—1221no—
263C4×D4—1221no—
264C4×D4—2442no—
265C4×D4—2442no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1222no—
272C2×C42—21no—
273C2×C42C22×D6111yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—612no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278#103—1222no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282C4×C8—21no—
283C4×C8—1221no—
284#95—612no—
285#95—1222no—
286#95—1221no—
287#95C22⋊D6111yescharacteristic
288#95C22⋊D6111yescharacteristic
289C24—1221no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—612no—
300C2×D4—612no—
301C2×D4—612no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—1221no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—4882no—
313C22×C4—4882no—
314C22×C4—2444no—
315C22×C4—2444no—
316C22×C4D4×D6111yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319SD16—2444no—
320D8—2444no—
321M16—2441no—
322M16—2441no—
323C2×C8—41no—
324C2×C8—1222no—
325C2×C8—2441no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—42no—
329C4⋊C4—612no—
330C4⋊C4—1221no—
331C4⋊C4—4882no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—1222no—
337C22⋊C4—1222no—
338C22⋊C4—2444no—
339C42—41no—
340C42—41no—
341C42C2×C4×D6111yescharacteristic
342C42#1175822211yescharacteristic
343C42D4×D6111yescharacteristic
344C42#1175822411yescharacteristic
345C23—41no—
346C23—1222no—
347C23—1222no—
348C23—1222no—
349C23—2442no—
350C23—2442no—
351C23#1451974511yescharacteristic
352Q8—41no—
353Q8—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—614no—
361D4—1222no—
362D4—1222no—
363D4—1222no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—1222no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4—4888no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4C22⋊C4×D6111yescharacteristic
384C2×C4#1451973711yescharacteristic
385C2×C4C4×C22⋊D6111yescharacteristic
386C2×C4#1451974811yescharacteristic
387C2×C4#1451975011yescharacteristic
388C2×C4#1451975911yescharacteristic
389C2×C4D8×D6111yescharacteristic
390C2×C4SD16×D6111yescharacteristic
391C2×C4#1451980111yescharacteristic
392C2×C4#1451980611yescharacteristic
393C8—81no—
394C8—4881no—
395C8—2442no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—1224no—
402C22—2444no—
403C22—2444no—
404C22—2442no—
405C22—2442no—
406C22#1683963811yescharacteristic
407C22#1683989411yescharacteristic
408C22#1683989911yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—81no—
415C4—81no—
416C4—4882no—
417C4—2442no—
418C4#102×D6111yescharacteristic
419C4#1683976011yescharacteristic
420C4#1684007411yescharacteristic
421C4#1684007911yescharacteristic
422C2—21no—
423C2—21no—
424C2—161no—
425C2—2444no—
426C2#2144613311yescharacteristic
427C2#2144630911yescharacteristic
428C2#2144632311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1502C2
socleC22×C61#386
prime radical#21455279C2
cosocleC2×C42×C61C23
hypercenter#1502D61
hypocenterC61#9179
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21448730C2semidirect
3. #21444379C2semidirect
4. #21444209C2semidirect
5. #21443722C2semidirect
6. #21443678C2semidirect
7. C61×#1502C2semidirect
12. #16839964C4semidirect
13. #16839954C4semidirect
50. C4⋊#92418C2×C4semidirect
59. C4⋊Q8×C61C2×C4semidirect
60. #127×C61C2×C4semidirect
211. #1502D61semidirect
243. C61#9179semidirect
253. C4⋊Q8C4×D61semidirect
254. #127C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214552792
#214487302
#214443792
#214442092
#214437222
#214436782
C61×#15022
#917961