Finite Group Databasebeta

← #29575946

Subgroups — #29575946

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#21455283C211yesmaximal, characteristic
2#21448729C211yesmaximal, characteristic, complemented
3#21444380C211yesmaximal, characteristic, complemented
4#21444315C211yesmaximal, characteristic, complemented
5#21443721C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1478C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10C2×D4×Dic61—21no—
11#16839977—22no—
12#16839967C412yescomplemented
13#16839957C412yescomplemented
14#16839786C2211yescharacteristic
15#16839761—21no—
16#16839661—21no—
17#16839637—22no—
18#16839628C2211yescharacteristic
19#16839622—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21C2×#14519684—21no—
22#16839497—22no—
23#16839497—21no—
24#16839497—21no—
25C2×C4×D4×C61C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C61×D4⋊C8—22no—
29C2×#11758222—21no—
30C23×Dic61—22no—
31#14519803—42no—
32#14519803—42no—
33C2×#11758200—22no—
34C2×#11758199—22no—
35C2×#11758194—21no—
36#14519777C2×C411yescharacteristic, complemented
37#14519774C2×C411yescharacteristic, complemented
38#14519765—21no—
39#14519751—21no—
40#14519751—21no—
41C4×D244—21no—
42C4×D244—21no—
43D4×Dic61—42no—
44D4×Dic61—42no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—22no—
48C2×C22⋊Dic61—22no—
49C2×C4×Dic61—21no—
50C4⋊#92418—21no—
51C4⋊#92418C2×C412yescomplemented
52#14519700—22no—
53#14519697—22no—
54#14519684—22no—
55#14519684—21no—
56#14519684D411yescharacteristic
57#14519684D411yescharacteristic
58#14519684D412yes—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61—21no—
61C4×D4×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61C2×C412yes—
64C4×D4×C61D412yes—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C22⋊C4×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—21no—
69C4⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C4×C8×C61—21no—
73C4×C8×C61—21no—
74C24×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—21no—
78C2×D4×C61C22⋊C412yes—
79C2×D4×C61D812yes—
80C2×D4×C61SD1612yes—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—82no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C2×C22⋊D61—41no—
89C22×Dic61—21no—
90C22×Dic61—22no—
91C22×Dic61—21no—
92C22×Dic61—44no—
93C22×Dic61—44no—
94C22×Dic61—82no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61C22⋊C412yes—
108C22⋊C4×C61—22no—
109C22⋊C4×C61—22no—
110C42×C61C22×C411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114C22⋊Dic61—44no—
115#11758200—44no—
116#11758199—44no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—42no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61—82no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127#11758194—41no—
128#11758194—41no—
129C2×#92418—22no—
130C23×C61—22no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61#12011yescharacteristic
136C22×D61—41no—
137D4×C61—22no—
138D4×C61—22no—
139D4×C61—22no—
140D4×C61#10214yes—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—82no—
146C2×C4×C61—82no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12311yescharacteristic
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C2×SD1611yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61#13711yescharacteristic
157C22⋊D61—81no—
158C22⋊D61—81no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—88no—
170D244—41no—
171D244—41no—
172D244—81no—
173D244—81no—
174C4×D61—81no—
175C4×D61—81no—
176Dic122—41no—
177Dic122—41no—
178C8×C61—81no—
179C8×C61—81no—
180#92418—42no—
181#9179—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—24no—
185C22×C61—44no—
186C22×C61—44no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61#38611yescharacteristic
190C22×C61#44711yescharacteristic
191C22×C61#44811yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—42no—
200C4×C61C2×#10211yescharacteristic
201C4×C61#41711yescharacteristic
202C4×C61#48011yescharacteristic
203C4×C61#48111yescharacteristic
204Dic61—42no—
205Dic61—81no—
206Dic61—82no—
207Dic61—81no—
208#3036—611no—
209#2349—611no—
210#1517—611no—
211#1502—611no—
212#1478D6111yescharacteristic, complemented
213#1303—611no—
214#1286—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—44no—
218C2×C61#187411yescharacteristic
219C2×C61#190411yescharacteristic
220C2×C61#190611yescharacteristic
221D61—161no—
222C2×AGL(1,ℤ/8)—1221no—
223#532—611no—
224C2×C4×D4—1221no—
225C2×C4×D4C2×D6111yescharacteristic
226#463—1222no—
227#461—612no—
228#459—612no—
229#431C2×D6111yescharacteristic
230#431C2×D6111yescharacteristic
231#423—611no—
232#417—1221no—
233#389—1221no—
234#385—1222no—
235#382—611no—
236#380—1221no—
237C4×C4⋊C4—611no—
238C2×#95—1221no—
239#327—1221no—
240#327—1221no—
241#327—1222no—
242D4⋊C8—22no—
243C61#917911yescharacteristic, complemented, Sylow
244C2×#41—1221no—
245C22×D4D24411yescharacteristic
246C23×C4—21no—
247C23×C4—1222no—
248AGL(1,ℤ/8)—2442no—
249AGL(1,ℤ/8)—2442no—
250C2×SD16—1222no—
251C2×D8—1222no—
252C2×M16—1221no—
253C4⋊Q8—611no—
254#127—611no—
255#124—1221no—
256#121—1221no—
257#121—1221no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—1221no—
261C4×D4—1221no—
262C4×D4—2442no—
263C4×D4—2442no—
264C4×D4C4×D6112yes—
265C4×D4C22⋊D6112yes—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4D24411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1222no—
272C2×C42—1221no—
273C2×C42C22×D6111yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—612no—
277C4⋊C8—1221no—
278#103—1222no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284#95—611no—
285#95—611no—
286#95—612no—
287#95—1222no—
288#95—1221no—
289C24—21no—
290#41—2441no—
291#41—2441no—
292#41—2441no—
293#41—2441no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—2441no—
302C2×D4#1175819012yes—
303C2×D4D48812yes—
304C2×D4#1175819812yes—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—82no—
309C22×C4—1221no—
310C22×C4—1222no—
311C22×C4—1221no—
312C22×C4—4882no—
313C22×C4—2444no—
314C22×C4—2444no—
315C22×C4—2441no—
316C22×C4C2×D24411yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319SD16—2444no—
320D8—2444no—
321M16—2441no—
322M16—2441no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—1222no—
326C4⋊C4—21no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—4882no—
330C4⋊C4—2442no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4#1175819812yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—2444no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—2441no—
340C42—2441no—
341C42C2×C4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42#1175822411yescharacteristic
344C42C2×C22⋊D6111yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—2441no—
351C23#1451974711yescharacteristic
352Q8—2441no—
353Q8—2441no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—4881no—
358D4—4881no—
359D4—4881no—
360D4—4881no—
361D4—2441no—
362D4—2441no—
363D4#1451969914yes—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—1222no—
371C2×C4—4888no—
372C2×C4—4881no—
373C2×C4—4881no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4C2×#1175819811yescharacteristic
384C2×C4#1451973711yescharacteristic
385C2×C4#1451973711yescharacteristic
386C2×C4#1451974911yescharacteristic
387C2×C4#1451974911yescharacteristic
388C2×C4#1451976111yescharacteristic
389C2×C4C2×D48811yescharacteristic
390C2×C4C2×#1175819011yescharacteristic
391C2×C4#1451980511yescharacteristic
392C2×C4#1451981011yescharacteristic
393C8—81no—
394C8—81no—
395C8—2442no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22#1683963911yescharacteristic
407C22#1683989811yescharacteristic
408C22#1683990311yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—4882no—
415C4—4881no—
416C4—4881no—
417C4—2442no—
418C4C2×#1451969911yescharacteristic
419C4#1683976411yescharacteristic
420C4#1684007611yescharacteristic
421C4#1684008111yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—9761no—
426C2#2144613411yescharacteristic
427C2#2144630411yescharacteristic
428C2#2144631811yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1478C2
socleC22×C61#386
prime radical#21455283C2
cosocleC2×C42×C61C23
hypercenter#1478D61
hypocenterC61#9179
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #21448729C2semidirect
3. #21444380C2semidirect
4. #21444315C2semidirect
5. #21443721C2semidirect
6. #21443678C2semidirect
7. C61×#1478C2semidirect
12. #16839967C4semidirect
13. #16839957C4semidirect
36. #14519777C2×C4semidirect
37. #14519774C2×C4semidirect
51. C4⋊#92418C2×C4semidirect
212. #1478D61semidirect
243. C61#9179semidirect

Maximal subgroups

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

subgroupindex
#214552832
#214487292
#214443802
#214443152
#214437212
#214436782
C61×#14782
#917961