Finite Group Databasebeta

← #29652784

Subgroups — #29652784

Order 15616.

1386 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic, complemented
2#21455910C212yesmaximal, complemented
3#21444838C214yesmaximal, complemented
4#21444736C214yesmaximal, complemented
5#21444326C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#2397C211yesmaximal, characteristic, complemented
8C2×#14519822—21no—
9#16840416—22no—
10#16840407—24no—
11D4×D244—22no—
12#16840343—21no—
13#16840324C2214yes—
14#16840312C2212yes—
15C2×#14519751—22no—
16#16840080C2214yescomplemented
17#16840052—22no—
18#16839957—22no—
19#16839896C2214yescomplemented
20#16839872—24no—
21#16839819C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C2×C4×D4×C61—22no—
25C2×C4×D4×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic, complemented
27C61×#399—21no—
28C61×#399C2212yescomplemented
29C61×#392—21no—
30C61×#392C2212yescomplemented
31C61×#376C2211yescharacteristic, complemented
32#14519822—48no—
33C2×#11758222—22no—
34C2×#11758224—24no—
35C22×D244—22no—
36C22×C22⋊D61—24no—
37C2×D4×D61—24no—
38C2×#11758199—24no—
39#14519777D412yescomplemented
40#14519774D412yescomplemented
41#14519765—22no—
42#14519761—28no—
43#14519757—24no—
44#14519752—24no—
45#14519751—24no—
46#14519751—24no—
47#14519751—22no—
48#14519751—24no—
49#14519751C2314yes—
50#14519747—28no—
51C4×D244—22no—
52C4×D244D414yescomplemented
53C2×C4⋊Dic61—22no—
54C2×#11758198—24no—
55#14519708—24no—
56C4⋊#92418—22no—
57#14519698—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61C22×D4×C61—22no—
62C22×D4×C61D411yescharacteristic
63C22×D4×C61C2311yescharacteristic
64C23×C4×C61—21no—
65C23×C4×C61D412yes—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C4×D4×C61—48no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61C2311yescharacteristic
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61C2312yes—
78C2×C42×C61—21no—
79C2×C42×C61C2311yescharacteristic
80#95×C61—21no—
81#95×C61C2312yes—
82C24×C61—21no—
83C24×C61C2×D412yes—
84C23×D61—24no—
85C2×D4×C61—24no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—44no—
89C2×D4×C61—42no—
90C2×D4×C61—44no—
91C2×D4×C61—44no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—44no—
97C22×C4×C61—44no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2411yescharacteristic
104C2×C22⋊D61—28no—
105C2×C22⋊D61—24no—
106C2×C22⋊D61—48no—
107C2×C22⋊D61—48no—
108C22×Dic61—24no—
109#11758224—416no—
110D4×D61—416no—
111#11758222—48no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—24no—
115C2×D244—84no—
116C2×D244C2×D414yes—
117C2×C4×D61—24no—
118C2×Dic122—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—44no—
122C22⋊C4×C61—24no—
123C22⋊C4×C61—44no—
124C22⋊C4×C61—44no—
125C22⋊C4×C61—42no—
126C42×C61—24no—
127C42×C61C2×D412yes—
128C42×C61C2×D412yes—
129#11758199—48no—
130#11758198—28no—
131#11758198—24no—
132#11758198—24no—
133C4⋊Dic61—24no—
134C4⋊Dic61—22no—
135C4⋊Dic61C2×D414yes—
136C2×#92418—24no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—22no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—42no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—42no—
146C23×C61AGL(1,ℤ/8)14yes—
147C23×C61C22×D411yescharacteristic
148C22×D61—24no—
149C22×D61—48no—
150C22×D61—84no—
151C22×D61—84no—
152D4×C61—28no—
153D4×C61—24no—
154D4×C61—42no—
155D4×C61—416no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—24no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—44no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—82no—
168C2×C4×C61—82no—
169C2×C4×C61—82no—
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12012yes—
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61C22×D412yes—
174C22⋊D61—416no—
175C22⋊D61—48no—
176C2×Dic61—24no—
177C2×Dic61—48no—
178D244—44no—
179D244—44no—
180D244—44no—
181D244—84no—
182C4×D61—48no—
183Dic122—44no—
184#92418—44no—
185#18178—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—24no—
189C22×C61—44no—
190C22×C61—44no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61C2×#12011yescharacteristic
199C22×C61C2×AGL(1,ℤ/8)12yes—
200C2×D61—44no—
201C2×D61—44no—
202C2×D61—44no—
203C2×D61—168no—
204C4×C61—22no—
205C4×C61—21no—
206C4×C61—22no—
207C4×C61—44no—
208C4×C61—42no—
209C4×C61—81no—
210C4×C61#45312yes—
211Dic61—44no—
212#3496—611no—
213#3109—612no—
214#2397D6111yescharacteristic, complemented
215#1613—614no—
216#1595—614no—
217#1505—612no—
218#1466—611no—
219C2×C61—22no—
220C2×C61—44no—
221C2×C61—82no—
222C2×C61#300311yescharacteristic
223C2×C61C2×#45311yescharacteristic
224C2×C61#302511yescharacteristic
225D61—84no—
226C2×H(5,𝔽2)—1221no—
227#547—1222no—
228#546—1224no—
229D42—1222no—
230#537—1221no—
231#534—614no—
232#532—612no—
233C2×#121—1222no—
234C2×C4×D4—22no—
235C2×C4×D4C2×D6111yescharacteristic
236C2×C4×D4C2×D6111yescharacteristic, complemented
237#481—614no—
238#476—1222no—
239#459—1222no—
240#447—614no—
241#442—1224no—
242#432—612no—
243#399—21no—
244#399C2×D6112yescomplemented
245#392—21no—
246#392C2×D6112yescomplemented
247#376C2×D6111yescharacteristic, complemented
248#327—618no—
249#323—614no—
250C61#1817811yescharacteristic, complemented, Sylow
251H(5,𝔽2)—2448no—
252C2×#41—1224no—
253C2×#41—1222no—
254C22×D4—22no—
255C22×D4—1222no—
256C22×D4—1224no—
257C22×D4—1224no—
258C22×D4C22⋊D6111yescharacteristic
259C22×D4C22×D6111yescharacteristic
260C23×C4—21no—
261C23×C4C22⋊D6112yes—
262C2×D8—1224no—
263C4⋊Q8—612no—
264#127—612no—
265#124—1222no—
266#123—1228no—
267C22⋊Q8—1224no—
268#121—614no—
269#121—1224no—
270#121—1224no—
271#121—1224no—
272#121—1222no—
273#121—1224no—
274#120—1228no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—48no—
280C4×D4—614no—
281C4×D4—1222no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4C22⋊D6111yescharacteristic
285C2×C4⋊C4C22×D6111yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1224no—
290C2×C22⋊C4C22×D6112yes—
291C2×C42—21no—
292C2×C42C22×D6111yescharacteristic
293#106—1224no—
294C4⋊C8—1222no—
295#102—1224no—
296#102—1224no—
297C22⋊C8—614no—
298#97—1222no—
299#95—21no—
300#95C22×D6112yes—
301C24—21no—
302C24—1224no—
303C24C2×C22⋊D6112yes—
304#41—2448no—
305#41—24416no—
306C2×Q8—1222no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—614no—
315C2×D4—1224no—
316C2×D4—1228no—
317C2×D4—1224no—
318C2×D4—1222no—
319C2×D4—1222no—
320C2×D4—4884no—
321C2×D4—24416no—
322C2×D4—2448no—
323C2×D4—2448no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—1224no—
333C22×C4—1224no—
334C22×C4C2×C22⋊D6111yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6112yes—
337C22×C4C23×D6111yescharacteristic
338D8—2448no—
339C2×C8—1224no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—44no—
343C4⋊C4—614no—
344C4⋊C4—1224no—
345C4⋊C4—1222no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—1228no—
351C22⋊C4—1224no—
352C22⋊C4—1224no—
353C42—24no—
354C42D4×D6112yes—
355C42D4×D6112yes—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—1224no—
366C23—4884no—
367C23—4884no—
368C23—2448no—
369C23#1451980214yes—
370C23C22×C22⋊D6111yescharacteristic
371Q8—2444no—
372D4—28no—
373D4—24no—
374D4—42no—
375D4—416no—
376D4—4884no—
377D4—2444no—
378D4—2444no—
379D4—24416no—
380D4—2448no—
381D4—2444no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—1224no—
397C2×C4—2448no—
398C2×C4—2448no—
399C2×C4#1451974512yes—
400C2×C4#1451974511yescharacteristic
401C2×C4#1451974511yescharacteristic
402C2×C4C2×D4×D6112yes—
403C8—2444no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—44no—
408C22—44no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22—81no—
416C22—9768no—
417C22—2444no—
418C22—2444no—
419C22—2444no—
420C22C2×#1451974511yescharacteristic
421C22C2×#1451980212yes—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—42no—
427C4—81no—
428C4—2444no—
429C4#1683992912yes—
430C2—22no—
431C2—44no—
432C2—82no—
433C2—4884no—
434C2#2145504511yescharacteristic
435C2C2×#1683992911yescharacteristic
436C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2397C2
socleC22×C61C2×#120
hypercenter#2397D61
hypocenterC61#18178
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #21459583C2semidirect
2. #21455910C2semidirect
3. #21444838C2semidirect
4. #21444736C2semidirect
5. #21444326C2semidirect
6. #21444160C2semidirect
7. C61×#2397C2semidirect
16. #16840080C22semidirect
19. #16839896C22semidirect
21. #16839819C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
26. C2×C4×D4×C61C22semidirect
28. C61×#399C22semidirect
30. C61×#392C22semidirect
31. C61×#376C22semidirect
39. #14519777D4semidirect
40. #14519774D4semidirect
52. C4×D244D4semidirect
214. #2397D61semidirect
236. C2×C4×D4C2×D61semidirect
244. #399C2×D61semidirect
246. #392C2×D61semidirect
247. #376C2×D61semidirect
250. C61#18178semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214559102
#214448382
#214447362
#214443262
#214441602
C61×#23972
#1817861