Finite Group Databasebeta

← #29653131

Subgroups — #29653131

Order 15616.

1154 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459590C211yesmaximal, characteristic
2#21455850C212yesmaximal, complemented
3#21444855C214yesmaximal, complemented
4#21444787C214yesmaximal, complemented
5#21444378C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#2581C211yesmaximal, characteristic, complemented
8C2×#14519825—21no—
9#16840434—22no—
10Q8×D244—22no—
11#16840424—24no—
12#16840343—21no—
13#16840343C2214yes—
14#16840312—22no—
15#16840312C2212yes—
16#16840080C2214yes—
17#16840047—22no—
18#16839962—22no—
19#16839901C2214yes—
20C4×#11758201—24no—
21#16839815C2212yes—
22#16839495C2218yes—
23#16839488C2214yes—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#400C2212yes—
27C61×#395C2212yes—
28C61×#391—21no—
29C61×#389—21no—
30C61×#387—22no—
31C61×#382C2211yescharacteristic
32#14519825—44no—
33#14519825—44no—
34C2×#11758226—24no—
35C2×#11758222—24no—
36C2×#11758222—22no—
37C2×Q8×D61—24no—
38C2×#11758201—24no—
39#14519777—22no—
40#14519777D412yes—
41#14519774—22no—
42#14519774D412yes—
43#14519765—24no—
44#14519765—22no—
45#14519757—24no—
46#14519757—28no—
47#14519752—24no—
48#14519752—28no—
49#14519751—22no—
50#14519751—24no—
51#14519751C2314yes—
52C4×D244—24no—
53C4×D244—24no—
54C4×D244—22no—
55C4×D244D414yes—
56#14519708—24no—
57C4⋊#92418—22no—
58#14519701—24no—
59#14519697—24no—
60C22⋊#92418C2314yes—
61C4×#92418—22no—
62C22×Q8×C61D411yescharacteristic
63C22×Q8×C61C2311yescharacteristic
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×Q8×C61—22no—
67C4×Q8×C61—22no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61C2312yes—
74C2×C42×C61—21no—
75C2×C42×C61D412yes—
76C2×C42×C61C2311yescharacteristic
77#95×C61—21no—
78#95×C61—22no—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61C2312yes—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—42no—
86C2×Q8×C61SD1614yescomplemented
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2411yescharacteristic
95C2×C22⋊D61—24no—
96#11758226—416no—
97Q8×D61—416no—
98#11758222—48no—
99#11758222—48no—
100#11758222—44no—
101#11758222—44no—
102C2×D244—24no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—24no—
106C2×D244C2×D414yes—
107C2×C4×D61—28no—
108C2×C4×D61—24no—
109C2×Dic122—24no—
110C2×Dic122—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—44no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—42no—
118C42×C61—22no—
119C42×C61—22no—
120C42×C61—42no—
121C42×C61C2×D412yes—
122C42×C61C2×D412yes—
123#11758201—48no—
124#11758198—28no—
125#11758198—24no—
126#11758198—24no—
127C4⋊Dic61—24no—
128C4⋊Dic61—22no—
129C4⋊Dic61C2×D414yes—
130C2×#92418—24no—
131C23×C61C22×D411yescharacteristic
132C22×D61—24no—
133Q8×C61—28no—
134Q8×C61—24no—
135Q8×C61—42no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61#12012yes—
153C2×C4×C61C2×SD1614yes—
154C2×C4×C61C2×SD1612yes—
155C2×C4×C61AGL(1,ℤ/8)12yes—
156C2×C4×C61C22×D412yes—
157C22⋊D61—44no—
158C22⋊D61—44no—
159C2×Dic61—24no—
160D244—44no—
161D244—44no—
162D244—48no—
163D244—44no—
164D244—84no—
165C4×D61—416no—
166C4×D61—48no—
167Dic122—48no—
168Dic122—44no—
169#92418—44no—
170#18219—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61C2×#12011yescharacteristic
174C22×C61C22×SD1611yescharacteristic
175C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
176C2×D61—44no—
177C2×D61—44no—
178C2×D61—44no—
179C4×C61—24no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—22no—
183C4×C61—42no—
184C4×C61—42no—
185C4×C61—82no—
186C4×C61—81no—
187C4×C61#45014yes—
188Dic61—44no—
189#3497—611no—
190#3103—612no—
191#2581D6111yescharacteristic, complemented
192#1616—614no—
193#1604—614no—
194#1517—612no—
195#1461—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61C2×#45011yescharacteristic
199C2×C61#302011yescharacteristic
200C2×C61#302511yescharacteristic
201D61—84no—
202C2×2−1+4—1221no—
203#550—1222no—
204D4×Q8—1222no—
205#548—1224no—
206#537—614no—
207#537—1221no—
208#532—612no—
209#532—1222no—
210C2×C4×Q8C2×D6111yescharacteristic
211C2×C4×Q8C2×D6111yescharacteristic
212#481—614no—
213#475—1222no—
214#460—1222no—
215#448—614no—
216C4×SD16—1224no—
217#431—612no—
218#400C2×D6112yes—
219#395C2×D6112yes—
220#391—21no—
221#389—21no—
222#387—22no—
223#382C2×D6111yescharacteristic
224#327—618no—
225#324—614no—
226C61#1821911yescharacteristic, complemented, Sylow
2272−1+4—2444no—
2282−1+4—2444no—
229C2×#41—1224no—
230C2×#41—1222no—
231C2×#41—1224no—
232C22×Q8—1224no—
233C22×Q8C22⋊D6111yescharacteristic
234C22×Q8C22×D6111yescharacteristic
235C2×SD16—1224no—
236C4⋊Q8—612no—
237C4⋊Q8—1222no—
238#127—612no—
239#127—1222no—
240#124—1224no—
241#124—1222no—
242C22⋊Q8—1228no—
243C22⋊Q8—1224no—
244#121—614no—
245#121—1228no—
246#121—1224no—
247#121—1222no—
248#121—1224no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×D4—614no—
254C4×D4—1224no—
255C4×D4—1224no—
256C4×D4—1222no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C4⋊C4C22×D6111yescharacteristic
262C2×C4⋊C4C22×D6112yes—
263C2×C42—21no—
264C2×C42C22⋊D6112yes—
265C2×C42C22×D6111yescharacteristic
266#106—1224no—
267C4⋊C8—1222no—
268#103—1224no—
269#102—1224no—
270C22⋊C8—614no—
271C4×C8—1222no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95C22×D6112yes—
277#41—2448no—
278#41—24416no—
279#41—2448no—
280#41—2444no—
281#41—2444no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—1224no—
287C2×Q8—1222no—
288C2×Q8—24416no—
289C2×Q8#1175820114yescomplemented
290C2×D4—614no—
291C2×D4—1224no—
292C2×D4—1224no—
293C2×D4—1224no—
294C2×D4—1222no—
295C2×D4—1222no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—1228no—
300C22×C4—1224no—
301C22×C4C2×C22⋊D6112yes—
302C22×C4C2×C22⋊D6112yes—
303C22×C4C2×C22⋊D6111yescharacteristic
304C22×C4C2×C22⋊D6111yescharacteristic
305C22×C4C23×D6111yescharacteristic
306SD16—2448no—
307C2×C8—1224no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—614no—
316C4⋊C4—1224no—
317C4⋊C4—1222no—
318C22⋊C4—1228no—
319C22⋊C4—1224no—
320C22⋊C4—1224no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42D4×D6112yes—
325C42D4×D6112yes—
326C23—1224no—
327C23C22×C22⋊D6111yescharacteristic
328Q8—28no—
329Q8—24no—
330Q8—42no—
331Q8—2448no—
332Q8—2444no—
333D4—4884no—
334D4—2444no—
335D4—2444no—
336D4—2448no—
337D4—2444no—
338D4—2444no—
339D4—2444no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—1224no—
355C2×C4—24416no—
356C2×C4—2448no—
357C2×C4#1451974511yescharacteristic
358C2×C4#1451974511yescharacteristic
359C2×C4#1451974512yes—
360C2×C4C2×#1175820114yes—
361C2×C4C2×#1175820112yes—
362C2×C4#1451980212yes—
363C2×C4C2×D4×D6112yes—
364C8—2444no—
365C22—21no—
366C22—21no—
367C22—2444no—
368C22—2444no—
369C22—2444no—
370C22C2×#1451974511yescharacteristic
371C22C22×#1175820111yescharacteristic
372C22C2×#1451980211yescharacteristic
373C4—21no—
374C4—21no—
375C4—22no—
376C4—24no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—81no—
381C4—2444no—
382C4#1683991214yes—
383C2—21no—
384C2—21no—
385C2—4884no—
386C2C2×#1683991211yescharacteristic
387C2#2145518211yescharacteristic
388C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2581C2
socleC22×C61C2×#120
prime radical#21459590C2
hypercenter#2581D61
hypocenterC61#18219
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. #21444855C2semidirect
4. #21444787C2semidirect
5. #21444378C2semidirect
6. #21444143C2semidirect
7. C61×#2581C2semidirect
86. C2×Q8×C61SD16semidirect
191. #2581D61semidirect
226. C61#18219semidirect
289. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214595902
#214558502
#214448552
#214447872
#214443782
#214441432
C61×#25812
#1821961