Finite Group Databasebeta

← #29654855

Subgroups — #29654855

Order 15616.

1174 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 191 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#21456744C212yesmaximal, complemented
3#21444787C218yesmaximal, complemented
4#21444313C212yesmaximal, complemented
5#21444143C211yesmaximal, characteristic, complemented
6C61×#2567C211yesmaximal, characteristic, complemented
7C2×#14519825—21no—
8#16840434—22no—
9Q8×D244—22no—
10#16840424—24no—
11#16840343—21no—
12#16840343C2214yes—
13#16840312—22no—
14#16840312C2212yes—
15#16840136—22no—
16#16840112—22no—
17#16839964—24no—
18#16839901C2218yes—
19#16839815C2212yes—
20#16839495C2218yes—
21#16839488C2214yes—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×C4×Q8×C61C2211yescharacteristic
24C61×#395C2214yes—
25C61×#391—21no—
26C61×#389—22no—
27C61×#384—21no—
28C4×C4⋊C4×C61C2211yescharacteristic
29#14519825—44no—
30#14519825—44no—
31C2×#11758226—24no—
32C2×#11758222—24no—
33C2×#11758222—22no—
34C2×Q8×D61—24no—
35C2×#11758201—28no—
36#14519777—22no—
37#14519777D412yes—
38#14519774—22no—
39#14519774D412yes—
40#14519765—24no—
41#14519765—22no—
42#14519757—24no—
43#14519757—28no—
44#14519752—24no—
45#14519752—28no—
46#14519751—22no—
47#14519751—24no—
48#14519751C2314yes—
49C4×D244—24no—
50C4×D244—24no—
51C4×D244—22no—
52C4×D244D414yes—
53C4⋊#92418—22no—
54#14519701—28no—
55C22⋊#92418C2314yes—
56C4×#92418—22no—
57C22×Q8×C61D411yescharacteristic
58C22×Q8×C61C2312yes—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4×Q8×C61—22no—
62C4×Q8×C61—22no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—24no—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61C2312yes—
69C2×C42×C61—21no—
70C2×C42×C61D412yes—
71C2×C42×C61C2311yescharacteristic
72#95×C61—24no—
73#95×C61C2312yes—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61—24no—
77C2×Q8×C61—82no—
78C2×Q8×C61SD1614yescomplemented
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2411yescharacteristic
86C2×C22⋊D61—24no—
87#11758226—416no—
88Q8×D61—416no—
89#11758222—48no—
90#11758222—48no—
91#11758222—44no—
92#11758222—44no—
93C2×D244—24no—
94C2×D244—22no—
95C2×D244—22no—
96C2×D244—24no—
97C2×D244C2×D414yes—
98C2×C4×D61—28no—
99C2×C4×D61—24no—
100C2×Dic122—24no—
101C2×Dic122—22no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—42no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—84no—
109C42×C61—22no—
110C42×C61—22no—
111C42×C61—42no—
112C42×C61C2×D412yes—
113C42×C61C2×D412yes—
114#11758201—88no—
115#11758198—28no—
116#11758198—24no—
117#11758198—24no—
118C4⋊Dic61—24no—
119C4⋊Dic61—22no—
120C4⋊Dic61C2×D414yes—
121C2×#92418—24no—
122C23×C61C22×D411yescharacteristic
123C22×D61—24no—
124Q8×C61—28no—
125Q8×C61—48no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—24no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—44no—
136C2×C4×C61—42no—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12012yes—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61C2×SD1614yes—
141C2×C4×C61C2×SD1612yes—
142C2×C4×C61AGL(1,ℤ/8)12yes—
143C2×C4×C61C22×D412yes—
144C22⋊D61—44no—
145C22⋊D61—44no—
146C2×Dic61—24no—
147D244—44no—
148D244—48no—
149D244—44no—
150D244—44no—
151D244—84no—
152C4×D61—416no—
153C4×D61—48no—
154Dic122—48no—
155Dic122—44no—
156#92418—44no—
157#18402—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61C2×#12011yescharacteristic
161C22×C61C22×SD1611yescharacteristic
162C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
163C2×D61—44no—
164C2×D61—44no—
165C2×D61—44no—
166C4×C61—24no—
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—44no—
170C4×C61—42no—
171C4×C61—82no—
172C4×C61#45014yes—
173C4×C61#45612yes—
174Dic61—44no—
175#3497—611no—
176#3206—612no—
177#2567D6111yescharacteristic, complemented
178#1604—618no—
179#1502—612no—
180#1461—611no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61C2×#45011yescharacteristic
184C2×C61C2×#45611yescharacteristic
185C2×C61#302711yescharacteristic
186D61—84no—
187C2×2−1+4—1221no—
188#550—1222no—
189D4×Q8—1222no—
190#548—1224no—
191#537—614no—
192#537—1221no—
193#532—612no—
194#532—1222no—
195C2×C4⋊Q8C2×D6111yescharacteristic
196C2×C4×Q8C2×D6111yescharacteristic
197#492—1222no—
198#487—1222no—
199#461—1224no—
200#448—618no—
201#431—612no—
202#395C2×D6114yes—
203#391—21no—
204#389—22no—
205#384—21no—
206C4×C4⋊C4C2×D6111yescharacteristic
207#327—618no—
208#324—614no—
209C61#1840211yescharacteristic, complemented, Sylow
2102−1+4—2444no—
2112−1+4—2444no—
212C2×#41—1224no—
213C2×#41—1222no—
214C2×#41—1224no—
215C22×Q8—1224no—
216C22×Q8C22⋊D6111yescharacteristic
217C22×Q8C22×D6112yes—
218C2×SD16—1228no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—612no—
222C4⋊Q8—1222no—
223#127—612no—
224#127—1222no—
225#124—1224no—
226#124—1222no—
227C22⋊Q8—1228no—
228C22⋊Q8—1224no—
229#121—614no—
230#121—1228no—
231#121—1224no—
232#121—1222no—
233#121—1224no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×D4—614no—
237C4×D4—1224no—
238C4×D4—1224no—
239C4×D4—1222no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4C22⋊D6111yescharacteristic
245C2×C4⋊C4C22×D6112yes—
246C2×C42—21no—
247C2×C42C22⋊D6112yes—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—1222no—
250#103—1228no—
251C22⋊C8—614no—
252C4×C8—1222no—
253#95—24no—
254#95C22×D6112yes—
255#41—2448no—
256#41—24416no—
257#41—2448no—
258#41—2444no—
259#41—2444no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—82no—
264C2×Q8—1224no—
265C2×Q8—1222no—
266C2×Q8—24416no—
267C2×Q8#1175820114yescomplemented
268C2×D4—614no—
269C2×D4—1224no—
270C2×D4—1224no—
271C2×D4—1224no—
272C2×D4—1222no—
273C2×D4—1222no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—1228no—
277C22×C4—1224no—
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C2×C22⋊D6112yes—
280C22×C4C2×C22⋊D6112yes—
281C22×C4C2×C22⋊D6111yescharacteristic
282C22×C4C23×D6111yescharacteristic
283SD16—4888no—
284C2×C8—1224no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—84no—
292C4⋊C4—614no—
293C4⋊C4—1224no—
294C4⋊C4—1222no—
295C22⋊C4—1228no—
296C22⋊C4—1224no—
297C22⋊C4—1224no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42D4×D6112yes—
302C42D4×D6112yes—
303C23—1224no—
304C23C22×C22⋊D6111yescharacteristic
305Q8—28no—
306Q8—48no—
307Q8—2448no—
308Q8—2444no—
309D4—4884no—
310D4—2444no—
311D4—2448no—
312D4—2444no—
313D4—2444no—
314D4—2444no—
315D4—2444no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—1224no—
328C2×C4—24416no—
329C2×C4—2448no—
330C2×C4#1451974511yescharacteristic
331C2×C4#1451974511yescharacteristic
332C2×C4#1451974512yes—
333C2×C4C2×#1175820114yes—
334C2×C4C2×#1175820112yes—
335C2×C4#1451980212yes—
336C2×C4C2×D4×D6112yes—
337C8—2444no—
338C22—21no—
339C22—21no—
340C22—2444no—
341C22—2444no—
342C22—2444no—
343C22C2×#1451974511yescharacteristic
344C22C22×#1175820111yescharacteristic
345C22C2×#1451980211yescharacteristic
346C4—21no—
347C4—22no—
348C4—24no—
349C4—44no—
350C4—42no—
351C4—82no—
352C4—2444no—
353C4#1683991214yes—
354C4#1683994412yes—
355C2—21no—
356C2—21no—
357C2—4884no—
358C2C2×#1683991211yescharacteristic
359C2C2×#1683994411yescharacteristic
360C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2567C2
socleC22×C61C2×#120
prime radical#21459590C2
hypercenter#2567D61
hypocenterC61#18402
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. #21456744C2semidirect
3. #21444787C2semidirect
4. #21444313C2semidirect
5. #21444143C2semidirect
6. C61×#2567C2semidirect
78. C2×Q8×C61SD16semidirect
177. #2567D61semidirect
209. C61#18402semidirect
267. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214595902
#214567442
#214447872
#214443132
#214441432
C61×#25672
#1840261