Finite Group Databasebeta

← #29581405

Subgroups — #29581405

Order 15616.

480 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 83 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#21457011C211yesmaximal, characteristic
2#21456215C211yesmaximal, characteristic
3#21451769C211yesmaximal, characteristic, complemented
4#21444422C211yesmaximal, characteristic, complemented
5#21444328C211yesmaximal, characteristic
6#21443711C211yesmaximal, characteristic, complemented
7C61×#1556C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×#14519769C2211yescharacteristic
10#16840171—21no—
11#16840127—21no—
12#16840124—22no—
13#16840114—21no—
14#16840110—21no—
15#16840081—21no—
16#16840076—21no—
17#16840063—22no—
18#16839761—22no—
19C4×#11758194C2211yescharacteristic
20#16839700—22no—
21#16839700—22no—
22#16839669—21no—
23#16839669—21no—
24#16839608C2211yescharacteristic
25#16839503—22no—
26#16839497—22no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C61×#432C2211yescharacteristic
30C61×#332—22no—
31C2×#11758222—21no—
32C2×#11758194—21no—
33#14519777D411yescharacteristic
34#14519774D411yescharacteristic
35#14519769—21no—
36#14519769—21no—
37#14519769—21no—
38#14519769—21no—
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×C4×Dic61—21no—
49#14519711—22no—
50#14519709—22no—
51#14519700—24no—
52#14519699—22no—
53#14519699—22no—
54#14519697—24no—
55#14519688D412yes—
56C4×#92418—21no—
57#14519684—24no—
58#14519684—24no—
59#14519684—21no—
60#14519684D412yes—
61C22×Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61D412yes—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C4⋊C8×C61D412yes—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73#97×C61—21no—
74#97×C61—21no—
75C2×Q8×C61—44no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C22×C4×C61—22no—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C2×C22⋊D61—41no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86#11758222—44no—
87C2×D244—21no—
88C2×D244—21no—
89C2×D244—21no—
90C2×C4×D61—41no—
91C2×Dic122—21no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61#4111yescharacteristic
102#11758198—41no—
103#11758198—41no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—41no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—42no—
115#11758194—42no—
116C2×#92418—22no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119Q8×C61—82no—
120Q8×C61—82no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—41no—
124C2×C4×C61—44no—
125C2×C4×C61—42no—
126C2×C4×C61#12011yescharacteristic
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61#12111yescharacteristic
130C2×C4×C61#12111yescharacteristic
131C2×C4×C61#12711yescharacteristic
132C22⋊D61—82no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141D244—42no—
142D244—42no—
143D244—81no—
144C4×D61—82no—
145Dic122—42no—
146C8×C61—42no—
147C8×C61—81no—
148#92418—42no—
149#10095—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#39211yescharacteristic
153C22×C61#45412yes—
154C2×D61—81no—
155C2×D61—81no—
156C2×D61—81no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—82no—
163C4×C61#45412yes—
164Dic61—82no—
165Dic61—81no—
166Dic61—81no—
167Dic61—81no—
168#3234—611no—
169#3141—611no—
170#2642—611no—
171#1556D6111yescharacteristic, complemented
172#1528—611no—
173#1505—611no—
174#1300—611no—
175C2×C61—22no—
176C2×C61#200311yescharacteristic
177C2×C61#200311yescharacteristic
178C2×C61#200911yescharacteristic
179D61—161no—
180#532—611no—
181C2×C4⋊Q8C2×D6111yescharacteristic
182C2×#125—611no—
183#501—1221no—
184#490—1221no—
185#489—1222no—
186#487—1221no—
187#486—1221no—
188#481—1221no—
189#480—1221no—
190#478—1222no—
191#432C2×D6111yescharacteristic
192#432C2×D6111yescharacteristic
193#417—1222no—
194C4×M16—611no—
195#400—1222no—
196#400—1222no—
197#391—1221no—
198#391—1221no—
199#376—611no—
200#332—22no—
201#330—1222no—
202#327—1222no—
203C61#1009511yescharacteristic, complemented, Sylow
204C2×#41—1221no—
205C22×Q8—21no—
206C2×M16—1221no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—611no—
212#127—611no—
213#125—1221no—
214#125—1221no—
215#125—1221no—
216#125—1221no—
217#124—1221no—
218#121—1221no—
219#121—1221no—
220C4×D4—1221no—
221C4×D4—1221no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4D24412yes—
228C2×C42—1221no—
229C2×C42C22×D6111yescharacteristic
230#107—1222no—
231#106—1222no—
232C4⋊C8—21no—
233C4⋊C8C22⋊D6112yes—
234#103—1224no—
235#102—1224no—
236#102—1222no—
237#102—1222no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#97—21no—
241#97—21no—
242#97—612no—
243C4×C8—1221no—
244#95—612no—
245#95—1224no—
246#95—1224no—
247#95—1221no—
248#41—2444no—
249C2×Q8—44no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—1221no—
253C2×D4—1221no—
254C2×D4—1221no—
255C2×D4—1221no—
256C2×D4—2441no—
257C22×C4—22no—
258C22×C4—1222no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—2441no—
262C22×C4C2×D24411yescharacteristic
263C22×C4D4×D6111yescharacteristic
264C22×C4D4×D6111yescharacteristic
265M16—2442no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—1222no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—1221no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C22⋊C4—2441no—
284C22⋊C4—2441no—
285C42—2442no—
286C42#1175822211yescharacteristic
287C42D4×D6111yescharacteristic
288C42D4×D6111yescharacteristic
289C42C2×C22⋊D6111yescharacteristic
290C23—2441no—
291C23#1451974711yescharacteristic
292Q8—82no—
293Q8—82no—
294Q8—2442no—
295D4—4881no—
296D4—4882no—
297D4—2442no—
298D4—2442no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—4882no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4#1451974511yescharacteristic
314C2×C4#1451974711yescharacteristic
315C2×C4#1451975011yescharacteristic
316C2×C4#1451975011yescharacteristic
317C2×C4#1451975111yescharacteristic
318C2×C4#1451977311yescharacteristic
319C8—42no—
320C8—81no—
321C8—2442no—
322C22—21no—
323C22—21no—
324C22—4881no—
325C22—4881no—
326C22—4881no—
327C22#1683967111yescharacteristic
328C22#1683993712yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—4882no—
336C4—4881no—
337C4—4881no—
338C4—4881no—
339C4#1683993312yes—
340C2—22no—
341C2—9761no—
342C2#2144684211yescharacteristic
343C2#2144684311yescharacteristic
344C2#2144688111yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1556C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#1556D61
hypocenterC61#10095
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
3. #21451769C2semidirect
4. #21444422C2semidirect
6. #21443711C2semidirect
7. C61×#1556C2semidirect
171. #1556D61semidirect
203. C61#10095semidirect

Maximal subgroups

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

subgroupindex
#214570112
#214562152
#214517692
#214444222
#214443282
#214437112
C61×#15562
#1009561