Finite Group Databasebeta

← #29572247

Subgroups — #29572247

Order 15616.

834 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 139 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#21446660C212yesmaximal, complemented
2#21444557C211yesmaximal, characteristic, complemented
3#21444414C211yesmaximal, characteristic, complemented
4#21444258C212yesmaximal, complemented
5C61×#3508C211yesmaximal, characteristic
6#16839786C2212yes—
7C2×#14519694—24no—
8C4×#11758194C2212yes—
9#16839508—22no—
10#16839504—22no—
11Q8⋊#92418—22no—
12D4⋊#92418—22no—
13C61×#559—22no—
14C61×#550—22no—
15C61×#546—24no—
16C61×#540—22no—
17C61×#535C412yes—
18C61×#533C2211yescharacteristic
19C61×#532—22no—
20C2×#127×C61C2211yescharacteristic
21C61×#518C412yes—
22C4×#41×C61C2211yescharacteristic
23C2×#11758194—22no—
24C2×#11758194—22no—
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27#14519694—48no—
28#14519688—22no—
29C4×#92418—22no—
30C2×#41×C61—24no—
31C2×#41×C61C2×C411yescharacteristic
32C22×D4×C61C2×C412yes—
33C22×D4×C61D414yes—
34C4⋊Q8×C61D412yes—
35#127×C61—22no—
36#127×C61—21no—
37#127×C61—21no—
38#127×C61—22no—
39#125×C61D412yes—
40#124×C61—24no—
41#124×C61—22no—
42#123×C61—28no—
43C22⋊Q8×C61—22no—
44#121×C61—24no—
45#121×C61—28no—
46#121×C61—24no—
47#120×C61—24no—
48C4×Q8×C61—22no—
49C4×Q8×C61D412yes—
50C4×D4×C61—24no—
51C4×D4×C61—22no—
52C4×D4×C61—24no—
53C4×D4×C61—24no—
54C4×D4×C61D412yes—
55#117×C61—21no—
56#117×C61C2×C412yes—
57#117×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—24no—
59C2×C42×C61—21no—
60C2×C42×C61C2311yescharacteristic
61C24×C61C22⋊C414yes—
62#41×C61—42no—
63#41×C61—84no—
64#41×C61—84no—
65C2×Q8×C61—22no—
66C2×Q8×C61C22⋊C411yescharacteristic
67C2×D4×C61—21no—
68C2×D4×C61—28no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61—24no—
72C2×D4×C61—24no—
73C2×D4×C61—44no—
74C2×D4×C61—44no—
75C2×D4×C61—44no—
76C2×D4×C61C22⋊C411yescharacteristic
77C22×C4×C61—28no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D412yes—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61C22⋊C412yes—
86C4⋊C4×C61C22⋊C414yes—
87C22⋊C4×C61—28no—
88C22⋊C4×C61—28no—
89C22⋊C4×C61—21no—
90C22⋊C4×C61—22no—
91C22⋊C4×C61—22no—
92C42×C61—21no—
93C42×C61—22no—
94C42×C61—22no—
95C42×C61C2×D411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61C2×D412yes—
98#11758194—24no—
99#11758194—42no—
100#11758194—44no—
101C2×#92418—42no—
102C2×#92418—42no—
103C23×C61—24no—
104C23×C61—21no—
105C23×C61—48no—
106C23×C61—44no—
107C23×C61C2×C22⋊C411yescharacteristic
108Q8×C61—22no—
109Q8×C61—42no—
110Q8×C61—42no—
111D4×C61—22no—
112D4×C61—44no—
113D4×C61—42no—
114D4×C61—44no—
115D4×C61—44no—
116D4×C61—44no—
117D4×C61—84no—
118D4×C61—88no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—88no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61#12012yes—
132C2×C4×C61#12012yes—
133#92418—44no—
134#92418—82no—
135#8472—611nomaximal, Sylow
136C22×C61—22no—
137C22×C61—21no—
138C22×C61—42no—
139C22×C61—48no—
140C22×C61—41no—
141C22×C61—44no—
142C22×C61—164no—
143C22×C61#37911yescharacteristic
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—42no—
147C4×C61—44no—
148C4×C61—41no—
149C4×C61#41312yes—
150C4×C61#45312yes—
151C4×C61#45512yes—
152#3508D6111yescharacteristic
153#1965—612no—
154#1561—611no—
155#1526—611no—
156#1489—612no—
157C2×C61—41no—
158C2×C61—81no—
159C2×C61—84no—
160C2×C61#178511yescharacteristic
161C2×C61#179212yes—
162#559—22no—
163#550—22no—
164#546—24no—
165#540—22no—
166#535Dic6112yes—
167#533C2×D6111yescharacteristic
168#532—22no—
169C2×#127C2×D6111yescharacteristic
170#518Dic6112yes—
171C4×#41C2×D6111yescharacteristic
172#423—612no—
173C2×#100—1224no—
174C4×M16—612no—
175#332—1222no—
176#330—1222no—
177Q8⋊C8—1222no—
178D4⋊C8—1222no—
179C61#847211yescharacteristic, complemented, Sylow
180C2×#41—24no—
181C2×#41C2×Dic6111yescharacteristic
182C22×D4C2×Dic6112yes—
183C22×D4C22⋊D6114yes—
184C2×M16—1222no—
185C2×M16—1222no—
186C4⋊Q8C22⋊D6112yes—
187#127—22no—
188#127—21no—
189#127—21no—
190#127—22no—
191#125C22⋊D6112yes—
192#124—24no—
193#124—22no—
194#123—28no—
195C22⋊Q8—22no—
196#121—24no—
197#121—28no—
198#121—24no—
199#120—24no—
200C4×Q8—22no—
201C4×Q8C22⋊D6112yes—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4C22⋊D6112yes—
207#117—21no—
208#117C2×Dic6111yescharacteristic
209#117C2×Dic6112yes—
210C2×C22⋊C4—24no—
211C2×C42—21no—
212C2×C42C22×D6111yescharacteristic
213C4⋊C8—1222no—
214C4⋊C8—1222no—
215#100—2448no—
216#97—1222no—
217C4×C8—1222no—
218C24C22⋊Dic6114yes—
219#41—42no—
220#41—84no—
221#41—84no—
222C2×Q8—22no—
223C2×Q8C22⋊Dic6111yescharacteristic
224C2×D4—21no—
225C2×D4—28no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4C22⋊Dic6111yescharacteristic
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4C22×Dic6111yescharacteristic
238C22×C4C2×C22⋊D6112yes—
239M16—1224no—
240M16—2444no—
241M16—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4C22⋊Dic6112yes—
248C4⋊C4C22⋊Dic6114yes—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—21no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42C2×C22⋊D6112yes—
258C42C2×C22⋊D6111yescharacteristic
259C42C2×C22⋊D6111yescharacteristic
260C23—24no—
261C23—21no—
262C23—44no—
263C23—48no—
264C23C2×C22⋊Dic6111yescharacteristic
265Q8—22no—
266Q8—42no—
267Q8—42no—
268D4—22no—
269D4—44no—
270D4—42no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—84no—
275D4—88no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4C2×C22⋊Dic6111yescharacteristic
287C2×C4C2×C22⋊Dic6111yescharacteristic
288C2×C4#1451974612yes—
289C2×C4#1451974612yes—
290C8—4882no—
291C8—2444no—
292C22—21no—
293C22—22no—
294C22—41no—
295C22—44no—
296C22—42no—
297C22—48no—
298C22—164no—
299C22#1683961811yescharacteristic
300C4—21no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—44no—
305C4#1683974612yes—
306C4#1683993012yes—
307C4#1683994012yes—
308C2—41no—
309C2—84no—
310C2—81no—
311C2#2144574711yescharacteristic
312C2#2144577312yes—
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3508D61
hypocenterC61#8472
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21446660C2semidirect
2. #21444557C2semidirect
3. #21444414C2semidirect
4. #21444258C2semidirect
179. C61#8472semidirect

Maximal subgroups

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

subgroupindex
#214466602
#214445572
#214444142
#214442582
C61×#35082
#847261