Finite Group Databasebeta

← #29600429

Subgroups — #29600429

Order 15616.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#21447514C211yesmaximal, characteristic, complemented
2#21447508C211yesmaximal, characteristic, complemented
3#21443992C212yesmaximal, complemented
4#21443986C212yesmaximal, complemented
5C61×#3221C211yesmaximal, characteristic
6#16839746—21no—
7C2×#14519696—21no—
8#16839561—21no—
9#16839561C2212yes—
10#16839561C2212yes—
11#16839559—21no—
12C61×#540C411yescharacteristic
13C61×#536C2211yescharacteristic
14C61×#496C411yescharacteristic
15C61×#492C2211yescharacteristic
16C61×#491C2211yescharacteristic
17C61×#488C411yescharacteristic
18C61×#463—21no—
19C61×#461—21no—
20C61×#449—21no—
21C61×#448—21no—
22C61×#433C411yescharacteristic
23C2×#11758194—41no—
24C2×#11758194—41no—
25#14519696—22no—
26#14519696—22no—
27#14519696—41no—
28#14519694—41no—
29C2×#41×C61D411yescharacteristic
30C22×Q8×C61D411yescharacteristic
31C2×SD16×C61—21no—
32C2×SD16×C61—21no—
33C2×SD16×C61—21no—
34C2×D8×C61—21no—
35C4⋊Q8×C61D411yescharacteristic
36C4⋊Q8×C61C2311yescharacteristic
37#127×C61C2×C411yescharacteristic
38#125×C61C2×C411yescharacteristic
39#124×C61C2×C411yescharacteristic
40#124×C61D411yescharacteristic
41#123×C61—41no—
42#123×C61—41no—
43C22⋊Q8×C61—41no—
44#121×C61—21no—
45#121×C61—41no—
46C4×D4×C61—21no—
47#117×C61C2×C411yescharacteristic
48C4⋊C8×C61—21no—
49#103×C61—21no—
50#103×C61—21no—
51#103×C61—21no—
52#102×C61—21no—
53C22⋊C8×C61—21no—
54#97×C61C2×C411yescharacteristic
55C4×C8×C61C2×C411yescharacteristic
56#41×C61—81no—
57C2×Q8×C61—21no—
58C2×Q8×C61—81no—
59C2×Q8×C61C22⋊C411yescharacteristic
60C2×Q8×C61C2×D411yescharacteristic
61C2×Q8×C61C2×D411yescharacteristic
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C2×D4×C61C22⋊C411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—41no—
70C22×C4×C61C22⋊C411yescharacteristic
71SD16×C61—81no—
72SD16×C61—81no—
73SD16×C61—81no—
74D8×C61—81no—
75C2×C8×C61—21no—
76C2×C8×C61—21no—
77C4⋊C4×C61—21no—
78C4⋊C4×C61—21no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C22⋊C4×C61—21no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C42×C61C22⋊C411yescharacteristic
88C42×C61C22×C411yescharacteristic
89#11758194—42no—
90#11758194—42no—
91#11758194—81no—
92#11758194—81no—
93C2×#92418—81no—
94C2×#92418—81no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61#9911yescharacteristic
98Q8×C61—41no—
99Q8×C61—41no—
100Q8×C61—41no—
101Q8×C61—81no—
102D4×C61—41no—
103D4×C61—41no—
104D4×C61—41no—
105D4×C61—81no—
106D4×C61—161no—
107D4×C61—161no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—161no—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C8×C61—41no—
123C8×C61—41no—
124#92418—82no—
125#92418—82no—
126#13611—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—81no—
131C22×C61—81no—
132C22×C61—81no—
133C22×C61—81no—
134C22×C61C2×#9911yescharacteristic
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—41no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61—41no—
142C4×C61—81no—
143#3221D6111yescharacteristic
144#2135—611no—
145#2133—611no—
146#1413—612no—
147#1410—612no—
148C2×C61—21no—
149C2×C61—41no—
150C2×C61—81no—
151C2×C61—161no—
152C2×C61#212711yescharacteristic
153#540Dic6111yescharacteristic
154#536C2×D6111yescharacteristic
155#496Dic6111yescharacteristic
156#492C2×D6111yescharacteristic
157#491C2×D6111yescharacteristic
158#488Dic6111yescharacteristic
159#463—21no—
160#461—21no—
161#449—21no—
162#448—21no—
163#433Dic6111yescharacteristic
164#413—1221no—
165C2×#101—1221no—
166#356—612no—
167#356—612no—
168#356—1221no—
169#355—1221no—
170C61#1361111yescharacteristic, complemented, Sylow
171C2×#41C22⋊D6111yescharacteristic
172C22×Q8C22⋊D6111yescharacteristic
173C2×SD16—21no—
174C2×SD16—21no—
175C2×SD16—21no—
176C2×D8—21no—
177C2×M16—2441no—
178C2×M16—2441no—
179C4⋊Q8C22⋊D6111yescharacteristic
180C4⋊Q8C22×D6111yescharacteristic
181#127C2×Dic6111yescharacteristic
182#125C2×Dic6111yescharacteristic
183#124C2×Dic6111yescharacteristic
184#124C22⋊D6111yescharacteristic
185#123—41no—
186#123—41no—
187C22⋊Q8—41no—
188#121—21no—
189#121—41no—
190C4×D4—21no—
191#117C2×Dic6111yescharacteristic
192C4⋊C8—21no—
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#101—1222no—
198#101—1222no—
199#101—2441no—
200#100—2441no—
201C22⋊C8—21no—
202#97C2×Dic6111yescharacteristic
203C4×C8C2×Dic6111yescharacteristic
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8C22⋊Dic6111yescharacteristic
208C2×Q8C2×C22⋊D6111yescharacteristic
209C2×Q8C2×C22⋊D6111yescharacteristic
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C22⋊Dic6111yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4C22⋊Dic6111yescharacteristic
219SD16—81no—
220SD16—81no—
221SD16—81no—
222D8—81no—
223M16—4881no—
224M16—4881no—
225M16—2442no—
226M16—2442no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—4881no—
230C2×C8—4881no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42C22⋊Dic6111yescharacteristic
242C42C22×Dic6111yescharacteristic
243C23—21no—
244C23—41no—
245C23#1451969211yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—161no—
255D4—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—161no—
268C2×C4#1451969211yescharacteristic
269C2×C4C2×C22⋊Dic6111yescharacteristic
270C8—41no—
271C8—41no—
272C8—4882no—
273C8—4882no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#1451969211yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2#2144748811yescharacteristic
subgroupG/H
centerC2#21447488
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
Fitting, prime radicalC61×#3221C2
socleC2×C61#2127
cosocleC4⋊Q8×C61C23
hypercenter#3221D61
hypocenterC61#13611
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3221

Complements

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

NH ≅ G/Ntype
1. #21447514C2semidirect
2. #21447508C2semidirect
3. #21443992C2semidirect
4. #21443986C2semidirect
170. C61#13611semidirect

Maximal subgroups

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

subgroupindex
#214475142
#214475082
#214439922
#214439862
C61×#32212
#1361161