Finite Group Databasebeta

← #29600430

Subgroups — #29600430

Order 15616.

316 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456880C211yesmaximal, characteristic
2#21447508C211yesmaximal, characteristic, complemented
3#21443991C212yesmaximal, complemented
4#21443985C212yesmaximal, complemented
5C61×#2135C211yesmaximal, characteristic, complemented
6#16840367C411yescharacteristic
7#16840152C411yescharacteristic
8#16840136C2211yescharacteristic
9#16840130C2211yescharacteristic
10#16840116C411yescharacteristic
11#16839974—21no—
12#16839964—21no—
13#16839907—21no—
14#16839901—21no—
15#16839823C411yescharacteristic
16C2×#14519696—21no—
17#16839561C2212yes—
18#16839559—21no—
19C61×#536C2211yescharacteristic
20C61×#413—21no—
21C61×#356—21no—
22C61×#356C2212yes—
23C2×#11758201—21no—
24C2×#11758201—21no—
25C2×#11758201—21no—
26C2×#11758199—21no—
27C2×#11758194—41no—
28#14519774C2×C411yescharacteristic
29#14519769C2×C411yescharacteristic
30#14519762—41no—
31#14519751—21no—
32#14519750—41no—
33C4×D244—21no—
34C4⋊#92418—21no—
35#14519701—21no—
36#14519701—21no—
37#14519701—21no—
38#14519698—21no—
39#14519696—22no—
40#14519696—41no—
41C22⋊#92418—21no—
42#14519688C2×C411yescharacteristic
43C4×#92418C2×C411yescharacteristic
44C2×#41×C61D411yescharacteristic
45C22×Q8×C61D411yescharacteristic
46C2×M16×C61—41no—
47C4⋊Q8×C61D411yescharacteristic
48C4⋊Q8×C61C2311yescharacteristic
49#124×C61C2×C411yescharacteristic
50#124×C61D411yescharacteristic
51#123×C61—41no—
52C22⋊Q8×C61—41no—
53#117×C61C2×C411yescharacteristic
54#101×C61—22no—
55#100×C61—41no—
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×C61C22⋊C411yescharacteristic
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C2×C22⋊D61—41no—
68C2×D244—21no—
69C2×D244—41no—
70C2×C4×D61—41no—
71M16×C61—42no—
72M16×C61—81no—
73C2×C8×C61—81no—
74C4⋊C4×C61—21no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—21no—
78C22⋊C4×C61—41no—
79C22⋊C4×C61—41no—
80C42×C61C22⋊C411yescharacteristic
81C42×C61C22×C411yescharacteristic
82#11758201—81no—
83#11758201—81no—
84#11758201—81no—
85#11758199—81no—
86#11758198—41no—
87#11758198—41no—
88C4⋊Dic61—21no—
89#11758196—41no—
90#11758194—42no—
91#11758194—81no—
92C2×#92418—21no—
93C2×#92418—21no—
94C2×#92418—81no—
95C23×C61—21no—
96C23×C61#9911yescharacteristic
97C22×D61—41no—
98Q8×C61—41no—
99Q8×C61—41no—
100Q8×C61—41no—
101Q8×C61—81no—
102D4×C61—41no—
103D4×C61—81no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61#9911yescharacteristic
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C22⋊D61—161no—
117C2×Dic61—41no—
118D244—41no—
119D244—41no—
120D244—161no—
121C4×D61—161no—
122C8×C61—82no—
123#92418—41no—
124#92418—41no—
125#92418—82no—
126#13611—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—81no—
131C22×C61C2×#9911yescharacteristic
132C2×D61—81no—
133C2×D61—81no—
134C2×D61—81no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—41no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61—41no—
142Dic61—81no—
143#3221—611no—
144#2135D6111yescharacteristic, complemented
145#2133—611no—
146#1413—612no—
147#1410—612no—
148C2×C61—21no—
149C2×C61—41no—
150C2×C61—81no—
151C2×C61#212711yescharacteristic
152D61—161no—
153#540—611no—
154#536C2×D6111yescharacteristic
155#496—611no—
156#492—611no—
157#491—611no—
158#488—611no—
159#463—1221no—
160#461—1221no—
161#449—1221no—
162#448—1221no—
163#433—611no—
164#413—21no—
165C2×#101—1221no—
166#356—21no—
167#356—612no—
168#356C2×D6112yes—
169#355—1221no—
170C61#1361111yescharacteristic, complemented, Sylow
171C2×#41C22⋊D6111yescharacteristic
172C22×Q8D24411yescharacteristic
173C2×SD16—1221no—
174C2×SD16—1221no—
175C2×SD16—1221no—
176C2×D8—1221no—
177C2×M16—41no—
178C2×M16—2441no—
179C4⋊Q8C22⋊D6111yescharacteristic
180C4⋊Q8C22×D6111yescharacteristic
181#127—611no—
182#125—611no—
183#124C4×D6111yescharacteristic
184#124D24411yescharacteristic
185#123—41no—
186#123—2441no—
187C22⋊Q8—41no—
188#121—1221no—
189#121—2441no—
190C4×D4—1221no—
191#117C4×D6111yescharacteristic
192C4⋊C8—1221no—
193#103—1221no—
194#103—1221no—
195#103—1221no—
196#102—1221no—
197#101—22no—
198#101—1222no—
199#101—2441no—
200#100—41no—
201C22⋊C8—1221no—
202#97—611no—
203C4×C8—611no—
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8#1175819811yescharacteristic
208C2×Q8C2×D24411yescharacteristic
209C2×Q8C2×C22⋊D6111yescharacteristic
210C2×D4—21no—
211C2×D4—1221no—
212C2×D4—2441no—
213C2×D4—2441no—
214C2×D4#1175819811yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2441no—
218C22×C4#1175819811yescharacteristic
219SD16—4881no—
220SD16—4881no—
221SD16—4881no—
222D8—4881no—
223M16—42no—
224M16—81no—
225M16—4881no—
226M16—2442no—
227C2×C8—81no—
228C2×C8—1221no—
229C2×C8—1221no—
230C2×C8—4881no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—1221no—
235C4⋊C4—2441no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—2441no—
240C22⋊C4—2441no—
241C42#1175819811yescharacteristic
242C42C2×C4×D6111yescharacteristic
243C23—21no—
244C23—2441no—
245C23#1451969111yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—81no—
252D4—9761no—
253D4—9761no—
254D4—2441no—
255D4—2441no—
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—9761no—
267C2×C4—2441no—
268C2×C4#1451969111yescharacteristic
269C2×C4C2×#1175819811yescharacteristic
270C8—82no—
271C8—4882no—
272C8—2441no—
273C8—2441no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—4881no—
279C22—4881no—
280C22—4881no—
281C22C2×#1451969111yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—4881no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—9761no—
294C2#2144748611yescharacteristic
subgroupG/H
centerC2#21447486
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#2135C2
socleC2×C61#2127
prime radical#21456880C2
cosocleC4⋊Q8×C61C23
hypercenter#2135D61
hypocenterC61#13611
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2135

Complements

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

NH ≅ G/Ntype
2. #21447508C2semidirect
3. #21443991C2semidirect
4. #21443985C2semidirect
5. C61×#2135C2semidirect
144. #2135D61semidirect
170. C61#13611semidirect

Maximal subgroups

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

subgroupindex
#214568802
#214475082
#214439912
#214439852
C61×#21352
#1361161