Finite Group Databasebeta

← #29593451

Subgroups — #29593451

Order 15616.

384 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 79 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#21448076C211yesmaximal, characteristic, complemented
2#21447859C211yesmaximal, characteristic, complemented
3#21445174C211yesmaximal, characteristic
4#21443868C211yesmaximal, characteristic, complemented
5#21443783C211yesmaximal, characteristic, complemented
6D4⋊#11758185C211yesmaximal, characteristic, complemented
7D4×Q16×C61C211yesmaximal, characteristic, complemented
8C2×#14519720—22no—
9C2×#14519720—21no—
10#16840139C2211yescharacteristic
11#16839957—21no—
12C4⋊#11758185C2211yescharacteristic
13#16839562—21no—
14C4×#11758185C2211yescharacteristic
15#16839515C2211yescharacteristic
16#16839505—21no—
17C22×Q16×C61—21no—
18D4×Q8×C61—21no—
19C61×#494C2211yescharacteristic
20C61×#467—21no—
21C61×#462—21no—
22C61×#451—22no—
23C4×Q16×C61C2211yescharacteristic
24C8×D4×C61C2211yescharacteristic
25C2×D488—21no—
26C2×D488—21no—
27C2×#11758199—41no—
28#14519774—21no—
29#14519720—44no—
30#14519720—42no—
31C2×#11758185—21no—
32C2×#11758185—21no—
33C4⋊#92418—21no—
34#14519697—41no—
35C22×Q8×C61—22no—
36C2×Q16×C61—22no—
37C2×Q16×C61—21no—
38C2×Q16×C61—42no—
39C2×Q16×C61—42no—
40C2×Q16×C61D411yescharacteristic
41C2×Q16×C61D412yes—
42C22×C8×C61—21no—
43C4⋊Q8×C61—21no—
44C4⋊Q8×C61—41no—
45C22⋊Q8×C61—22no—
46C22⋊Q8×C61—42no—
47C4×Q8×C61—21no—
48C4×D4×C61—41no—
49C4×D4×C61D411yescharacteristic
50#107×C61D411yescharacteristic
51C4⋊C8×C61D411yescharacteristic
52#103×C61—22no—
53#103×C61—21no—
54C22⋊C8×C61—21no—
55C4×C8×C61C2311yescharacteristic
56C2×Q8×C61—21no—
57C2×Q8×C61—22no—
58C2×Q8×C61—42no—
59C2×Q8×C61—44no—
60C2×Q8×C61—82no—
61C2×D4×C61D811yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244—41no—
67Q16×C61—21no—
68Q16×C61—42no—
69Q16×C61—44no—
70Q16×C61—41no—
71Q16×C61—41no—
72Q16×C61D812yescomplemented
73C2×C8×C61—21no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C2×C8×C61C2×D411yescharacteristic
77C2×C8×C61C2×D411yescharacteristic
78C4⋊C4×C61—21no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61D811yescharacteristic
84C22⋊C4×C61—21no—
85C22⋊C4×C61—42no—
86C42×C61—41no—
87C42×C61C2×D411yescharacteristic
88#11758199—82no—
89C2×#92418—41no—
90D488—42no—
91D488—41no—
92D488—41no—
93#11758185—22no—
94#11758185—41no—
95C23×C61—21no—
96C22×D61—81no—
97Q8×C61—22no—
98Q8×C61—41no—
99Q8×C61—42no—
100Q8×C61—42no—
101Q8×C61—84no—
102D4×C61—21no—
103D4×C61SD3212yescomplemented
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—84no—
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61C2×D811yescharacteristic
113C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
114D244—81no—
115D244—81no—
116D244—81no—
117D244—81no—
118D244—161no—
119C8×C61—21no—
120C8×C61—41no—
121C8×C61C2×D811yescharacteristic
122C8×C61AGL(1,ℤ/8)11yescharacteristic
123#92418—81no—
124#12318—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61#44711yescharacteristic
129C2×D61—161no—
130C2×D61—161no—
131C2×D61—161no—
132C4×C61—21no—
133C4×C61—41no—
134C4×C61—42no—
135C4×C61—82no—
136C4×C61—81no—
137C4×C61#44711yescharacteristic
138C4×C61#45311yescharacteristic
139C4×C61C2×SD3211yescharacteristic
140C4×C61#51011yescharacteristic
141D4×Q16D6111yescharacteristic, complemented
142#2248—611no—
143#2207—611no—
144#1667—611no—
145#1362—611no—
146#1332—611no—
147D4⋊C16—611no—
148C2×C61—42no—
149C2×C61#161911yescharacteristic
150C2×C61#218511yescharacteristic
151C2×C61#219311yescharacteristic
152D61—321no—
153C22×Q16—21no—
154D4×Q8—21no—
155C2×SD32—1221no—
156C2×SD32—1222no—
157#494C2×D6111yescharacteristic
158#493—611no—
159#467—21no—
160#462—21no—
161#459—1221no—
162#451—22no—
163C4×Q16C2×D6111yescharacteristic
164C8×D4C2×D6111yescharacteristic
165C4⋊C16—611no—
166#357—1221no—
167C4×C16—611no—
168#335—611no—
169#331—1221no—
170C61#1231811yescharacteristic, complemented, Sylow
171C22×Q8—22no—
172C2×Q16—22no—
173C2×Q16—21no—
174C2×Q16—42no—
175C2×Q16—42no—
176C2×Q16C22⋊D6111yescharacteristic
177C2×Q16C22⋊D6112yes—
178C2×D8—1221no—
179C2×D8—1221no—
180C2×D8—2441no—
181C22×C8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—41no—
184#127—1221no—
185C22⋊Q8—22no—
186C22⋊Q8—42no—
187C4×Q8—21no—
188C4×D4—41no—
189C4×D4C22⋊D6111yescharacteristic
190SD32—2444no—
191SD32—2442no—
192C2×C16—1221no—
193C2×C16—1221no—
194#107C22⋊D6111yescharacteristic
195C4⋊C8—1221no—
196C4⋊C8C22⋊D6111yescharacteristic
197#103—22no—
198#103—21no—
199#102—2441no—
200C22⋊C8—21no—
201C4×C8C22×D6111yescharacteristic
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—82no—
207C2×D4—2441no—
208C2×D4—2441no—
209C2×D4—2441no—
210C2×D4#1175819911yescharacteristic
211C22×C4—21no—
212C22×C4—42no—
213Q16—21no—
214Q16—42no—
215Q16—44no—
216Q16—41no—
217Q16—41no—
218Q16#1175819912yescomplemented
219D8—4882no—
220D8—2442no—
221D8—2441no—
222D8—2441no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—2441no—
227C2×C8C2×C22⋊D6111yescharacteristic
228C2×C8C2×C22⋊D6111yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4#1175819911yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—42no—
237C42—41no—
238C42C2×C22⋊D6111yescharacteristic
239C16—1222no—
240C16—2441no—
241C23—21no—
242C23—4881no—
243Q8—22no—
244Q8—41no—
245Q8—42no—
246Q8—42no—
247Q8—84no—
248D4—21no—
249D4—9761no—
250D4—4881no—
251D4—4881no—
252D4—4881no—
253D4—4881no—
254D4#1451972012yescomplemented
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4#1451974611yescharacteristic
263C2×C4C2×#1175819911yescharacteristic
264C2×C4#1451980311yescharacteristic
265C8—21no—
266C8—41no—
267C8—4881no—
268C8C2×#1175819911yescharacteristic
269C8#1451980311yescharacteristic
270C22—22no—
271C22—41no—
272C22—41no—
273C22—9761no—
274C22—9761no—
275C22—9761no—
276C22#1683989611yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4#1683989611yescharacteristic
283C4#1683993011yescharacteristic
284C4C2×#1451972011yescharacteristic
285C4#1684020311yescharacteristic
286C2—42no—
287C2—19521no—
288C2#2144487311yescharacteristic
289C2#2144773611yescharacteristic
290C2#2144778311yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingD4×Q16×C61C2
socleC22×C61#447
prime radical#21445174C2
hypercenterD4×Q16D61
hypocenterC61#12318
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. #21448076C2semidirect
2. #21447859C2semidirect
4. #21443868C2semidirect
5. #21443783C2semidirect
6. D4⋊#11758185C2semidirect
7. D4×Q16×C61C2semidirect
72. Q16×C61D8semidirect
103. D4×C61SD32semidirect
141. D4×Q16D61semidirect
170. C61#12318semidirect
218. Q16#11758199semidirect
254. D4#14519720semidirect

Maximal subgroups

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

subgroupindex
#214480762
#214478592
#214451742
#214438682
#214437832
D4⋊#117581852
D4×Q16×C612
#1231861