Finite Group Databasebeta

← #29599122

Subgroups — #29599122

Order 15616.

402 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16840160C211yesmaximal, characteristic, complemented
2C2×#16839573C212yesmaximal
3#21446707C211yesmaximal, characteristic
4#21443918C212yesmaximal, complemented
5C61×#1376C211yesmaximal, characteristic, complemented
6C2×#14519799C2211yescharacteristic
7C2×#14519777—21no—
8#16840160—24no—
9C2×#14519713C2211yescharacteristic
10C2×#14519709—21no—
11C2×#14519709C2212yes—
12C2×#14519705—21no—
13#16839728—21no—
14#16839573—24no—
15C2×M32×C61C2212yes—
16C2×C4×C8×C61C2211yescharacteristic
17C2×#11758222—21no—
18C22×Dic122—41no—
19#14519799—22no—
20#14519799—22no—
21C2×Dic244Q811yescharacteristic, complemented
22C2×#11758190—21no—
23C2×D488Q811yescharacteristic, complemented
24C2×#11758194—21no—
25#14519777—24no—
26#14519777—42no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—41no—
29#14519713—22no—
30#14519709—24no—
31#14519709—22no—
32#14519709D414yes—
33#14519705—42no—
34#14519689—42no—
35C22×C8×C61D411yescharacteristic
36C22×C8×C61C2311yescharacteristic
37C2×C42×C61D411yescharacteristic
38M32×C61—22no—
39M32×C61—22no—
40C2×C16×C61—22no—
41C4×C8×C61—22no—
42C22×C4×C61—21no—
43C22×C4×C61C2×D411yescharacteristic
44C2×C22⋊D61—41no—
45C22×Dic61—42no—
46#11758222—41no—
47#11758222—41no—
48#11758222—41no—
49#11758222—41no—
50C2×D244—21no—
51C2×C4×D61—41no—
52C2×Dic122—21no—
53C2×Dic122—44no—
54C2×Dic122—84no—
55C2×C8×C61—21no—
56C2×C8×C61—21no—
57C2×C8×C61SD1612yes—
58C2×C8×C61C2×D412yes—
59C2×C8×C61C2×Q811yescharacteristic
60C2×C8×C61#4112yes—
61C2×C8×C61#4111yescharacteristic
62C42×C61—21no—
63C42×C61D812yes—
64C4⋊Dic61—24no—
65C4⋊Dic61—44no—
66C4⋊Dic61—42no—
67#11758194—41no—
68#11758194—41no—
69C2×#92418—41no—
70Dic244—22no—
71D488—22no—
72#11758190—42no—
73C16×C61—42no—
74C23×C61AGL(1,ℤ/8)11yescharacteristic
75C22×D61—41no—
76C2×C4×C61—24no—
77C2×C4×C61—22no—
78C2×C4×C61#12011yescharacteristic
79C2×C4×C61C22⋊Q811yescharacteristic
80C2×C4×C61#12411yescharacteristic
81C2×C4×C61C2×D811yescharacteristic
82C2×C4×C61C2×SD1611yescharacteristic
83C2×C4×C61#13711yescharacteristic
84C22⋊D61—81no—
85C22⋊D61—81no—
86C2×Dic61—44no—
87C2×Dic61—41no—
88C2×Dic61—84no—
89D244—41no—
90D244—41no—
91C4×D61—81no—
92C4×D61—81no—
93Dic122—41no—
94Dic122—41no—
95Dic122—88no—
96C8×C61—22no—
97C8×C61C22⋊Q812yes—
98C8×C61#12412yes—
99#92418—81no—
100#13371—611nomaximal, Sylow
101C22×C61—22no—
102C22×C61#44711yescharacteristic
103C22×C61#47611yescharacteristic
104C22×C61#48911yescharacteristic
105C2×D61—82no—
106C2×D61—81no—
107C4×C61—24no—
108C4×C61#39311yescharacteristic
109C4×C61#44811yescharacteristic
110C4×C61#47411yescharacteristic
111C4×C61#48611yescharacteristic
112Dic61—84no—
113Dic61—81no—
114C2×#498—611no—
115C2×#360—612no—
116#1974—611no—
117#1383—612no—
118#1376D6111yescharacteristic, complemented
119C2×C61—21no—
120C2×C61—21no—
121C2×C61#202311yescharacteristic
122C2×C61#223012yes—
123D61—161no—
124C2×#135—611no—
125C2×C4⋊Q8—1221no—
126C2×M32C2×D6112yes—
127#498—1224no—
128C2×#108—611no—
129C2×#106—612no—
130C2×#106—1221no—
131C2×#104—1221no—
132#408—1221no—
133C2×C4×C8C2×D6111yescharacteristic
134#360—1224no—
135C61#1337111yescharacteristic, complemented, Sylow
136C2×#41—1221no—
137C22×Q8—2441no—
138#135—1222no—
139#135—1222no—
140C2×Q16—611no—
141C2×SD16—1221no—
142C2×D8—611no—
143C2×M16—1221no—
144C22×C8D24411yescharacteristic
145C22×C8C22×D6111yescharacteristic
146C4⋊Q8—1224no—
147C4⋊Q8—2442no—
148C2×C4⋊C4—1222no—
149C2×C4⋊C4—2441no—
150C2×C42D24411yescharacteristic
151M32—22no—
152M32—22no—
153C2×C16—22no—
154#108—1222no—
155#106—614no—
156#106—1224no—
157#106—1222no—
158#104—2442no—
159C22⋊C8—2442no—
160C4×C8—22no—
161#41—2441no—
162#41—2441no—
163#41—2441no—
164#41—2441no—
165C2×Q8—1221no—
166C2×Q8—4884no—
167C2×Q8—2444no—
168C2×D4—1221no—
169C2×D4—2441no—
170C22×C4—21no—
171C22×C4—2442no—
172C22×C4—2441no—
173C22×C4C2×D24411yescharacteristic
174Q16—1222no—
175SD16—2442no—
176D8—1222no—
177M16—2441no—
178M16—2441no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—2441no—
182C2×C8#1175819012yes—
183C2×C8#1175822212yes—
184C2×C8D4×D6112yes—
185C2×C8#1175822411yescharacteristic
186C2×C8Q8×D6111yescharacteristic
187C4⋊C4—1224no—
188C4⋊C4—2444no—
189C4⋊C4—2442no—
190C42—21no—
191C42D48812yes—
192C16—42no—
193C23—2441no—
194C23#1451980511yescharacteristic
195Q8—4888no—
196Q8—2441no—
197Q8—2441no—
198D4—4881no—
199D4—4881no—
200D4—2441no—
201D4—2441no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—4884no—
205C2×C4—4881no—
206C2×C4—4881no—
207C2×C4—2441no—
208C2×C4—2444no—
209C2×C4#1451974711yescharacteristic
210C2×C4#1451975711yescharacteristic
211C2×C4#1451976511yescharacteristic
212C2×C4C2×D48811yescharacteristic
213C2×C4C2×#1175819011yescharacteristic
214C2×C4#1451981011yescharacteristic
215C8—22no—
216C8—4881no—
217C8#1451975512yes—
218C8#1451976312yes—
219C22—22no—
220C22—4881no—
221C22—4882no—
222C22#1683989811yescharacteristic
223C22#1684005311yescharacteristic
224C22#1684012211yescharacteristic
225C4—24no—
226C4—4884no—
227C4—4881no—
228C4#1683967511yescharacteristic
229C4#1683990311yescharacteristic
230C4#1684004311yescharacteristic
231C4#1684010911yescharacteristic
232C2—21no—
233C2—21no—
234C2—9761no—
235C2#2144695611yescharacteristic
236C2#2144799112yes—
subgroupG/H
centerC22#16840053
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1376C2
socleC22×C61#476
prime radicalC2×#14519799C22
hypercenter#1376D61
hypocenterC61#13371
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#16840160C2semidirect
4. #21443918C2semidirect
5. C61×#1376C2semidirect
21. C2×Dic244Q8semidirect
23. C2×D488Q8semidirect
118. #1376D61semidirect
135. C61#13371semidirect

Maximal subgroups

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

subgroupindex
C2×#168401602
C2×#168395732
#214467072
#214439182
C61×#13762
#1337161