Finite Group Databasebeta

← #29837661

Subgroups — #29837661

Order 15616.

506 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×D488C211yesmaximal, characteristic
2#21448046C213yesmaximal, complemented
3#21447961C213yesmaximal, complemented
4#21447861C213yesmaximal
5Q8×#11758185C211yesmaximal, characteristic, complemented
6C4×#14519720C213yesmaximal
7C61×#3395C211yesmaximal, characteristic, complemented
8Q8×D244—21no—
9C2×#14519720C2213yes—
10#16840167C2213yes—
11#16840043—23no—
12C4×D488C2213yes—
13#16839589C2213yes—
14C4⋊#11758185C2213yes—
15#16839566C2216yes—
16#16839566C2213yes—
17#16839562C2211yescharacteristic
18C4×#11758185C2213yes—
19C61×#557—21no—
20C61×#494C2213yes—
21C61×#462—23no—
22C8×Q8×C61C2211yescharacteristic
23C4×Q16×C61C2213yes—
24C2×Q8×D61—41no—
25C2×D488C2311yescharacteristic
26#14519777—23no—
27#14519757—43no—
28C4×D244—23no—
29#14519720—26no—
30C2×#11758185C2311yescharacteristic
31C2×#11758185C2313yes—
32#14519711—23no—
33#14519711C2313yes—
34#14519699—23no—
35C2×Q16×C61—23no—
36C2×Q16×C61C2313yes—
37C4⋊Q8×C61—23no—
38#125×C61—43no—
39C4×Q8×C61—23no—
40C4×Q8×C61—41no—
41C4×Q8×C61D411yescharacteristic
42#107×C61C2311yescharacteristic
43C4⋊C8×C61D413yes—
44#103×C61—23no—
45C4×C8×C61C2313yes—
46C2×Q8×C61—23no—
47C2×Q8×C61D811yescharacteristic
48Q8×D61—84no—
49C2×D244—21no—
50C2×C4×D61—43no—
51C2×Dic122—43no—
52Q16×C61—23no—
53Q16×C61—23no—
54Q16×C61—83no—
55C2×C8×C61C2×D413yes—
56C2×C8×C61C2411yescharacteristic
57C4⋊C4×C61—21no—
58C4⋊C4×C61—43no—
59C4⋊C4×C61—43no—
60C4⋊C4×C61—43no—
61C4⋊C4×C61D813yes—
62C42×C61—43no—
63C42×C61C2×D413yes—
64#11758198—43no—
65C4⋊Dic61—23no—
66C4⋊Dic61—43no—
67D488—21no—
68D488#4112yes—
69#11758185—23no—
70#11758185—21no—
71C22×D61—41no—
72Q8×C61—43no—
73Q8×C61—43no—
74Q8×C61SD3214yescomplemented
75C2×C4×C61—41no—
76C2×C4×C61—43no—
77C2×C4×C61C2×D813yes—
78C2×C4×C61C2×D813yes—
79C2×C4×C61C22×D411yescharacteristic
80C2×Dic61—43no—
81D244—22no—
82D244—41no—
83C4×D61—86no—
84Dic122—86no—
85C8×C61—23no—
86C8×C61C2×#4111yescharacteristic
87C8×C612−1+411yescharacteristic
88#33926—611nomaximal, Sylow
89C22×C61C22×D811yescharacteristic
90C2×D61—42no—
91C2×D61—81no—
92C2×D61—81no—
93C4×C61—23no—
94C4×C61—83no—
95C4×C61—81no—
96C4×C61C2×SD3216yes—
97C4×C61#54811yescharacteristic
98C4×C61#57711yescharacteristic
99Dic61—83no—
100#3395D6111yescharacteristic, complemented
101Q8×D8—611no—
102#2241—613no—
103#2225—613no—
104#2208—613no—
105Q8×C16—611no—
106C4×SD32—613no—
107C2×C61#335811yescharacteristic
108C2×C61C22×SD3211yescharacteristic
109C2×C61#341711yescharacteristic
110D61—82no—
111#557—21no—
112D4×Q8—1221no—
113C2×SD32—613no—
114#500—613no—
115#494C2×D6113yes—
116#474—1223no—
117#462—23no—
118C8×Q8C2×D6111yescharacteristic
119C4×Q16C2×D6113yes—
120C4×D8—613no—
121#367—613no—
122C4⋊C16—613no—
123#358—613no—
124#358—616no—
125#357—611no—
126C4×C16—613no—
127C61#3392611yescharacteristic, complemented, Sylow
128C22×Q8—2441no—
129C2×Q16—23no—
130C2×Q16C22×D6113yes—
131C2×D8—611no—
132C4⋊Q8—23no—
133C4⋊Q8—1223no—
134#125—43no—
135C22⋊Q8—2443no—
136C4×Q8—23no—
137C4×Q8—41no—
138C4×Q8C22⋊D6111yescharacteristic
139C4×D4—1223no—
140SD32—1226no—
141C2×C16—611no—
142C2×C16—613no—
143#107—613no—
144#107—1223no—
145#107C22×D6111yescharacteristic
146C4⋊C8C22⋊D6113yes—
147#103—23no—
148#102—1223no—
149C4×C8C22×D6113yes—
150C2×Q8—23no—
151C2×Q8—4884no—
152C2×Q8—2443no—
153C2×Q8#1175819911yescharacteristic
154C2×D4—1221no—
155C22×C4—2443no—
156Q16—23no—
157Q16—23no—
158Q16—83no—
159D8—612no—
160D8—1221no—
161C2×C8C2×C22⋊D6113yes—
162C2×C8C23×D6111yescharacteristic
163C4⋊C4—21no—
164C4⋊C4—43no—
165C4⋊C4—43no—
166C4⋊C4—43no—
167C4⋊C4—1223no—
168C4⋊C4—2443no—
169C4⋊C4#1175819913yes—
170C22⋊C4—2443no—
171C42—43no—
172C42C2×C22⋊D6113yes—
173C16—1223no—
174C16—1221no—
175C23—2441no—
176Q8—43no—
177Q8—43no—
178Q8—4886no—
179Q8#1451972014yescomplemented
180D4—1222no—
181D4—2441no—
182C2×C4—41no—
183C2×C4—43no—
184C2×C4—4886no—
185C2×C4—2443no—
186C2×C4C2×#1175819913yes—
187C2×C4C2×#1175819913yes—
188C2×C4C22×C22⋊D6111yescharacteristic
189C8—23no—
190C8#41×D6111yescharacteristic
191C8#1451982411yescharacteristic
192C22—4881no—
193C22—4881no—
194C22—2442no—
195C22C22×#1175819911yescharacteristic
196C4—23no—
197C4—83no—
198C4—81no—
199C4—4883no—
200C4C2×#1451972016yes—
201C4#1684041911yescharacteristic
202C4#1684057311yescharacteristic
203C2—4882no—
204C2#2145845011yescharacteristic
205C2C22×#1451972011yescharacteristic
206C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3395C2
socleC22×C61C22×D8
prime radicalC2×D488C23
hypercenter#3395D61
hypocenterC61#33926
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
2. #21448046C2semidirect
3. #21447961C2semidirect
5. Q8×#11758185C2semidirect
7. C61×#3395C2semidirect
74. Q8×C61SD32semidirect
100. #3395D61semidirect
127. C61#33926semidirect
179. Q8#14519720semidirect

Maximal subgroups

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

subgroupindex
Q8×D4882
#214480462
#214479612
#214478612
Q8×#117581852
C4×#145197202
C61×#33952
#3392661