Finite Group Databasebeta

← #16020560

Subgroups — #16020560

Order 3328.

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

normal subgroups only

This group has 562 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×D104C211yesmaximal, characteristic, complemented
2#13117059C213yesmaximal, complemented
3#13116973C213yesmaximal, complemented
4#13116868C213yesmaximal, complemented
5Q8×#6090C211yesmaximal, characteristic, complemented
6C4×#88066C213yesmaximal, complemented
7C13×#3391C211yesmaximal, characteristic, complemented
8C13×#550—21no—
9Q8×D52—21no—
10C13×#493C2213yes—
11C13×#459—23no—
12C8×Q8×C13C2211yescharacteristic
13C4×C13×D8C2213yes—
14C2×#88066C2213yes—
15#11718507—23no—
16C4×D104C2213yes—
17#11718343C2213yes—
18#11718154C2213yes—
19#11718154C2216yes—
20#11718083C2211yescharacteristic
21#11718081C2213yes—
22C4⋊#6090C2213yes—
23C4×#6090C2213yes—
24C2×C13×#41—41no—
25C2×Q8×D13—41no—
26C2×C13×D8—23no—
27C2×C13×D8C2313yes—
28C13×#127—23no—
29C13×#121—43no—
30C4×Q8×C13D411yescharacteristic
31C4×D4×C13—23no—
32C2×D104C2311yescharacteristic
33#88151—43no—
34C4×D52—23no—
35#88123—23no—
36C13×#107C2311yescharacteristic
37C13×C4⋊C8D413yes—
38C13×#102—23no—
39C4×C8×C13C2313yes—
40#88066—26no—
41#88061—23no—
42#88058—23no—
43#88058C2313yes—
44C2×#6090C2311yescharacteristic
45C2×#6090C2313yes—
46#33925—131nomaximal, Sylow
47C13×#41—84no—
48C2×Q8×C13D811yescharacteristic
49C2×D4×C13—23no—
50C2×D4×C13—43no—
51C2×D4×C13—43no—
52C22×C4×C13—43no—
53Q8×D13—84no—
54C2×D52—21no—
55C2×C4×D13—43no—
56C2×Dic26—43no—
57C13×D8—23no—
58C13×D8—23no—
59C13×D8—83no—
60C2×C8×C13C2×D413yes—
61C2×C8×C13C2411yescharacteristic
62C13×C4⋊C4—21no—
63C13×C4⋊C4D813yes—
64C13×C22⋊C4—43no—
65C42×C13C2×D413yes—
66#6103—43no—
67C4⋊Dic13—23no—
68C4⋊Dic13—43no—
69D104—21no—
70D104#4112yescomplemented
71#6090—23no—
72#6090—21no—
73#3391D1311yescharacteristic, complemented
74Q8×D8—131no—
75#2240—133no—
76#2224—133no—
77#2206—133no—
78Q8×C16—131no—
79C4×D16—133no—
80C23×C13—43no—
81C22×D13—41no—
82Q8×C13D1614yescomplemented
83D4×C13—43no—
84D4×C13—43no—
85D4×C13—86no—
86D4×C13—163no—
87C2×C4×C13—41no—
88C2×C4×C13—86no—
89C2×C4×C13C2×D813yes—
90C2×C4×C13C2×D813yes—
91C2×C4×C13C22×D411yescharacteristic
92C2×Dic13—43no—
93D52—22no—
94D52—41no—
95C4×D13—86no—
96Dic26—86no—
97C8×C13—23no—
98C8×C13C2×#4111yescharacteristic
99C8×C132−1+411yescharacteristic
100#550—21no—
101D4×Q8—261no—
102C2×D16—133no—
103#500—133no—
104#493C2×D1313yes—
105#474—263no—
106#459—23no—
107C8×Q8C2×D1311yescharacteristic
108C4×D8—133no—
109C4×D8C2×D1313yes—
110#366—133no—
111C4⋊C16—133no—
112#357—133no—
113#357—136no—
114#357—131no—
115C4×C16—133no—
116C22×C13—83no—
117C22×C13—83no—
118C22×C13—83no—
119C22×C13C22×D811yescharacteristic
120C2×D13—42no—
121C2×D13—81no—
122C2×D13—81no—
123C4×C13—23no—
124C4×C13—81no—
125C4×C13C2×D1616yes—
126C4×C13#54811yescharacteristic
127C4×C13#57711yescharacteristic
128Dic13—83no—
129C2×#41—41no—
130C22×Q8—521no—
131C2×D8—23no—
132C2×D8—131no—
133C2×D8C22×D1313yes—
134C4⋊Q8—263no—
135#127—23no—
136C22⋊Q8—523no—
137#121—43no—
138C4×Q8C22⋊D1311yescharacteristic
139C4×D4—23no—
140C4×D4—263no—
141D16—266no—
142C2×C16—131no—
143C2×C16—133no—
144#107—133no—
145#107—263no—
146#107C22×D1311yescharacteristic
147C4⋊C8C22⋊D1313yes—
148#102—23no—
149#102—263no—
150C4×C8C22×D1313yes—
151C2×C13—163no—
152C2×C13#335811yescharacteristic
153C2×C13C22×D1611yescharacteristic
154C2×C13#341611yescharacteristic
155D13—82no—
156#41—84no—
157C2×Q8—523no—
158C2×Q8—1044no—
159C2×Q8#610411yescharacteristic
160C2×D4—23no—
161C2×D4—43no—
162C2×D4—43no—
163C2×D4—261no—
164C22×C4—43no—
165C22×C4—523no—
166D8—23no—
167D8—23no—
168D8—83no—
169D8—132no—
170D8—261no—
171C2×C8C2×C22⋊D1313yes—
172C2×C8C23×D1311yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—263no—
175C4⋊C4—523no—
176C4⋊C4#610413yes—
177C22⋊C4—43no—
178C22⋊C4—523no—
179C42C2×C22⋊D1313yes—
180C16—263no—
181C16—261no—
182C13#3392511yescharacteristic, complemented, Sylow
183C23—43no—
184C23—521no—
185Q8—1046no—
186Q8#8806614yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—163no—
191D4—262no—
192D4—521no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—523no—
196C2×C4—1046no—
197C2×C4C2×#610413yes—
198C2×C4C2×#610413yes—
199C2×C4C22×C22⋊D1311yescharacteristic
200C8—23no—
201C8#41×D1311yescharacteristic
202C8#8825711yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—522no—
207C22—1041no—
208C22—1041no—
209C22C22×#610411yescharacteristic
210C4—23no—
211C4—81no—
212C4—1043no—
213C4C2×#8806616yes—
214C4#1171953611yescharacteristic
215C4#1171954111yescharacteristic
216C2—163no—
217C2—1042no—
218C2#1312746711yescharacteristic
219C2C22×#8806611yescharacteristic
220C2#1312798311yescharacteristic
subgroupG/H
centerC22C22×#6104
commutator, cosocleC2×C8×C13C24
FrattiniC2×C8C23×D13
FittingC13×#3391C2
socleC22×C13C22×D8
hypercenter#3391D13
hypocenterC13#33925
LCSG ▷ C2×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×D104C2semidirect
2. #13117059C2semidirect
3. #13116973C2semidirect
4. #13116868C2semidirect
5. Q8×#6090C2semidirect
6. C4×#88066C2semidirect
7. C13×#3391C2semidirect
70. D104#41semidirect
73. #3391D13semidirect
82. Q8×C13D16semidirect
182. C13#33925semidirect
186. Q8#88066semidirect

Maximal subgroups

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

subgroupindex
Q8×D1042
#131170592
#131169732
#131168682
Q8×#60902
C4×#880662
C13×#33912
#3392513