Finite Group Databasebeta

← #37546655

Subgroups — #37546655

Order 23168.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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×Dic362C211yesmaximal, characteristic
2#26042007C211yesmaximal, characteristic
3#26041688C212yesmaximal
4#26041677C211yesmaximal, characteristic
5#26041587C212yesmaximal
6#26041575C211yesmaximal, characteristic
7#26041565C211yesmaximal, characteristic
8#26041565C212yesmaximal
9#26041387C212yesmaximal
10C4×Dic724C211yesmaximal, characteristic
11#446×C181C211yesmaximal, characteristic
12C2×Dic724C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153816C2211yescharacteristic
15#19153816C2212yes—
16#19153815—21no—
17#19153815—22no—
18#19153814—21no—
19#19153808C2212yes—
20#19153806—22no—
21Q8×Dic181—21no—
22C4×Dic362C2211yescharacteristic
23C4×Dic362C2211yescharacteristic
24C4×Dic362C2212yes—
25C4×Dic362C2212yes—
26#19153781—21no—
27#19153750C2211yescharacteristic
28#19153750C2212yes—
29#19153748C2212yes—
30#19153748C2212yes—
31#19153748C2212yes—
32#19153741C2211yescharacteristic
33#19153741C2211yescharacteristic
34#19153741C2212yes—
35#19153741C2212yes—
36C4×Q8×C181C2211yescharacteristic
37C4⋊C8×C181C2211yescharacteristic
38C4⋊C8×C181C2212yes—
39#97×C181C2212yes—
40C4×C8×C181C2211yescharacteristic
41C2×Dic362—21no—
42C2×Dic362—22no—
43C2×Dic362—21no—
44C2×Dic362C2311yescharacteristic
45C2×Dic362C2311yescharacteristic
46Dic724Q814yescomplemented
47C4⋊Dic181—22no—
48C4⋊Dic181—21no—
49C4⋊Dic181—21no—
50C4⋊Dic181C2312yes—
51C4⋊Dic181C2311yescharacteristic
52C4⋊Dic181C2312yes—
53C4⋊Dic181C2311yescharacteristic
54#15668428—22no—
55#15668428—21no—
56#15668428—22no—
57#15668428—22no—
58#15668428—21no—
59#15668428—22no—
60C4×Dic181—22no—
61C4×Dic181—21no—
62C4×Dic181—22no—
63C4×Dic181—21no—
64C2×Q8×C181D411yescharacteristic
65C2×C8×C181C2311yescharacteristic
66C2×C8×C181C2312yes—
67C2×C8×C181C2311yescharacteristic
68C4⋊C4×C181D411yescharacteristic
69C4⋊C4×C181D412yes—
70C42×C181C2311yescharacteristic
71C42×C181C2312yes—
72Dic362—21no—
73Dic362—21no—
74Dic362—42no—
75Dic362—82no—
76Dic362—81no—
77Dic362C2×Q812yes—
78Dic362C2×Q812yes—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—21no—
82C2×Dic181—22no—
83C2×Dic181—22no—
84C2×Dic181—21no—
85Q8×C181—22no—
86C2×C4×C181C2×D412yes—
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2×D412yes—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2411yescharacteristic
91C8×C181—21no—
92C8×C181—22no—
93C8×C181C2×Q812yes—
94C22×C181C22×D411yescharacteristic
95C4×C181—22no—
96C4×C181—22no—
97C4×C181—21no—
98C4×C181#13712yes—
99C4×C181C22×Q811yescharacteristic
100C4×C1812−1+411yescharacteristic
101Dic181—22no—
102Dic181—22no—
103Dic181—42no—
104Dic181—41no—
105Dic181—42no—
106Dic181—41no—
107C2×C181D4×Q811yescharacteristic
108C2×C181C2×#13711yescharacteristic
109C2×C181#57711yescharacteristic
110C181#338711yescharacteristic, complemented, Sylow
111#3387—1811nomaximal, Sylow
112Q82—1811no—
113#557—1811no—
114#501—1812no—
115#498—1811no—
116#479—1812no—
117#477—1811no—
118#475—1812no—
119#475—1811no—
120#446D18111yescharacteristic
121#441—1812no—
122C4×Q16—1811no—
123C2×Q16—1811no—
124C4⋊Q8—3621no—
125C4⋊Q8—3621no—
126C4⋊Q8—3622no—
127C4⋊Q8—1811no—
128C4⋊Q8—1812no—
129C4⋊Q8—1811no—
130#125—3622no—
131#125—1812no—
132C4×Q8—3621no—
133C4×Q8—3621no—
134C4×Q8—1811no—
135C4×Q8—1812no—
136C4×Q8—1811no—
137C4×Q8—1812no—
138C4×Q8C2×D18111yescharacteristic
139#107—1811no—
140#107—1812no—
141#106—1812no—
142#106—1812no—
143#106—1812no—
144C4⋊C8C2×D18111yescharacteristic
145C4⋊C8C2×D18112yes—
146#103—1811no—
147#103—1811no—
148#103—1812no—
149#103—1812no—
150#97C2×D18112yes—
151C4×C8C2×D18111yescharacteristic
152C2×Q8—3622no—
153C2×Q8—3621no—
154C2×Q8—3621no—
155C2×Q8—1811no—
156C2×Q8—1811no—
157C2×Q8D72411yescharacteristic
158Q16—1814no—
159C2×C8C22×D18111yescharacteristic
160C2×C8C22×D18112yes—
161C2×C8C22×D18111yescharacteristic
162C4⋊C4—3622no—
163C4⋊C4—3622no—
164C4⋊C4—3621no—
165C4⋊C4—3621no—
166C4⋊C4—3622no—
167C4⋊C4—3622no—
168C4⋊C4—3621no—
169C4⋊C4—3621no—
170C4⋊C4—3622no—
171C4⋊C4—1812no—
172C4⋊C4—1811no—
173C4⋊C4—1812no—
174C4⋊C4—1811no—
175C4⋊C4D72411yescharacteristic
176C4⋊C4D72412yes—
177C42—3622no—
178C42—3621no—
179C42—3622no—
180C42—3621no—
181C42C22×D18111yescharacteristic
182C42C22×D18112yes—
183Q8—22no—
184Q8—3621no—
185Q8—3621no—
186Q8—14482no—
187Q8—14481no—
188Q8—1812no—
189Q8—1812no—
190Q8—7242no—
191C2×C4—3621no—
192C2×C4—3621no—
193C2×C4—3621no—
194C2×C4—3622no—
195C2×C4—3622no—
196C2×C4—3621no—
197C2×C4C2×D72412yes—
198C2×C4C2×D72412yes—
199C2×C4C2×D72411yescharacteristic
200C2×C4C2×D72411yescharacteristic
201C2×C4C23×D18111yescharacteristic
202C8—21no—
203C8—22no—
204C8Q8×D18112yes—
205C22C22×D72411yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—3622no—
210C4—3622no—
211C4—7242no—
212C4—7242no—
213C4—7241no—
214C4—7241no—
215C4#1915384912yes—
216C4C2×Q8×D18111yescharacteristic
217C4#1915386311yescharacteristic
218C2Q8×D72411yescharacteristic
219C2C2×#1915384911yescharacteristic
220C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#446×C181C2
socle, prime radicalC22×C181C22×D4
hypercenter#446D181
hypocenterC181#3387
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
46. Dic724Q8semidirect
110. C181#3387semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3622
#260420072
#260416882
#260416772
#260415872
#260415752
#260415652
#260413872
C4×Dic7242
#446×C1812
#3387181