Finite Group Databasebeta

← #11779954

Subgroups — #11779954

Order 1152.

624 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 131 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#10596050C211yesmaximal, characteristic
2#10592730C211yesmaximal, characteristic, complemented
3#10592545C211yesmaximal, characteristic
4#10592380C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589347C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic
8#68270—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10#64155—21no—
11C2×#3756—21no—
12#64105—22no—
13C4⋊C4×C3⋊D3—21no—
14C4×C4⋊(C3⋊D3)—21no—
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16#64079—21no—
17#64078C2211yescharacteristic
18#64078C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×C4×C8C2211yescharacteristic
21#63648—21no—
22#63645—21no—
23#63643—22no—
24#63637—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5579—34no—
27#5031—34no—
28#4846—34no—
29#4681—34no—
30C3×Q8⋊C8D314yescomplemented
31#4481—34no—
32#4459—34no—
33C2×C32×Q8D411yescharacteristic
34C2×C4⋊(C3⋊D3)D411yescharacteristic
35C2×C4×C3⋊D3—21no—
36C2×C4×C3⋊D3—22no—
37C2×#634—21no—
38C2×#634D412yes—
39C2×C32×C8—21no—
40C2×C32×C8—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4D411yescharacteristic
43C32×C42—21no—
44C32×C42C2311yescharacteristic
45#3756—42no—
46#3752—22no—
47#3752—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)D411yescharacteristic
53#3750—22no—
54#3750—22no—
55C2×C3⋊(C3⋊C8)—21no—
56#1644—91noSylow
57C3×C4×Q8C2×D314yes—
58#946—64no—
59C2×#216—64no—
60#896—68no—
61D3×C4⋊C4—64no—
62C4×D12—34no—
63C4×D12—64no—
64#870—64no—
65#869—34no—
66#869—34no—
67C3×C4⋊C8C2×D314yes—
68C3×C4×C8C2×D314yes—
69#821—64no—
70#818—64no—
71#816—68no—
72#810—64no—
73C4⋊(C3⋊C8)—34no—
74C22×C3⋊D3—21no—
75C32×Q8—21no—
76C32×Q8—21no—
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4C2×D411yescharacteristic
80C2×C32×C4C2×D411yescharacteristic
81C2×C32×C4C2×D411yescharacteristic
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3—41no—
85C4⋊(C3⋊D3)—21no—
86C4⋊(C3⋊D3)Q1612yescomplemented
87C4×C3⋊D3—41no—
88C4×C3⋊D3—41no—
89C4×C3⋊D3—82no—
90#634—42no—
91#634—41no—
92#634—41no—
93C32×C8—41no—
94C32×C8—41no—
95C3⋊(C3⋊C8)—41no—
96#554—91no—
97#487—91no—
98#474—91no—
99#462—91no—
100#333—91no—
101Q8⋊C8C3⋊D311yescharacteristic, complemented
102D4⋊C8—91no—
103C2×C3×Q8D1214yes—
104C2×D12—34no—
105C2×C4×D3—64no—
106C2×C4×D3—68no—
107C2×Dic6—38no—
108C2×Dic6—64no—
109C2×C3×C8—24no—
110C2×C3×C8—24no—
111C3×C4⋊C4—24no—
112C3×C4⋊C4D1214yes—
113C3×C42—24no—
114C3×C42C22×D314yes—
115#216—128no—
116(C2×D3)⋊C4—68no—
117(C2×D3)⋊C4—64no—
118C4⋊Dic3—34no—
119C4⋊Dic3—64no—
120C4⋊Dic3—64no—
121C4⋊Dic3—64no—
122C4⋊Dic3—64no—
123Dic3⋊C4—68no—
124Dic3⋊C4—68no—
125C2×C3⋊C8—64no—
126C22×C32#12011yescharacteristic
127C2×C3⋊D3—22no—
128C2×C3⋊D3—41no—
129C2×C3⋊D3—41no—
130C32×C4—21no—
131C32×C4—41no—
132C32×C4—41no—
133C32×C4C2×Q1611yescharacteristic
134C32×C4#13511yescharacteristic
135C32×C4AGL(1,ℤ/8)11yescharacteristic
136C32×C4#13711yescharacteristic
137C3⋊Dic3—41no—
138C3⋊Dic3—42no—
139C3⋊Dic3—81no—
140C2×Q16—181no—
141C4⋊Q8—91no—
142C4⋊Q8—91no—
143#125—181no—
144C22⋊Q8—182no—
145C22⋊Q8—181no—
146C4×Q8C2×C3⋊D311yescharacteristic
147C4×D4—91no—
148C4×D4—181no—
149C2×C4⋊C4—181no—
150#107—181no—
151C4⋊C8—91no—
152C4⋊C8C2×C3⋊D311yescharacteristic
153#103—182no—
154#103—181no—
155#102—181no—
156C4×C8C2×C3⋊D311yescharacteristic
157C22×D3—64no—
158C3×Q8—24no—
159C3×Q8—24no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4C2×D1214yes—
163C2×C3×C4D3×D414yes—
164C2×C3×C4D3×D414yes—
165C2×Dic3—64no—
166C2×Dic3—68no—
167C2×Dic3—124no—
168D12—38no—
169D12—64no—
170C4×D3—124no—
171C4×D3—124no—
172C4×D3—248no—
173Dic6—128no—
174Dic6—124no—
175Dic6—124no—
176C3×C8—44no—
177C3×C8—44no—
178C3⋊C8—124no—
179C2×C32#44911yescharacteristic
180C2×C32#45111yescharacteristic
181C2×C32#45511yescharacteristic
182C3⋊D3—42no—
183C2×Q8—92no—
184C2×Q8—181no—
185C2×Q8C4⋊(C3⋊D3)11yescharacteristic
186C2×D4—91no—
187C22×C4—182no—
188C22×C4—181no—
189Q16—362no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—181no—
193C4⋊C4—21no—
194C4⋊C4—91no—
195C4⋊C4—182no—
196C4⋊C4—182no—
197C4⋊C4—181no—
198C4⋊C4—181no—
199C4⋊C4—181no—
200C4⋊C4—181no—
201C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
202C22⋊C4—182no—
203C22⋊C4—181no—
204C42—21no—
205C42C22×C3⋊D311yescharacteristic
206C22×C3#88214yes—
207C2×D3—68no—
208C2×D3—124no—
209C2×D3—124no—
210C3×C4—24no—
211C3×C4—44no—
212C3×C4—44no—
213C3×C4#90414yes—
214C3×C4#90814yes—
215C3×C4#91514yes—
216C3×C4D3×Q1614yes—
217Dic3—124no—
218Dic3—128no—
219Dic3—244no—
220C32#164411yescharacteristic, complemented, Sylow
221C23—181no—
222Q8—21no—
223Q8—21no—
224Q8—362no—
225Q8—361no—
226Q8—361no—
227D4—92no—
228D4—181no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—181no—
232C2×C4—182no—
233C2×C4—361no—
234C2×C4—361no—
235C2×C4—361no—
236C2×C4—722no—
237C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
238C2×C4D4×C3⋊D311yescharacteristic
239C2×C4D4×C3⋊D311yescharacteristic
240C8—41no—
241C8—41no—
242C8—361no—
243C2×C3#473414yes—
244C2×C3#480914yes—
245C2×C3#482514yes—
246D3—128no—
247C22—182no—
248C22—361no—
249C22—361no—
250C22#6409111yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—362no—
255C4—361no—
256C4—721no—
257C4#6411311yescharacteristic
258C4#6411711yescharacteristic
259C4#6412411yescharacteristic
260C4Q16×C3⋊D311yescharacteristic
261C3#6827014yescomplemented
262C2—362no—
263C2#1059243311yescharacteristic
264C2#1059250811yescharacteristic
265C2#1059252411yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1644
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592730C2semidirect
4. #10592380C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589347C2semidirect
30. C3×Q8⋊C8D3semidirect
86. C4⋊(C3⋊D3)Q16semidirect
101. Q8⋊C8C3⋊D3semidirect
220. C32#1644semidirect
261. C3#68270semidirect

Maximal subgroups

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

subgroupindex
#105960502
#105927302
#105925452
#105923802
C32×Q8⋊C82
#105893472
#105893252
#682703