Finite Group Databasebeta

← #11780001

Subgroups — #11780001

Order 1152.

600 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 143 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×#634C211yesmaximal, characteristic
2#10592730C211yesmaximal, characteristic
3#10592550C211yesmaximal, characteristic
4#10592380C211yesmaximal, characteristic
5C32×Q8⋊C8C211yesmaximal, characteristic
6#10589347C211yesmaximal, characteristic
7#10589322C211yesmaximal, characteristic
8#68271—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10Q8×C3⋊Dic3—21no—
11C2×#3756—21no—
12#64097—22no—
13#64097—21no—
14#64078—21no—
15#64078C2211yescharacteristic
16#64078C2211yescharacteristic
17C4×#634—21no—
18C4×#634C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×C4×C8C2211yescharacteristic
21#63644—21no—
22#63643—21no—
23#63643—22no—
24#63637—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26Q8×Dic6—34no—
27#5031—34no—
28#4851—34no—
29#4681—34no—
30C3×Q8⋊C8D314yes—
31#4481—34no—
32#4456—34no—
33C2×C32×Q8D411yescharacteristic
34C2×#634—21no—
35C2×#634—22no—
36C2×#634D411yescharacteristic
37C2×#634D412yes—
38C2×C32×C8—21no—
39C2×C32×C8—21no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4D411yescharacteristic
42C32×C42—21no—
43C32×C42C2311yescharacteristic
44#3756—42no—
45C4⋊(C3⋊Dic3)—22no—
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)D411yescharacteristic
51#3750—22no—
52#3750—21no—
53C4×C3⋊Dic3—22no—
54C4×C3⋊Dic3—21no—
55C2×C3⋊(C3⋊C8)—21no—
56#1645—91noSylow
57C3×C4×Q8C2×D314yes—
58Q8×Dic3—64no—
59C2×#216—64no—
60#888—68no—
61#888—64no—
62#869—34no—
63#869—34no—
64#869—64no—
65C4×Dic6—34no—
66C4×Dic6—64no—
67C3×C4⋊C8C2×D314yes—
68C3×C4×C8C2×D314yes—
69#817—64no—
70#816—68no—
71#816—64no—
72#810—64no—
73C4⋊(C3⋊C8)—34no—
74C32×Q8—21no—
75C32×Q8SD1612yes—
76C2×C32×C4—21no—
77C2×C32×C4—21no—
78C2×C32×C4C2×D411yescharacteristic
79C2×C32×C4C2×D411yescharacteristic
80C2×C32×C4C2×D411yescharacteristic
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—41no—
85#634—21no—
86#634—42no—
87#634—41no—
88#634—41no—
89#634—82no—
90#634Q1612yescomplemented
91C32×C8—41no—
92C32×C8—41no—
93C3⋊(C3⋊C8)—41no—
94Q82—91no—
95#487—91no—
96#475—91no—
97#462—91no—
98#333—91no—
99Q8⋊C8—91no—
100Q8⋊C8C3⋊D311yescharacteristic
101C2×C3×Q8D1214yes—
102C2×Dic6—34no—
103C2×Dic6—38no—
104C2×Dic6—64no—
105C2×Dic6—68no—
106C2×C3×C8—24no—
107C2×C3×C8—24no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4D1214yes—
110C3×C42—24no—
111C3×C42C22×D314yes—
112#216—128no—
113C4⋊Dic3—34no—
114C4⋊Dic3—68no—
115C4⋊Dic3—64no—
116C4⋊Dic3—64no—
117C4⋊Dic3—64no—
118C4⋊Dic3—64no—
119Dic3⋊C4—68no—
120Dic3⋊C4—64no—
121C4×Dic3—68no—
122C4×Dic3—64no—
123C2×C3⋊C8—64no—
124C22×C32#12011yescharacteristic
125C32×C4—22no—
126C32×C4—21no—
127C32×C4—41no—
128C32×C4C2×SD1611yescharacteristic
129C32×C4C2×Q1611yescharacteristic
130C32×C4#13711yescharacteristic
131C32×C4#13711yescharacteristic
132C3⋊Dic3—22no—
133C3⋊Dic3—42no—
134C3⋊Dic3—41no—
135C3⋊Dic3—81no—
136C2×Q16—181no—
137C4⋊Q8—91no—
138C4⋊Q8—91no—
139C4⋊Q8—181no—
140C4⋊Q8—182no—
141C4⋊Q8—181no—
142C4×Q8—91no—
143C4×Q8—181no—
144C4×Q8—181no—
145C4×Q8C2×C3⋊D311yescharacteristic
146#106—181no—
147C4⋊C8—91no—
148C4⋊C8C2×C3⋊D311yescharacteristic
149#103—181no—
150#103—182no—
151#103—181no—
152C4×C8C2×C3⋊D311yescharacteristic
153C3×Q8—24no—
154C3×Q8#20418yes—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4C2×D1214yes—
158C2×C3×C4D3×D414yes—
159C2×C3×C4D3×D414yes—
160C2×Dic3—64no—
161C2×Dic3—64no—
162C2×Dic3—68no—
163C2×Dic3—124no—
164Dic6—38no—
165Dic6—64no—
166Dic6—128no—
167Dic6—124no—
168Dic6—124no—
169Dic6—248no—
170C3×C8—44no—
171C3×C8—44no—
172C3⋊C8—124no—
173C2×C32#44811yescharacteristic
174C2×C32#45111yescharacteristic
175C2×C32#45511yescharacteristic
176C2×Q8—91no—
177C2×Q8—92no—
178C2×Q8—182no—
179C2×Q8—181no—
180C2×Q8C4⋊(C3⋊D3)11yescharacteristic
181Q16—362no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—181no—
185C4⋊C4—21no—
186C4⋊C4—91no—
187C4⋊C4—182no—
188C4⋊C4—182no—
189C4⋊C4—181no—
190C4⋊C4—181no—
191C4⋊C4—181no—
192C4⋊C4—181no—
193C4⋊C4—181no—
194C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
195C42—21no—
196C42—182no—
197C42—181no—
198C42C22×C3⋊D311yescharacteristic
199C22×C3#88214yes—
200C3×C4—28no—
201C3×C4—24no—
202C3×C4—44no—
203C3×C4C2×#20414yes—
204C3×C4#90914yes—
205C3×C4#91514yes—
206C3×C4D3×Q1614yes—
207Dic3—68no—
208Dic3—128no—
209Dic3—124no—
210Dic3—244no—
211C32#164511yescharacteristic, complemented, Sylow
212Q8—21no—
213Q8—92no—
214Q8—181no—
215Q8—362no—
216Q8—361no—
217Q8—361no—
218Q8—722no—
219Q8#374412yes—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—182no—
225C2×C4—361no—
226C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
227C2×C4D4×C3⋊D311yescharacteristic
228C2×C4D4×C3⋊D311yescharacteristic
229C8—41no—
230C8—41no—
231C8—361no—
232C2×C3#473814yes—
233C2×C3#480914yes—
234C2×C3#482514yes—
235C22#6409111yescharacteristic
236C4—21no—
237C4—22no—
238C4—41no—
239C4—182no—
240C4—362no—
241C4—361no—
242C4—721no—
243C4C2×#374411yescharacteristic
244C4#6411811yescharacteristic
245C4#6412411yescharacteristic
246C4Q16×C3⋊D311yescharacteristic
247C3#6827114yescomplemented
248C2#1059243711yescharacteristic
249C2#1059250811yescharacteristic
250C2#1059252411yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socle, prime radicalC22×C32#120
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1645
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
90. #634Q16semidirect
211. C32#1645semidirect
247. C3#68271semidirect

Maximal subgroups

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

subgroupindex
Q8×#6342
#105927302
#105925502
#105923802
C32×Q8⋊C82
#105893472
#105893222
#682713