Finite Group Databasebeta

← #11780184

Subgroups — #11780184

Order 1152.

702 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C32×D4×Q8C211yesmaximal, characteristic
2#10592777C211yesmaximal, characteristic, complemented
3#10592730C211yesmaximal, characteristic, complemented
4#10592706C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589401C211yesmaximal, characteristic, complemented
7D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#71916—34nomaximal
9C22×C32×Q8—21no—
10C32×C4⋊Q8—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C32×C22⋊Q8—21no—
13C32×C22⋊Q8—22no—
14C32×C4×Q8C2211yescharacteristic
15C32×C4×D4—21no—
16C32×C4×D4C2211yescharacteristic
17C2×#3756—21no—
18#64078C2211yescharacteristic
19#63659—21no—
20#63637—21no—
21#63634—22no—
22#63634—21no—
23C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4×C3⋊(C3⋊C8)C2211yescharacteristic
26C3×D4×Q8D314yes—
27#5078—34no—
28#5031—34no—
29#5007—34no—
30#4547—34no—
31#4535—34no—
32D4⋊(C3⋊C8)—34no—
33C2×C32×Q8—22no—
34C2×C32×Q8—21no—
35C2×C32×Q8—42no—
36C2×C32×Q8—42no—
37C2×C32×Q8D411yescharacteristic
38C2×C32×D4D411yescharacteristic
39C22×C32×C4—22no—
40C22×C32×C4—21no—
41C2×#634—21no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4D411yescharacteristic
46C32×C4⋊C4D412yes—
47C32×C4⋊C4D411yescharacteristic
48C32×C22⋊C4—22no—
49C32×C22⋊C4—21no—
50C32×C42—21no—
51C32×C42C2311yescharacteristic
52#3756—42no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C2×C3⋊(C3⋊C8)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58#1649—91noSylow
59C22×C3×Q8—24no—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8C2×D314yes—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—28no—
64C3×C4×Q8C2×D314yes—
65C3×C4×D4—24no—
66C3×C4×D4C2×D314yes—
67C2×#216—64no—
68#869—34no—
69#832—64no—
70#810—64no—
71#807—64no—
72#807—68no—
73C4⋊(C3⋊C8)—34no—
74C4⋊(C3⋊C8)—34no—
75C4×C3⋊C8—34no—
76C23×C32—21no—
77C32×Q8—21no—
78C32×Q8—44no—
79C32×Q8—41no—
80C32×Q8—41no—
81C32×Q8D812yes—
82C32×D4—21no—
83C32×D4Q1612yescomplemented
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—44no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—41no—
95#634—41no—
96#634—41no—
97C3⋊(C3⋊C8)—22no—
98C3⋊(C3⋊C8)—41no—
99C3⋊(C3⋊C8)—41no—
100D4×Q8C3⋊D311yescharacteristic
101#500—91no—
102#474—91no—
103#462—91no—
104#332—91no—
105Q8⋊C8—91no—
106D4⋊C8—91no—
107C2×C3×Q8—28no—
108C2×C3×Q8—24no—
109C2×C3×Q8—48no—
110C2×C3×Q8—48no—
111C2×C3×Q8#6714yes—
112C2×C3×D4#6714yes—
113C22×C3×C4—28no—
114C22×C3×C4—24no—
115C2×Dic6—64no—
116C3×C4⋊C4—28no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4#6714yes—
120C3×C4⋊C4#6718yes—
121C3×C4⋊C4#6714yes—
122C3×C22⋊C4—28no—
123C3×C22⋊C4—24no—
124C3×C42—24no—
125C3×C42C22×D314yes—
126#216—128no—
127C4⋊Dic3—64no—
128C4⋊Dic3—64no—
129C2×C3⋊C8—64no—
130C2×C3⋊C8—64no—
131C2×C3⋊C8—64no—
132C22×C32—22no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32#12011yescharacteristic
136C32×C4—22no—
137C32×C4—21no—
138C32×C4—42no—
139C32×C4—41no—
140C32×C4—41no—
141C32×C4C2×D811yescharacteristic
142C32×C4C2×Q1611yescharacteristic
143C32×C4AGL(1,ℤ/8)11yescharacteristic
144C32×C4#13711yescharacteristic
145C3⋊Dic3—81no—
146C22×Q8—21no—
147C2×Q16—181no—
148C4⋊Q8—21no—
149C4⋊Q8—91no—
150C4⋊Q8C2×C3⋊D311yescharacteristic
151C22⋊Q8—21no—
152C22⋊Q8—22no—
153C4×Q8C2×C3⋊D311yescharacteristic
154C4×D4—21no—
155C4×D4C2×C3⋊D311yescharacteristic
156#107—181no—
157#107—182no—
158C4⋊C8—91no—
159C4⋊C8—91no—
160#103—181no—
161#102—181no—
162C4×C8—91no—
163C23×C3—24no—
164C3×Q8—24no—
165C3×Q8—416no—
166C3×Q8—44no—
167C3×Q8—44no—
168C3×Q8#21318yes—
169C3×D4—24no—
170C3×D4#21618yescomplemented
171C2×C3×C4—24no—
172C2×C3×C4—24no—
173C2×C3×C4—28no—
174C2×C3×C4—24no—
175C2×C3×C4—416no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×C3×C4C2×#6714yes—
179C2×C3×C4C2×#6714yes—
180C2×C3×C4C2×#6714yes—
181C2×Dic3—124no—
182Dic6—124no—
183Dic6—124no—
184C3⋊C8—68no—
185C3⋊C8—124no—
186C3⋊C8—124no—
187C2×C32—42no—
188C2×C32#44711yescharacteristic
189C2×C32#45111yescharacteristic
190C2×C32#45611yescharacteristic
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—181no—
196C2×Q8#63811yescharacteristic
197C2×D4#63811yescharacteristic
198C22×C4—22no—
199C22×C4—21no—
200Q16—362no—
201C2×C8—181no—
202C2×C8—181no—
203C2×C8—181no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—181no—
208C4⋊C4—181no—
209C4⋊C4#63811yescharacteristic
210C4⋊C4#63811yescharacteristic
211C4⋊C4#63812yes—
212C22⋊C4—22no—
213C22⋊C4—21no—
214C42—21no—
215C42C22×C3⋊D311yescharacteristic
216C22×C3—28no—
217C22×C3—44no—
218C22×C3—44no—
219C22×C3#95314yes—
220C3×C4—28no—
221C3×C4—24no—
222C3×C4—48no—
223C3×C4—44no—
224C3×C4—44no—
225C3×C4C2×#21314yes—
226C3×C4#93214yes—
227C3×C4#94214yes—
228C3×C4C2×#21614yes—
229Dic3—244no—
230C32#164911yescharacteristic, complemented, Sylow
231C23—21no—
232Q8—21no—
233Q8—44no—
234Q8—41no—
235Q8—41no—
236Q8—361no—
237Q8—361no—
238Q8#375312yes—
239D4—21no—
240D4#375612yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—361no—
249C2×C4C2×#63811yescharacteristic
250C2×C4C2×#63811yescharacteristic
251C2×C4C2×#63811yescharacteristic
252C8—182no—
253C8—361no—
254C8—361no—
255C2×C3—48no—
256C2×C3#521714yes—
257C2×C3#522814yes—
258C2×C3#524614yes—
259C22—22no—
260C22—41no—
261C22—41no—
262C22#6416211yescharacteristic
263C4—22no—
264C4—21no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—721no—
269C4C2×#375311yescharacteristic
270C4#6414111yescharacteristic
271C4#6415111yescharacteristic
272C4C2×#375611yescharacteristic
273C3#7191614yescomplemented
274C2—42no—
275C2#1059291611yescharacteristic
276C2#1059292711yescharacteristic
277C2#1059294511yescharacteristic
subgroupG/H
centerC22#64162
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×D4×Q8C2
socleC22×C32#120
prime radicalC2×C32×D4D4
hypercenterD4×Q8C3⋊D3
hypocenterC32#1649
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #10592777C2semidirect
3. #10592730C2semidirect
5. #10589413C2semidirect
6. #10589401C2semidirect
83. C32×D4Q16semidirect
170. C3×D4#216semidirect
230. C32#1649semidirect
240. D4#3756semidirect
273. C3#71916semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Q82
#105927772
#105927302
#105927062
#105894132
#105894012
D4⋊(C3⋊(C3⋊C8))2
#719163