Finite Group Databasebeta

← #11779496

Subgroups — #11779496

Order 1152.

497 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C3×#5734C211yesmaximal, characteristic
2#10592791C211yesmaximal, characteristic, complemented
3#10590268C212yesmaximal, complemented
4C3×#4546C211yesmaximal, characteristic, complemented
5#10589080C212yesmaximal, complemented
6#71928—31nomaximal
7#68493—31nomaximal
8C32×C4⋊Q8C2211yescharacteristic
9C2×#3756—22no—
10#64078C2211yescharacteristic
11C3×#947—22no—
12C3×#895—22no—
13C3×#892—22no—
14C3×C4×D12C2212yes—
15C4×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C4⋊(C3⋊C8)C2212yes—
17#63573—22no—
18#63563—22no—
19C3×#555—31no—
20#5734D311yescharacteristic
21#5092—31no—
22#5092—31no—
23#5092—61no—
24#5008—32no—
25#4842—32no—
26C3×#333—31no—
27#4546D311yescharacteristic, complemented
28D4⋊(C3⋊C8)—32no—
29#4483—32no—
30C2×C32×Q8D412yes—
31C2×#634—21no—
32C2×C3×D12—22no—
33C2×C3×D12D412yes—
34C2×C3×C4×D3—22no—
35C32×C4⋊C4—22no—
36C32×C42C2311yescharacteristic
37#3756—44no—
38C4⋊(C3⋊Dic3)—22no—
39C2×C3⋊(C3⋊C8)—21no—
40C3×(C2×D3)⋊C4—24no—
41C3×(C2×D3)⋊C4—22no—
42C3×C4⋊Dic3D412yes—
43C3×C4×Dic3—22no—
44C2×C3×C3⋊C8—22no—
45#1634—91noSylow
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8C2×D311yescharacteristic
48C3×C4⋊Q8C2×D311yescharacteristic
49C3×#124—62no—
50C3×#121—62no—
51C3×C4×D4—32no—
52C3×#117—62no—
53#947—22no—
54C2×#216—64no—
55C2×#216—62no—
56C2×#216—62no—
57#895—22no—
58#892—22no—
59C4×D12C2×D312yes—
60#869—31no—
61#869—31no—
62#869—61no—
63C3×C4⋊C8—32no—
64#832—62no—
65#821—62no—
66#817—62no—
67#808—62no—
68C4⋊(C3⋊C8)C2×D312yes—
69C4×C3⋊C8—31no—
70C4×C3⋊C8—31no—
71C4×C3⋊C8—61no—
72C22×C3×D3—22no—
73C32×Q8—42no—
74C2×C32×C4—22no—
75C2×C32×C4C2×D411yescharacteristic
76C2×C32×C4C2×D412yes—
77C2×C3⋊Dic3—41no—
78#634—42no—
79C2×C3×Dic3—22no—
80C3×D12—22no—
81C3×D12—22no—
82C3×D12—82no—
83C3×C4×D3—42no—
84C3×C4×D3—42no—
85C3⋊(C3⋊C8)—22no—
86C3×C3⋊C8—42no—
87#555—31no—
88#494—91no—
89#476—92no—
90#333—31no—
91D4⋊C8—92no—
92C2×C3×Q8—22no—
93C2×C3×Q8D1212yes—
94C2×C3×Q8#6712yes—
95C2×C3×D4—32no—
96C2×C3×D4—62no—
97C22×C3×C4—62no—
98C2×D12—22no—
99C2×D12#6712yes—
100C2×C4×D3—22no—
101C2×Dic6—62no—
102C2×Dic6—61no—
103C2×Dic6—61no—
104C2×C3×C8—62no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—42no—
109C3×C22⋊C4—64no—
110C3×C22⋊C4—62no—
111C3×C42—21no—
112C3×C42—62no—
113C3×C42C22×D311yescharacteristic
114C3×C42C22×D311yescharacteristic
115#216—124no—
116#216—124no—
117#216—128no—
118(C2×D3)⋊C4—24no—
119(C2×D3)⋊C4—22no—
120C4⋊Dic3—62no—
121C4⋊Dic3—62no—
122C4⋊Dic3—122no—
123C4⋊Dic3#6712yes—
124C4×Dic3—22no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—62no—
127C2×C3⋊C8—61no—
128C2×C3⋊C8—61no—
129C22×C32#12011yescharacteristic
130C2×C3×D3—42no—
131C2×C3×D3—42no—
132C2×C3×D3—42no—
133C32×C4—21no—
134C32×C4—42no—
135C32×C4#13512yes—
136C32×C4#13712yes—
137C3⋊Dic3—81no—
138C3×Dic3—42no—
139C2×Q16—182no—
140C4⋊Q8—91no—
141C4⋊Q8D3211yescharacteristic
142#124—62no—
143#121—62no—
144C4×D4—32no—
145#117—62no—
146#106—182no—
147C4⋊C8—32no—
148#102—182no—
149C4×C8—91no—
150C23×C3—62no—
151C22×D3—22no—
152C3×Q8—42no—
153C3×Q8—42no—
154C3×Q8—44no—
155C3×D4—62no—
156C3×D4—62no—
157C3×D4—242no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—42no—
163C2×C3×C4—62no—
164C2×C3×C4—122no—
165C2×C3×C4—122no—
166C2×C3×C4C2×D1211yescharacteristic
167C2×C3×C4D3×D412yes—
168C2×C3×C4C2×#6711yescharacteristic
169C2×C3×C4C2×#6712yes—
170C2×Dic3—22no—
171C2×Dic3—122no—
172C2×Dic3—121no—
173C2×Dic3—121no—
174D12—22no—
175D12—22no—
176D12—82no—
177C4×D3—42no—
178C4×D3—42no—
179Dic6—124no—
180Dic6—122no—
181Dic6—122no—
182C3×C8—122no—
183C3⋊C8—42no—
184C3⋊C8—64no—
185C3⋊C8—62no—
186C3⋊C8—62no—
187C2×C32#45212yes—
188C2×C32#45311yescharacteristic
189C3×D3—82no—
190C2×Q8—181no—
191C2×Q8#62612yes—
192C2×D4—32no—
193C2×D4—62no—
194C22×C4—62no—
195Q16—364no—
196C2×C8—62no—
197C2×C8—181no—
198C4⋊C4—22no—
199C4⋊C4—32no—
200C4⋊C4—182no—
201C22⋊C4—64no—
202C22⋊C4—62no—
203C42—62no—
204C42C2×D3211yescharacteristic
205C22×C3—21no—
206C22×C3—122no—
207C22×C3—122no—
208C22×C3—122no—
209C22×C3#88211yescharacteristic
210C22×C3#95311yescharacteristic
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C3×C4—22no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—42no—
220C3×C4—42no—
221C3×C4—82no—
222C3×C4—122no—
223C3×C4#91612yes—
224C3×C4#91812yes—
225C3×C4#95012yes—
226C3×C4#95112yes—
227Dic3—42no—
228Dic3—242no—
229Dic3—241no—
230Dic3—241no—
231C32#163411yescharacteristic, complemented, Sylow
232C23—62no—
233Q8—42no—
234Q8—362no—
235D4—62no—
236D4—62no—
237D4—242no—
238C2×C4—22no—
239C2×C4—62no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—361no—
243C2×C4C2×#62611yescharacteristic
244C2×C4D3×#6712yes—
245C8—122no—
246C8—182no—
247C2×C3—21no—
248C2×C3—22no—
249C2×C3—242no—
250C2×C3#474711yescharacteristic
251C2×C3#480812yes—
252C2×C3#523912yes—
253C2×C3#524111yescharacteristic
254D3—82no—
255C22—122no—
256C22—122no—
257C22—122no—
258C22#6397711yescharacteristic
259C4—21no—
260C4—42no—
261C4—122no—
262C4—721no—
263C4#6394312yes—
264C4#6394712yes—
265C3—21no—
266C3#6849311yescharacteristic, complemented
267C3#7192811yescharacteristic, complemented
268C2—242no—
269C2#1059151011yescharacteristic
270C2#1059159912yes—
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC3×#5734C2
hypercenterC4⋊Q8D32
hypocenterC32#1634
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10592791C2semidirect
3. #10590268C2semidirect
4. C3×#4546C2semidirect
5. #10589080C2semidirect
27. #4546D3semidirect
231. C32#1634semidirect
266. C3#68493semidirect
267. C3#71928semidirect

Maximal subgroups

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

subgroupindex
C3×#57342
#105927912
#105902682
C3×#45462
#105890802
#719283
#684933