Finite Group Databasebeta

← #11713637

Subgroups — #11713637

Order 768.

785 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×#5427C214yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25759—31nomaximal, Sylow
6C22×#41⋊C3C2212yes—
7C22×GL(2,𝔽3)C2212yes—
8C4⋊(#41⋊C3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10#5427—28no—
11C2×#859C2212yes—
12C4⋊GL(2,𝔽3)—22no—
13#4988—41nomaximal
14C2×#550D311yescharacteristic, complemented
15C2×#493—31no—
16C2×#460—31no—
17C2×#449—34no—
18#2541—31no—
19#1999—34no—
20#1982—31no—
21#1954—31no—
22C4×#103—31no—
23C22×D12—41no—
24C2×#41⋊C3—24no—
25C2×#41⋊C3—22no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×GL(2,𝔽3)—22no—
28C2×GL(2,𝔽3)—24no—
29C2×C3×C4⋊C4—41no—
30C2×(C2×D3)⋊C4—44no—
31C2×C4×Dic3—41no—
32C4×SL(2,𝔽3)D412yescomplemented
33#859D414yescomplemented
34C22×#41C2×D312yes—
35C22×SD16—32no—
36C22×D8—34no—
37#550—22no—
38#550—62no—
39C2×#127—31no—
40C2×#127—31no—
41C2×#127—61no—
42C2×#121—32no—
43C2×#121—34no—
44C2×C4×Q8C2×D311yescharacteristic
45C2×C4×D4—31no—
46#493—64no—
47#460—62no—
48#460—62no—
49#449—68no—
50#449—68no—
51C2×C4⋊C8—31no—
52C2×#103—32no—
53C2×#102—34no—
54C2×#102—32no—
55C2×C22⋊C8—34no—
56C2×C4×C8—31no—
57#392—62no—
58#390—61no—
59C4×C4⋊C4—31no—
60#340—32no—
61C23×D3—42no—
62C22×C3×C4—41no—
63C22×C3×C4—42no—
64C22×Dic3—42no—
65C2×D12—44no—
66C2×D12—82no—
67C2×D12—82no—
68#41⋊C3—42no—
69C2×SL(2,𝔽3)C2×D412yes—
70C2×SL(2,𝔽3)C2×D411yescharacteristic
71GL(2,𝔽3)—42no—
72C3×C4⋊C4—82no—
73(C2×D3)⋊C4—88no—
74C4×Dic3—82no—
75C2×#41—24no—
76C2×#41—22no—
77C2×#41—62no—
78C2×#41—64no—
79C22×Q8C22×D311yescharacteristic
80C22×D4—32no—
81C22×D4—32no—
82C22×D4—31no—
83C22×D4—61no—
84C22×D4—61no—
85C22×D4—64no—
86C22×D4—61no—
87C23×C4—32no—
88C2×SD16—64no—
89C2×SD16—62no—
90C2×SD16—64no—
91C2×SD16—62no—
92C2×D8—68no—
93C2×D8—68no—
94C2×D8—124no—
95C22×C8—32no—
96C22×C8—32no—
97#127—34no—
98#127—62no—
99#127—62no—
100#127—62no—
101#127—124no—
102#121—64no—
103#121—64no—
104#121—68no—
105#121—124no—
106C4×Q8—32no—
107C4×Q8#6712yes—
108C4×D4—62no—
109C4×D4—62no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4S411yescharacteristic
113C2×C22⋊C4—32no—
114C2×C22⋊C4—64no—
115C2×C42—31no—
116C2×C42—61no—
117C4⋊C8—62no—
118#103—34no—
119#103—34no—
120#102—68no—
121#102—64no—
122C22⋊C8—68no—
123C4×C8—62no—
124#95—61no—
125C23×C3—41no—
126C22×D3—44no—
127C22×D3—84no—
128C22×D3—82no—
129C22×D3—82no—
130C22×D3—84no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—84no—
135C2×C3×C4—82no—
136C2×Dic3—44no—
137C2×Dic3—82no—
138C2×Dic3—82no—
139D12—88no—
140SL(2,𝔽3)#12711yescharacteristic, complemented
141C24—32no—
142C24—62no—
143#41—42no—
144#41—124no—
145#41—124no—
146#41—122no—
147C2×Q8—32no—
148C2×Q8—31no—
149C2×Q8D3×D412yes—
150C2×Q8C2×#6711yescharacteristic
151C2×D4—34no—
152C2×D4—64no—
153C2×D4—64no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4—64no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—64no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—122no—
164C2×D4—128no—
165C2×D4—128no—
166C2×D4—122no—
167C2×D4—122no—
168C2×D4—122no—
169C2×D4—242no—
170C2×D4—242no—
171C2×D4—244no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—64no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4—64no—
178C22×C4—62no—
179C22×C4C2×S412yes—
180C22×C4C2×S411yescharacteristic
181SD16—124no—
182SD16—122no—
183SD16—122no—
184D8—1216no—
185C2×C8—34no—
186C2×C8—62no—
187C2×C8—62no—
188C2×C8—64no—
189C2×C8—62no—
190C4⋊C4—22no—
191C4⋊C4—32no—
192C4⋊C4—34no—
193C4⋊C4—61no—
194C4⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—128no—
197C42—32no—
198C42—32no—
199C42—122no—
200C22×C3—42no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—41no—
205C2×D3—84no—
206C2×D3—88no—
207C2×D3—162no—
208C2×D3—164no—
209C2×D3—162no—
210C3×C4—44no—
211C3×C4—162no—
212Dic3—84no—
213C23—64no—
214C23—62no—
215C23—62no—
216C23—64no—
217C23—62no—
218C23—124no—
219C23—122no—
220C23—122no—
221C23—124no—
222C23—122no—
223C23C22×S411yescharacteristic
224Q8—32no—
225Q8—31no—
226Q8—61no—
227Q8#94011yescharacteristic
228D4—68no—
229D4—122no—
230D4—122no—
231D4—124no—
232D4—122no—
233D4—122no—
234D4—124no—
235D4—244no—
236D4—244no—
237D4—2416no—
238D4—244no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—32no—
242C2×C4—32no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—32no—
246C2×C4—62no—
247C2×C4—64no—
248C2×C4—122no—
249C2×C4—124no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4#98512yes—
254C2×C4#98612yes—
255C2×C4GL(2,ℤ/4)12yes—
256C8—64no—
257C8—122no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—41no—
263D3—164no—
264C22—122no—
265C22—122no—
266C22—124no—
267C22—124no—
268C22—122no—
269C22—242no—
270C22—242no—
271C22—244no—
272C22—244no—
273C22—242no—
274C22D4×S412yes—
275C22C2×#98511yescharacteristic
276C22C2×#98611yescharacteristic
277C22#592512yes—
278C22C2×GL(2,ℤ/4)11yescharacteristic
279C4—32no—
280C4—31no—
281C4—31no—
282C4—42no—
283C4—62no—
284C4—124no—
285C4#542714yescomplemented
286C3—41noSylow
287C2—242no—
288C2—482no—
289C2#8540212yes—
290C2#8541012yes—
291C2#8549611yescharacteristic
292C2C2×#542711yescharacteristic
293C2#8553011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(#41⋊C3)C2semidirect
2. C2×#5427C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. C4×#859C2semidirect
14. C2×#550D3semidirect
32. C4×SL(2,𝔽3)D4semidirect
33. #859D4semidirect
140. SL(2,𝔽3)#127semidirect
285. C4#5427semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#54272
C2×C4⋊GL(2,𝔽3)2
C4×#8592
#257593
#49884