Finite Group Databasebeta

← #11713680

Subgroups — #11713680

Order 768.

438 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 59 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×#557⋊C3C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73062C212yesmaximal
5#73059C212yesmaximal
6#25846—31nomaximal, Sylow
7#557⋊C3—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12C2×#859C2212yes—
13#5389—22no—
14#4983—41nomaximal
15C2×#557D311yescharacteristic
16C2×#501—31no—
17C2×#479—31no—
18#2584—31no—
19#2059—32no—
20#2059—32no—
21#1987—31no—
22#1985—31no—
23#1865—32no—
24#1863—32no—
25#1763—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×C3×C4⋊C4—41no—
28C2×Dic3⋊C4—41no—
29C2×Dic3⋊C4—42no—
30C4×SL(2,𝔽3)—22no—
31C4×SL(2,𝔽3)D412yes—
32#859—21no—
33#859—21no—
34#859Q814yescomplemented
35C3⋊#95—42no—
36C3⋊#95—41no—
37#557—22no—
38#557—62no—
39C2×C4⋊Q8—31no—
40C2×#125—32no—
41C2×#125—31no—
42C2×C4×Q8—31no—
43C2×C4×Q8C2×D312yes—
44C2×C4×Q8C2×D311yescharacteristic
45#501—64no—
46#479—62no—
47#479—62no—
48C2×#107—31no—
49C2×#107—32no—
50C2×#106—31no—
51C2×#106—32no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—31no—
55C2×#103—31no—
56C2×#97—31no—
57#398—61no—
58#395—61no—
59#389—32no—
60#389—32no—
61#389—62no—
62C4×C4⋊C4—31no—
63#340—32no—
64#340—32no—
65#340—32no—
66#336—32no—
67#336—32no—
68C22×C3×C4—41no—
69C22×C3×C4—42no—
70C22×Dic3—42no—
71C22×Dic3—41no—
72C22×Dic3—41no—
73C2×SL(2,𝔽3)C2×D411yescharacteristic
74C2×SL(2,𝔽3)C2×Q812yes—
75C3×C4⋊C4—82no—
76Dic3⋊C4—82no—
77Dic3⋊C4—84no—
78C22×Q8—31no—
79C22×Q8C22×D311yescharacteristic
80C22×C8—32no—
81C22×C8—31no—
82C22×C8—31no—
83C4⋊Q8—34no—
84C4⋊Q8—62no—
85#125—62no—
86#125—62no—
87#125—64no—
88#125—64no—
89C4×Q8—22no—
90C4×Q8—32no—
91C4×Q8—62no—
92C4×Q8—62no—
93C4×Q8—62no—
94C4×Q8#6712yes—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—61no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—62no—
106C2×C4⋊C4S411yescharacteristic
107C2×C42—31no—
108C2×C42—32no—
109C2×C42—61no—
110#107—62no—
111#107—64no—
112#106—62no—
113#106—64no—
114C4⋊C8—62no—
115#103—34no—
116#103—34no—
117#103—61no—
118#103—61no—
119#103—61no—
120#103—61no—
121#97—62no—
122#95—62no—
123#95—61no—
124#95—62no—
125C23×C3—41no—
126C2×C3×C4—42no—
127C2×C3×C4—82no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×C3×C4—82no—
131C2×Dic3—44no—
132C2×Dic3—42no—
133C2×Dic3—81no—
134C2×Dic3—82no—
135C2×Dic3—82no—
136C2×Dic3—81no—
137C2×Dic3—81no—
138C2×Dic3—82no—
139SL(2,𝔽3)C4⋊Q811yescharacteristic
140C2×Q8—32no—
141C2×Q8—31no—
142C2×Q8—62no—
143C2×Q8—122no—
144C2×Q8D3×Q812yes—
145C2×Q8C2×#6711yescharacteristic
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—62no—
150C22×C4—61no—
151C22×C4—61no—
152C22×C4C2×S412yes—
153C22×C4C2×S411yescharacteristic
154C2×C8—34no—
155C2×C8—62no—
156C2×C8—62no—
157C2×C8—61no—
158C2×C8—61no—
159C2×C8—62no—
160C2×C8—62no—
161C4⋊C4—22no—
162C4⋊C4—32no—
163C4⋊C4—34no—
164C4⋊C4—34no—
165C4⋊C4—64no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—64no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—122no—
175C4⋊C4—124no—
176C4⋊C4—122no—
177C4⋊C4—242no—
178C42—32no—
179C42—32no—
180C42—62no—
181C42—62no—
182C42—241no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C3×C4—82no—
189C3×C4—162no—
190Dic3—84no—
191Dic3—82no—
192Dic3—161no—
193C23C22×S411yescharacteristic
194Q8—32no—
195Q8—31no—
196Q8—61no—
197Q8—124no—
198Q8#94411yescharacteristic
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—64no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—121no—
218C2×C4—121no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4GL(2,ℤ/4)12yes—
222C8—64no—
223C8—121no—
224C8—121no—
225C2×C3—41no—
226C2×C3—41no—
227C2×C3—42no—
228C2×C3—42no—
229C2×C3—41no—
230C22Q8×S412yes—
231C22#592312yes—
232C22#592611yescharacteristic
233C22#592711yescharacteristic
234C22C2×GL(2,ℤ/4)11yescharacteristic
235C4—22no—
236C4—32no—
237C4—31no—
238C4—31no—
239C4—42no—
240C4—62no—
241C4—122no—
242C4—124no—
243C4—241no—
244C4—241no—
245C3—41noSylow
246C2#8541612yes—
247C2#8542312yes—
248C2#8550211yescharacteristic
249C2#8552211yescharacteristic
250C2#8553811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4⋊Q8
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
34. #859Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53892
#730632
#730622
#730592
#258463
#49834