Finite Group Databasebeta

← #11713627

Subgroups — #11713627

Order 768.

641 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#3499⋊C3C211yesmaximal, characteristic
2C2×#5430C214yesmaximal, complemented
3#85336C211yesmaximal, characteristic, complemented
4C4×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5#25746—31nomaximal, Sylow
6C2×2−1+4⋊C3C2212yes—
7C2×#984C2212yes—
8D4×SL(2,𝔽3)—22no—
9C4×#41⋊C3C2211yescharacteristic
10#5430—28no—
11C2×GU(2,𝔽3)C2212yes—
12C4⋊#226—22no—
13#5076—41nomaximal
14#3499D311yescharacteristic
15#3152—31no—
16#3039—31no—
17C2×#456—34no—
18#2868—31no—
19#2005—34no—
20#1966—31no—
21#1962—31no—
22C4×#104—31no—
232−1+4⋊C3—24no—
24C2×#41⋊C3C2311yescharacteristic
25C22×SL(2,𝔽3)—22no—
26#984—24no—
27C2×#226—22no—
28C3×#127—41no—
29C2×#214—44no—
30#869—41no—
31C4×SL(2,𝔽3)D412yes—
32GU(2,𝔽3)D414yescomplemented
33C4×C3⋊C8—41no—
34C2×2−1+4C2×D312yes—
35C2×#137—32no—
36C2×#137—34no—
37D4×Q8—22no—
38#547—62no—
39#536—32no—
40#532—31no—
41C2×C4⋊Q8—31no—
42C2×C4⋊Q8—31no—
43C2×C4⋊Q8—34no—
44C4×#41C2×D311yescharacteristic
45#497—62no—
46#494—61no—
47#492—61no—
48#462—62no—
49#461—62no—
50#456—68no—
51#456—68no—
52#423—31no—
53C2×#104—32no—
54C2×#104—34no—
55#419—32no—
56C2×#101—34no—
57C4×M16—31no—
58#384—64no—
59C43—31no—
60#339—32no—
61C2×C3×D4—42no—
62C2×C3×D4—82no—
63C2×Dic6—42no—
64#41⋊C3C2×D412yes—
65C2×SL(2,𝔽3)—42no—
66C2×SL(2,𝔽3)C2×D411yescharacteristic
67#226—42no—
68C3×C42—41no—
69#214—88no—
70C4⋊Dic3—82no—
71C2×C3⋊C8—42no—
722−1+4—24no—
732−1+4—64no—
74C2×#41—32no—
75C2×#41—62no—
76C2×#41C22×D311yescharacteristic
77C22×Q8—22no—
78C22×Q8—32no—
79C22×Q8—32no—
80#137—68no—
81#137—64no—
82#137—64no—
83#137—124no—
84C2×Q16—64no—
85C2×Q16—62no—
86C2×SD16—62no—
87C2×SD16—64no—
88C2×M16—32no—
89C2×M16—32no—
90C4⋊Q8—31no—
91C4⋊Q8—61no—
92C4⋊Q8—62no—
93C4⋊Q8—68no—
94C4⋊Q8—62no—
95C4⋊Q8—61no—
96C4⋊Q8—61no—
97C4⋊Q8—124no—
98#127S411yescharacteristic
99#124—32no—
100#123—64no—
101C22⋊Q8—64no—
102#121—62no—
103C4×Q8#6712yes—
104C4×D4—32no—
105C4×D4—62no—
106#117—31no—
107C2×C4⋊C4—34no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—68no—
110C2×C42—31no—
111C2×C42—32no—
112C2×C42—62no—
113C4⋊C8—62no—
114#104—34no—
115#104—34no—
116#104—68no—
117#103—64no—
118#101—68no—
119#97—61no—
120C4×C8—61no—
121C23×C3—82no—
122C3×D4—84no—
123C3×D4—164no—
124C2×C3×C4—42no—
125C2×C3×C4—41no—
126C2×Dic3—82no—
127Dic6—84no—
128SL(2,𝔽3)#12711yescharacteristic
129C3⋊C8—44no—
130#41—32no—
131#41—64no—
132#41—64no—
133#41—124no—
134#41—124no—
135#41D3×D412yes—
136C2×Q8—32no—
137C2×Q8—42no—
138C2×Q8—64no—
139C2×Q8—64no—
140C2×Q8—62no—
141C2×Q8—62no—
142C2×Q8—64no—
143C2×Q8—64no—
144C2×Q8—62no—
145C2×Q8—122no—
146C2×Q8C2×#6711yescharacteristic
147C2×D4—22no—
148C2×D4—31no—
149C2×D4—62no—
150C2×D4C2×S412yes—
151C22×C4—31no—
152C22×C4—32no—
153C22×C4—34no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4—62no—
157Q16—128no—
158Q16—122no—
159Q16—122no—
160SD16—128no—
161SD16—124no—
162M16—34no—
163M16—62no—
164M16—64no—
165C2×C8—62no—
166C2×C8—62no—
167C4⋊C4—32no—
168C4⋊C4—64no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C4⋊C4—1216no—
172C4⋊C4—122no—
173C22⋊C4—61no—
174C22⋊C4—64no—
175C42—32no—
176C42—31no—
177C42—34no—
178C42—62no—
179C42—68no—
180C42C2×S411yescharacteristic
181C22×C3—41no—
182C22×C3—162no—
183C22×C3—164no—
184C3×C4—42no—
185C3×C4—44no—
186Dic3—162no—
187C23—22no—
188C23—31no—
189Q8—31no—
190Q8—64no—
191Q8—122no—
192Q8—122no—
193Q8—122no—
194Q8—122no—
195Q8—124no—
196Q8#94011yescharacteristic
197D4—24no—
198D4—32no—
199D4—44no—
200D4—61no—
201D4—124no—
202C2×C4—31no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—64no—
207C2×C4—68no—
208C2×C4—64no—
209C2×C4—68no—
210C2×C4—62no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—124no—
214C2×C4—122no—
215C2×C4GL(2,ℤ/4)12yes—
216C2×C4C22×S411yescharacteristic
217C8—64no—
218C8—122no—
219C2×C3—42no—
220C2×C3—41no—
221C2×C3—322no—
222C22—32no—
223C22—44no—
224C22—42no—
225C22—62no—
226C22C2×GL(2,ℤ/4)11yescharacteristic
227C4—31no—
228C4—31no—
229C4—68no—
230C4—62no—
231C4—122no—
232C4—242no—
233C4#543014yescomplemented
234C4D4×S412yes—
235C3—41noSylow
236C2—62no—
237C2—82no—
238C2#8549611yescharacteristic
239C2C2×#543012yes—
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3499D3
prime radical#3499⋊C3C2
hypercenter#127S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. C2×#5430C2semidirect
3. #85336C2semidirect
4. C4×GU(2,𝔽3)C2semidirect
32. GU(2,𝔽3)D4semidirect
233. C4#5430semidirect

Maximal subgroups

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

subgroupindex
#3499⋊C32
C2×#54302
#853362
C4×GU(2,𝔽3)2
#257463
#50764