Finite Group Databasebeta

← #11711512

Subgroups — #11711512

Order 768.

322 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 31 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
2#73156C212yesmaximal, complemented
3#12717—31nomaximal, Sylow
4C2×2−1+4⋊C3C411yescharacteristic, complemented
5C2×2−1+4⋊C3C2211yescharacteristic
6D4×SL(2,𝔽3)—22no—
7C4×#41⋊C3C411yescharacteristic, complemented
8#5420—22no—
9#4545—41nomaximal
10#3499D311yescharacteristic
11#2132—31no—
12#2126—31no—
13#2123—32no—
14#1796—32no—
152−1+4⋊C3—41no—
162−1+4⋊C3—41no—
17C2×#41⋊C3C2×C411yescharacteristic
18C22×SL(2,𝔽3)—21no—
19C22×SL(2,𝔽3)D412yes—
20C3×#127—41no—
21C4×SL(2,𝔽3)—21no—
22#859—42no—
23#834—42no—
24C2×2−1+4Dic311yescharacteristic, complemented
25C2×2−1+4C2×D311yescharacteristic
26D4×Q8—22no—
27#547—62no—
28#536—31no—
29#536—31no—
30#532—31no—
31C2×C4⋊Q8—31no—
32C4×#41Dic311yescharacteristic, complemented
33#421—62no—
34#419—62no—
35#413—32no—
36#410—32no—
37#356—61no—
38#355—61no—
39#354—61no—
40#353—61no—
41#352—64no—
42C2×C3×D4—41no—
43C2×C3×D4—41no—
44C2×C3×D4—82no—
45#41⋊C3—21no—
46C2×SL(2,𝔽3)—22no—
47C2×SL(2,𝔽3)—41no—
48C2×SL(2,𝔽3)C22⋊C411yescharacteristic
49C3×C42—41no—
50#217—82no—
512−1+4—41no—
522−1+4—41no—
532−1+4—121no—
542−1+4—121no—
55C2×#41—31no—
56C2×#41—32no—
57C2×#41—31no—
58C2×#41—61no—
59C2×#41C2×Dic311yescharacteristic
60C22×Q8—21no—
61C22×Q8—31no—
62C22×Q8—31no—
63C22×Q8#6712yes—
64C2×M16—62no—
65C4⋊Q8—31no—
66C4⋊Q8—61no—
67C4⋊Q8—62no—
68#127S411yescharacteristic
69#124—31no—
70#124—31no—
71#123—34no—
72#123—62no—
73C22⋊Q8—62no—
74C22⋊Q8—62no—
75#121—62no—
76C4×Q8—21no—
77C4×D4—61no—
78C4×D4—62no—
79#117—31no—
80#117—62no—
81C2×C4⋊C4—62no—
82C2×C42—31no—
83#104—124no—
84#103—122no—
85#103—122no—
86#101—62no—
87#100—62no—
88#99—62no—
89#99—62no—
90C23×C3—42no—
91C23×C3—81no—
92C3×D4—161no—
93C3×D4—161no—
94C3×D4—164no—
95C2×C3×C4—41no—
96C2×C3×C4—81no—
97C2×Dic3—162no—
98SL(2,𝔽3)#9911yescharacteristic
99#41—21no—
100#41—61no—
101#41—64no—
102#41—121no—
103#41—121no—
104#41—122no—
105#41—121no—
106#41—121no—
107#41—122no—
108#41—122no—
109C2×Q8—22no—
110C2×Q8—31no—
111C2×Q8—32no—
112C2×Q8—31no—
113C2×Q8—41no—
114C2×Q8—61no—
115C2×Q8—62no—
116C2×Q8—121no—
117C2×Q8—121no—
118C2×Q8—121no—
119C2×Q8—121no—
120C2×Q8—121no—
121C2×Q8#21711yescharacteristic
122C2×D4—22no—
123C2×D4—31no—
124C2×D4—32no—
125C2×D4—61no—
126C2×D4SL(2,ℤ/4)11yescharacteristic
127C2×D4C2×S411yescharacteristic
128C22×C4—32no—
129C22×C4—31no—
130C22×C4—32no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—61no—
134M16—122no—
135M16—122no—
136C2×C8—122no—
137C4⋊C4—62no—
138C4⋊C4—61no—
139C4⋊C4—64no—
140C4⋊C4—122no—
141C22⋊C4—32no—
142C22⋊C4—62no—
143C22⋊C4—62no—
144C22⋊C4—122no—
145C42—31no—
146C42—61no—
147C42—122no—
148C42SL(2,ℤ/4)11yescharacteristic
149C22×C3—41no—
150C22×C3—82no—
151C22×C3—161no—
152C22×C3—162no—
153C22×C3—162no—
154C3×C4—81no—
155C3×C4—82no—
156Dic3—322no—
157C23—21no—
158C23—31no—
159C23GL(2,ℤ/4)12yes—
160Q8—31no—
161Q8—62no—
162Q8—62no—
163Q8—121no—
164Q8—121no—
165Q8—121no—
166Q8—121no—
167Q8#83411yescharacteristic
168D4—32no—
169D4—41no—
170D4—41no—
171D4—44no—
172D4—61no—
173D4—122no—
174D4—122no—
175C2×C4—21no—
176C2×C4—31no—
177C2×C4—32no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—242no—
190C2×C4C2×SL(2,ℤ/4)11yescharacteristic
191C8—242no—
192C2×C3—41no—
193C2×C3—81no—
194C2×C3—162no—
195C2×C3—321no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—61no—
201C22—121no—
202C22#543211yescharacteristic
203C4—22no—
204C4—21no—
205C4—31no—
206C4—31no—
207C4—62no—
208C4—121no—
209C4—121no—
210C4—482no—
211C3—41noSylow
212C2—21no—
213C2—42no—
214C2—81no—
215C2—121no—
216C2#7314511yescharacteristic
subgroupG/H
center, socleC2#73145
commutatorC2×#41⋊C3C2×C4
FrattiniC2×D4C2×S4
Fitting#3499D3
prime radical#3499⋊C3C2
cosocleC2×2−1+4⋊C3C22
hypercenter#127S4
hypocenterSL(2,𝔽3)#99
LCSG ▷ C2×#41⋊C3 ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #73156C2semidirect
4. C2×2−1+4⋊C3C4semidirect
7. C4×#41⋊C3C4semidirect
24. C2×2−1+4Dic3semidirect
32. C4×#41Dic3semidirect

Maximal subgroups

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

subgroupindex
#3499⋊C32
#731562
#127173
#45454