Finite Group Databasebeta

← #11715659

Subgroups — #11715659

Order 768.

965 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 965 conjugacy classes of subgroups — too many to draw. Only its 36 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×#5950C211yesmaximal, characteristic
2#85564C212yesmaximal, complemented
3#20610—31nomaximal, Sylow
4#5950C412yescomplemented
5C2×#997C2211yescharacteristic
6C4⋊SL(2,ℤ/4)—43nomaximal
7C2×#585D311yescharacteristic
8C2×#410—33no—
9#2311—33no—
10#1790—36no—
11#1790—32no—
12#997C2×C411yescharacteristic
13C2×#41⋊C3—81no—
14C2×C4×A4—43no—
15C2×SL(2,ℤ/4)—46no—
16#585—36no—
17#585—62no—
18#585Dic312yescomplemented
19C2×2−1+4—61no—
20C2×H(5,𝔽2)—33no—
21C2×H(5,𝔽2)C2×D311yescharacteristic
22C22×#41—33no—
23C22×#41—61no—
24C2×#117—33no—
25C22×C4⋊C4—31no—
26#410—66no—
27#410—66no—
28C2×#99—36no—
29C2×#99—36no—
30#391—63no—
31#390—63no—
32#386—36no—
33#386—36no—
34#380—63no—
35#380—63no—
36C22×A4—43no—
37#41⋊C3—82no—
38C2×SL(2,𝔽3)—81no—
39C4×A4—46no—
40SL(2,ℤ/4)—86no—
41C4⋊Dic3—161no—
422−1+4—62no—
432−1+4—66no—
442−1+4—66no—
452−1+4—62no—
46H(5,𝔽2)—36no—
47H(5,𝔽2)—33no—
48H(5,𝔽2)—36no—
49H(5,𝔽2)—61no—
50H(5,𝔽2)—63no—
51H(5,𝔽2)—612no—
52H(5,𝔽2)—63no—
53H(5,𝔽2)C2×Dic311yescharacteristic
54C2×#41—21no—
55C2×#41—36no—
56C2×#41—36no—
57C2×#41—36no—
58C2×#41—36no—
59C2×#41—61no—
60C2×#41—62no—
61C2×#41—63no—
62C2×#41—66no—
63C2×#41—612no—
64C2×#41—66no—
65C2×#41—63no—
66C2×#41—66no—
67C2×#41—62no—
68C22×Q8—33no—
69C22×Q8—61no—
70C22×D4—33no—
71C22×D4—33no—
72C22×D4—33no—
73C22×D4—63no—
74C23×C4—31no—
75C23×C4—33no—
76C23×C4—32no—
77C23×C4S413yes—
78#117—612no—
79C2×C4⋊C4—63no—
80C2×C4⋊C4—63no—
81C2×C4⋊C4—63no—
82C2×C22⋊C4—36no—
83C2×C22⋊C4—66no—
84C2×C22⋊C4—66no—
85C2×C42—63no—
86#99—612no—
87#99—66no—
88#99—66no—
89#95—66no—
90#95—66no—
91C2×A4—43no—
92C2×A4—43no—
93C2×A4—43no—
94C2×C3×C4—161no—
95C2×Dic3—162no—
96SL(2,𝔽3)—81no—
97C24—31no—
98C24—33no—
99C24C2×S413yes—
100#41—22no—
101#41—61no—
102#41—66no—
103#41—612no—
104#41—66no—
105#41—612no—
106#41—612no—
107#41—63no—
108#41—612no—
109#41—612no—
110#41—66no—
111#41—66no—
112#41—63no—
113#41—61no—
114#41—66no—
115#41—66no—
116#41—62no—
117C2×Q8—21no—
118C2×Q8—36no—
119C2×Q8—36no—
120C2×Q8—63no—
121C2×Q8—66no—
122C2×Q8—62no—
123C2×Q8—66no—
124C2×D4—36no—
125C2×D4—33no—
126C2×D4—36no—
127C2×D4—33no—
128C2×D4—36no—
129C2×D4—33no—
130C2×D4—33no—
131C2×D4—36no—
132C2×D4—66no—
133C2×D4—66no—
134C2×D4—63no—
135C2×D4—66no—
136C2×D4—66no—
137C2×D4—66no—
138C2×D4—612no—
139C2×D4—61no—
140C2×D4—66no—
141C22×C4—33no—
142C22×C4—36no—
143C22×C4—32no—
144C22×C4—33no—
145C22×C4—33no—
146C22×C4—36no—
147C22×C4—33no—
148C22×C4—36no—
149C22×C4—36no—
150C22×C4—63no—
151C22×C4—66no—
152C22×C4—61no—
153C22×C4—66no—
154C22×C4—66no—
155C22×C4—242no—
156C22×C4SL(2,ℤ/4)16yescomplemented
157C4⋊C4—123no—
158C4⋊C4—126no—
159C4⋊C4—121no—
160C22⋊C4—612no—
161C22⋊C4—246no—
162C22⋊C4—246no—
163C42—126no—
164C22×C3—161no—
165A4—43no—
166C3×C4—162no—
167Dic3—322no—
168C23—33no—
169C23—31no—
170C23—33no—
171C23—33no—
172C23—36no—
173C23—43no—
174C23—43no—
175C23—63no—
176C23—61no—
177C23—121no—
178C23—121no—
179C23—123no—
180C23—123no—
181C23C2×SL(2,ℤ/4)13yes—
182Q8—21no—
183Q8—66no—
184Q8—66no—
185Q8—61no—
186D4—612no—
187D4—612no—
188D4—63no—
189D4—63no—
190D4—66no—
191D4—62no—
192D4—62no—
193C2×C4—36no—
194C2×C4—36no—
195C2×C4—33no—
196C2×C4—36no—
197C2×C4—36no—
198C2×C4—36no—
199C2×C4—33no—
200C2×C4—33no—
201C2×C4—63no—
202C2×C4—66no—
203C2×C4—66no—
204C2×C4—63no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—126no—
209C2×C4—126no—
210C2×C4—126no—
211C2×C4—126no—
212C2×C4—126no—
213C2×C4—122no—
214C2×C4#102011yescharacteristic
215C2×C3—161no—
216C2×C3—161no—
217C2×C3—161no—
218C22—33no—
219C22—33no—
220C22—43no—
221C22—63no—
222C22—63no—
223C22—62no—
224C22—123no—
225C22—123no—
226C22—121no—
227C22—126no—
228C22—121no—
229C22—121no—
230C22—243no—
231C22C2×#102011yescharacteristic
232C4—36no—
233C4—61no—
234C4—61no—
235C4—246no—
236C4—242no—
237C4#593612yes—
238C3—161noSylow
239C2—63no—
240C2—63no—
241C2—122no—
242C2#8751711yescharacteristic
243C2#8752111yescharacteristic
244C2C2×#593611yescharacteristic
subgroupG/H
center, Frattini, socleC22C2×#1020
commutator, hypocenter#997C2×C4
FittingC2×#585D3
prime radicalC2×#5950C2
cosocleC2×#997C22
hypercenterC2×C4#1020
LCSG ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #85564C2semidirect
4. #5950C4semidirect
18. #585Dic3semidirect
156. C22×C4SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×#59502
#855642
#206103
C4⋊SL(2,ℤ/4)4