Finite Group Databasebeta

← #11714053

Subgroups — #11714053

Order 768.

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

normal subgroups only

This group has 511 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
1C2×#520⋊C3C211yesmaximal, characteristic
2C2×#5425C211yesmaximal, characteristic
3#73151C211yesmaximal, characteristic
4#73061C212yesmaximal, complemented
5#73059C212yesmaximal, complemented
6#33016—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×#226C2211yescharacteristic
9#520⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11#5425—22no—
12C2×#859C2211yescharacteristic
13C2×#859C2212yes—
14#4983—41nomaximal
15C2×#520D311yescharacteristic
16C2×#483—31no—
17C2×#452—31no—
18#2625—31no—
19#1878—32no—
20#1877—32no—
21#1865—32no—
22#1864—32no—
23#1824—31no—
24#1818—31no—
25#1788—31no—
26C2×#41⋊C3—22no—
27C2×#41⋊C3D412yes—
28C22×SL(2,𝔽3)C2311yescharacteristic
29C2×#226—21no—
30C2×#226—22no—
31C2×C3×C4⋊C4—41no—
32C2×Dic3⋊C4—41no—
33C2×Dic3⋊C4—42no—
34C4×SL(2,𝔽3)—22no—
35#859—21no—
36#859—22no—
37C3⋊#95—42no—
38C3⋊#95—41no—
39C22×#41C2×D311yescharacteristic
40C22×Q16—31no—
41C22×SD16—31no—
42C2×C22⋊Q8—31no—
43#520—22no—
44#520—62no—
45C2×C4×Q8C2×D312yes—
46C2×C4×D4—32no—
47C2×#117—31no—
48C22×C4⋊C4—31no—
49#483—64no—
50#452—62no—
51#452—62no—
52C2×#106—32no—
53C2×#103—32no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#102—31no—
57C2×#102—32no—
58C2×C22⋊C8—31no—
59C2×C22⋊C8—31no—
60#400—61no—
61#398—61no—
62#397—62no—
63#391—32no—
64#389—32no—
65#380—31no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C22×C3×C4—41no—
71C22×C3×C4—42no—
72C22×Dic3—42no—
73C22×Dic3—41no—
74C22×Dic3—41no—
75#41⋊C3—22no—
76C2×SL(2,𝔽3)C2×D411yescharacteristic
77C2×SL(2,𝔽3)#4112yes—
78#226—41no—
79C3×C4⋊C4—82no—
80Dic3⋊C4—82no—
81Dic3⋊C4—84no—
82C2×#41—22no—
83C2×#41—32no—
84C2×#41—62no—
85C2×#41#6712yes—
86C22×Q8—31no—
87C22×Q8C22×D311yescharacteristic
88C22×D4—31no—
89C23×C4—31no—
90C23×C4—61no—
91C2×Q16—62no—
92C2×Q16—61no—
93C2×Q16—62no—
94C2×Q16—61no—
95C2×SD16—64no—
96C2×SD16—62no—
97C22×C8—31no—
98C22×C8—31no—
99C22×C8—32no—
100C22⋊Q8—62no—
101C22⋊Q8—62no—
102C4×Q8—22no—
103C4×Q8—62no—
104C4×D4—64no—
105C4×D4—64no—
106#117—62no—
107#117—62no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—34no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4S411yescharacteristic
117C2×C22⋊C4—32no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—61no—
120C2×C42—32no—
121#106—64no—
122#103—64no—
123#103—62no—
124#103—61no—
125#103—62no—
126#103—61no—
127#102—64no—
128#102—62no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—62no—
132#95—62no—
133#95—61no—
134#95—62no—
135C23×C3—41no—
136C2×C3×C4—42no—
137C2×C3×C4—82no—
138C2×C3×C4—82no—
139C2×C3×C4—82no—
140C2×C3×C4—82no—
141C2×Dic3—42no—
142C2×Dic3—44no—
143C2×Dic3—81no—
144C2×Dic3—82no—
145C2×Dic3—82no—
146C2×Dic3—81no—
147C2×Dic3—81no—
148C2×Dic3—82no—
149SL(2,𝔽3)#12311yescharacteristic
150C24—31no—
151#41—22no—
152#41—64no—
153#41—62no—
154#41—64no—
155C2×Q8—31no—
156C2×Q8—32no—
157C2×Q8—61no—
158C2×Q8—61no—
159C2×Q8—62no—
160C2×Q8—62no—
161C2×Q8#23512yes—
162C2×Q8C2×#6711yescharacteristic
163C2×D4—32no—
164C2×D4—34no—
165C2×D4—61no—
166C2×D4—62no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—32no—
170C22×C4—32no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4—64no—
174C22×C4—62no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—242no—
178C22×C4C2×S412yes—
179C22×C4C2×S411yescharacteristic
180Q16—122no—
181Q16—121no—
182Q16—121no—
183SD16—124no—
184C2×C8—61no—
185C2×C8—62no—
186C2×C8—61no—
187C2×C8—62no—
188C2×C8—62no—
189C2×C8—62no—
190C2×C8—62no—
191C4⋊C4—22no—
192C4⋊C4—34no—
193C4⋊C4—62no—
194C4⋊C4—61no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—61no—
198C4⋊C4—62no—
199C4⋊C4—122no—
200C4⋊C4—242no—
201C22⋊C4—64no—
202C22⋊C4—122no—
203C22⋊C4—241no—
204C42—62no—
205C42—62no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C3×C4—82no—
212C3×C4—162no—
213Dic3—82no—
214Dic3—84no—
215Dic3—161no—
216C23—32no—
217C23—62no—
218C23—62no—
219C23—61no—
220C23—61no—
221C23C22×S411yescharacteristic
222Q8—32no—
223Q8—31no—
224Q8—61no—
225Q8—121no—
226Q8—121no—
227Q8—122no—
228Q8#92911yescharacteristic
229D4—34no—
230D4—62no—
231D4—64no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—64no—
243C2×C4—64no—
244C2×C4—62no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—244no—
254C2×C4GL(2,ℤ/4)12yes—
255C8—122no—
256C8—121no—
257C8—121no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—41no—
263C22—62no—
264C22—62no—
265C22—62no—
266C22—122no—
267C22—122no—
268C22#591212yes—
269C22#592412yes—
270C22#592611yescharacteristic
271C22#592711yescharacteristic
272C22C2×GL(2,ℤ/4)11yescharacteristic
273C4—22no—
274C4—31no—
275C4—31no—
276C4—32no—
277C4—42no—
278C4—122no—
279C4—241no—
280C4—242no—
281C4—241no—
282C3—41noSylow
283C2—122no—
284C2#8536212yes—
285C2#8536912yes—
286C2#8548511yescharacteristic
287C2#8552711yescharacteristic
288C2#8553811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
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
4. #73061C2semidirect
5. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#54252
#731512
#730612
#730592
#330163
#49834