Finite Group Databasebeta

← #11714051

Subgroups — #11714051

Order 768.

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

normal subgroups only

This group has 550 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, complemented
2C2×#5427C211yesmaximal, characteristic
3#73151C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5#73059C212yesmaximal, complemented
6#33013—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×GL(2,𝔽3)C2211yescharacteristic
9#520⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11#5427—22no—
12C2×#859C2211yescharacteristic
13C2×#859C2212yes—
14#4993—41nomaximal
15C2×#520D311yescharacteristic, complemented
16C2×#481—31no—
17C2×#449—31no—
18#2624—31no—
19#1880—32no—
20#1879—32no—
21#1865—32no—
22#1864—32no—
23#1823—31no—
24#1818—31no—
25#1788—31no—
26C2×#41⋊C3—22no—
27C2×#41⋊C3D412yes—
28C22×SL(2,𝔽3)C2311yescharacteristic
29C2×GL(2,𝔽3)—21no—
30C2×GL(2,𝔽3)—22no—
31C2×C3×C4⋊C4—41no—
32C2×(C2×D3)⋊C4—41no—
33C2×(C2×D3)⋊C4—42no—
34C2×Dic3⋊C4—42no—
35C4×SL(2,𝔽3)—22no—
36#859—21no—
37#859—22no—
38C3⋊#95—41no—
39C22×#41C2×D311yescharacteristic
40C22×SD16—31no—
41C22×D8—31no—
42C2×#121—31no—
43#520—22no—
44#520—62no—
45C2×C4×Q8C2×D312yes—
46C2×C4×D4—32no—
47C2×#117—31no—
48C22×C4⋊C4—31no—
49#481—64no—
50#449—62no—
51#449—62no—
52C2×#106—32no—
53C2×#103—31no—
54C2×#103—32no—
55C2×#102—32no—
56C2×#102—31no—
57C2×#102—32no—
58C2×C22⋊C8—31no—
59C2×C22⋊C8—31no—
60#399—61no—
61#397—61no—
62#394—62no—
63#390—32no—
64#389—32no—
65#380—31no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C23×D3—41no—
71C22×C3×C4—41no—
72C22×C3×C4—42no—
73C22×Dic3—42no—
74C22×Dic3—41no—
75#41⋊C3—22no—
76C2×SL(2,𝔽3)C2×D411yescharacteristic
77C2×SL(2,𝔽3)#4112yes—
78GL(2,𝔽3)—41no—
79C3×C4⋊C4—82no—
80(C2×D3)⋊C4—82no—
81(C2×D3)⋊C4—84no—
82Dic3⋊C4—84no—
83C2×#41—22no—
84C2×#41—32no—
85C2×#41—62no—
86C2×#41#6712yes—
87C22×Q8C22×D311yescharacteristic
88C22×D4—31no—
89C22×D4—31no—
90C22×D4—61no—
91C23×C4—31no—
92C23×C4—61no—
93C2×SD16—62no—
94C2×SD16—61no—
95C2×SD16—62no—
96C2×SD16—61no—
97C2×D8—64no—
98C2×D8—62no—
99C22×C8—31no—
100C22×C8—31no—
101C22×C8—32no—
102#121—62no—
103#121—62no—
104C4×Q8—22no—
105C4×Q8—62no—
106C4×D4—64no—
107C4×D4—64no—
108#117—62no—
109#117—62no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—34no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4S411yescharacteristic
118C2×C22⋊C4—32no—
119C2×C22⋊C4—61no—
120C2×C22⋊C4—62no—
121C2×C22⋊C4—61no—
122C2×C22⋊C4—62no—
123C2×C42—32no—
124#106—64no—
125#103—62no—
126#103—61no—
127#103—62no—
128#103—61no—
129#102—64no—
130#102—64no—
131#102—62no—
132C22⋊C8—62no—
133C22⋊C8—62no—
134#95—62no—
135#95—61no—
136C23×C3—41no—
137C22×D3—42no—
138C22×D3—81no—
139C22×D3—82no—
140C22×D3—82no—
141C22×D3—81no—
142C2×C3×C4—42no—
143C2×C3×C4—82no—
144C2×C3×C4—82no—
145C2×C3×C4—82no—
146C2×C3×C4—82no—
147C2×Dic3—44no—
148C2×Dic3—82no—
149C2×Dic3—82no—
150C2×Dic3—81no—
151C2×Dic3—82no—
152SL(2,𝔽3)#12311yescharacteristic
153C24—31no—
154C24—61no—
155#41—22no—
156#41—64no—
157#41—62no—
158#41—64no—
159C2×Q8—31no—
160C2×Q8—32no—
161C2×Q8#23512yes—
162C2×Q8C2×#6711yescharacteristic
163C2×D4—32no—
164C2×D4—34no—
165C2×D4—61no—
166C2×D4—62no—
167C2×D4—61no—
168C2×D4—61no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—122no—
172C2×D4—242no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—64no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—241no—
183C22×C4—241no—
184C22×C4C2×S412yes—
185C22×C4C2×S411yescharacteristic
186SD16—122no—
187SD16—121no—
188SD16—121no—
189D8—124no—
190C2×C8—62no—
191C2×C8—61no—
192C2×C8—61no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—62no—
197C4⋊C4—22no—
198C4⋊C4—34no—
199C4⋊C4—62no—
200C4⋊C4—61no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—61no—
204C4⋊C4—62no—
205C4⋊C4—242no—
206C22⋊C4—64no—
207C22⋊C4—122no—
208C22⋊C4—242no—
209C22⋊C4—242no—
210C22⋊C4—241no—
211C42—62no—
212C42—62no—
213C22×C3—42no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—41no—
218C2×D3—82no—
219C2×D3—84no—
220C2×D3—161no—
221C2×D3—162no—
222C2×D3—161no—
223C3×C4—82no—
224C3×C4—162no—
225Dic3—84no—
226Dic3—161no—
227C23—32no—
228C23—62no—
229C23—61no—
230C23—61no—
231C23—62no—
232C23—122no—
233C23—121no—
234C23—121no—
235C23—122no—
236C23—121no—
237C23C22×S411yescharacteristic
238Q8—32no—
239Q8—31no—
240Q8—61no—
241Q8#92911yescharacteristic
242D4—34no—
243D4—62no—
244D4—64no—
245D4—121no—
246D4—121no—
247D4—122no—
248D4—244no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—64no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—121no—
268C2×C4—244no—
269C2×C4GL(2,ℤ/4)12yes—
270C8—122no—
271C8—121no—
272C8—121no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—42no—
276C2×C3—42no—
277C2×C3—41no—
278D3—162no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—122no—
283C22—122no—
284C22—241no—
285C22—241no—
286C22—242no—
287C22—242no—
288C22—241no—
289C22#591212yes—
290C22#592411yescharacteristic
291C22#592511yescharacteristic
292C22#592612yes—
293C22C2×GL(2,ℤ/4)11yescharacteristic
294C4—22no—
295C4—31no—
296C4—31no—
297C4—32no—
298C4—42no—
299C4—122no—
300C4—241no—
301C4—242no—
302C3—41noSylow
303C2—122no—
304C2—481no—
305C2#8536712yes—
306C2#8536912yes—
307C2#8548511yescharacteristic
308C2#8552911yescharacteristic
309C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC2×#5427C2
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
1. C2×#520⋊C3C2semidirect
3. #73151C2semidirect
4. #73065C2semidirect
5. #73059C2semidirect
15. C2×#520D3semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#54272
#731512
#730652
#730592
#330133
#49934