Finite Group Databasebeta

← #11713239

Subgroups — #11713239

Order 768.

559 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 57 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×#548⋊C3C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3#73151C211yesmaximal, characteristic, complemented
4#73148C212yesmaximal, complemented
5#73059C212yesmaximal, complemented
6#24052—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C23×SL(2,𝔽3)C2211yescharacteristic
9#548⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11#5426—22no—
12C2×#859C2212yes—
13C2×#859C2212yes—
14#4949—41nomaximal
15C2×#548D311yescharacteristic
16C2×#483—31no—
17C2×#417—31no—
18#2487—31no—
19#2042—32no—
20#2034—32no—
21#1894—31no—
22#1870—31no—
23#1865—32no—
24#1788—31no—
25#1787—32no—
26C2×#41⋊C3—22no—
27C2×#41⋊C3D412yes—
28C22×SL(2,𝔽3)—21no—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×C3×C22⋊C4—41no—
32C2×#217—42no—
33C2×C4⋊Dic3—41no—
34C2×Dic3⋊C4—42no—
35C4×SL(2,𝔽3)—21no—
36#859—22no—
37#859—22no—
38C3⋊#95—41no—
39C22×#41C2×D311yescharacteristic
40C23×Q8C2×D311yescharacteristic
41C22×M16—31no—
42#548—22no—
43#548—62no—
44C2×#124—31no—
45C2×C22⋊Q8—31no—
46C2×#121—31no—
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×D4—31no—
49C22×C4⋊C4—31no—
50#483—64no—
51C2×#106—32no—
52#417—62no—
53#417—62no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×C22⋊C8—31no—
61C2×C22⋊C8—32no—
62#400—61no—
63#399—61no—
64#389—32no—
65#386—61no—
66#385—61no—
67#380—31no—
68#380—32no—
69#340—32no—
70#336—32no—
71C24×C3—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76#41⋊C3—22no—
77C2×SL(2,𝔽3)—41no—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C2×SL(2,𝔽3)#4112yes—
80C3×C22⋊C4—82no—
81#217—84no—
82C4⋊Dic3—82no—
83Dic3⋊C4—84no—
84C2×#41—22no—
85C2×#41—32no—
86C2×#41—62no—
87C2×#41#6712yes—
88C22×Q8—22no—
89C22×Q8—21no—
90C22×Q8—31no—
91C22×Q8—61no—
92C22×Q8—62no—
93C22×Q8C22×D311yescharacteristic
94C22×D4—31no—
95C22×D4—31no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—61no—
99C2×M16—62no—
100C2×M16—122no—
101C22×C8—32no—
102C22×C8—32no—
103#124—62no—
104#124—62no—
105C22⋊Q8—34no—
106C22⋊Q8—62no—
107#121—62no—
108#121—62no—
109C4×Q8—21no—
110C4×Q8—61no—
111C4×D4—62no—
112C4×D4—62no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4—62no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—61no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4S411yescharacteristic
126C2×C42—31no—
127#106—64no—
128#103—62no—
129#103—62no—
130#103—64no—
131#103—62no—
132#103—62no—
133#103—122no—
134#102—64no—
135#102—122no—
136C22⋊C8—62no—
137C22⋊C8—122no—
138#95—62no—
139#95—62no—
140#95—61no—
141C23×C3—42no—
142C23×C3—41no—
143C23×C3—82no—
144C23×C3—81no—
145C23×C3—82no—
146C23×C3—81no—
147C2×C3×C4—42no—
148C2×C3×C4—82no—
149C2×C3×C4—81no—
150C2×C3×C4—82no—
151C2×Dic3—44no—
152C2×Dic3—82no—
153C2×Dic3—82no—
154C2×Dic3—82no—
155C2×Dic3—82no—
156C2×Dic3—82no—
157SL(2,𝔽3)#12311yescharacteristic
158C24—31no—
159C24C2×S411yescharacteristic
160#41—22no—
161#41—62no—
162#41—64no—
163#41—122no—
164C2×Q8—31no—
165C2×Q8—32no—
166C2×Q8—32no—
167C2×Q8—41no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×Q8—61no—
171C2×Q8—122no—
172C2×Q8—122no—
173C2×Q8—121no—
174C2×Q8#23712yes—
175C2×Q8C2×#6711yescharacteristic
176C2×D4—32no—
177C2×D4—62no—
178C2×D4—62no—
179C2×D4—61no—
180C2×D4—62no—
181C2×D4—62no—
182C2×D4—121no—
183C22×C4—31no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4—62no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—122no—
197C22×C4—242no—
198C22×C4C2×S411yescharacteristic
199C22×C4C2×S411yescharacteristic
200M16—124no—
201C2×C8—62no—
202C2×C8—62no—
203C2×C8—62no—
204C2×C8—62no—
205C2×C8—62no—
206C2×C8—62no—
207C4⋊C4—32no—
208C4⋊C4—61no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—122no—
215C4⋊C4—242no—
216C22⋊C4—22no—
217C22⋊C4—62no—
218C22⋊C4—64no—
219C22⋊C4—122no—
220C22⋊C4—242no—
221C42—61no—
222C42—61no—
223C22×C3—42no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—82no—
229C22×C3—84no—
230C22×C3—161no—
231C22×C3—162no—
232C22×C3—161no—
233C3×C4—82no—
234C3×C4—161no—
235Dic3—84no—
236Dic3—162no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—32no—
242C23—62no—
243C23—61no—
244C23—61no—
245C23—62no—
246C23#98412yes—
247C23C22×S411yescharacteristic
248Q8—32no—
249Q8—31no—
250Q8—61no—
251Q8—64no—
252Q8—121no—
253Q8—121no—
254Q8#93511yescharacteristic
255D4—62no—
256D4—62no—
257D4—122no—
258D4—124no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—64no—
269C2×C4—62no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—244no—
284C2×C4GL(2,ℤ/4)12yes—
285C8—122no—
286C8—122no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—42no—
290C2×C3—42no—
291C2×C3—41no—
292C2×C3—162no—
293C22—22no—
294C22—24no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—62no—
299C22—62no—
300C22—62no—
301C22—122no—
302C22—122no—
303C22#591412yes—
304C22C2×#98411yescharacteristic
305C22#592311yescharacteristic
306C22#592612yes—
307C22C2×GL(2,ℤ/4)11yescharacteristic
308C4—22no—
309C4—31no—
310C4—32no—
311C4—31no—
312C4—41no—
313C4—121no—
314C4—121no—
315C4—242no—
316C4—242no—
317C3—41noSylow
318C2—42no—
319C2—122no—
320C2#8537712yes—
321C2#8538712yes—
322C2#8549111yescharacteristic
323C2#8551711yescharacteristic
324C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#548D3
prime radicalC2×#548⋊C3C2
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. #73151C2semidirect
4. #73148C2semidirect
5. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#548⋊C32
C2×#54262
#731512
#731482
#730592
#240523
#49494