Finite Group Databasebeta

← #11714042

Subgroups — #11714042

Order 768.

492 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 49 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
2#85517C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic
4#73077C212yesmaximal, complemented
5#73076C212yesmaximal, complemented
6#33000—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C2×#984C2211yescharacteristic
9#520⋊C3—21no—
10#520⋊C3C2212yes—
11C2×C4×SL(2,𝔽3)—21no—
12#5426C2211yescharacteristic
13#5425—21no—
14C2×GU(2,𝔽3)C2212yes—
15#5419—21no—
16#4971—41nomaximal
17C2×#520D311yescharacteristic
18#3095—31no—
19#3007—31no—
20#2502—31no—
21#1882—32no—
22#1882—32no—
23#1859—32no—
24#1859—32no—
25#1826—31no—
26#1821—31no—
27#1789—31no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3D411yescharacteristic
30C2×#41⋊C3C2311yescharacteristic
31C22×SL(2,𝔽3)D411yescharacteristic
32#984—22no—
33C2×#226—21no—
34C2×C3×C4⋊C4—41no—
35#924—41no—
36C4×SL(2,𝔽3)—22no—
37GU(2,𝔽3)—22no—
38#859—21no—
39#830—41no—
40#810—42no—
41#808—42no—
42C22×#41C2×D311yescharacteristic
43C2×#137—31no—
44C2×#137—31no—
45C2×C22⋊Q8—31no—
46#520—21no—
47#520—32no—
48#520—61no—
49#520C2×D312yes—
50C2×C4×Q8—21no—
51C2×C4×D4—61no—
52C2×#117—31no—
53C22×C4⋊C4—31no—
54#485—62no—
55#484—61no—
56#483—61no—
57#452—61no—
58#452—61no—
59#451—61no—
60#450—61no—
61#428—32no—
62C2×#104—32no—
63C2×#104—32no—
64#419—32no—
65#417—31no—
66#417—31no—
67#407—31no—
68#407—31no—
69#398—62no—
70#391—32no—
71#385—62no—
72#384—32no—
73C4×C22⋊C4—31no—
74#343—32no—
75#343—32no—
76#339—32no—
77#339—32no—
78C22×C3×C4—41no—
79C22×C3×C4—81no—
80C2×Dic6—41no—
81#41⋊C3—21no—
82#41⋊C3#4112yes—
83C2×SL(2,𝔽3)—21no—
84C2×SL(2,𝔽3)C2×D411yescharacteristic
85#226—41no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—81no—
88#217—81no—
89C4⋊Dic3—41no—
90Dic3⋊C4—81no—
91C2×C3⋊C8—42no—
92C2×#41—22no—
93C2×#41—31no—
94C2×#41—31no—
95C2×#41—62no—
96C2×#41#6711yescharacteristic
97C2×#41C22×D311yescharacteristic
98C22×Q8—31no—
99C22×Q8#6711yescharacteristic
100C22×D4—31no—
101C23×C4—31no—
102C23×C4—32no—
103#137—62no—
104#137—64no—
105#137—62no—
106C2×Q16—61no—
107C2×Q16—61no—
108C2×SD16—61no—
109C2×SD16—61no—
110C2×M16—31no—
111C2×M16—31no—
112C2×M16—32no—
113C22⋊Q8—61no—
114C22⋊Q8—61no—
115C22⋊Q8—61no—
116C22⋊Q8—61no—
117C4×Q8—22no—
118C4×Q8—62no—
119C4×D4—64no—
120C4×D4—62no—
121C4×D4—62no—
122#117—32no—
123#117—32no—
124#117—61no—
125#117—61no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—32no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—64no—
135C2×C4⋊C4S411yescharacteristic
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C42—32no—
140C2×C42—61no—
141#107—62no—
142#106—62no—
143#104—62no—
144#104—64no—
145#104—62no—
146#103—62no—
147#103—61no—
148#103—61no—
149#103—62no—
150#102—61no—
151#102—61no—
152C22⋊C8—61no—
153C22⋊C8—61no—
154C22⋊C8—61no—
155C22⋊C8—61no—
156#95—61no—
157#95—64no—
158C23×C3—41no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—82no—
162C2×C3×C4—82no—
163C2×C3×C4—162no—
164C2×Dic3—81no—
165C2×Dic3—81no—
166Dic6—82no—
167SL(2,𝔽3)#12311yescharacteristic
168C3⋊C8—82no—
169C24—31no—
170#41—21no—
171#41—32no—
172#41—34no—
173#41—62no—
174#41—62no—
175#41—61no—
176#41—62no—
177#41#23512yes—
178C2×Q8—21no—
179C2×Q8—31no—
180C2×Q8—61no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—61no—
184C2×Q8—61no—
185C2×Q8C2×#6711yescharacteristic
186C2×D4—31no—
187C2×D4—31no—
188C2×D4—61no—
189C2×D4—62no—
190C2×D4—62no—
191C22×C4—21no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—34no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—62no—
201C22×C4—61no—
202C22×C4—62no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—242no—
206C22×C4C2×S411yescharacteristic
207Q16—121no—
208Q16—122no—
209Q16—121no—
210SD16—122no—
211SD16—122no—
212M16—62no—
213M16—62no—
214M16—62no—
215M16—62no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—62no—
219C4⋊C4—21no—
220C4⋊C4—32no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—244no—
229C4⋊C4C2×S412yes—
230C22⋊C4—62no—
231C22⋊C4—62no—
232C22⋊C4—64no—
233C22⋊C4—121no—
234C22⋊C4—121no—
235C42—62no—
236C42—62no—
237C42—62no—
238C42—62no—
239C22×C3—41no—
240C22×C3—42no—
241C22×C3—81no—
242C22×C3—81no—
243C3×C4—42no—
244C3×C4—81no—
245C3×C4—162no—
246Dic3—161no—
247Dic3—161no—
248C23—31no—
249C23—31no—
250C23—62no—
251C23—62no—
252C23—121no—
253C23GL(2,ℤ/4)11yescharacteristic
254Q8—31no—
255Q8—32no—
256Q8—61no—
257Q8—121no—
258Q8—121no—
259Q8—122no—
260Q8#92911yescharacteristic
261D4—32no—
262D4—32no—
263D4—61no—
264D4—61no—
265D4—64no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—31no—
272C2×C4—42no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—64no—
280C2×C4—62no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—124no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—242no—
290C2×C4GL(2,ℤ/4)11yescharacteristic
291C2×C4C22×S411yescharacteristic
292C8—121no—
293C8—121no—
294C8—122no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—82no—
299C22—21no—
300C22—21no—
301C22—32no—
302C22—61no—
303C22—61no—
304C22—61no—
305C22—61no—
306C22—61no—
307C22—124no—
308C22#543012yes—
309C22C2×GL(2,ℤ/4)11yescharacteristic
310C4—21no—
311C4—31no—
312C4—31no—
313C4—42no—
314C4—61no—
315C4—124no—
316C4—122no—
317C4—241no—
318C4—241no—
319C4#591212yes—
320C3—41noSylow
321C2—22no—
322C2—62no—
323C2—121no—
324C2#8548511yescharacteristic
325C2#8552611yescharacteristic
326C2C2×#543011yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #73077C2semidirect
5. #73076C2semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
#855172
#731492
#730772
#730762
#330003
#49714