Finite Group Databasebeta

← #11712607

Subgroups — #11712607

Order 768.

1177 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1177 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C23×#41⋊C3C211yesmaximal, characteristic
2C2×#5426C212yesmaximal, complemented
3#73151C212yesmaximal, complemented
4#73148C212yesmaximal, complemented
5#20601—31nomaximal, Sylow
6C22×#41⋊C3—24no—
7C22×#41⋊C3C414yescomplemented
8C22×#41⋊C3C2211yescharacteristic
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11#5426—24no—
12C2×#859C2214yescomplemented
13#5212—41nomaximal
14C23×#41D311yescharacteristic
15C2×#417—32no—
16C2×#417—32no—
17C2×#407—31no—
18C2×#380—31no—
19#1788—32no—
20#1788—32no—
21#1787—32no—
22#1786—32no—
23C23×C3×C4—41no—
24C2×#41⋊C3—28no—
25C2×#41⋊C3—22no—
26C2×#41⋊C3C2×C412yes—
27C2×#41⋊C3C2×C412yescomplemented
28C2×#41⋊C3D414yes—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)C2×C412yescomplemented
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×#217—42no—
33C2×C4⋊Dic3—42no—
34#859—24no—
35C3⋊#95—42no—
36C22×#41—24no—
37C22×#41—31no—
38C22×#41—34no—
39C22×#41—31no—
40C22×#41—64no—
41C22×#41Dic314yescomplemented
42C22×#41C2×D311yescharacteristic
43C22×#41C2×D311yescharacteristic
44C23×Q8C2×D311yescharacteristic
45C23×D4—31no—
46C24×C4—31no—
47C22×M16—32no—
48C22×C4⋊C4—32no—
49C22×C22⋊C4—61no—
50#417—64no—
51#417—68no—
52#417—64no—
53C2×#103—34no—
54C2×#103—34no—
55C2×#102—34no—
56C2×#102—34no—
57#407—64no—
58C2×C22⋊C8—32no—
59C2×C22⋊C8—32no—
60#380—32no—
61#380—32no—
62#380—61no—
63#380—61no—
64C2×#95—61no—
65C24×C3—41no—
66C22×C3×C4—44no—
67C22×C3×C4—41no—
68C22×C3×C4—41no—
69C22×C3×C4—84no—
70C22×Dic3—44no—
71#41⋊C3—22no—
72#41⋊C3C22⋊C414yescomplemented
73C2×SL(2,𝔽3)—22no—
74C2×SL(2,𝔽3)C22×C411yescharacteristic
75C2×SL(2,𝔽3)C2×D412yes—
76#217—84no—
77C4⋊Dic3—84no—
78C25—31no—
79C2×#41—28no—
80C2×#41—22no—
81C2×#41—34no—
82C2×#41—32no—
83C2×#41—34no—
84C2×#41—34no—
85C2×#41—32no—
86C2×#41—38no—
87C2×#41—62no—
88C2×#41—62no—
89C2×#41—64no—
90C2×#41—64no—
91C2×#41—616no—
92C2×#41—62no—
93C2×#41—64no—
94C2×#41—68no—
95C2×#41—616no—
96C2×#41C2×Dic312yes—
97C2×#41C2×Dic312yes—
98C2×#41#6714yes—
99C22×Q8—22no—
100C22×Q8—31no—
101C22×Q8—32no—
102C22×Q8—62no—
103C22×Q8C2×Dic312yes—
104C22×Q8C22×D311yescharacteristic
105C22×D4—31no—
106C22×D4—31no—
107C22×D4—34no—
108C22×D4—62no—
109C22×D4—61no—
110C22×D4—64no—
111C22×D4—64no—
112C23×C4—31no—
113C23×C4—31no—
114C23×C4—34no—
115C23×C4—34no—
116C23×C4—31no—
117C23×C4—31no—
118C23×C4—31no—
119C23×C4—64no—
120C23×C4—64no—
121C23×C4—62no—
122C23×C4S411yescharacteristic
123C2×M16—64no—
124C2×M16—64no—
125C2×M16—64no—
126C22×C8—34no—
127C2×C4⋊C4—34no—
128C2×C4⋊C4—64no—
129C2×C4⋊C4—64no—
130C2×C4⋊C4—62no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—124no—
134#103—64no—
135#103—68no—
136#103—64no—
137#102—68no—
138#102—68no—
139C22⋊C8—64no—
140C22⋊C8—64no—
141#95—62no—
142#95—62no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—41no—
147C23×C3—82no—
148C23×C3—82no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—44no—
152C2×C3×C4—88no—
153C2×C3×C4—82no—
154C2×Dic3—84no—
155C2×Dic3—84no—
156C2×Dic3—84no—
157SL(2,𝔽3)C2×C22⋊C411yescharacteristic, complemented
158C24—31no—
159C24—31no—
160C24—34no—
161C24—64no—
162C24—62no—
163C24—62no—
164C24—122no—
165C24C2×S411yescharacteristic
166#41—22no—
167#41—34no—
168#41—38no—
169#41—316no—
170#41—64no—
171#41—616no—
172#41—68no—
173#41—64no—
174#41—68no—
175#41—616no—
176#41—62no—
177#41—68no—
178#41—64no—
179#41#21714yescomplemented
180C2×Q8—22no—
181C2×Q8—32no—
182C2×Q8—32no—
183C2×Q8—31no—
184C2×Q8—34no—
185C2×Q8—61no—
186C2×Q8—62no—
187C2×Q8—64no—
188C2×Q8—62no—
189C2×Q8—64no—
190C2×Q8C22×Dic311yescharacteristic
191C2×Q8C2×#6712yes—
192C2×D4—34no—
193C2×D4—32no—
194C2×D4—32no—
195C2×D4—34no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—62no—
199C2×D4—64no—
200C2×D4—62no—
201C2×D4—68no—
202C2×D4—68no—
203C2×D4—68no—
204C2×D4—616no—
205C22×C4—24no—
206C22×C4—31no—
207C22×C4—32no—
208C22×C4—32no—
209C22×C4—34no—
210C22×C4—34no—
211C22×C4—32no—
212C22×C4—32no—
213C22×C4—31no—
214C22×C4—32no—
215C22×C4—32no—
216C22×C4—34no—
217C22×C4—34no—
218C22×C4—62no—
219C22×C4—68no—
220C22×C4—62no—
221C22×C4—616no—
222C22×C4—68no—
223C22×C4—64no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4—64no—
227C22×C4—64no—
228C22×C4—124no—
229C22×C4—124no—
230C22×C4—124no—
231C22×C4SL(2,ℤ/4)14yes—
232C22×C4C2×S411yescharacteristic
233C22×C4C2×S411yescharacteristic
234M16—128no—
235C2×C8—64no—
236C2×C8—64no—
237C2×C8—64no—
238C4⋊C4—64no—
239C4⋊C4—64no—
240C4⋊C4—124no—
241C22⋊C4—124no—
242C22⋊C4—124no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—42no—
246C22×C3—41no—
247C22×C3—42no—
248C22×C3—44no—
249C22×C3—82no—
250C22×C3—84no—
251C22×C3—82no—
252C22×C3—84no—
253C3×C4—44no—
254C3×C4—82no—
255Dic3—164no—
256C23—22no—
257C23—22no—
258C23—32no—
259C23—34no—
260C23—32no—
261C23—68no—
262C23—61no—
263C23—61no—
264C23—62no—
265C23—61no—
266C23—61no—
267C23—62no—
268C23—62no—
269C23—68no—
270C23—124no—
271C23—128no—
272C23—128no—
273C23#98412yes—
274C23#98612yes—
275C23C2×SL(2,ℤ/4)12yes—
276C23C22×S411yescharacteristic
277Q8—31no—
278Q8—32no—
279Q8—34no—
280Q8—61no—
281Q8—62no—
282Q8—64no—
283Q8C2×#21711yescharacteristic
284D4—34no—
285D4—34no—
286D4—64no—
287D4—62no—
288D4—62no—
289D4—64no—
290D4—616no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—38no—
298C2×C4—34no—
299C2×C4—34no—
300C2×C4—32no—
301C2×C4—62no—
302C2×C4—64no—
303C2×C4—62no—
304C2×C4—68no—
305C2×C4—62no—
306C2×C4—64no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—68no—
310C2×C4—124no—
311C2×C4—124no—
312C2×C4—124no—
313C2×C4—248no—
314C2×C4C2×SL(2,ℤ/4)12yes—
315C2×C4C2×SL(2,ℤ/4)12yes—
316C2×C4GL(2,ℤ/4)14yes—
317C8—124no—
318C2×C3—42no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—42no—
322C2×C3—41no—
323C2×C3—84no—
324C22—24no—
325C22—22no—
326C22—24no—
327C22—22no—
328C22—34no—
329C22—64no—
330C22—62no—
331C22—62no—
332C22—64no—
333C22—62no—
334C22—62no—
335C22—62no—
336C22—1216no—
337C22—122no—
338C22—122no—
339C22#542614yescomplemented
340C22C2×#98411yescharacteristic
341C22C2×#98611yescharacteristic
342C22#592312yes—
343C22C22×SL(2,ℤ/4)11yescharacteristic
344C22C2×GL(2,ℤ/4)12yes—
345C4—22no—
346C4—32no—
347C4—31no—
348C4—31no—
349C4—62no—
350C4—244no—
351C4#543214yes—
352C3—41noSylow
353C2—24no—
354C2—64no—
355C2—122no—
356C2#8551712yes—
357C2C2×#542611yescharacteristic
358C2#8551911yescharacteristic
359C2#8552312yes—
360C2C2×#543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC23×#41D3
prime radicalC23×#41⋊C3C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)C2×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. #73151C2semidirect
4. #73148C2semidirect
7. C22×#41⋊C3C4semidirect
12. C2×#859C22semidirect
27. C2×#41⋊C3C2×C4semidirect
30. C22×SL(2,𝔽3)C2×C4semidirect
41. C22×#41Dic3semidirect
72. #41⋊C3C22⋊C4semidirect
157. SL(2,𝔽3)C2×C22⋊C4semidirect
179. #41#217semidirect
339. C22#5426semidirect

Maximal subgroups

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

subgroupindex
C23×#41⋊C32
C2×#54262
#731512
#731482
#206013
#52124