Finite Group Databasebeta

← #11713943

Subgroups — #11713943

Order 768.

1120 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 75 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×#5425C212yesmaximal, complemented
3C2×#5419C211yesmaximal, characteristic, complemented
4#73151C212yesmaximal, complemented
5#73148C211yesmaximal, characteristic, complemented
6#32883—31nomaximal, Sylow
7C22×#41⋊C3—24no—
8C22×#41⋊C3—22no—
9C22×#41⋊C3C2212yes—
10C23×SL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic, complemented
12#5425—24no—
13#5419—22no—
14C2×#859C2212yescomplemented
15C2×#859C2211yescharacteristic, complemented
16#5210—41nomaximal
17C23×#41D311yescharacteristic
18C2×#452—32no—
19C2×#452—32no—
20C2×#451—31no—
21C2×#450—31no—
22#2848—31no—
23#1788—32no—
24#1788—32no—
25#1787—31no—
26#1786—31no—
27#1779—31no—
28C23×C3×C4—41no—
29C2×#41⋊C3—28no—
30C2×#41⋊C3—22no—
31C2×#41⋊C3—22no—
32C2×#41⋊C3D414yes—
33C22×SL(2,𝔽3)—22no—
34C22×SL(2,𝔽3)D412yes—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×#226—21no—
37C2×#226—22no—
38C2×#217—41no—
39C2×#217—41no—
40C2×Dic3⋊C4—42no—
41#859—21no—
42#859—22no—
43C3⋊#95—42no—
44C22×#41—24no—
45C22×#41—22no—
46C22×#41—32no—
47C22×#41—62no—
48C22×#41—64no—
49C22×#41C2×D312yes—
50C23×Q8C2×D311yescharacteristic
51C23×D4—31no—
52C24×C4—31no—
53C22×Q16—31no—
54C22×Q16—31no—
55C22×SD16—32no—
56C2×C22⋊Q8—31no—
57C2×C22⋊Q8—32no—
58#452—64no—
59#452—68no—
60#452—64no—
61#451—62no—
62#451—62no—
63#450—64no—
64C2×#103—31no—
65C2×#103—32no—
66C2×#103—31no—
67C2×#103—32no—
68C2×#102—32no—
69C2×#102—32no—
70C2×#102—32no—
71C2×C22⋊C8—31no—
72C2×C22⋊C8—31no—
73C2×C22⋊C8—32no—
74C2×C22⋊C8—32no—
75#385—62no—
76#385—61no—
77#380—31no—
78#380—32no—
79#379—61no—
80C24×C3—41no—
81C22×C3×C4—42no—
82C22×C3×C4—84no—
83C22×C3×C4—82no—
84C22×Dic3—42no—
85C22×Dic3—41no—
86C22×Dic3—41no—
87#41⋊C3—24no—
88C2×SL(2,𝔽3)—22no—
89C2×SL(2,𝔽3)C2×D412yes—
90C2×SL(2,𝔽3)C2×D411yescharacteristic
91#226—41no—
92#217—44no—
93#217—82no—
94Dic3⋊C4—84no—
95C25—31no—
96C2×#41—28no—
97C2×#41—22no—
98C2×#41—22no—
99C2×#41—34no—
100C2×#41—38no—
101C2×#41—64no—
102C2×#41—64no—
103C2×#41—64no—
104C2×#41—616no—
105C2×#41—62no—
106C2×#41—62no—
107C2×#41—64no—
108C2×#41—64no—
109C2×#41—616no—
110C2×#41—68no—
111C2×#41#6714yes—
112C22×Q8—22no—
113C22×Q8—31no—
114C22×Q8—32no—
115C22×Q8—31no—
116C22×Q8—62no—
117C22×Q8#6712yes—
118C22×Q8C22×D311yescharacteristic
119C22×D4—32no—
120C22×D4—34no—
121C22×D4—61no—
122C22×D4—62no—
123C22×D4—64no—
124C22×D4—64no—
125C23×C4—31no—
126C23×C4—32no—
127C23×C4—32no—
128C23×C4—62no—
129C23×C4—64no—
130C23×C4—62no—
131C23×C4—64no—
132C23×C4S411yescharacteristic
133C2×Q16—62no—
134C2×Q16—61no—
135C2×Q16—62no—
136C2×Q16—64no—
137C2×Q16—61no—
138C2×Q16—62no—
139C2×SD16—64no—
140C2×SD16—64no—
141C2×SD16—64no—
142C22×C8—32no—
143C22×C8—31no—
144C22×C8—31no—
145C22⋊Q8—62no—
146C22⋊Q8—62no—
147C22⋊Q8—64no—
148C22⋊Q8—64no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—32no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157#103—62no—
158#103—61no—
159#103—62no—
160#103—64no—
161#103—61no—
162#103—62no—
163#102—64no—
164#102—64no—
165#102—64no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168C22⋊C8—64no—
169C22⋊C8—64no—
170#95—61no—
171#95—62no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—41no—
176C23×C3—82no—
177C23×C3—82no—
178C2×C3×C4—44no—
179C2×C3×C4—82no—
180C2×C3×C4—82no—
181C2×C3×C4—88no—
182C2×Dic3—44no—
183C2×Dic3—82no—
184C2×Dic3—82no—
185C2×Dic3—81no—
186C2×Dic3—82no—
187C2×Dic3—81no—
188C2×Dic3—82no—
189SL(2,𝔽3)#12011yescharacteristic
190C24—32no—
191C24—64no—
192C24—62no—
193C24—62no—
194C24—62no—
195C24—122no—
196C24C2×S411yescharacteristic
197#41—24no—
198#41—68no—
199#41—616no—
200#41—616no—
201#41—616no—
202#41—64no—
203#41—64no—
204#41—64no—
205#41—616no—
206C2×Q8—22no—
207C2×Q8—32no—
208C2×Q8—31no—
209C2×Q8—32no—
210C2×Q8—34no—
211C2×Q8—61no—
212C2×Q8—62no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—64no—
218C2×Q8—62no—
219C2×Q8—64no—
220C2×Q8C2×#6712yes—
221C2×Q8C2×#6711yescharacteristic
222C2×D4—34no—
223C2×D4—34no—
224C2×D4—34no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—64no—
229C2×D4—616no—
230C2×D4—68no—
231C2×D4—68no—
232C2×D4—68no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—31no—
239C22×C4—34no—
240C22×C4—34no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—68no—
248C22×C4—616no—
249C22×C4—68no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—64no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4C2×S412yes—
257Q16—122no—
258Q16—121no—
259Q16—124no—
260Q16—121no—
261SD16—128no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—61no—
268C2×C8—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—122no—
274C4⋊C4—124no—
275C22⋊C4—124no—
276C22⋊C4—124no—
277C22⋊C4—122no—
278C22⋊C4—122no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—41no—
284C22×C3—44no—
285C22×C3—82no—
286C22×C3—84no—
287C22×C3—84no—
288C22×C3—82no—
289C3×C4—84no—
290Dic3—84no—
291Dic3—161no—
292Dic3—161no—
293C23—22no—
294C23—22no—
295C23—34no—
296C23—68no—
297C23—62no—
298C23—62no—
299C23—62no—
300C23—62no—
301C23—62no—
302C23—62no—
303C23—128no—
304C23—128no—
305C23—124no—
306C23—124no—
307C23#98412yes—
308C23#98512yes—
309C23GL(2,ℤ/4)12yes—
310C23C22×S411yescharacteristic
311Q8—32no—
312Q8—31no—
313Q8—34no—
314Q8—61no—
315Q8—62no—
316Q8—64no—
317Q8—122no—
318Q8—121no—
319Q8—121no—
320Q8#95311yescharacteristic
321D4—38no—
322D4—64no—
323D4—64no—
324D4—64no—
325D4—616no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—64no—
337C2×C4—64no—
338C2×C4—616no—
339C2×C4—62no—
340C2×C4—64no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4GL(2,ℤ/4)14yes—
351C8—122no—
352C8—121no—
353C8—121no—
354C2×C3—41no—
355C2×C3—42no—
356C2×C3—42no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—84no—
360C22—24no—
361C22—24no—
362C22—22no—
363C22—22no—
364C22—64no—
365C22—64no—
366C22—64no—
367C22—1216no—
368C22—122no—
369C22—122no—
370C22—122no—
371C22—122no—
372C22#542514yescomplemented
373C22C2×#98411yescharacteristic
374C22C2×#98511yescharacteristic
375C22#592312yes—
376C22C2×GL(2,ℤ/4)11yescharacteristic
377C22C2×GL(2,ℤ/4)12yes—
378C4—24no—
379C4—32no—
380C4—31no—
381C4—31no—
382C4—62no—
383C4—242no—
384C4—241no—
385C4—241no—
386C3—41noSylow
387C2—24no—
388C2—124no—
389C2C2×#542511yescharacteristic
390C2#8551712yes—
391C2#8552112yes—
392C2#8552211yescharacteristic
393C2#8554611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC23×#41D3
prime radicalC23×#41⋊C3C2
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#120
LCSG ▷ C22×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×#5425C2semidirect
3. C2×#5419C2semidirect
4. #73151C2semidirect
5. #73148C2semidirect
11. C22×#226C22semidirect
14. C2×#859C22semidirect
15. C2×#859C22semidirect
372. C22#5425semidirect

Maximal subgroups

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

subgroupindex
C23×#41⋊C32
C2×#54252
C2×#54192
#731512
#731482
#328833
#52104