Finite Group Databasebeta

← #10698046

Subgroups — #10698046

Order 768.

1066 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#78611C211yesmaximal, characteristic, complemented
2#74048C212yesmaximal, complemented
3#69955C211yesmaximal, characteristic, complemented
4#69509C212yesmaximal, complemented
5C3×#1289C211yesmaximal, characteristic, complemented
6#10320—31nomaximal, Sylow
7C23×C4×D3C2211yescharacteristic
8C2×#930C2212yes—
9C2×#895—22no—
10C2×#882—22no—
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2212yes—
13C2×#830C2211yescharacteristic
14#5115—22no—
15C2×(C2×D3)⋊C8—22no—
16#4987—22no—
17#4958—22no—
18#4736—22no—
19#4732—22no—
20#4665—24no—
21#4451—24no—
22#3186—31no—
23#2847—32no—
24#1779—31no—
25#1519—32no—
26#1289D311yescharacteristic, complemented
27C24×D3D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C22×#67—22no—
30C23×Dic3D411yescharacteristic
31C22×D12—24no—
32C22×D12D412yescomplemented
33C22×C4×D3—22no—
34C22×C4×D3—21no—
35C22×C4×D3—22no—
36C22×C4×D3—21no—
37C22×C4×D3—44no—
38C22×C3×C8D412yescomplemented
39C2×C3×C4⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41#930—22no—
42#930—22no—
43#930—22no—
44#930—22no—
45C2×(C2×D3)⋊C4—24no—
46C2×(C2×D3)⋊C4—24no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3D412yescomplemented
49C22×C3⋊C8—21no—
50C2×D24—22no—
51#895—48no—
52#882—44no—
53#882—44no—
54C3×C22⋊C8—22no—
55C3×#95D414yescomplemented
56C3⋊#95—24no—
57#830—22no—
58#821—24no—
59(C2×D3)⋊C8—44no—
60#818—22no—
61C24×C4—31no—
62C2×#121—32no—
63C2×#121—62no—
64C2×#120—62no—
65#480—62no—
66#466—62no—
67#447—62no—
68C2×C22⋊C8—31no—
69C2×C22⋊C8—62no—
70C2×C22⋊C8C2×D311yescharacteristic
71#386—62no—
72#386—64no—
73#380—62no—
74#380C2×D312yes—
75#327—64no—
76C24×C3C2×D411yescharacteristic
77C23×D3—22no—
78C23×D3—21no—
79C23×D3—21no—
80C23×D3—22no—
81C23×D3—22no—
82C23×D3—42no—
83C23×D3—42no—
84C23×D3D812yescomplemented
85C22×C3×C4—24no—
86C22×C3×C4—42no—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D412yes—
90C22×C3×C4#4111yescharacteristic
91C2×#67—42no—
92C2×#67—42no—
93C2×#67—42no—
94C2×#67—42no—
95C2×#67—42no—
96C2×#67—42no—
97C22×Dic3—22no—
98C22×Dic3—21no—
99C22×Dic3—22no—
100C22×Dic3—42no—
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—44no—
104C2×D12—48no—
105C2×D12—48no—
106C2×D12—48no—
107C2×C4×D3—24no—
108C2×C4×D3—28no—
109C2×C4×D3—44no—
110C2×C4×D3—42no—
111C2×C4×D3—42no—
112C2×C4×D3—44no—
113C2×C4×D3—44no—
114C2×C4×D3—42no—
115C2×C4×D3—41no—
116C2×C4×D3—42no—
117C2×C4×D3—41no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C3×C4⋊C4—44no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—42no—
123(C2×D3)⋊C4—48no—
124(C2×D3)⋊C4—48no—
125(C2×D3)⋊C4—42no—
126(C2×D3)⋊C4—42no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C2×C3⋊C8—42no—
130C2×C3⋊C8—41no—
131D24—82no—
132C25—31no—
133C22×D4—32no—
134C22×D4—64no—
135C22×D4—62no—
136C23×C4—31no—
137C23×C4—61no—
138C23×C4—62no—
139C23×C4—62no—
140C23×C4—61no—
141C23×C4—124no—
142C23×C4C22×D311yescharacteristic
143C2×D8—62no—
144C22×C8—61no—
145C22×C8D1212yescomplemented
146#121—62no—
147#121—62no—
148#121—62no—
149#121—62no—
150#121—128no—
151#120—124no—
152#120—124no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—32no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—64no—
157C2×C22⋊C4—64no—
158C2×C22⋊C4—62no—
159#107—62no—
160#102—64no—
161C22⋊C8—22no—
162C22⋊C8—62no—
163C22⋊C8—124no—
164#95—64no—
165#95D1214yescomplemented
166C23×C3—41no—
167C23×C3—41no—
168C23×C3#12012yes—
169C23×C3#12111yescharacteristic
170C23×C3C2×D811yescharacteristic
171C23×C3#13511yescharacteristic
172C23×C3AGL(1,ℤ/8)12yes—
173C22×D3—24no—
174C22×D3—22no—
175C22×D3—24no—
176C22×D3—22no—
177C22×D3—24no—
178C22×D3—22no—
179C22×D3—44no—
180C22×D3—48no—
181C22×D3—44no—
182C22×D3—42no—
183C22×D3—41no—
184C22×D3—42no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—42no—
188C22×D3—41no—
189C22×D3—44no—
190C22×D3—44no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—44no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12112yes—
203C2×C3×C4#12711yescharacteristic
204#67—82no—
205#67—82no—
206#67—82no—
207#67—82no—
208C2×Dic3—24no—
209C2×Dic3—22no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216D12—44no—
217D12—82no—
218D12—816no—
219C4×D3—44no—
220C4×D3—48no—
221C4×D3—44no—
222C3×C8—42no—
223C3⋊C8—81no—
224C24—32no—
225C24—62no—
226C24—61no—
227C24—62no—
228C24—61no—
229C24—62no—
230C24—122no—
231C24—122no—
232C24D3×D411yescharacteristic
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—124no—
236C2×D4—128no—
237C2×D4—128no—
238C2×D4—128no—
239C2×D4—122no—
240C2×D4—122no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—122no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—64no—
248C22×C4—68no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—124no—
253C22×C4—122no—
254C22×C4—124no—
255C22×C4—122no—
256C22×C4—124no—
257C22×C4—122no—
258C22×C4—122no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4—122no—
262C22×C4C2×D1211yescharacteristic
263C22×C4C2×D1212yes—
264C22×C4D3×D412yes—
265C22×C4#23911yescharacteristic
266D8—242no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—122no—
270C2×C8—121no—
271C4⋊C4—44no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—128no—
277C22⋊C4—128no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—42no—
287C22×C3#39211yescharacteristic
288C22×C3#44712yes—
289C22×C3#44912yes—
290C22×C3#46611yescharacteristic
291C22×C3#46911yescharacteristic
292C2×D3—28no—
293C2×D3—24no—
294C2×D3—44no—
295C2×D3—42no—
296C2×D3—44no—
297C2×D3—44no—
298C2×D3—42no—
299C2×D3—42no—
300C2×D3—42no—
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—84no—
304C2×D3—84no—
305C2×D3—82no—
306C2×D3—82no—
307C2×D3—82no—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—84no—
311C3×C4B(4,𝔽2)12yes—
312Dic3—42no—
313Dic3—82no—
314C23—41no—
315C23—41no—
316C23—64no—
317C23—62no—
318C23—64no—
319C23—62no—
320C23—62no—
321C23—64no—
322C23—122no—
323C23—121no—
324C23—122no—
325C23—121no—
326C23—124no—
327C23—124no—
328C23—124no—
329C23—128no—
330C23—124no—
331C23—121no—
332C23—121no—
333C23—122no—
334C23—122no—
335C23—122no—
336C23—122no—
337C23#88212yes—
338C23#89511yescharacteristic
339C23D3×D811yescharacteristic
340C23#91412yes—
341C23#91911yescharacteristic
342D4—124no—
343D4—242no—
344D4—2416no—
345D4—242no—
346D4—242no—
347D4—242no—
348D4—242no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—64no—
357C2×C4—62no—
358C2×C4—124no—
359C2×C4—128no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—124no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4#87411yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#89512yes—
370C8—42no—
371C8—241no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3#199911yescharacteristic
377C2×C3#201011yescharacteristic
378C2×C3#202111yescharacteristic
379D3—44no—
380D3—162no—
381C22—22no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—68no—
389C22—64no—
390C22—124no—
391C22—122no—
392C22—124no—
393C22—124no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22—122no—
398C22—122no—
399C22—244no—
400C22—244no—
401C22—242no—
402C22—242no—
403C22—242no—
404C22#467111yescharacteristic
405C22#477312yes—
406C22#481012yes—
407C22#486511yescharacteristic
408C22#488311yescharacteristic
409C4—21no—
410C4—22no—
411C4—84no—
412C4—122no—
413C4—242no—
414C4#474112yes—
415C3#1032011yescharacteristic, complemented, Sylow
416C2—22no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—124no—
421C2—482no—
422C2#6870311yescharacteristic
423C2#6888611yescharacteristic
424C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#392
hypercenter#1289D3
hypocenterC3#10320
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
1. #78611C2semidirect
2. #74048C2semidirect
3. #69955C2semidirect
4. #69509C2semidirect
5. C3×#1289C2semidirect
26. #1289D3semidirect
32. C22×D12D4semidirect
38. C22×C3×C8D4semidirect
48. C2×C4⋊Dic3D4semidirect
55. C3×#95D4semidirect
84. C23×D3D8semidirect
145. C22×C8D12semidirect
165. #95D12semidirect
415. C3#10320semidirect

Maximal subgroups

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

subgroupindex
#786112
#740482
#699552
#695092
C3×#12892
#103203