Finite Group Databasebeta

← #10826861

Subgroups — #10826861

Order 768.

1866 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 211 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#87251C211yesmaximal, characteristic, complemented
2C22×#947C211yesmaximal, characteristic, complemented
3C2×#4753C212yesmaximal, complemented
4#78637C212yesmaximal, complemented
5C3×#1784C211yesmaximal, characteristic, complemented
6#70116C214yesmaximal, complemented
7#70104C212yesmaximal, complemented
8#69965C212yesmaximal, complemented
9#23910—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×#1007C2211yescharacteristic, complemented
12C23×D12C2211yescharacteristic
13#5817—24no—
14#5816—22no—
15#5816C2212yescomplemented
16Q8×#67—24no—
17C2×#947—22no—
18C2×#947—28no—
19C2×#947C2212yes—
20C2×#947C2212yes—
21#5812C2211yescharacteristic, complemented
22#5791—22no—
23#5791C2212yes—
24C22×(C2×D3)⋊C4—22no—
25C22×C4×Dic3C2211yescharacteristic
26C2×#908C2214yes—
27C2×C3×#101C2212yescomplemented
28C2×C3×#101C2212yescomplemented
29C3×#407C2212yes—
30C2×#825C2214yescomplemented
31#5131C2214yes—
32#4753—28no—
33#4732—24no—
34#4731—24no—
35#4555C2214yescomplemented
36#3486—31no—
37C22×#124—31no—
38C2×#454—32no—
39#3003—32no—
40#2003—34no—
41#1896—32no—
42#1888—32no—
43#1784D311yescharacteristic, complemented
44C24×D3—21no—
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—42no—
48C22×C3×Q8D412yes—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×Q8C2311yescharacteristic
51C23×C3×C4D412yes—
52C23×C3×C4C2311yescharacteristic
53C23×Dic3—21no—
54#1007—24no—
55#1007—42no—
56C2×#239—22no—
57C2×D3×Q8—22no—
58C2×#235—22no—
59C2×#235C2312yes—
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12—42no—
64C22×D12C2312yes—
65C2×C3×M16C2314yes—
66#953—22no—
67#947—216no—
68#947—24no—
69#947—24no—
70#947—24no—
71#947—22no—
72#947—24no—
73#947D418yescomplemented
74#946—28no—
75#946—24no—
76Q8×Dic3—24no—
77#944—24no—
78#944—22no—
79#930—24no—
80#930D414yes—
81#929—24no—
82#929—22no—
83C4×#67—28no—
84C4×#67—24no—
85C2×(C2×D3)⋊C4—216no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88#926—22no—
89#926C2312yes—
90#924—24no—
91#924D414yes—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3C2312yes—
95#908—28no—
96#908—44no—
97C2×D24—24no—
98C2×#204—24no—
99C3×#101—24no—
100C3×#101—42no—
101C3×C22⋊C8—24no—
102#825—28no—
103#821—28no—
104C2×2−1+4—31no—
105C23×Q8C2×D311yescharacteristic
106C23×D4—31no—
107C2×AGL(1,ℤ/8)—34no—
108D4×Q8—64no—
109#548—64no—
110#536—32no—
111#536—62no—
112C2×#124—32no—
113C2×#124—32no—
114C2×#124—68no—
115C2×#124—62no—
116#525—32no—
117#525—62no—
118#519—31no—
119C22×C22⋊C4—62no—
120C22×C42—31no—
121#454—68no—
122#450—64no—
123#447—64no—
124C2×#104—34no—
125#418—34no—
126C2×#101C2×D312yescomplemented
127C2×#101C2×D312yescomplemented
128#407C2×D312yes—
129#339—34no—
130C24×C3C2×D411yescharacteristic
131C23×D3—24no—
132C23×D3—22no—
133C23×D3—42no—
134C23×D3—48no—
135C23×D3—42no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—42no—
144C2×C3×Q8—416no—
145C2×C3×Q8C2×D414yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2411yescharacteristic
156C2×#67—24no—
157C2×#67—22no—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—44no—
161#239—48no—
162D3×Q8—48no—
163#235—24no—
164#235—24no—
165#235—44no—
166#235—44no—
167C2×D12—24no—
168C2×D12—24no—
169C2×D12—22no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—416no—
173C2×D12—42no—
174C2×D12—42no—
175C2×D12C2×D412yes—
176C2×D12C2×D414yes—
177C2×C4×D3—24no—
178C2×C4×D3—22no—
179C2×Dic6—22no—
180C2×Dic6C2×D412yes—
181C3×M16—24no—
182C3×M16—24no—
183C2×C3×C8—24no—
184#217—24no—
185#217—22no—
186(C2×D3)⋊C4—216no—
187(C2×D3)⋊C4—28no—
188(C2×D3)⋊C4—216no—
189(C2×D3)⋊C4—24no—
190C4⋊Dic3—24no—
191C4⋊Dic3C2×D414yes—
192Dic3⋊C4—28no—
193Dic3⋊C4—24no—
194C4×Dic3—24no—
195C4×Dic3—24no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3C2×D414yes—
199D24—48no—
200#204—48no—
201C25—61no—
2022−1+4—64no—
2032−1+4—122no—
204C2×#41—32no—
205C2×#41—62no—
206C2×#41—62no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—62no—
211C22×Q8D1212yes—
212C22×Q8C22×D311yescharacteristic
213C22×Q8C22×D311yescharacteristic
214C22×D4—32no—
215C22×D4—64no—
216C22×D4—61no—
217C22×D4—64no—
218C22×D4—122no—
219C23×C4—61no—
220C23×C4D1212yes—
221C23×C4C22×D311yescharacteristic
222AGL(1,ℤ/8)—68no—
223AGL(1,ℤ/8)—124no—
224C2×SD16—64no—
225C2×D8—64no—
226C2×M16C22×D314yes—
227C4⋊Q8—62no—
228C4⋊Q8—64no—
229#124—38no—
230#124—64no—
231#124—64no—
232#124—62no—
233#124—616no—
234#124—64no—
235#124—64no—
236#123—62no—
237#123—64no—
238C22⋊Q8—34no—
239C22⋊Q8—64no—
240C22⋊Q8—64no—
241C22⋊Q8—68no—
242#121—34no—
243#121—64no—
244#120—62no—
245C4×Q8—64no—
246C4×D4—68no—
247C4×D4—64no—
248#117—32no—
249#117—62no—
250C2×C22⋊C4—616no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C42—32no—
254C2×C42—64no—
255C2×C42—61no—
256#104—68no—
257#102—68no—
258#101—24no—
259#101—42no—
260C22⋊C8—24no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—24no—
264C23×C3#12012yes—
265C23×C3C22×D411yescharacteristic
266C22×D3—24no—
267C22×D3—22no—
268C22×D3—48no—
269C22×D3—416no—
270C22×D3—48no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—88no—
274C22×D3—84no—
275C22×D3—84no—
276C3×Q8—28no—
277C3×Q8—48no—
278C3×Q8—44no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—28no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—48no—
288C2×C3×C4#12012yes—
289C2×C3×C4#12014yes—
290C2×C3×C4C22×D412yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C4H(5,𝔽2)12yes—
293#67—42no—
294#67—42no—
295#67—48no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—22no—
299C2×Dic3—48no—
300C2×Dic3—44no—
301D12—28no—
302D12—24no—
303D12—48no—
304D12—44no—
305D12—44no—
306C4×D3—48no—
307C4×D3—44no—
308Dic6—24no—
309Dic6—44no—
310C3×C8—44no—
311C24—64no—
312C24—62no—
313C24—122no—
314C24—128no—
315C24—122no—
316C24C2×D1211yescharacteristic
317#41—64no—
318#41—64no—
319#41—124no—
320#41—128no—
321#41—124no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—32no—
330C2×Q8—42no—
331C2×Q8—416no—
332C2×Q8—62no—
333C2×Q8—128no—
334C2×Q8D3×D414yes—
335C2×D4—32no—
336C2×D4—34no—
337C2×D4—62no—
338C2×D4—64no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—64no—
342C2×D4—62no—
343C2×D4—64no—
344C2×D4—1216no—
345C2×D4—122no—
346C2×D4—122no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—42no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—124no—
357C22×C4C2×D1212yes—
358C22×C4C2×D1211yescharacteristic
359C22×C4C2×D1212yes—
360C22×C4D3×D414yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364M16—24no—
365M16—24no—
366C2×C8—24no—
367C4⋊C4—34no—
368C4⋊C4—64no—
369C4⋊C4—68no—
370C4⋊C4—64no—
371C22⋊C4—616no—
372C22⋊C4—68no—
373C22⋊C4—64no—
374C22⋊C4—62no—
375C22⋊C4—616no—
376C22⋊C4—64no—
377C42—34no—
378C42—64no—
379C42—64no—
380C42—62no—
381C42—62no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3#45414yescomplemented
388C22×C3C2×#12011yescharacteristic
389C22×C3C2×#12011yescharacteristic
390C22×C3#53411yescharacteristic
391C2×D3—48no—
392C2×D3—42no—
393C2×D3—44no—
394C2×D3—44no—
395C2×D3—84no—
396C2×D3—816no—
397C2×D3—84no—
398C3×C4—24no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4D4214yes—
402Dic3—44no—
403Dic3—42no—
404Dic3—44no—
405C23—21no—
406C23—21no—
407C23—24no—
408C23—64no—
409C23—62no—
410C23—128no—
411C23—1216no—
412C23—128no—
413C23—122no—
414C23—122no—
415C23—248no—
416C23—244no—
417C23—244no—
418C23#88212yes—
419C23C22×D1211yescharacteristic
420Q8—28no—
421Q8—48no—
422Q8—44no—
423Q8—64no—
424Q8—124no—
425D4—64no—
426D4—68no—
427D4—128no—
428D4—124no—
429D4—122no—
430D4—122no—
431D4—128no—
432D4—124no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—48no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—62no—
445C2×C4—128no—
446C2×C4—128no—
447C2×C4—124no—
448C2×C4—124no—
449C2×C4#88214yes—
450C2×C4#88212yes—
451C2×C4C2×D3×D412yes—
452C2×C4C2×D3×D412yes—
453C2×C4#100412yes—
454C8—44no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3#240311yescharacteristic
459C2×C3C2×#45412yes—
460D3—82no—
461D3—84no—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—128no—
468C22—124no—
469C22—122no—
470C22—124no—
471C22—244no—
472C22—2416no—
473C22—244no—
474C22#475314yescomplemented
475C22C2×#88211yescharacteristic
476C22C2×#88211yescharacteristic
477C22#549311yescharacteristic
478C4—22no—
479C4—24no—
480C4—42no—
481C4—124no—
482C4—124no—
483C4—122no—
484C4#559014yes—
485C3#2391011yescharacteristic, complemented, Sylow
486C2—21no—
487C2—21no—
488C2—24no—
489C2—244no—
490C2—242no—
491C2#7385211yescharacteristic
492C2C2×#475312yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C2×#120
hypercenter#1784D3
hypocenterC3#23910
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
1. #87251C2semidirect
2. C22×#947C2semidirect
3. C2×#4753C2semidirect
4. #78637C2semidirect
5. C3×#1784C2semidirect
6. #70116C2semidirect
7. #70104C2semidirect
8. #69965C2semidirect
11. C2×#1007C22semidirect
15. #5816C22semidirect
21. #5812C22semidirect
27. C2×C3×#101C22semidirect
28. C2×C3×#101C22semidirect
30. C2×#825C22semidirect
35. #4555C22semidirect
43. #1784D3semidirect
73. #947D4semidirect
126. C2×#101C2×D3semidirect
127. C2×#101C2×D3semidirect
387. C22×C3#454semidirect
474. C22#4753semidirect
485. C3#23910semidirect

Maximal subgroups

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

subgroupindex
#872512
C22×#9472
C2×#47532
#786372
C3×#17842
#701162
#701042
#699652
#239103