Finite Group Databasebeta

← #10831019

Subgroups — #10831019

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#214C211yesmaximal, characteristic, complemented
2C3×#3003C211yesmaximal, characteristic
3C2×#5200C212yesmaximal, complemented
4#82933C211yesmaximal, characteristic, complemented
5#82916C211yesmaximal, characteristic, complemented
6#72129C211yesmaximal, characteristic, complemented
7#72097C212yesmaximal, complemented
8#70320C212yesmaximal, complemented
9#70315C212yesmaximal, complemented
10#69990C211yesmaximal, characteristic, complemented
11#69982C211yesmaximal, characteristic, complemented
12#24278—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C3×#525C2211yescharacteristic
17C2×#951C2212yescomplemented
18C2×#948C2211yescharacteristic, complemented
19C22×#214—28no—
20C22×#214—22no—
21C22×#214—22no—
22C22×#214C2212yes—
23C22×#214C2212yes—
24C23×C3⋊C8C2211yescharacteristic
25C3×#450—22no—
26C3×#447—22no—
27C3×#418C2212yes—
28C3×#407C2211yescharacteristic
29#5234C2211yescharacteristic, complemented
30C2×#833C2212yes—
31C2×#833C2212yes—
32#5207C2211yescharacteristic
33#5207C2211yescharacteristic
34#5200—28no—
35C2×#824C2212yescomplemented
36C2×#824C2212yescomplemented
37C2×#822C2212yescomplemented
38#5050—22no—
39#5040—22no—
40#5011—22no—
41D4×C3⋊C8—22no—
42#5004C2212yescomplemented
43#4999C2211yescharacteristic, complemented
44#4997C2211yescharacteristic, complemented
45#4558C2212yescomplemented
46C23×SD16—31no—
47C2×#471—32no—
48#3004—31no—
49#3003D311yescharacteristic
50#2929—31no—
51#2046—32no—
52#2019—32no—
53#1977—32no—
54#1810—31no—
55#1784—31no—
56#1783—31no—
57C25×C3—21no—
58C2×C3×#41C2311yescharacteristic
59C22×C3×D4—24no—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—42no—
64C22×C3×D4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C22×Dic6—22no—
68C22×Dic6—24no—
69C22×Dic6—22no—
70C22×Dic6—21no—
71C22×Dic6—42no—
72C22×Dic6C2312yes—
73C3×AGL(1,ℤ/8)—24no—
74C3×AGL(1,ℤ/8)—42no—
75C2×C3×SD16—22no—
76C2×C3×D8—22no—
77C2×C3×M16C2312yes—
78C3×#123—21no—
79C3×C22⋊Q8D412yes—
80C3×#121D412yes—
81C3×#120—21no—
82C3×C4×D4—22no—
83C3×#117C2311yescharacteristic
84#951—24no—
85#951—42no—
86#948—24no—
87C2×#216—22no—
88C2×#214—216no—
89C2×#214—216no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—22no—
93C2×#214—24no—
94C2×#214—22no—
95C2×#214—22no—
96C2×#214—24no—
97C2×#214D418yescomplemented
98C2×#208—21no—
99C2×#208C2312yes—
100C2×#208C2311yescharacteristic
101C2×#208C2311yescharacteristic
102C22×C3⋊C8—24no—
103C22×C3⋊C8—21no—
104C22×C3⋊C8D412yes—
105C22×C3⋊C8D412yes—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C3×#102—24no—
109C3×C22⋊C8—22no—
110#833—24no—
111#833—42no—
112#830—22no—
113#830—22no—
114#830—22no—
115#824—24no—
116#824—42no—
117#822—24no—
118#810—24no—
119C4⋊(C3⋊C8)—22no—
120#803—21no—
121C4×C3⋊C8—21no—
122C23×Q8—31no—
123C23×D4C2×D311yescharacteristic
124C2×#137—32no—
125C2×AGL(1,ℤ/8)C2×D312yes—
126C22×SD16—32no—
127C22×SD16—32no—
128C22×SD16—62no—
129C22×SD16—62no—
130C22×SD16—68no—
131C2×#131—31no—
132C23×C8—31no—
133#525C2×D311yescharacteristic
134#471—68no—
135#451—62no—
136#450—22no—
137#448—62no—
138#447—22no—
139#435—62no—
140C8×D4—62no—
141C2×#108—32no—
142#424—31no—
143#419—32no—
144#418C2×D312yes—
145C2×#101—32no—
146C2×#101—32no—
147C2×#100—32no—
148C2×#100—32no—
149#408—31no—
150#407—31no—
151#407—31no—
152#407C2×D311yescharacteristic
153#405—31no—
154#341—32no—
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—42no—
158C24×C3—84no—
159C24×C3#4111yescharacteristic
160C3×#41—22no—
161C3×#41—22no—
162C2×C3×Q8C2×D411yescharacteristic
163C2×C3×D4—28no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—48no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—42no—
175C2×C3×D4C2×D414yes—
176C2×C3×D4C2×D411yescharacteristic
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—22no—
185C22×Dic3—44no—
186C22×Dic3—42no—
187C2×Dic6—28no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—48no—
194C2×Dic6—44no—
195C2×Dic6—44no—
196C2×Dic6—42no—
197C2×Dic6C2×D414yes—
198C3×SD16—44no—
199C3×D8—44no—
200C3×M16—22no—
201C3×M16—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—22no—
208C3×C42—21no—
209#216—44no—
210#214—216no—
211#214—28no—
212#214—28no—
213#214—44no—
214#208—22no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—42no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—24no—
221C2×C3⋊C8—24no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—42no—
226C2×C3⋊C8C2×D412yes—
227C2×C3⋊C8C2×D411yescharacteristic
228C2×C3⋊C8C2×D411yescharacteristic
229C2×C3⋊C8C2×D414yes—
230C25—21no—
231C2×#41C22×D311yescharacteristic
232C22×Q8—32no—
233C22×Q8—64no—
234C22×Q8—61no—
235C22×Q8—62no—
236C22×Q8—62no—
237C22×Q8—122no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—42no—
243C22×D4C22×D312yes—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246#137—64no—
247#137—122no—
248AGL(1,ℤ/8)—24no—
249AGL(1,ℤ/8)—42no—
250C2×Q16—62no—
251C2×SD16—22no—
252C2×SD16—38no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×SD16—64no—
256C2×SD16—64no—
257C2×SD16—64no—
258C2×SD16—616no—
259C2×SD16—616no—
260C2×SD16—64no—
261C2×SD16—62no—
262C2×D8—22no—
263#131—64no—
264C2×M16—31no—
265C2×M16—32no—
266C2×M16—31no—
267C2×M16—61no—
268C2×M16C22×D312yes—
269C22×C8—31no—
270C22×C8—32no—
271C22×C8—32no—
272C22×C8—31no—
273C22×C8—64no—
274C22×C8—61no—
275#123—21no—
276C22⋊Q8#6712yes—
277#121#6712yes—
278#120—21no—
279C4×D4—22no—
280#117C22×D311yescharacteristic
281#108—64no—
282C4⋊C8—62no—
283#103—64no—
284#102—24no—
285#101—64no—
286#101—122no—
287#100—64no—
288#100—122no—
289C22⋊C8—22no—
290C22⋊C8—62no—
291C22⋊C8—62no—
292C22⋊C8—62no—
293#97—61no—
294C4×C8—61no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—21no—
298C23×C3—24no—
299C23×C3—48no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—84no—
303C23×C3—816no—
304C23×C3—84no—
305C23×C3—84no—
306C23×C3#12112yes—
307C23×C3C2×#4111yescharacteristic
308C3×Q8—22no—
309C3×D4—24no—
310C3×D4—22no—
311C3×D4—24no—
312C3×D4—48no—
313C3×D4—42no—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—42no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4#12112yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D412yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4H(5,𝔽2)12yes—
330C2×Dic3—24no—
331C2×Dic3—48no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334Dic6—24no—
335Dic6—24no—
336Dic6—48no—
337Dic6—42no—
338Dic6—42no—
339C3×C8—42no—
340C3⋊C8—22no—
341C3⋊C8—41no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C3⋊C8#12012yes—
345C3⋊C8#12012yes—
346C24—22no—
347C24—44no—
348C24—42no—
349C24—84no—
350C24#23711yescharacteristic
351#41—22no—
352#41—22no—
353C2×Q8—34no—
354C2×Q8—68no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×Q8—124no—
361C2×Q8—128no—
362C2×Q8—124no—
363C2×Q8—122no—
364C2×Q8C2×#6711yescharacteristic
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4D3×D414yes—
378C2×D4C2×#6711yescharacteristic
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—62no—
383C22×C4—124no—
384C22×C4—122no—
385C22×C4D3×D412yes—
386C22×C4#23711yescharacteristic
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389Q16—124no—
390SD16—44no—
391SD16—68no—
392SD16—68no—
393SD16—616no—
394SD16—124no—
395D8—44no—
396M16—22no—
397M16—22no—
398M16—62no—
399M16—62no—
400M16—62no—
401M16—62no—
402M16—122no—
403C2×C8—22no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—34no—
407C2×C8—32no—
408C2×C8—64no—
409C2×C8—64no—
410C2×C8—62no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—62no—
414C2×C8—122no—
415C4⋊C4—22no—
416C4⋊C4C2×#6712yes—
417C22⋊C4—22no—
418C22⋊C4—21no—
419C22⋊C4—22no—
420C42—21no—
421C22×C3—28no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—41no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—84no—
429C22×C3—84no—
430C22×C3—816no—
431C22×C3—84no—
432C22×C3#47114yescomplemented
433C22×C3C2×#12111yescharacteristic
434C22×C3#54612yes—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4C2×#12011yescharacteristic
439C3×C4#53511yescharacteristic
440C3×C4D4212yes—
441Dic3—44no—
442C23—24no—
443C23—22no—
444C23—21no—
445C23—24no—
446C23—48no—
447C23—42no—
448C23—42no—
449C23—84no—
450C23—816no—
451C23—84no—
452C23—84no—
453C23#93912yes—
454C23C2×#23711yescharacteristic
455Q8—22no—
456Q8—64no—
457Q8—64no—
458Q8—128no—
459Q8—122no—
460Q8—122no—
461D4—22no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—42no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—64no—
478C2×C4—128no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4#93912yes—
482C2×C4C2×D3×D412yes—
483C2×C4C2×D3×D411yescharacteristic
484C2×C4#100412yes—
485C2×C4C22×#6711yescharacteristic
486C8—32no—
487C8—32no—
488C8—42no—
489C8—62no—
490C8—122no—
491C8—121no—
492C8—121no—
493C2×C3—22no—
494C2×C3—24no—
495C2×C3—81no—
496C2×C3—84no—
497C2×C3#240811yescharacteristic
498C2×C3C2×#47112yes—
499C22—28no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—44no—
505C22—42no—
506C22—84no—
507C22—816no—
508C22—84no—
509C22—84no—
510C22#520014yescomplemented
511C22#559112yes—
512C22C2×#93911yescharacteristic
513C4—22no—
514C4—41no—
515C4—42no—
516C4—124no—
517C4D3×#12011yescharacteristic
518C4#559211yescharacteristic
519C4D4×#6712yes—
520C3#2427811yescharacteristic, complemented, Sylow
521C2—24no—
522C2—22no—
523C2—81no—
524C2—84no—
525C2#8258211yescharacteristic
526C2C2×#520012yes—
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3003C2
socleC22×C3C2×#121
hypercenter#3003D3
hypocenterC3#24278
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3003

Complements

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

NH ≅ G/Ntype
1. C23×#214C2semidirect
3. C2×#5200C2semidirect
4. #82933C2semidirect
5. #82916C2semidirect
6. #72129C2semidirect
7. #72097C2semidirect
8. #70320C2semidirect
9. #70315C2semidirect
10. #69990C2semidirect
11. #69982C2semidirect
17. C2×#951C22semidirect
18. C2×#948C22semidirect
29. #5234C22semidirect
35. C2×#824C22semidirect
36. C2×#824C22semidirect
37. C2×#822C22semidirect
42. #5004C22semidirect
43. #4999C22semidirect
44. #4997C22semidirect
45. #4558C22semidirect
97. C2×#214D4semidirect
432. C22×C3#471semidirect
510. C22#5200semidirect
520. C3#24278semidirect

Maximal subgroups

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

subgroupindex
C23×#2142
C3×#30032
C2×#52002
#829332
#829162
#721292
#720972
#703202
#703152
#699902
#699822
#242783