Finite Group Databasebeta

← #10831016

Subgroups — #10831016

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×C3×SD16C211yesmaximal, characteristic, complemented
2#84059C211yesmaximal, characteristic, complemented
3#84038C211yesmaximal, characteristic
4C2×#5177C212yesmaximal, complemented
5#83734C211yesmaximal, characteristic, complemented
6#72133C211yesmaximal, characteristic, complemented
7#72129C211yesmaximal, characteristic, complemented
8#72059C212yesmaximal, complemented
9#72053C212yesmaximal, complemented
10#72052C212yesmaximal, complemented
11#71977C211yesmaximal, characteristic, complemented
12#24278—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16—28no—
16C22×C3×SD16—22no—
17C22×C3×SD16—22no—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2212yes—
20C23×C3×C8C2211yescharacteristic
21C2×#942C2212yescomplemented
22C2×#932C2212yes—
23#5791C2211yescharacteristic
24C2×#901C2211yescharacteristic, complemented
25#5228—22no—
26#5227—22no—
27C2×#836C2212yescomplemented
28C2×#836C2212yescomplemented
29#5225C2212yescomplemented
30#5218—22no—
31#5217—22no—
32C2×#833C2212yes—
33C2×#833C2212yes—
34#5215C2212yes—
35#5207C2211yescharacteristic
36#5207C2211yescharacteristic
37#5207C2211yescharacteristic
38#5177—28no—
39#5111—22no—
40#5110C2211yescharacteristic, complemented
41C8×#67—22no—
42C2×#819C2212yescomplemented
43#5102C2211yescharacteristic, complemented
44#5101C2211yescharacteristic, complemented
45#4552C2212yescomplemented
46C23×SD16D311yescharacteristic, complemented
47C2×#471—32no—
48#3004—31no—
49#3003—31no—
50#2929—31no—
51#2046—32no—
52#2019—32no—
53#1977—32no—
54#1810—31no—
55#1784—31no—
56#1783—31no—
57C25×C3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—42no—
63C22×C3×Q8C2312yes—
64C22×C3×D4—24no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—42no—
69C22×C3×D4C2312yes—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C2×C3×SD16—216no—
74C2×C3×SD16—216no—
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—24no—
81C2×C3×SD16D418yescomplemented
82C22×C3×C8—24no—
83C22×C3×C8D412yes—
84C22×C3×C8D412yes—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87#953—21no—
88C2×#216—22no—
89#942—24no—
90#942—42no—
91C2×#215—22no—
92C2×#214—22no—
93#932—24no—
94#932—42no—
95C2×#213—22no—
96#930D412yes—
97#929—21no—
98C4×#67—22no—
99#926C2311yescharacteristic
100#924D412yes—
101C2×#208C2312yes—
102C2×#208C2312yes—
103C2×#208C2311yescharacteristic
104C2×#208C2311yescharacteristic
105#901—24no—
106C2×ΓL(1,𝔽25)—21no—
107C2×D3×C8—21no—
108#836—24no—
109#836—42no—
110#835—24no—
111#833—24no—
112#833—42no—
113#832—24no—
114#830—22no—
115#830—22no—
116#830—22no—
117(C2×D3)⋊C8—22no—
118#819—24no—
119#815—21no—
120Dic3⋊C8—22no—
121C8×Dic3—21no—
122C23×Q8C2×D311yescharacteristic
123C23×D4C2×D311yescharacteristic
124C2×#137—32no—
125C2×AGL(1,ℤ/8)—32no—
126C22×SD16—28no—
127C22×SD16—22no—
128C22×SD16—22no—
129C22×SD16C2×D312yes—
130C22×SD16C2×D312yes—
131C2×#131—31no—
132C23×C8C2×D311yescharacteristic
133#525—31no—
134#471—68no—
135#451—62no—
136#450—62no—
137#448—62no—
138#447—62no—
139#435—62no—
140C8×D4—62no—
141C2×#108—32no—
142#424—31no—
143#419—32no—
144#418—32no—
145C2×#101—32no—
146C2×#101—32no—
147C2×#100—32no—
148C2×#100—32no—
149#408—31no—
150#407—31no—
151#407—31no—
152#407—31no—
153#405—31no—
154#341—32no—
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—42no—
158C24×C3—84no—
159C24×C3#4111yescharacteristic
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—44no—
167C2×C3×Q8—48no—
168C2×C3×Q8—44no—
169C2×C3×Q8—42no—
170C2×C3×Q8C2×D414yes—
171C2×C3×D4—28no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—48no—
178C2×C3×D4—44no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4C2×D414yes—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—44no—
186C22×C3×C4—42no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—22no—
192C2×#67—21no—
193#235—22no—
194#235—22no—
195C2×D12C2×D411yescharacteristic
196C2×C4×D3—21no—
197C2×Dic6C2×D411yescharacteristic
198C3×SD16—216no—
199C3×SD16—28no—
200C3×SD16—28no—
201C2×C3×C8—22no—
202C2×C3×C8—24no—
203C2×C3×C8—24no—
204C2×C3×C8C2×D412yes—
205C2×C3×C8C2×D411yescharacteristic
206C2×C3×C8C2×D411yescharacteristic
207C2×C3×C8C2×D414yes—
208#217—22no—
209#217—21no—
210#216—44no—
211#215—44no—
212#214—44no—
213#213—44no—
214(C2×D3)⋊C4—22no—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—22no—
217C4×Dic3—21no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228ΓL(1,𝔽25)—42no—
229D3×C8—42no—
230C25—21no—
231C2×#41—31no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—42no—
237C22×Q8C22×D312yes—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—42no—
243C22×D4C22×D312yes—
244C23×C4—21no—
245C23×C4C22×D311yescharacteristic
246#137—64no—
247#137—122no—
248AGL(1,ℤ/8)—64no—
249AGL(1,ℤ/8)—122no—
250C2×Q16—62no—
251C2×SD16—216no—
252C2×SD16—216no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—22no—
257C2×SD16—22no—
258C2×SD16—24no—
259C2×SD16—62no—
260C2×SD16—62no—
261C2×SD16#6718yescomplemented
262C2×D8—62no—
263#131—64no—
264C2×M16—31no—
265C2×M16—32no—
266C2×M16—32no—
267C2×M16—31no—
268C2×M16—61no—
269C22×C8—24no—
270C22×C8—61no—
271C22×C8#6712yes—
272C22×C8#6712yes—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275#123—61no—
276C22⋊Q8—32no—
277#121—32no—
278#120—61no—
279C4×D4—62no—
280#117—31no—
281#108—64no—
282C4⋊C8—62no—
283#103—64no—
284#102—64no—
285#101—64no—
286#101—122no—
287#100—64no—
288#100—122no—
289C22⋊C8—62no—
290C22⋊C8—62no—
291C22⋊C8—62no—
292C22⋊C8—62no—
293#97—61no—
294C4×C8—61no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—48no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—84no—
302C23×C3—816no—
303C23×C3—84no—
304C23×C3—84no—
305C23×C3#12112yes—
306C23×C3C2×#4111yescharacteristic
307C22×D3—21no—
308C3×Q8—24no—
309C3×Q8—24no—
310C3×Q8—48no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×D4—24no—
314C3×D4—24no—
315C3×D4—48no—
316C3×D4—42no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—48no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4#12112yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D412yes—
330C2×C3×C4H(5,𝔽2)12yes—
331#67—42no—
332#67—44no—
333C2×Dic3—22no—
334C2×Dic3—21no—
335D12—22no—
336C4×D3—41no—
337C4×D3—41no—
338Dic6—22no—
339C3×C8—22no—
340C3×C8#12012yes—
341C3×C8#12012yes—
342C3⋊C8—41no—
343C3⋊C8—41no—
344C3⋊C8—42no—
345C3⋊C8—42no—
346C24—22no—
347C24—44no—
348C24—42no—
349C24—84no—
350C24#23711yescharacteristic
351#41—62no—
352#41—62no—
353C2×Q8—22no—
354C2×Q8—28no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—31no—
360C2×Q8—44no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—42no—
364C2×Q8C2×#6714yes—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—31no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—62no—
377C2×D4—61no—
378C2×D4C2×#6714yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—61no—
385C22×C4D3×D412yes—
386C22×C4#23711yescharacteristic
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389Q16—124no—
390SD16—216no—
391SD16—28no—
392SD16—28no—
393SD16—124no—
394SD16—124no—
395D8—124no—
396M16—62no—
397M16—62no—
398M16—62no—
399M16—62no—
400M16—62no—
401M16—62no—
402M16—122no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—122no—
411C2×C8C2×#6712yes—
412C2×C8C2×#6711yescharacteristic
413C2×C8C2×#6711yescharacteristic
414C2×C8C2×#6714yes—
415C4⋊C4—32no—
416C4⋊C4—62no—
417C22⋊C4—62no—
418C22⋊C4—61no—
419C22⋊C4—62no—
420C42—61no—
421C22×C3—28no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—44no—
426C22×C3—84no—
427C22×C3—84no—
428C22×C3—816no—
429C22×C3—84no—
430C22×C3#47114yescomplemented
431C22×C3C2×#12111yescharacteristic
432C22×C3#54612yes—
433C2×D3—41no—
434C2×D3—42no—
435C3×C4—22no—
436C3×C4—44no—
437C3×C4C2×#12011yescharacteristic
438C3×C4#53511yescharacteristic
439C3×C4D4212yes—
440Dic3—41no—
441Dic3—42no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—48no—
446C23—42no—
447C23—42no—
448C23—61no—
449C23—84no—
450C23—816no—
451C23—84no—
452C23—84no—
453C23#93812yes—
454C23C2×#23711yescharacteristic
455Q8—24no—
456Q8—24no—
457Q8—48no—
458Q8—42no—
459Q8—42no—
460Q8—62no—
461D4—24no—
462D4—24no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—62no—
467D4—124no—
468D4—122no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—48no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4#93812yes—
482C2×C4C2×D3×D411yescharacteristic
483C2×C4#100412yes—
484C2×C4C22×#6712yes—
485C2×C4C22×#6711yescharacteristic
486C8—22no—
487C8—122no—
488C8—122no—
489C8—121no—
490C8—121no—
491C8#95312yes—
492C8#95312yes—
493C2×C3—22no—
494C2×C3—24no—
495C2×C3—84no—
496C2×C3#240811yescharacteristic
497C2×C3C2×#47112yes—
498D3—81no—
499C22—28no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—44no—
504C22—84no—
505C22—816no—
506C22—84no—
507C22—84no—
508C22—121no—
509C22—122no—
510C22#517714yescomplemented
511C22C2×#93811yescharacteristic
512C22#580612yes—
513C4—22no—
514C4—44no—
515C4—121no—
516C4—122no—
517C4D4×#6712yes—
518C4#580511yescharacteristic
519C4C2×#95311yescharacteristic
520C3#2427811yescharacteristic, complemented, Sylow
521C2—24no—
522C2—22no—
523C2—84no—
524C2—241no—
525C2C2×#517712yes—
526C2#8405311yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×SD16C2
socleC22×C3C2×#121
prime radical#84038C2
hypercenterC23×SD16D3
hypocenterC3#24278
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. #84059C2semidirect
4. C2×#5177C2semidirect
5. #83734C2semidirect
6. #72133C2semidirect
7. #72129C2semidirect
8. #72059C2semidirect
9. #72053C2semidirect
10. #72052C2semidirect
11. #71977C2semidirect
21. C2×#942C22semidirect
24. C2×#901C22semidirect
27. C2×#836C22semidirect
28. C2×#836C22semidirect
29. #5225C22semidirect
40. #5110C22semidirect
42. C2×#819C22semidirect
43. #5102C22semidirect
44. #5101C22semidirect
45. #4552C22semidirect
46. C23×SD16D3semidirect
81. C2×C3×SD16D4semidirect
261. C2×SD16#67semidirect
430. C22×C3#471semidirect
510. C22#5177semidirect
520. C3#24278semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SD162
#840592
#840382
C2×#51772
#837342
#721332
#721292
#720592
#720532
#720522
#719772
#242783