Finite Group Databasebeta

← #10901278

Subgroups — #10901278

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic, complemented
2#82526C212yesmaximal, complemented
3#82378C211yesmaximal, characteristic, complemented
4#71903C212yesmaximal, complemented
5#71879C212yesmaximal, complemented
6#71856C214yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71617C211yesmaximal, characteristic, complemented
9#29678—31nomaximal, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#540—22no—
13C3×#538—22no—
14C3×#537C2211yescharacteristic
15C3×#535C2212yes—
16C3×#532—22no—
17C3×#529—22no—
18C3×#525—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#949C2214yes—
22C2×#940C2212yes—
23C2×#930—22no—
24C2×#874C2211yescharacteristic
25C2×#837C2214yes—
26C2×#833C2212yes—
27#5027—22no—
28#5027—22no—
29#5015—22no—
30#5015—22no—
31#5001C2214yes—
32#4988C2212yes—
33#4960—22no—
34C4×C4⋊Dic3C2211yescharacteristic
35#4924C2212yes—
36#4532C2214yes—
37#4530C2212yes—
38#3460D311yescharacteristic, complemented
39#3038—32no—
40#2541—31no—
41#2027—32no—
42#2006—32no—
43#2002—34no—
44#1806—32no—
45#1800—31no—
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4D412yes—
50C22×#67—24no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12C2312yes—
54C3×C4⋊Q8—22no—
55C3×#127—22no—
56C3×#126—24no—
57C3×#125—22no—
58C3×#124—24no—
59C3×#124—22no—
60C3×#123—24no—
61C3×#123—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×#121—22no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—22no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#120—24no—
72C3×C4×Q8D412yescomplemented
73C3×C4×Q8D412yescomplemented
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4D412yescomplemented
80C3×C4×D4D412yescomplemented
81C3×#117—22no—
82C3×#117—22no—
83C3×#117C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4D412yes—
86C2×C3×C42—22no—
87C2×C3×C42C2311yescharacteristic
88#949—44no—
89#949—44no—
90C2×#215—24no—
91#940—44no—
92C2×#213—24no—
93#930—48no—
94C2×(C2×D3)⋊C4—24no—
95C2×C4⋊Dic3—21no—
96C2×C4×Dic3C2312yes—
97C2×#208C2314yes—
98#874—22no—
99#874—22no—
100#837—44no—
101#833—42no—
102C3⋊#95—21no—
103#809—24no—
104#809—24no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×AGL(1,ℤ/8)—34no—
108#544C2×D311yescharacteristic
109#542—22no—
110#540—22no—
111#538—22no—
112#537C2×D311yescharacteristic
113#535C2×D312yes—
114#532—22no—
115C2×#127—32no—
116C2×#127—31no—
117#529—22no—
118#525—22no—
119C2×#121—62no—
120C4×#41C2×D312yes—
121C2×C4×D4C2×D311yescharacteristic
122#460—62no—
123#460—62no—
124#459—62no—
125#459—62no—
126#423—32no—
127C2×#104—34no—
128#418—34no—
129C2×#100—32no—
130#392—62no—
131#390—32no—
132C4×C4⋊C4—31no—
133#343—32no—
134#339—34no—
135C24×C3C2×D412yes—
136C23×D3—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—84no—
142C2×C3×Q8—22no—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4C2×D412yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—44no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C2×#67—48no—
164C2×#67—48no—
165C2×#67—84no—
166C22×Dic3—22no—
167C2×D12—24no—
168C2×D12—24no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—44no—
172C2×D12—41no—
173C2×D12—42no—
174C2×D12—41no—
175C2×D12—82no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#215—48no—
197#213—48no—
198(C2×D3)⋊C4—48no—
199C4⋊Dic3—24no—
200C4×Dic3—24no—
201#208—24no—
202#208—24no—
203C2×C3⋊C8—24no—
204C2×#41—22no—
205C2×#41C22×D312yes—
206C22×D4—32no—
207C22×D4—64no—
208C22×D4—61no—
209C22×D4—61no—
210C22×D4C22×D311yescharacteristic
211C23×C4#6712yes—
212AGL(1,ℤ/8)—124no—
213AGL(1,ℤ/8)—124no—
214C2×SD16—64no—
215C2×D8—64no—
216C2×M16—34no—
217C4⋊Q8—22no—
218#127—22no—
219#127—62no—
220#127—62no—
221#127—124no—
222#126—24no—
223#125—22no—
224#124—24no—
225#124—22no—
226#123—24no—
227#123—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231#121—22no—
232#121—24no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—128no—
238#120—24no—
239C4×Q8#6712yescomplemented
240C4×Q8#6712yescomplemented
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4#6712yescomplemented
247C4×D4#6712yescomplemented
248#117—22no—
249#117—22no—
250#117C22×D312yes—
251C2×C4⋊C4—61no—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C22⋊C4—64no—
254C2×C22⋊C4#6712yes—
255C2×C42—22no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258C4⋊C8—62no—
259C4⋊C8—62no—
260#104—124no—
261#102—64no—
262#102—64no—
263#100—122no—
264#95—61no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C22×D3—44no—
273C22×D3—44no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—82no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—84no—
287C3×D4—82no—
288C3×D4—84no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—82no—
301C2×C3×C4—82no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12011yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C2×C3×C4H(5,𝔽2)12yes—
308C2×C3×C42−1+411yescharacteristic
309#67—816no—
310C2×Dic3—24no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313D12—44no—
314D12—44no—
315D12—82no—
316D12—84no—
317D12—82no—
318C3⋊C8—44no—
319C24—62no—
320C24C2×#6712yes—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—84no—
326C2×Q8—22no—
327C2×Q8C2×#6712yes—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—64no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—64no—
340C2×D4—124no—
341C2×D4—128no—
342C2×D4—128no—
343C2×D4—122no—
344C2×D4—121no—
345C2×D4—121no—
346C2×D4—242no—
347C2×D4—244no—
348C2×D4C2×#6712yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—62no—
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6712yes—
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360SD16—128no—
361D8—128no—
362M16—64no—
363M16—64no—
364C2×C8—64no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—64no—
372C4⋊C4C2×#6712yes—
373C4⋊C4C2×#6712yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—128no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—64no—
386C42C2×#6712yes—
387C42C2×#6712yes—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—82no—
398C22×C3C2×#12011yescharacteristic
399C22×C3#53511yescharacteristic
400C22×C3#53711yescharacteristic
401C2×D3—84no—
402C2×D3—84no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4#45312yes—
413C3×C4#53512yes—
414C3×C4#53712yes—
415Dic3—44no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—124no—
423C23—124no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23C22×#6711yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—82no—
431D4—22no—
432D4—22no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—82no—
438D4—84no—
439D4—84no—
440D4—124no—
441D4—124no—
442D4—242no—
443D4—244no—
444D4—2416no—
445D4—242no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—64no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4#95312yes—
463C2×C4#95311yescharacteristic
464C2×C4#95311yescharacteristic
465C2×C4#100911yescharacteristic
466C2×C4#100912yes—
467C2×C4#101011yescharacteristic
468C2×C4C22×#6712yes—
469C8—124no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—82no—
473C2×C3—82no—
474C2×C3#261511yescharacteristic
475C2×C3C2×#45311yescharacteristic
476C2×C3#302111yescharacteristic
477D3—162no—
478C22—21no—
479C22—21no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—82no—
488C22—244no—
489C22—244no—
490C22—242no—
491C22—242no—
492C22—242no—
493C22#583011yescharacteristic
494C22#583311yescharacteristic
495C22C2×#95311yescharacteristic
496C4—21no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—124no—
503C4#524112yes—
504C4#583012yes—
505C4#583312yes—
506C3#2967811yescharacteristic, complemented, Sylow
507C2—21no—
508C2—21no—
509C2—82no—
510C2—82no—
511C2—482no—
512C2#8409811yescharacteristic
513C2C2×#524111yescharacteristic
514C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3460C2
socleC22×C3C2×#120
hypercenter#3460D3
hypocenterC3#29678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3460

Complements

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

NH ≅ G/Ntype
1. C3×#3460C2semidirect
2. #82526C2semidirect
3. #82378C2semidirect
4. #71903C2semidirect
5. #71879C2semidirect
6. #71856C2semidirect
7. #71646C2semidirect
8. #71617C2semidirect
38. #3460D3semidirect
72. C3×C4×Q8D4semidirect
73. C3×C4×Q8D4semidirect
79. C3×C4×D4D4semidirect
80. C3×C4×D4D4semidirect
239. C4×Q8#67semidirect
240. C4×Q8#67semidirect
246. C4×D4#67semidirect
247. C4×D4#67semidirect
506. C3#29678semidirect

Maximal subgroups

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

subgroupindex
C3×#34602
#825262
#823782
#719032
#718792
#718562
#716462
#716172
#296783