Finite Group Databasebeta

← #10763237

Subgroups — #10763237

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5668C211yesmaximal, characteristic
2#79331C212yesmaximal, complemented
3C3×#1533C211yesmaximal, characteristic, complemented
4#71719C211yesmaximal, characteristic, complemented
5#70808C212yesmaximal, complemented
6#68349C214yesmaximal, complemented
7#68342C214yesmaximal, complemented
8#17663—31nomaximal, Sylow
9C22×#239C411yescharacteristic
10C22×#237C411yescharacteristic
11C2×C3×#124C2211yescharacteristic
12C2×#944C411yescharacteristic
13C2×#940C411yescharacteristic
14#5668—22no—
15#5668—22no—
16#5668C412yes—
17#5668C412yes—
18#5668C2212yes—
19#5668C2212yes—
20C2×#884—22no—
21C2×#67⋊C4—22no—
22C2×#875C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C3×#432C2212yes—
25#4926C2212yes—
26#4837—22no—
27#4833—22no—
28#4726—22no—
29#4718—22no—
30#4707C2214yescomplemented
31#4705C2214yescomplemented
32C3×#329C2214yescomplemented
33#4540C2214yescomplemented
34#4464C2218yescomplemented
35C2×#532—31no—
36#2920—32no—
37#1536—34no—
38#1533—32no—
39#1533—31no—
40#1533D311yescharacteristic, complemented
41#1525—34no—
42C22×C3×Q8C2×C411yescharacteristic
43C22×C3×D4C2×C411yescharacteristic
44C22×#67—21no—
45C22×#67—22no—
46C23×Dic3—21no—
47C2×#239—21no—
48C2×#239—21no—
49C2×#239—44no—
50C2×#239C2×C412yes—
51C2×#239C2×C412yes—
52C2×#237—21no—
53C2×#237—21no—
54C2×#237—44no—
55C2×#237C2×C412yes—
56C2×#237C2×C412yes—
57C22×D12—21no—
58C22×D12C2×C411yescharacteristic
59C22×C4×D3—21no—
60C22×C4×D3C2×C411yescharacteristic
61C22×C4×D3D412yes—
62C22×Dic6C2×C411yescharacteristic
63C3×#124—22no—
64C3×#124C2×C412yes—
65C3×#124C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42C2311yescharacteristic
68#944—22no—
69#944C2×C412yes—
70#944C2×C412yes—
71#940—22no—
72#940C2×C412yes—
73#940C2×C412yes—
74C2×(C2×D3)⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×C4×Dic3C2×C411yescharacteristic
77C2×C4×Dic3D412yes—
78C2×ΓL(1,𝔽25)—24no—
79#884—24no—
80#884—24no—
81#884—24no—
82#884—24no—
83#67⋊C4—24no—
84#67⋊C4—24no—
85#67⋊C4—24no—
86#67⋊C4—24no—
87#875—22no—
88#875C2×C412yes—
89#875C2312yes—
90C42×D3C2×C411yescharacteristic
91C42×D3C2×C411yescharacteristic
92C42×D3D412yes—
93C42×D3D412yes—
94C42×D3C2311yescharacteristic
95C42×D3C2311yescharacteristic
96C3×C4⋊C8—22no—
97C3×C22⋊C8—22no—
98C3×#97C2314yes—
99#830—22no—
100(C2×D3)⋊C8—24no—
101Dic3⋊C8—24no—
102C8×Dic3—24no—
103C4⋊(C3⋊C8)—22no—
104#803C2314yes—
105C22×#41—31no—
106C22×#41—31no—
107#532—32no—
108#532—32no—
109#532—32no—
110#532—32no—
111#532—62no—
112#532—62no—
113C2×C4⋊Q8—31no—
114C2×#127—31no—
115C2×#124—31no—
116C2×#124C2×D311yescharacteristic
117C2×#121—62no—
118C2×C4×D4—62no—
119C22×C42—31no—
120#432—32no—
121#432—34no—
122#432C2×D312yes—
123#431—62no—
124#431—62no—
125#423—62no—
126#407—62no—
127C4×M16—34no—
128#329—34no—
129#329—38no—
130#329C2×D314yescomplemented
131C24×C3—21no—
132C23×D3—21no—
133C23×D3C22⋊C411yescharacteristic
134C2×C3×Q8—21no—
135C2×C3×Q8—42no—
136C2×C3×Q8C22×C412yes—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4C22×C412yes—
140C22×C3×C4—21no—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C2×D412yes—
143C2×#67—22no—
144C2×#67—22no—
145C2×#67—24no—
146C2×#67—24no—
147C2×#67—48no—
148C2×#67—44no—
149C22×Dic3—21no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—44no—
153C22×Dic3C22⋊C412yes—
154C22×Dic3C22⋊C411yescharacteristic
155#239—44no—
156#239—44no—
157#239—44no—
158#239—44no—
159#237—44no—
160#237—44no—
161#237—44no—
162#237—44no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—21no—
166C2×D12—42no—
167C2×D12—44no—
168C2×D12C22×C412yes—
169C2×C4×D3—24no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—44no—
173C2×C4×D3—44no—
174C2×C4×D3C22⋊C412yes—
175C2×C4×D3C22⋊C411yescharacteristic
176C2×C4×D3C22⋊C411yescharacteristic
177C2×C4×D3C22⋊C412yes—
178C2×C4×D3C22×C411yescharacteristic
179C2×C4×D3C22×C411yescharacteristic
180C2×C4×D3C2×D412yes—
181C2×C4×D3C2×D412yes—
182C2×Dic6—21no—
183C2×Dic6—42no—
184C2×Dic6C22×C412yes—
185C2×C3×C8—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—24no—
188C3×C42C22×C411yescharacteristic
189C3×C42C2×D412yes—
190C3×C42C2411yescharacteristic
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4—24no—
193Dic3⋊C4—24no—
194Dic3⋊C4—24no—
195C4×Dic3C22⋊C412yes—
196C4×Dic3C22⋊C412yes—
197C4×Dic3C22⋊C412yes—
198C4×Dic3C22×C411yescharacteristic
199C4×Dic3C22×C411yescharacteristic
200C4×Dic3C2×D412yes—
201C4×Dic3C2×D412yes—
202C2×C3⋊C8—24no—
203ΓL(1,𝔽25)—48no—
204C2×#41—32no—
205C2×#41—32no—
206C2×#41—32no—
207C2×#41—32no—
208C2×#41—61no—
209C2×#41—61no—
210C2×#41—61no—
211C2×#41—61no—
212C2×#41—124no—
213C2×#41—124no—
214C22×Q8—31no—
215C22×Q8C4×D311yescharacteristic
216C22×D4—31no—
217C22×D4—61no—
218C22×D4—62no—
219C22×D4—61no—
220C22×D4C4×D311yescharacteristic
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—61no—
224C23×C4—61no—
225C2×M16—64no—
226C4⋊Q8—32no—
227C4⋊Q8—32no—
228C4⋊Q8—62no—
229#127—32no—
230#127—32no—
231#127—62no—
232#124—22no—
233#124—32no—
234#124—32no—
235#124—62no—
236#124C4×D312yes—
237#124C22×D312yes—
238#121—64no—
239#121—64no—
240#121—64no—
241#121—64no—
242C4×D4—64no—
243C4×D4—64no—
244C4×D4—64no—
245C4×D4—64no—
246C2×C4⋊C4—62no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—62no—
249C2×C42—31no—
250C2×C42—32no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—32no—
254C2×C42—32no—
255C2×C42—31no—
256C2×C42—31no—
257C2×C42C22×D311yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—64no—
260C4⋊C8—62no—
261C22⋊C8—22no—
262C22⋊C8—64no—
263C22⋊C8—62no—
264#97—34no—
265#97C22×D314yes—
266C4×C8—64no—
267C23×C3—22no—
268C23×C3—44no—
269C23×C3—42no—
270C23×C3C2×C22⋊C411yescharacteristic
271C22×D3—22no—
272C22×D3—22no—
273C22×D3—42no—
274C22×D3—44no—
275C22×D3—42no—
276C22×D3C2×C22⋊C411yescharacteristic
277C22×D3C2×C22⋊C411yescharacteristic
278C3×Q8—42no—
279C3×Q8—42no—
280C3×D4—42no—
281C3×D4—42no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4C2×C22⋊C412yes—
287C2×C3×C4C2×C22⋊C411yescharacteristic
288C2×C3×C4C23×C411yescharacteristic
289C2×C3×C4C22×D412yes—
290#67—48no—
291#67—44no—
292#67—44no—
293#67—48no—
294C2×Dic3—24no—
295C2×Dic3—22no—
296C2×Dic3—42no—
297C2×Dic3—44no—
298C2×Dic3—44no—
299C2×Dic3#10014yes—
300C2×Dic3C2×C22⋊C412yes—
301C2×Dic3C2×C22⋊C411yescharacteristic
302C2×Dic3C2×C22⋊C411yescharacteristic
303C2×Dic3C2×C22⋊C412yes—
304D12—42no—
305D12—42no—
306D12—44no—
307D12—44no—
308C4×D3—22no—
309C4×D3—22no—
310C4×D3—22no—
311C4×D3—22no—
312C4×D3—24no—
313C4×D3—24no—
314C4×D3—42no—
315C4×D3—42no—
316C4×D3—44no—
317C4×D3—44no—
318Dic6—42no—
319Dic6—42no—
320C3×C8—44no—
321C3⋊C8—44no—
322C24—21no—
323C24—31no—
324C24—61no—
325#41—124no—
326#41—124no—
327#41—124no—
328#41—124no—
329#41—124no—
330#41—124no—
331#41—124no—
332#41—124no—
333C2×Q8—21no—
334C2×Q8—32no—
335C2×Q8—42no—
336C2×Q8—61no—
337C2×Q8—122no—
338C2×Q8C2×C4×D312yes—
339C2×D4—21no—
340C2×D4—32no—
341C2×D4—42no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—64no—
345C2×D4—64no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—61no—
349C2×D4—122no—
350C2×D4—124no—
351C2×D4—128no—
352C2×D4—124no—
353C2×D4C2×C4×D312yes—
354C22×C4—21no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4—124no—
372C22×C4—124no—
373C22×C4—124no—
374C22×C4C2×C4×D311yescharacteristic
375C22×C4D3×D412yes—
376M16—128no—
377C2×C8—24no—
378C2×C8—64no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C42—32no—
386C42—32no—
387C42—32no—
388C42—31no—
389C42—31no—
390C42—32no—
391C42—32no—
392C42C2×C4×D311yescharacteristic
393C42D3×D412yes—
394C42C23×D311yescharacteristic
395C22×C3—22no—
396C22×C3—42no—
397C22×C3—44no—
398C22×C3—84no—
399C22×C3C2×#10012yes—
400C22×C3C22×C22⋊C411yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—44no—
406C2×D3—42no—
407C2×D3—44no—
408C2×D3—84no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—42no—
414Dic3—22no—
415Dic3—42no—
416Dic3C2×#10012yes—
417Dic3C2×#10012yes—
418Dic3#42114yes—
419C23—22no—
420C23—31no—
421C23—31no—
422C23—44no—
423C23—42no—
424C23—62no—
425C23—62no—
426C23—122no—
427C23—124no—
428C23—122no—
429C23D3×C22⋊C411yescharacteristic
430Q8—42no—
431Q8—42no—
432Q8—122no—
433Q8—122no—
434D4—42no—
435D4—42no—
436D4—122no—
437D4—122no—
438D4—128no—
439D4—124no—
440D4—124no—
441D4—128no—
442D4—124no—
443D4—124no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—32no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—34no—
450C2×C4—32no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—124no—
466C2×C4—124no—
467C2×C4—122no—
468C2×C4D3×C22⋊C412yes—
469C2×C4D3×C22⋊C411yescharacteristic
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C2×D3×D412yes—
472C8—44no—
473C8—124no—
474C2×C3—22no—
475C2×C3—82no—
476C2×C3C22×#10011yescharacteristic
477C2×C3C2×#42112yes—
478D3—41no—
479D3—41no—
480D3—82no—
481C22—22no—
482C22—42no—
483C22—44no—
484C22—61no—
485C22—61no—
486C22—62no—
487C22—62no—
488C22—84no—
489C22—124no—
490C22—122no—
491C22—124no—
492C22—244no—
493C22D3×#10012yes—
494C22C2×D3×C22⋊C411yescharacteristic
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—32no—
500C4—32no—
501C4—34no—
502C4—42no—
503C4—62no—
504C4—122no—
505C3#1766311yescharacteristic, complemented, Sylow
506C2—22no—
507C2—82no—
508C2—121no—
509C2—121no—
510C2—242no—
511C2C2×D3×#10011yescharacteristic
512C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1533C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5668C2
cosocleC3×C42C24
hypercenter#1533D3
hypocenterC3#17663
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C3×#1533C2semidirect
4. #71719C2semidirect
5. #70808C2semidirect
6. #68349C2semidirect
7. #68342C2semidirect
30. #4707C22semidirect
31. #4705C22semidirect
32. C3×#329C22semidirect
33. #4540C22semidirect
34. #4464C22semidirect
40. #1533D3semidirect
130. #329C2×D3semidirect
505. C3#17663semidirect

Maximal subgroups

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

subgroupindex
C2×#56682
#793312
C3×#15332
#717192
#708082
#683492
#683422
#176633