Finite Group Databasebeta

← #10816036

Subgroups — #10816036

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5061C211yesmaximal, characteristic
2#78774C212yesmaximal, complemented
3#78742C212yesmaximal
4C3×#1958C211yesmaximal, characteristic, complemented
5#71724C211yesmaximal, characteristic, complemented
6#70286C212yesmaximal, complemented
7#70195C212yesmaximal, complemented
8#69789C212yesmaximal, complemented
9#69562C212yesmaximal, complemented
10#22896—31nomaximal, Sylow
11C2×#1004C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13#5812C2211yescharacteristic, complemented
14C2×#942C2212yes—
15C2×#901C2212yes—
16#5504C2212yes—
17C2×C3×#103C2212yescomplemented
18C2×C3×#102C2212yes—
19C2×C3×C4×C8C2211yescharacteristic, complemented
20C3×#382C2211yescharacteristic, complemented
21C2×#836C2211yescharacteristic, complemented
22C2×#833C2211yescharacteristic
23#5215C2212yes—
24#5113C2212yes—
25#5101C2212yescomplemented
26#5061C418yescomplemented
27#4966C2212yes—
28#4964C2212yescomplemented
29C2×#805C2212yes—
30#4817—22no—
31#4790—22no—
32#4789—22no—
33#4788—22no—
34#4772—22no—
35#4771—22no—
36#4770—22no—
37D3×#102—22no—
38#4552C2212yescomplemented
39#4530C2212yescomplemented
40#4523C2212yescomplemented
41C2×#455—31no—
42#2947—32no—
43#2897—32no—
44#1958D311yescharacteristic, complemented
45#1931—32no—
46#1908—32no—
47#1888—31no—
48#1860—32no—
49#1764—32no—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×D4C2311yescharacteristic
52C22×#67—41no—
53#1004—24no—
54#1004—41no—
55#1004C2×C414yes—
56C2×#237—22no—
57C2×D3×D4—22no—
58C2×#235C2312yes—
59C22×C3×C8C2312yes—
60C3×#124—41no—
61C3×#124C2×C414yescomplemented
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—41no—
64C2×C3×C42C2311yescharacteristic
65Q8×Dic3—22no—
66Q8×Dic3—22no—
67C2×#216—22no—
68#942—24no—
69#942C2×C418yescomplemented
70C2×#215—22no—
71C4×#67—42no—
72#926—41no—
73#926C2312yes—
74C2×#208C2312yes—
75C2×#208C2312yes—
76#901—24no—
77#901—42no—
78C2×ΓL(1,𝔽25)—22no—
79C2×D3×C8—22no—
80#893—22no—
81#892—22no—
82D3×C4⋊C4—22no—
83#887—22no—
84C3×#103—22no—
85C3×#103—22no—
86C3×#102—22no—
87C3×#102—22no—
88C3×C4×C8—41no—
89C3×#95—41no—
90#836—41no—
91#833C2×C414yescomplemented
92#832—22no—
93#832—22no—
94#821D412yes—
95#821D412yes—
96#816D412yes—
97#816D412yes—
98#815—22no—
99C8×Dic3—22no—
100#810—22no—
101#809—22no—
102#808—22no—
103#807—22no—
104#805C2×C418yescomplemented
105C2×H(5,𝔽2)—31no—
106C2×#137—32no—
107C2×#131—32no—
108C2×#124C2×D311yescharacteristic
109#520—32no—
110#519—31no—
111#455—38no—
112#441—62no—
113#440—62no—
114C4×Q16—62no—
115C4×SD16—62no—
116#428—32no—
117C2×#104—32no—
118#418—32no—
119#418—62no—
120#417—32no—
121#417—62no—
122#416—32no—
123#416—62no—
124C2×#103C2×D312yescomplemented
125C2×#102—62no—
126C2×#102C2×D312yes—
127C2×#101—31no—
128C2×#100—31no—
129#405—32no—
130C2×C4×C8C2×D311yescharacteristic, complemented
131#382C2×D311yescharacteristic, complemented
132#343—32no—
133#341—32no—
134#339—32no—
135C24×C3—21no—
136C23×D3—41no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C22×C412yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4C22×C412yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4C2411yescharacteristic
149C2×#67—22no—
150C2×#67—42no—
151C2×#67—44no—
152C2×#67—44no—
153C22×Dic3—41no—
154#237—28no—
155#237—44no—
156D3×D4—28no—
157D3×D4—44no—
158#235—22no—
159#235—22no—
160#235C22×C414yes—
161#235#4114yes—
162C2×D12C2×D412yes—
163C2×C4×D3—22no—
164C2×C4×D3—42no—
165C2×Dic6C2×D412yes—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8C2×D412yes—
169C2×C3×C8C2×D412yes—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C22⋊C4—44no—
173C3×C42—21no—
174C3×C42C22×C412yes—
175#217—22no—
176#216—28no—
177#215—28no—
178(C2×D3)⋊C4—42no—
179C4⋊Dic3—42no—
180C4⋊Dic3C2×D412yes—
181C4⋊Dic3C2×D412yes—
182Dic3⋊C4—42no—
183C4×Dic3—22no—
184C4×Dic3—42no—
185C4×Dic3—41no—
186C4×Dic3—41no—
187#208—22no—
188#208—22no—
189#208—22no—
190#208C22×C414yes—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193ΓL(1,𝔽25)—44no—
194D3×C8—44no—
195H(5,𝔽2)—34no—
196H(5,𝔽2)—64no—
197H(5,𝔽2)—121no—
198C2×#41—32no—
199C2×#41—62no—
200C22×Q8C22×D311yescharacteristic
201C22×D4—62no—
202C22×D4—121no—
203C22×D4C22×D311yescharacteristic
204#137—38no—
205#137—64no—
206C2×Q16—62no—
207C2×SD16—62no—
208#131—64no—
209#131—122no—
210C2×M16—32no—
211C2×M16—32no—
212C2×M16—62no—
213C22×C8—62no—
214C22×C8C22×D312yes—
215#124—41no—
216#124C4×D314yescomplemented
217C4×Q8—62no—
218C4×Q8—62no—
219C4×Q8—62no—
220C4×D4—62no—
221C4×D4—122no—
222#117—32no—
223#117—62no—
224#117—121no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4C22×D312yes—
227C2×C22⋊C4—41no—
228C2×C42C22×D311yescharacteristic
229#107—62no—
230#106—62no—
231#104—38no—
232#103—22no—
233#103—22no—
234#103—32no—
235#103—32no—
236#103—62no—
237#102—22no—
238#102—22no—
239#102—32no—
240#102—32no—
241#102—62no—
242#102—62no—
243#102—62no—
244#101—121no—
245#100—34no—
246#97—62no—
247C4×C8—41no—
248C4×C8—62no—
249#95—41no—
250C23×C3—22no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—82no—
254C23×C3H(5,𝔽2)11yescharacteristic
255C22×D3—22no—
256C22×D3—44no—
257C22×D3—162no—
258C3×Q8—24no—
259C3×Q8—42no—
260C3×D4—24no—
261C3×D4—42no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—82no—
269C2×C3×C4C23×C411yescharacteristic
270C2×C3×C4C22×D412yes—
271C2×C3×C4C2×#4112yes—
272C2×C3×C4H(5,𝔽2)11yescharacteristic
273#67—24no—
274#67—24no—
275#67—48no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—44no—
280D12—22no—
281D12C4×D414yes—
282C4×D3—24no—
283C4×D3—24no—
284C4×D3—162no—
285Dic6—22no—
286Dic6C4×D414yes—
287C3×C8—22no—
288C3×C8—22no—
289C3⋊C8—24no—
290C3⋊C8—42no—
291C24—21no—
292C24—121no—
293#41—34no—
294#41—34no—
295#41—62no—
296#41—62no—
297#41—68no—
298#41—124no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—32no—
303C2×Q8C2×C4×D312yes—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—32no—
308C2×D4—68no—
309C2×D4—62no—
310C2×D4—124no—
311C2×D4—122no—
312C2×D4—124no—
313C2×D4—124no—
314C2×D4C2×C4×D312yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—62no—
319C22×C4—122no—
320C22×C4—121no—
321C22×C4C23×D311yescharacteristic
322Q16—68no—
323SD16—68no—
324M16—34no—
325M16—62no—
326M16—62no—
327M16—62no—
328M16—124no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—124no—
334C2×C8D3×D412yes—
335C2×C8D3×D412yes—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—32no—
339C4⋊C4—32no—
340C4⋊C4—122no—
341C4⋊C4—122no—
342C22⋊C4—44no—
343C22⋊C4—62no—
344C22⋊C4—122no—
345C42—21no—
346C42—62no—
347C42—122no—
348C42—121no—
349C42—121no—
350C42C2×C4×D312yes—
351C22×C3—22no—
352C22×C3—42no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—161no—
356C22×C3#51811yescharacteristic
357C22×C3D4211yescharacteristic
358C22×C3#55111yescharacteristic
359C2×D3—24no—
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—164no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—82no—
366C3×C4C2×C4×D412yes—
367C3×C4#54612yes—
368Dic3—24no—
369Dic3—42no—
370Dic3—42no—
371C23—22no—
372C23—41no—
373C23—41no—
374C23—62no—
375C23—82no—
376C23—124no—
377C23—482no—
378C23#100411yescharacteristic
379Q8—24no—
380Q8—34no—
381Q8—42no—
382Q8—62no—
383D4—24no—
384D4—34no—
385D4—42no—
386D4—64no—
387D4—64no—
388D4—62no—
389D4—128no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—62no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—82no—
402C2×C4—124no—
403C2×C4—482no—
404C2×C4C22×C4×D311yescharacteristic
405C2×C4C2×D3×D412yes—
406C2×C4#100411yescharacteristic
407C2×C4D3×#4112yes—
408C8—22no—
409C8—22no—
410C8—64no—
411C8—122no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—82no—
415C2×C3#235811yescharacteristic
416C2×C3#329012yes—
417D3—44no—
418C22—22no—
419C22—42no—
420C22—64no—
421C22—82no—
422C22—82no—
423C22—122no—
424C22—122no—
425C22—161no—
426C22—484no—
427C22#548611yescharacteristic
428C22#567911yescharacteristic
429C22#571711yescharacteristic
430C4—42no—
431C4—42no—
432C4—64no—
433C4—82no—
434C4—122no—
435C4—122no—
436C4C4×D3×D412yes—
437C4#561012yes—
438C3#2289611yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—82no—
442C2—124no—
443C2#7643211yescharacteristic
444C2#8183512yes—
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1958C2
socleC22×C3D42
prime radicalC2×#1004C22
cosocleC22×C3×C4C24
hypercenter#1958D3
hypocenterC3#22896
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #78774C2semidirect
4. C3×#1958C2semidirect
5. #71724C2semidirect
6. #70286C2semidirect
7. #70195C2semidirect
8. #69789C2semidirect
9. #69562C2semidirect
13. #5812C22semidirect
17. C2×C3×#103C22semidirect
19. C2×C3×C4×C8C22semidirect
20. C3×#382C22semidirect
21. C2×#836C22semidirect
25. #5101C22semidirect
26. #5061C4semidirect
28. #4964C22semidirect
38. #4552C22semidirect
39. #4530C22semidirect
40. #4523C22semidirect
44. #1958D3semidirect
61. C3×#124C2×C4semidirect
69. #942C2×C4semidirect
91. #833C2×C4semidirect
104. #805C2×C4semidirect
124. C2×#103C2×D3semidirect
130. C2×C4×C8C2×D3semidirect
131. #382C2×D3semidirect
216. #124C4×D3semidirect
438. C3#22896semidirect

Maximal subgroups

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

subgroupindex
C2×#50612
#787742
#787422
C3×#19582
#717242
#702862
#701952
#697892
#695622
#228963