Finite Group Databasebeta

← #10816027

Subgroups — #10816027

Order 768.

1008 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5062C211yesmaximal, characteristic
2#78790C212yesmaximal, complemented
3#78726C212yesmaximal, complemented
4C3×#1958C211yesmaximal, characteristic, complemented
5#71723C211yesmaximal, characteristic, complemented
6#70263C212yesmaximal, complemented
7#70218C212yesmaximal, complemented
8#69789C212yesmaximal, complemented
9#69562C212yesmaximal, complemented
10#22895—31nomaximal, Sylow
11C2×#1007C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13#5798C2211yescharacteristic
14C2×#932C2212yes—
15C2×#901C2212yes—
16#5504C2212yes—
17C2×C3×#103C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20C3×#382C2211yescharacteristic
21C2×#836C2211yescharacteristic
22#5225C2212yes—
23C2×#833C2211yescharacteristic
24#5113C2212yes—
25#5101C2212yes—
26#5062C418yescomplemented
27#4966C2212yes—
28#4964C2212yes—
29C2×#805C2212yes—
30#4804—22no—
31#4803—22no—
32#4802—22no—
33D3×#103—22no—
34#4785—22no—
35#4758—22no—
36#4757—22no—
37#4756—22no—
38#4552C2212yes—
39#4530C2212yes—
40#4523C2212yes—
41C2×#454—31no—
42#2946—32no—
43#2898—32no—
44#1958D311yescharacteristic, complemented
45#1930—32no—
46#1907—32no—
47#1885—31no—
48#1860—32no—
49#1764—32no—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×D4C2311yescharacteristic
52#1007—24no—
53#1007—41no—
54#1007C2×C414yescomplemented
55C2×#239—22no—
56C2×D3×Q8—22no—
57C2×#235—41no—
58C2×#235C2312yes—
59C22×C3×C8C2312yes—
60C3×#124—41no—
61C3×#124C2×C414yescomplemented
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—41no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—41no—
66D4×Dic3—22no—
67D4×Dic3—22no—
68C2×#214—22no—
69#932—24no—
70#932C2×C418yescomplemented
71C2×#213—22no—
72C4×#67—42no—
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#836C2×C414yescomplemented
91#835—22no—
92#835—22no—
93#833—41no—
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
105C2×2−1+4—31no—
106C2×AGL(1,ℤ/8)—32no—
107C2×#131—32no—
108C2×#124C2×D311yescharacteristic
109#520—32no—
110#518—31no—
111#454—38no—
112#442—62no—
113#440—62no—
114C4×SD16—62no—
115C4×D8—62no—
116#428—32no—
117C2×#104—32no—
118#419—32no—
119#419—62no—
120#417—32no—
121#417—62no—
122#416—32no—
123#416—62no—
124C2×#103—62no—
125C2×#103C2×D312yes—
126C2×#102C2×D312yes—
127C2×#101—31no—
128C2×#100—31no—
129#405—32no—
130C2×C4×C8C2×D311yescharacteristic
131#382C2×D311yescharacteristic
132#343—32no—
133#341—32no—
134#339—32no—
135C24×C3—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8C22×C412yes—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4C22×C412yes—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4C2411yescharacteristic
148C2×#67—22no—
149C22×Dic3—42no—
150#239—28no—
151#239—44no—
152D3×Q8—28no—
153D3×Q8—44no—
154#235—22no—
155#235—22no—
156#235—44no—
157#235—44no—
158#235C22×C414yes—
159#235#4114yes—
160C2×D12—41no—
161C2×D12C2×D412yes—
162C2×C4×D3—22no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×Dic6—41no—
166C2×Dic6C2×D412yes—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8C2×D412yes—
170C2×C3×C8C2×D412yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C22⋊C4—44no—
174C3×C42—21no—
175C3×C42C22×C412yes—
176#217—22no—
177#217—41no—
178#217—41no—
179#214—28no—
180#213—28no—
181(C2×D3)⋊C4—42no—
182C4⋊Dic3C2×D412yes—
183C4⋊Dic3C2×D412yes—
184Dic3⋊C4—42no—
185C4×Dic3—22no—
186C4×Dic3—42no—
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—
1952−1+4—34no—
1962−1+4—64no—
1972−1+4—121no—
198C2×#41—32no—
199C2×#41—62no—
200C2×#41—121no—
201C22×Q8—62no—
202C22×Q8C22×D311yescharacteristic
203C22×D4C22×D311yescharacteristic
204AGL(1,ℤ/8)—38no—
205AGL(1,ℤ/8)—64no—
206C2×SD16—62no—
207C2×D8—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×D4—62no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4—122no—
222#117—32no—
223#117—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—121no—
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#103—62no—
238#103—62no—
239#102—22no—
240#102—22no—
241#102—32no—
242#102—32no—
243#102—62no—
244#101—34no—
245#100—121no—
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—
256C3×Q8—24no—
257C3×Q8—42no—
258C3×D4—24no—
259C3×D4—42no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—82no—
267C2×C3×C4C23×C411yescharacteristic
268C2×C3×C4C22×D412yes—
269C2×C3×C4C2×#4112yes—
270C2×C3×C4H(5,𝔽2)11yescharacteristic
271#67—24no—
272#67—24no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—162no—
277D12—22no—
278D12—44no—
279D12C4×D414yes—
280C4×D3—24no—
281C4×D3—24no—
282C4×D3—48no—
283C4×D3—162no—
284Dic6—22no—
285Dic6—44no—
286Dic6C4×D414yes—
287C3×C8—22no—
288C3×C8—22no—
289C3⋊C8—24no—
290C3⋊C8—42no—
291C24—21no—
292#41—34no—
293#41—34no—
294#41—62no—
295#41—62no—
296#41—68no—
297#41—124no—
298#41—124no—
299#41—124no—
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—32no—
304C2×Q8—68no—
305C2×Q8—121no—
306C2×Q8—124no—
307C2×Q8C2×C4×D312yes—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—32no—
312C2×D4—62no—
313C2×D4—121no—
314C2×D4C2×C4×D312yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—62no—
319C22×C4—122no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4C23×D311yescharacteristic
323SD16—68no—
324D8—68no—
325M16—34no—
326M16—62no—
327M16—62no—
328M16—62no—
329M16—124no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—124no—
335C2×C8D3×D412yes—
336C2×C8D3×D412yes—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—32no—
340C4⋊C4—32no—
341C4⋊C4—122no—
342C22⋊C4—44no—
343C22⋊C4—62no—
344C22⋊C4—122no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C42—21no—
348C42—62no—
349C42—122no—
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—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—82no—
365C3×C4C2×C4×D412yes—
366C3×C4#54612yes—
367Dic3—24no—
368Dic3—42no—
369Dic3—42no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—62no—
374C23—82no—
375C23#100411yescharacteristic
376Q8—24no—
377Q8—34no—
378Q8—42no—
379Q8—62no—
380Q8—124no—
381D4—24no—
382D4—34no—
383D4—42no—
384D4—64no—
385D4—64no—
386D4—62no—
387D4—124no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—62no—
395C2×C4—64no—
396C2×C4—64no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—82no—
400C2×C4—128no—
401C2×C4—482no—
402C2×C4—482no—
403C2×C4C22×C4×D311yescharacteristic
404C2×C4C2×D3×D412yes—
405C2×C4#100411yescharacteristic
406C2×C4D3×#4112yes—
407C8—22no—
408C8—22no—
409C8—64no—
410C8—122no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—82no—
414C2×C3#235811yescharacteristic
415C2×C3#329012yes—
416D3—44no—
417C22—22no—
418C22—42no—
419C22—64no—
420C22—82no—
421C22—82no—
422C22—122no—
423C22—122no—
424C22—161no—
425C22#548611yescharacteristic
426C22#567911yescharacteristic
427C22#571711yescharacteristic
428C4—42no—
429C4—42no—
430C4—64no—
431C4—82no—
432C4—122no—
433C4—122no—
434C4C4×D3×D412yes—
435C4#561012yes—
436C3#2289511yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—82no—
440C2—124no—
441C2#7643211yescharacteristic
442C2#8183512yes—
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1958C2
socleC22×C3D42
prime radicalC2×#5062C2
cosocleC22×C3×C4C24
hypercenter#1958D3
hypocenterC3#22895
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. #78790C2semidirect
3. #78726C2semidirect
4. C3×#1958C2semidirect
5. #71723C2semidirect
6. #70263C2semidirect
7. #70218C2semidirect
8. #69789C2semidirect
9. #69562C2semidirect
26. #5062C4semidirect
44. #1958D3semidirect
54. #1007C2×C4semidirect
61. C3×#124C2×C4semidirect
70. #932C2×C4semidirect
90. #836C2×C4semidirect
104. #805C2×C4semidirect
216. #124C4×D3semidirect
436. C3#22895semidirect

Maximal subgroups

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

subgroupindex
C2×#50622
#787902
#787262
C3×#19582
#717232
#702632
#702182
#697892
#695622
#228953