Finite Group Databasebeta

← #10819940

Subgroups — #10819940

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5073C211yesmaximal, characteristic
2#82538C212yesmaximal, complemented
3#78915C212yesmaximal, complemented
4C3×#1887C211yesmaximal, characteristic, complemented
5#71763C211yesmaximal, characteristic, complemented
6#70316C212yesmaximal, complemented
7#70074C212yesmaximal, complemented
8#68760C212yesmaximal, complemented
9#68741C212yesmaximal, complemented
10#23201—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C3×#518C2211yescharacteristic
13#5798C2211yescharacteristic
14C22×#213C2212yes—
15C22×#213C2212yes—
16#5521C2212yes—
17C2×#874C2211yescharacteristic
18C3×#418C2212yes—
19C2×C3×#100C2211yescharacteristic
20C2×#833C2211yescharacteristic
21#5215C2212yes—
22#5131C2212yes—
23C2×#823C2212yes—
24C2×#822C2212yes—
25#5122C2212yes—
26#5073C418yescomplemented
27#5014—24no—
28C4×#213—24no—
29#5001C2212yes—
30#4998C2212yes—
31#4997C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4785—24no—
34#4756—24no—
35C3×#339C2212yes—
36#4523C2212yes—
37C2×#493—31no—
38#2948—32no—
39#2948—32no—
40#1978—32no—
41#1978—32no—
42#1887—32no—
43#1887—31no—
44#1887D311yescharacteristic, complemented
45#1772—32no—
46C22×C3×D4—41no—
47C22×C3×D4C2311yescharacteristic
48C22×C3×D4C2311yescharacteristic
49C22×D12—41no—
50C22×D12C2312yes—
51C2×C3×M16C2312yes—
52C3×#127—41no—
53C3×#127C2×C414yescomplemented
54C3×C4×D4—24no—
55C3×#117C2312yes—
56C2×C3×C22⋊C4—41no—
57C2×C3×C42C2311yescharacteristic
58C2×#217—41no—
59D4×Dic3—24no—
60C2×#213—22no—
61C2×#213—22no—
62C2×#213—22no—
63C2×#213—22no—
64C2×#213—24no—
65C2×#213—24no—
66C2×#213C2×C418yescomplemented
67C2×#213C2×C418yescomplemented
68#926C2312yes—
69C2×#208C2312yes—
70C22×C3⋊C8C2312yes—
71#893—24no—
72D3×C22⋊C4—42no—
73#874—41no—
74#874C2×C414yescomplemented
75C4×D12—24no—
76C3×#102—24no—
77C3×#100—41no—
78#833—41no—
79#832—24no—
80#823—42no—
81#822—24no—
82#821—24no—
83#815—22no—
84C8×Dic3—22no—
85#809—24no—
86#808D414yescomplemented
87#807D414yescomplemented
88#803—22no—
89C4×C3⋊C8—22no—
90C4×C3⋊C8C2×C414yescomplemented
91C22×D8—32no—
92C22×D8—32no—
93C2×#127—31no—
94C2×#127C2×D311yescharacteristic
95#518—31no—
96#518—32no—
97#518C2×D311yescharacteristic
98#493—38no—
99#442—64no—
100#442—64no—
101C4×D8—64no—
102C4×D8—64no—
103C2×#108—32no—
104#427—32no—
105#418—32no—
106#418—32no—
107#418—32no—
108#418C2×D312yes—
109C2×#100—31no—
110C2×#100—32no—
111C2×#100C2×D311yescharacteristic
112#405—32no—
113#405—32no—
114C2×C4×C8—31no—
115#339—32no—
116#339C2×D312yes—
117C24×C3—21no—
118C24×C3—21no—
119C23×D3—22no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—44no—
127C2×C3×D4—84no—
128C2×C3×D4C22×C412yes—
129C2×C3×D4C22×C412yes—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—42no—
134C2×D12—22no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—44no—
140C2×D12—84no—
141C2×D12C22×C414yes—
142C2×C4×D3—44no—
143C3×M16—22no—
144C3×M16—22no—
145C2×C3×C8—22no—
146C3×C4⋊C4C2×D414yes—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—42no—
149C3×C42—22no—
150C3×C42—21no—
151C3×C42C22×C412yes—
152#217—22no—
153#217—42no—
154#213—28no—
155#213—28no—
156#213—28no—
157#213—28no—
158(C2×D3)⋊C4—44no—
159C4⋊Dic3C2×D414yes—
160C4×Dic3—22no—
161#208—22no—
162#208—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8C22×C414yes—
165C2×C3⋊C8C2×D412yes—
166C2×C3⋊C8C2×D412yes—
167C2×C3⋊C8#4114yes—
168C22×D4—32no—
169C22×D4—41no—
170C22×D4—121no—
171C22×D4C22×D311yescharacteristic
172C22×D4C22×D311yescharacteristic
173C2×D8—38no—
174C2×D8—38no—
175C2×D8—64no—
176C2×D8—64no—
177C2×D8—62no—
178C2×D8—62no—
179C2×D8—62no—
180C2×D8—62no—
181C2×M16—32no—
182C2×M16C22×D312yes—
183C22×C8—32no—
184#127—34no—
185#127—41no—
186#127—121no—
187#127C4×D314yescomplemented
188C4×D4—24no—
189C4×D4—64no—
190C4×D4—64no—
191C4×D4—64no—
192#117—32no—
193#117C22×D312yes—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—122no—
196C2×C22⋊C4—121no—
197C2×C42C22×D311yescharacteristic
198#108—64no—
199#107—34no—
200#106—34no—
201#102—24no—
202#102—64no—
203#102—64no—
204#102—64no—
205#100—41no—
206#100—122no—
207#100—121no—
208#97—62no—
209#97—62no—
210C4×C8—34no—
211C4×C8—62no—
212C4×C8—62no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—82no—
220C23×C3—82no—
221C23×C3H(5,𝔽2)11yescharacteristic
222C22×D3—24no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—84no—
226C3×D4—24no—
227C3×D4—24no—
228C3×D4—42no—
229C3×D4—42no—
230C3×D4—88no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—162no—
235C2×C3×C4C23×C411yescharacteristic
236C2×C3×C4C22×D412yes—
237C2×C3×C4C2×#4112yes—
238C2×C3×C4H(5,𝔽2)11yescharacteristic
239C2×Dic3—24no—
240C2×Dic3—162no—
241D12—24no—
242D12—24no—
243D12—42no—
244D12—42no—
245D12—88no—
246C4×D3—164no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8C4×D414yes—
250C3⋊C8C4×D414yes—
251C24—21no—
252C24—21no—
253C24—62no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—34no—
261C2×D4—44no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—84no—
268C2×D4—124no—
269C2×D4—244no—
270C2×D4C2×C4×D312yes—
271C2×D4C2×C4×D312yes—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—124no—
275C22×C4—122no—
276C22×C4C23×D311yescharacteristic
277D8—68no—
278D8—68no—
279D8—68no—
280D8—68no—
281M16—22no—
282M16—22no—
283M16—62no—
284M16—62no—
285C2×C8—22no—
286C2×C8—34no—
287C2×C8—32no—
288C2×C8—32no—
289C2×C8—34no—
290C2×C8—62no—
291C4⋊C4—34no—
292C4⋊C4D3×D414yes—
293C22⋊C4—22no—
294C22⋊C4—42no—
295C22⋊C4—62no—
296C22⋊C4—124no—
297C22⋊C4—122no—
298C42—22no—
299C42—21no—
300C42—62no—
301C42C2×C4×D312yes—
302C22×C3—22no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—161no—
310C22×C3—161no—
311C22×C3#51811yescharacteristic
312C22×C3D4211yescharacteristic
313C22×C3#55911yescharacteristic
314C2×D3—44no—
315C2×D3—84no—
316C2×D3—84no—
317C2×D3—162no—
318C3×C4—24no—
319C3×C4—44no—
320C3×C4C2×C4×D412yes—
321C3×C4#55012yes—
322Dic3—44no—
323C23—22no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—64no—
330C23—82no—
331C23—82no—
332C23—122no—
333C23—122no—
334C23—244no—
335C23#100411yescharacteristic
336D4—24no—
337D4—24no—
338D4—42no—
339D4—42no—
340D4—64no—
341D4—64no—
342D4—88no—
343D4—122no—
344D4—122no—
345D4—248no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—64no—
350C2×C4—162no—
351C2×C4—484no—
352C2×C4—482no—
353C2×C4C22×C4×D311yescharacteristic
354C2×C4C2×D3×D412yes—
355C2×C4#100411yescharacteristic
356C2×C4D3×#4112yes—
357C8—34no—
358C8—34no—
359C8—42no—
360C8—122no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—82no—
364C2×C3—82no—
365C2×C3#236611yescharacteristic
366C2×C3#329212yes—
367D3—84no—
368C22—22no—
369C22—42no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—124no—
376C22—161no—
377C22—161no—
378C22—244no—
379C22—244no—
380C22—482no—
381C22#548611yescharacteristic
382C22#559011yescharacteristic
383C22#566411yescharacteristic
384C4—24no—
385C4—44no—
386C4—124no—
387C4C4×D3×D412yes—
388C4#559912yes—
389C3#2320111yescharacteristic, complemented, Sylow
390C2—21no—
391C2—21no—
392C2—82no—
393C2—82no—
394C2—244no—
395C2#7467911yescharacteristic
396C2#8028212yes—
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1887C2
socleC22×C3D42
prime radicalC2×#5073C2
cosocleC22×C3×C4C24
hypercenter#1887D3
hypocenterC3#23201
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1887

Complements

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

NH ≅ G/Ntype
2. #82538C2semidirect
3. #78915C2semidirect
4. C3×#1887C2semidirect
5. #71763C2semidirect
6. #70316C2semidirect
7. #70074C2semidirect
8. #68760C2semidirect
9. #68741C2semidirect
26. #5073C4semidirect
44. #1887D3semidirect
53. C3×#127C2×C4semidirect
66. C2×#213C2×C4semidirect
67. C2×#213C2×C4semidirect
74. #874C2×C4semidirect
86. #808D4semidirect
87. #807D4semidirect
90. C4×C3⋊C8C2×C4semidirect
187. #127C4×D3semidirect
389. C3#23201semidirect

Maximal subgroups

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

subgroupindex
C2×#50732
#825382
#789152
C3×#18872
#717632
#703162
#700742
#687602
#687412
#232013