Finite Group Databasebeta

← #10948665

Subgroups — #10948665

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83823C211yesmaximal, characteristic, complemented
2C2×#5147C211yesmaximal, characteristic
3#79249C211yesmaximal, characteristic, complemented
4C2×C3×#361C211yesmaximal, characteristic, complemented
5#71450C211yesmaximal, characteristic, complemented
6#71425C211yesmaximal, characteristic, complemented
7#70685C211yesmaximal, characteristic, complemented
8#69254C218yesmaximal, complemented
9#33471—31nomaximal, Sylow
10C22×C3×D8C2211yescharacteristic
11#5798—21no—
12#5745—21no—
13C2×#904C2211yescharacteristic
14C2×C3×M32C2211yescharacteristic
15C2×C3×#108C2211yescharacteristic
16#5215—21no—
17#5152C2214yescomplemented
18Dic3×D8—22no—
19C2×#827C412yescomplemented
20#5147C414yescomplemented
21#5147C2214yes—
22C2×#826C412yescomplemented
23#5139—21no—
24#5106C2211yescharacteristic
25#5101C2211yescharacteristic
26C2×#812C2211yescharacteristic
27#4895—22no—
28#4894C2214yescomplemented
29C3×#361C2214yescomplemented
30#4518C2214yescomplemented
31#4512C2214yescomplemented
32#4494C2214yescomplemented
33C2×#509—31no—
34#2957—31no—
35#2948—31no—
36AGL(1,ℤ/16)—38no—
37#2155—31no—
38C2×#361D311yescharacteristic, complemented
39#2145—31no—
40#2110—31no—
41C22×C3×D4—21no—
42C2×#235—21no—
43C2×C3×D8—21no—
44C2×C3×D8—21no—
45C2×C3×D8—21no—
46C2×C3×D8C2×C412yes—
47C2×C3×D8C2×C412yes—
48C2×C3×D8C2312yes—
49C2×C3×M16—21no—
50C22×C3×C8C2311yescharacteristic
51C2×#217—42no—
52D4×Dic3—24no—
53D4×Dic3—42no—
54#926—21no—
55#926D411yescharacteristic
56C2×#208D411yescharacteristic
57#907—24no—
58D3×M16—42no—
59C2×Dic12C2×C411yescharacteristic, complemented
60#904—21no—
61#904—21no—
62#904C2×C412yescomplemented
63#904C2312yes—
64C2×D24C2×C411yescharacteristic, complemented
65C2×#204—21no—
66#901—42no—
67C3×M32—21no—
68C3×M32C2312yes—
69C2×C3×C16—21no—
70C3×#108—21no—
71C3×#108C2312yes—
72#832—24no—
73#827C2×C418yescomplemented
74#826C2×C418yescomplemented
75#825—24no—
76#818—21no—
77#817C2312yes—
78#815C2312yes—
79C8×Dic3D412yes—
80#812C2×C412yescomplemented
81#812C2312yes—
82C2×C3⋊C16C2×C412yescomplemented
83C2×#135—31no—
84C22×D8C2×D311yescharacteristic
85#568—61no—
86#518—61no—
87#509—34no—
88#509—34no—
89C2×SD32—32no—
90C2×D16—32no—
91C2×M32—31no—
92C2×M32C2×D311yescharacteristic
93#444—34no—
94#443—62no—
95#442—34no—
96C4×D8—62no—
97C2×#108C2×D311yescharacteristic
98#427—31no—
99#421—61no—
100#418—61no—
101#405—31no—
102#365—34no—
103#361C2×D314yescomplemented
104#360—34no—
105#347—34no—
106C24×C3—41no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—41no—
115C22×Dic3—44no—
116#235—22no—
117#235—22no—
118#235—41no—
119#235—41no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C3×D8—22no—
124C3×D8—22no—
125C3×D8—22no—
126C3×D8C22×C414yes—
127C3×M16—22no—
128C3×M16—41no—
129C2×C3×C8—41no—
130C2×C3×C8C22×C411yescharacteristic
131C2×C3×C8C22×C411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8#4111yescharacteristic
134C2×C3×C8#4111yescharacteristic
135C2×C3×C8C2411yescharacteristic
136#217—21no—
137#217—44no—
138#217—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4×Dic3—41no—
142C4×Dic3C2×D412yes—
143#208—21no—
144#208—21no—
145C2×C3⋊C8C2×D412yes—
146Dic12—21no—
147Dic12C22×C412yes—
148D24—21no—
149D24C22×C412yes—
150#204—22no—
151#204—22no—
152ΓL(1,𝔽25)—44no—
153D3×C8—44no—
154C3×C16—22no—
155C3⋊C16C22×C414yes—
156C2×#41—61no—
157C22×D4—21no—
158#135—32no—
159#135—32no—
160#135—61no—
161#135—61no—
162C2×Q16—31no—
163C2×SD16—61no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—31no—
168C2×D8C4×D312yes—
169C2×D8C4×D312yes—
170C2×D8C22×D312yes—
171#131—64no—
172#131—122no—
173C2×M16—21no—
174C2×M16—31no—
175C2×M16—122no—
176C22×C8C22×D311yescharacteristic
177C4×D4—64no—
178C4×D4—122no—
179#117—31no—
180#117—61no—
181C2×C22⋊C4—122no—
182SD32—38no—
183D16—38no—
184M32—21no—
185M32—32no—
186M32—32no—
187M32C22×D312yes—
188C2×C16—21no—
189C2×C16—32no—
190#108—21no—
191#108C22×D312yes—
192#107—61no—
193#106—32no—
194#104—64no—
195#102—64no—
196#97—32no—
197C4×C8—32no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—162no—
202C23×C3AGL(1,ℤ/8)11yescharacteristic
203C22×D3—41no—
204C3×D4—24no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—42no—
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4C22×D411yescharacteristic
214#67—42no—
215#67—42no—
216C2×Dic3—22no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—164no—
220D12—22no—
221D12—41no—
222C4×D3—42no—
223C4×D3—42no—
224Dic6—22no—
225Dic6—41no—
226C3×C8—42no—
227C3×C8C4×D411yescharacteristic
228C3×C8C4×D411yescharacteristic
229C3×C8C23×C411yescharacteristic
230C3×C8C2×#4111yescharacteristic
231C3⋊C8—22no—
232C24—41no—
233#41—62no—
234#41—62no—
235#41—121no—
236#41—121no—
237C2×Q8—61no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—61no—
245C2×D4—121no—
246C22×C4—124no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249Q16—32no—
250Q16—61no—
251SD16—62no—
252SD16—62no—
253D8—22no—
254D8—22no—
255D8—22no—
256D8—32no—
257D8—61no—
258D8C2×C4×D314yes—
259M16—22no—
260M16—41no—
261M16—61no—
262M16—61no—
263M16—124no—
264C2×C8—32no—
265C2×C8—41no—
266C2×C8—124no—
267C2×C8C2×C4×D311yescharacteristic
268C2×C8C2×C4×D311yescharacteristic
269C2×C8D3×D411yescharacteristic
270C2×C8#23711yescharacteristic
271C2×C8#23711yescharacteristic
272C2×C8C23×D311yescharacteristic
273C4⋊C4—61no—
274C4⋊C4—62no—
275C22⋊C4—61no—
276C22⋊C4—124no—
277C22⋊C4—121no—
278C42—32no—
279C42—121no—
280C16—22no—
281C16—34no—
282C22×C3—41no—
283C22×C3—44no—
284C22×C3—84no—
285C22×C3—162no—
286C22×C3—162no—
287C22×C3#44211yescharacteristic
288C22×C3#44211yescharacteristic
289C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
290C2×D3—42no—
291C2×D3—161no—
292C3×C4#44211yescharacteristic
293C3×C4#44211yescharacteristic
294C3×C4C2×C4×D411yescharacteristic
295C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
296Dic3—42no—
297Dic3—42no—
298Dic3—82no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—121no—
303C23—162no—
304C23#91411yescharacteristic
305Q8—62no—
306Q8—121no—
307D4—24no—
308D4—42no—
309D4—42no—
310D4—42no—
311D4—62no—
312D4—122no—
313D4—122no—
314D4—121no—
315C2×C4—62no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—484no—
321C2×C4#67⋊C411yescharacteristic
322C2×C4#67⋊C411yescharacteristic
323C2×C4#91111yescharacteristic
324C2×C4#91111yescharacteristic
325C2×C4#91411yescharacteristic
326C2×C4C2×D3×D411yescharacteristic
327C8—42no—
328C8—62no—
329C8#67⋊C411yescharacteristic
330C8#67⋊C411yescharacteristic
331C8C22×C4×D311yescharacteristic
332C8C2×#23711yescharacteristic
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—84no—
337C2×C3C2×#44211yescharacteristic
338D3—82no—
339C22—41no—
340C22—44no—
341C22—84no—
342C22—122no—
343C22—162no—
344C22—162no—
345C22—481no—
346C22#478511yescharacteristic
347C22#478511yescharacteristic
348C22C2×#91411yescharacteristic
349C4—122no—
350C4—122no—
351C4—242no—
352C4#478511yescharacteristic
353C4#478511yescharacteristic
354C4C2×#67⋊C411yescharacteristic
355C4C2×#91111yescharacteristic
356C3#3347111yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—84no—
361C2—242no—
362C2C2×#478511yescharacteristic
subgroupG/H
centerC2C2×#4785
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#361C2
socleC2×C3C2×#442
prime radicalC2×#5147C2
cosocleC2×C3×C8C24
hypercenterC2×#361D3
hypocenterC3#33471
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
1. #83823C2semidirect
3. #79249C2semidirect
4. C2×C3×#361C2semidirect
5. #71450C2semidirect
6. #71425C2semidirect
7. #70685C2semidirect
8. #69254C2semidirect
17. #5152C22semidirect
19. C2×#827C4semidirect
20. #5147C4semidirect
22. C2×#826C4semidirect
28. #4894C22semidirect
29. C3×#361C22semidirect
30. #4518C22semidirect
31. #4512C22semidirect
32. #4494C22semidirect
38. C2×#361D3semidirect
59. C2×Dic12C2×C4semidirect
62. #904C2×C4semidirect
64. C2×D24C2×C4semidirect
73. #827C2×C4semidirect
74. #826C2×C4semidirect
80. #812C2×C4semidirect
82. C2×C3⋊C16C2×C4semidirect
103. #361C2×D3semidirect
356. C3#33471semidirect

Maximal subgroups

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

subgroupindex
#838232
C2×#51472
#792492
C2×C3×#3612
#714502
#714252
#706852
#692542
#334713