Finite Group Databasebeta

← #10948664

Subgroups — #10948664

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
1C2×#4908C211yesmaximal, characteristic
2#78580C211yesmaximal, characteristic, complemented
3#78561C211yesmaximal, characteristic, complemented
4C3×#2110C211yesmaximal, characteristic, complemented
5#71450C211yesmaximal, characteristic, complemented
6C2×#4516C211yesmaximal, characteristic, complemented
7#71427C211yesmaximal, characteristic, complemented
8#69091C218yesmaximal, complemented
9#33471—31nomaximal, Sylow
10#5745—21no—
11C2×#904C2211yescharacteristic
12C22×D24C2211yescharacteristic
13#5521—21no—
14C2×C3×M32C2211yescharacteristic
15C2×C3×M32C2211yescharacteristic
16C3×#405C2211yescharacteristic
17#5131—21no—
18C2×#819C2211yescharacteristic
19#5106C2211yescharacteristic
20#5005—21no—
21#4908C414yescomplemented
22#4908C2214yes—
23C2×#800C412yescomplemented
24C2×D48C412yescomplemented
25#4717C2214yescomplemented
26#4711C2214yescomplemented
27#4700—22no—
28C4×D24—22no—
29C3×#347C2214yescomplemented
30#4518C2214yescomplemented
31#4516C2214yescomplemented
32#4513C2214yescomplemented
33C2×#509—31no—
34#2957—31no—
35#2948—31no—
36AGL(1,ℤ/16)—38no—
37#2155—31no—
38C2×#361—31no—
39#2145—31no—
40#2110D311yescharacteristic, complemented
41C2×#235—21no—
42C22×D12—21no—
43C2×C3×M16D411yescharacteristic
44C22×C3×C8C2311yescharacteristic
45C3×#117D411yescharacteristic
46#926—21no—
47C2×#208—21no—
48#907—42no—
49D3×M16—42no—
50C2×Dic12C2×C411yescharacteristic, complemented
51#904—21no—
52#904—21no—
53#904C2×C412yescomplemented
54#904C2312yes—
55C2×D24—21no—
56C2×D24—21no—
57C2×D24—21no—
58C2×D24C2×C412yes—
59C2×D24C2×C412yes—
60C2×D24C2×C411yescharacteristic, complemented
61C2×D24C2312yes—
62C2×#204—21no—
63#901—24no—
64#893—42no—
65D3×C22⋊C4—42no—
66C4×D12—24no—
67C3×M32—21no—
68C3×M32C2×C412yescomplemented
69C3×M32C2312yes—
70C3×M32C2312yes—
71C2×C3×C16—21no—
72C2×C3×C16C2×C412yescomplemented
73C3×#97C2312yes—
74C3×C4×C8D412yes—
75#821—24no—
76#819—21no—
77#819C2312yes—
78#818—21no—
79#817C2312yes—
80#805—24no—
81#800C2×C418yescomplemented
82D48C2×C418yescomplemented
83C2×#135—31no—
84C22×D8—31no—
85#568—61no—
86#518—61no—
87#509—34no—
88#509—34no—
89C2×SD32—32no—
90C2×D16—32no—
91C2×M32C2×D311yescharacteristic
92C2×M32C2×D311yescharacteristic
93#444—34no—
94#443—62no—
95#442—34no—
96C4×D8—62no—
97C2×#108—31no—
98#427—31no—
99#421—61no—
100#418—61no—
101#405C2×D311yescharacteristic
102#365—34no—
103#361—34no—
104#360—34no—
105#347C2×D314yescomplemented
106C23×D3—41no—
107C22×C3×C4C2×D411yescharacteristic
108C2×#67—41no—
109#235—22no—
110#235—22no—
111#235—41no—
112#235—41no—
113C2×D12—21no—
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×C4×D3—41no—
121C2×C4×D3—44no—
122C2×Dic6—21no—
123C3×M16—21no—
124C3×M16—21no—
125C2×C3×C8C22×C411yescharacteristic
126C2×C3×C8C22×C411yescharacteristic
127C2×C3×C8C2×D412yes—
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8#4111yescharacteristic
130C2×C3×C8#4111yescharacteristic
131C2×C3×C8C2411yescharacteristic
132C3×C4⋊C4—21no—
133C3×C22⋊C4—21no—
134C3×C42C2×D412yes—
135#217—41no—
136(C2×D3)⋊C4—44no—
137C4⋊Dic3—22no—
138C4×Dic3—41no—
139#208—22no—
140#208—41no—
141C2×C3⋊C8—41no—
142Dic12—21no—
143Dic12C22×C412yes—
144D24—22no—
145D24—22no—
146D24—21no—
147D24—22no—
148D24C22×C412yes—
149D24C22×C414yes—
150#204—22no—
151#204—22no—
152ΓL(1,𝔽25)—44no—
153D3×C8—44no—
154C3×C16—22no—
155C3×C16C22×C414yes—
156C2×#41—61no—
157C22×D4—61no—
158#135—32no—
159#135—32no—
160#135—61no—
161#135—61no—
162C2×Q16—31no—
163C2×SD16—61no—
164C2×D8—32no—
165C2×D8—32no—
166C2×D8—31no—
167C2×D8—32no—
168C2×D8—61no—
169C2×D8—61no—
170C2×D8—61no—
171#131—64no—
172#131—122no—
173C2×M16—61no—
174C2×M16—122no—
175C2×M16D1211yescharacteristic
176C22×C8C22×D311yescharacteristic
177C4×D4—64no—
178C4×D4—122no—
179#117—61no—
180#117D1211yescharacteristic
181C2×C22⋊C4—122no—
182SD32—38no—
183D16—38no—
184M32—21no—
185M32C4×D312yescomplemented
186M32C22×D312yes—
187M32C22×D312yes—
188C2×C16—21no—
189C2×C16C4×D312yescomplemented
190#108—32no—
191#108—61no—
192#107—61no—
193#106—32no—
194#104—64no—
195#102—64no—
196#97C22×D312yes—
197C4×C8D1212yes—
198C23×C3AGL(1,ℤ/8)11yescharacteristic
199C22×D3—42no—
200C22×D3—41no—
201C22×D3—42no—
202C22×D3—42no—
203C22×D3—162no—
204C2×C3×C4—22no—
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C3×C4C22×D411yescharacteristic
211#67—42no—
212#67—42no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215D12—22no—
216D12—24no—
217D12—42no—
218D12—42no—
219D12—42no—
220D12—41no—
221C4×D3—42no—
222C4×D3—42no—
223C4×D3—164no—
224Dic6—22no—
225Dic6—41no—
226C3×C8—22no—
227C3×C8C4×D411yescharacteristic
228C3×C8C4×D411yescharacteristic
229C3×C8C23×C411yescharacteristic
230C3×C8C2×#4111yescharacteristic
231C3⋊C8—42no—
232C24—121no—
233#41—62no—
234#41—62no—
235#41—121no—
236#41—121no—
237C2×Q8—61no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—61no—
241C2×D4—62no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4—121no—
246C22×C4—124no—
247C22×C4—121no—
248C22×C4C2×D1211yescharacteristic
249Q16—32no—
250Q16—61no—
251SD16—62no—
252SD16—62no—
253D8—32no—
254D8—34no—
255D8—62no—
256D8—62no—
257D8—61no—
258D8—62no—
259M16—21no—
260M16—21no—
261M16—62no—
262M16—124no—
263M16—121no—
264C2×C8—124no—
265C2×C8—121no—
266C2×C8C2×C4×D311yescharacteristic
267C2×C8C2×C4×D311yescharacteristic
268C2×C8C2×D1212yes—
269C2×C8C2×D1211yescharacteristic
270C2×C8#23511yescharacteristic
271C2×C8#23511yescharacteristic
272C2×C8C23×D311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—62no—
275C22⋊C4—21no—
276C22⋊C4—124no—
277C22⋊C4—121no—
278C42—121no—
279C42C2×D1212yes—
280C16—22no—
281C16C2×C4×D314yes—
282C22×C3—41no—
283C22×C3#44211yescharacteristic
284C22×C3#44211yescharacteristic
285C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
286C2×D3—42no—
287C2×D3—44no—
288C2×D3—84no—
289C2×D3—161no—
290C2×D3—162no—
291C2×D3—162no—
292C3×C4—42no—
293C3×C4#44211yescharacteristic
294C3×C4#44211yescharacteristic
295C3×C4C2×C4×D411yescharacteristic
296C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
297Dic3—42no—
298Dic3—82no—
299C23—122no—
300C23—121no—
301C23—122no—
302C23—122no—
303C23—482no—
304C23#90811yescharacteristic
305Q8—62no—
306Q8—121no—
307D4—64no—
308D4—62no—
309D4—122no—
310D4—122no—
311D4—122no—
312D4—122no—
313D4—121no—
314D4—122no—
315C2×C4—22no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—484no—
321C2×C4C4×D1211yescharacteristic
322C2×C4C4×D1211yescharacteristic
323C2×C4#90811yescharacteristic
324C2×C4#90811yescharacteristic
325C2×C4#90811yescharacteristic
326C2×C4C22×D1211yescharacteristic
327C8—22no—
328C8—122no—
329C8C4×D1211yescharacteristic
330C8C4×D1211yescharacteristic
331C8C22×C4×D311yescharacteristic
332C8C2×#23511yescharacteristic
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3C2×#44211yescharacteristic
337D3—84no—
338D3—82no—
339C22—41no—
340C22—122no—
341C22—124no—
342C22—244no—
343C22—482no—
344C22—482no—
345C22—481no—
346C22#471111yescharacteristic
347C22#471111yescharacteristic
348C22C2×#90811yescharacteristic
349C4—42no—
350C4—122no—
351C4—242no—
352C4#471111yescharacteristic
353C4#471111yescharacteristic
354C4C2×C4×D1211yescharacteristic
355C4C2×#90811yescharacteristic
356C3#3347111yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—244no—
361C2—242no—
362C2C2×#471111yescharacteristic
subgroupG/H
centerC2C2×#4711
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2110C2
socleC2×C3C2×#442
prime radicalC2×#4908C2
cosocleC2×C3×C8C24
hypercenter#2110D3
hypocenterC3#33471
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #78580C2semidirect
3. #78561C2semidirect
4. C3×#2110C2semidirect
5. #71450C2semidirect
6. C2×#4516C2semidirect
7. #71427C2semidirect
8. #69091C2semidirect
21. #4908C4semidirect
23. C2×#800C4semidirect
24. C2×D48C4semidirect
25. #4717C22semidirect
26. #4711C22semidirect
29. C3×#347C22semidirect
30. #4518C22semidirect
31. #4516C22semidirect
32. #4513C22semidirect
40. #2110D3semidirect
50. C2×Dic12C2×C4semidirect
53. #904C2×C4semidirect
60. C2×D24C2×C4semidirect
68. C3×M32C2×C4semidirect
72. C2×C3×C16C2×C4semidirect
81. #800C2×C4semidirect
82. D48C2×C4semidirect
105. #347C2×D3semidirect
185. M32C4×D3semidirect
189. C2×C16C4×D3semidirect
356. C3#33471semidirect

Maximal subgroups

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

subgroupindex
C2×#49082
#785802
#785612
C3×#21102
#714502
C2×#45162
#714272
#690912
#334713