Finite Group Databasebeta

← #10951975

Subgroups — #10951975

Order 768.

646 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#82014C212yesmaximal, complemented
2#79891C212yesmaximal, complemented
3C3×#2170C211yesmaximal, characteristic, complemented
4#71455C212yesmaximal, complemented
5#69304C212yesmaximal, complemented
6#69129C212yesmaximal, complemented
7#69126C212yesmaximal, complemented
8#69067C211yesmaximal, characteristic, complemented
9#69062C211yesmaximal, characteristic, complemented
10#33857—31nomaximal, Sylow
11#5770—22no—
12#5753C2212yes—
13#5752C2212yescomplemented
14C3×#504C2212yescomplemented
15C3×#443C2211yescharacteristic
16#5143C2212yescomplemented
17#5142C2212yes—
18#4909—22no—
19#4909C2212yescomplemented
20#4908—22no—
21#4908C2212yes—
22C2×Dic24C2211yescharacteristic, complemented
23#4904—22no—
24#4904C2211yescharacteristic
25C2×#800—22no—
26C2×#800C2212yescomplemented
27C2×D48C2211yescharacteristic
28#4898—22no—
29#4827—22no—
30#4825—22no—
31#4700C2212yescomplemented
32#4696C2212yescomplemented
33C3×#364C2211yescharacteristic, complemented
34C3×#350C2212yescomplemented
35C3×C4×C16C2211yescharacteristic, complemented
36#4517C2212yescomplemented
37#4516C2212yescomplemented
38#4510C2212yescomplemented
39#4506C2211yescharacteristic, complemented
40#3416—32no—
41#3290—32no—
42#2249—31no—
43#2223—32no—
44#2212—32no—
45#2195—32no—
46#2192—32no—
47#2178—31no—
48#2170D311yescharacteristic, complemented
49#1010—22no—
50#1009—22no—
51C3×#135D411yescharacteristic
52C3×#131C2312yes—
53#951—22no—
54#949—22no—
55#918—42no—
56#916—42no—
57D3×SD16—42no—
58#911—42no—
59#909—22no—
60#909—42no—
61#908—22no—
62#908—42no—
63C2×Dic12—22no—
64C2×Dic12C2312yes—
65#904—22no—
66#904—22no—
67#904C2312yes—
68C2×D24—22no—
69C2×D24C2312yes—
70C2×#204—22no—
71#901—22no—
72#875—22no—
73C3×M32—22no—
74C3×M32C2312yes—
75C2×C3×C16—22no—
76C2×C3×C16C2311yescharacteristic
77C2×C3×C16C2311yescharacteristic
78C3×#108D411yescharacteristic
79C3×#104D412yes—
80C3×C4×C8C2311yescharacteristic
81#824—22no—
82#823—22no—
83#819C2312yes—
84#805—22no—
85Dic24—22no—
86Dic24—41no—
87Dic24D411yescharacteristic
88#800—22no—
89#800—22no—
90#800—22no—
91#800—22no—
92#800D412yes—
93D48—22no—
94D48—41no—
95D48D411yescharacteristic
96#578—32no—
97#577—62no—
98#576—32no—
99#510—32no—
100#510—62no—
101#509—32no—
102#509—62no—
103#508—31no—
104#508—62no—
105C2×Q32—31no—
106C2×SD32—32no—
107C2×SD32—62no—
108C2×D16—31no—
109#504C2×D312yescomplemented
110#495—32no—
111#473—32no—
112#472—32no—
113#471—62no—
114#455—62no—
115#454—62no—
116#443—32no—
117#443C2×D311yescharacteristic
118#368—31no—
119#364C2×D311yescharacteristic, complemented
120#362—32no—
121#361—32no—
122#359—32no—
123#350C2×D312yescomplemented
124C4×C16C2×D311yescharacteristic, complemented
125C3×#41C2×D412yes—
126#239—42no—
127D3×Q8—42no—
128#237—42no—
129#237—82no—
130D3×D4—42no—
131D3×D4—82no—
132#235—22no—
133#235—42no—
134#235—42no—
135C2×D12—22no—
136C2×D12—42no—
137C2×Dic6—22no—
138C2×Dic6—42no—
139C3×Q16D811yescharacteristic
140C3×SD16D812yes—
141C3×D8D811yescharacteristic
142C3×M16—22no—
143C3×M16C2×D412yes—
144C2×C3×C8—22no—
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8C2411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#216—42no—
149#215—42no—
150#214—42no—
151#213—42no—
152(C2×D3)⋊C4—82no—
153#208—22no—
154Dic12—22no—
155Dic12—22no—
156Dic12—42no—
157Dic12C2×D412yes—
158D24—22no—
159D24—22no—
160D24—42no—
161D24C2×D412yes—
162#204—22no—
163#204—42no—
164#204—42no—
165#204—42no—
166#204—42no—
167ΓL(1,𝔽25)—42no—
168D3×C8—42no—
169C3×C16—22no—
170C3×C16—21no—
171C3×C16C2×D411yescharacteristic
172C3×C16C2×D411yescharacteristic
1732−1+4—62no—
174H(5,𝔽2)—62no—
175#137—62no—
176#137—62no—
177#137—122no—
178#137—122no—
179AGL(1,ℤ/8)—62no—
180AGL(1,ℤ/8)—62no—
181AGL(1,ℤ/8)—122no—
182AGL(1,ℤ/8)—122no—
183#135—32no—
184#135—62no—
185#135—62no—
186#135—122no—
187#135D1211yescharacteristic
188C2×Q16—32no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×SD16—122no—
192C2×D8—32no—
193C2×D8—62no—
194#131—62no—
195#131C22×D312yes—
196#124—62no—
197Q32—31no—
198Q32—62no—
199Q32—121no—
200SD32—32no—
201SD32—62no—
202SD32—62no—
203SD32—62no—
204SD32—62no—
205D16—31no—
206D16—62no—
207D16—121no—
208M32—22no—
209M32C22×D312yes—
210C2×C16—22no—
211C2×C16C22×D311yescharacteristic
212C2×C16C22×D311yescharacteristic
213#108—32no—
214#108D1211yescharacteristic
215#104—62no—
216#104D1212yes—
217#101—62no—
218#100—62no—
219C4×C8C22×D311yescharacteristic
220C22×D3—42no—
221C22×D3—82no—
222C3×Q8C2×D812yes—
223C3×D4—22no—
224C3×D4C2×D812yes—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4C22×D411yescharacteristic
228#67—42no—
229#67—82no—
230#67—82no—
231C2×Dic3—42no—
232C2×Dic3—82no—
233D12—22no—
234D12—42no—
235D12—42no—
236D12—82no—
237C4×D3—42no—
238C4×D3—82no—
239C4×D3—82no—
240Dic6—22no—
241Dic6—42no—
242Dic6—42no—
243Dic6—82no—
244C3×C8—22no—
245C3×C8C2×D811yescharacteristic
246C3×C8C2×D811yescharacteristic
247C3×C8C22×D411yescharacteristic
248C3×C8H(5,𝔽2)11yescharacteristic
249C3⋊C8—42no—
250#41—62no—
251#41—122no—
252#41—122no—
253#41—122no—
254#41—122no—
255#41—242no—
256#41C2×D1212yes—
257C2×Q8—62no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×D4—62no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—242no—
264Q16—32no—
265Q16—62no—
266Q16—62no—
267Q16—122no—
268Q16—122no—
269Q16D2411yescharacteristic
270SD16—62no—
271SD16—122no—
272SD16—122no—
273SD16—122no—
274SD16—122no—
275SD16—122no—
276SD16—122no—
277SD16D2412yes—
278D8—32no—
279D8—62no—
280D8—62no—
281D8—122no—
282D8—122no—
283D8D2411yescharacteristic
284M16—22no—
285M16—62no—
286M16—122no—
287M16C2×D1212yes—
288C2×C8—22no—
289C2×C8—122no—
290C2×C8C2×D1211yescharacteristic
291C2×C8C23×D311yescharacteristic
292C22⋊C4—242no—
293C42C2×D1211yescharacteristic
294C16—22no—
295C16—21no—
296C16D3×D411yescharacteristic
297C16D3×D411yescharacteristic
298C22×C3—22no—
299C22×C3#57611yescharacteristic
300C2×D3—42no—
301C2×D3—82no—
302C2×D3—162no—
303C2×D3—322no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4D4211yescharacteristic
307C3×C4C22×D811yescharacteristic
308Dic3—42no—
309Dic3—82no—
310C23—122no—
311C23—242no—
312Q8—62no—
313Q8—122no—
314Q8—122no—
315Q8—242no—
316Q8C2×D2412yes—
317D4—22no—
318D4—62no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—242no—
323D4—242no—
324D4—242no—
325D4C2×D2412yes—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—242no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4C22×D1211yescharacteristic
334C8—22no—
335C8—122no—
336C8C2×D2411yescharacteristic
337C8C2×D2411yescharacteristic
338C8C2×D3×D411yescharacteristic
339C8#100911yescharacteristic
340C2×C3—21no—
341C2×C3—42no—
342C2×C3D4×D811yescharacteristic
343D3—82no—
344D3—162no—
345C22—22no—
346C22—122no—
347C22—242no—
348C22—482no—
349C22—962no—
350C22#575211yescharacteristic
351C4—22no—
352C4—41no—
353C4—122no—
354C4—242no—
355C4D4×D1211yescharacteristic
356C4C22×D2411yescharacteristic
357C3#3385711yescharacteristic, complemented, Sylow
358C2—21no—
359C2—42no—
360C2—242no—
361C2—482no—
362C2D4×D2411yescharacteristic
subgroupG/H
centerC2D4×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2170C2
socleC2×C3D4×D8
hypercenter#2170D3
hypocenterC3#33857
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170

Complements

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

NH ≅ G/Ntype
1. #82014C2semidirect
2. #79891C2semidirect
3. C3×#2170C2semidirect
4. #71455C2semidirect
5. #69304C2semidirect
6. #69129C2semidirect
7. #69126C2semidirect
8. #69067C2semidirect
9. #69062C2semidirect
13. #5752C22semidirect
14. C3×#504C22semidirect
16. #5143C22semidirect
19. #4909C22semidirect
22. C2×Dic24C22semidirect
26. C2×#800C22semidirect
31. #4700C22semidirect
32. #4696C22semidirect
33. C3×#364C22semidirect
34. C3×#350C22semidirect
35. C3×C4×C16C22semidirect
36. #4517C22semidirect
37. #4516C22semidirect
38. #4510C22semidirect
39. #4506C22semidirect
48. #2170D3semidirect
109. #504C2×D3semidirect
119. #364C2×D3semidirect
123. #350C2×D3semidirect
124. C4×C16C2×D3semidirect
357. C3#33857semidirect

Maximal subgroups

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

subgroupindex
#820142
#798912
C3×#21702
#714552
#693042
#691292
#691262
#690672
#690622
#338573