Finite Group Databasebeta

← #10951972

Subgroups — #10951972

Order 768.

646 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82024C212yesmaximal, complemented
2#81835C212yesmaximal, complemented
3C3×#2249C211yesmaximal, characteristic, complemented
4#71506C212yesmaximal, complemented
5#71491C212yesmaximal, complemented
6#71471C212yesmaximal, complemented
7#71325C212yesmaximal, complemented
8#69067C211yesmaximal, characteristic, complemented
9#69056C211yesmaximal, characteristic, complemented
10#33857—31nomaximal, Sylow
11#5766C2212yes—
12#5762—22no—
13#5757C2212yescomplemented
14C2×C3×Q32C2211yescharacteristic, complemented
15C2×C3×SD32C2212yescomplemented
16C2×C3×D16C2211yescharacteristic
17C3×#495C2212yescomplemented
18#5192C2212yescomplemented
19#5179C2212yescomplemented
20#5177—22no—
21#5160C2212yes—
22#5147C2212yes—
23#5062—22no—
24#5061—22no—
25#4916—22no—
26#4915—22no—
27#4914—22no—
28D3×SD32—22no—
29#4904C2211yescharacteristic
30#4901C2212yescomplemented
31#4700C2211yescharacteristic
32#4700C2212yescomplemented
33C3×C4×C16C2211yescharacteristic, complemented
34#4565C2212yescomplemented
35#4563C2212yescomplemented
36#4510C2212yescomplemented
37#4508C2212yescomplemented
38#4506C2211yescharacteristic, complemented
39#4463C2211yescharacteristic, complemented
40#3416—32no—
41#3290—32no—
42#2249D311yescharacteristic, complemented
43#2223—32no—
44#2212—32no—
45#2195—32no—
46#2192—32no—
47#2178—31no—
48#2170—31no—
49#1007—22no—
50#1004—22no—
51C2×C3×Q16C2312yes—
52C2×C3×SD16—22no—
53C2×C3×D8C2312yes—
54C3×#124—22no—
55#942—22no—
56#942—22no—
57#932—22no—
58#932—22no—
59#919—22no—
60#918—42no—
61D3×Q16—22no—
62#916—42no—
63#915—42no—
64#914—42no—
65D3×SD16—42no—
66#912—22no—
67#911—42no—
68D3×D8—22no—
69#904D411yescharacteristic
70#904C2312yes—
71#901—22no—
72#901C2312yes—
73C3×Q32—41no—
74C3×SD32—22no—
75C3×SD32—22no—
76C3×D16—41no—
77C2×C3×C16C2311yescharacteristic
78C2×C3×C16C2311yescharacteristic
79C3×C4×C8C2311yescharacteristic
80#836—22no—
81#833—22no—
82#829—22no—
83#828—22no—
84#827—22no—
85#826—22no—
86#819D411yescharacteristic
87#819C2312yes—
88#812C2312yes—
89#805—22no—
90#805D412yes—
91Dic24D411yescharacteristic
92#800D412yes—
93D48D411yescharacteristic
94#798—22no—
95D3×C16—22no—
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×Q32C2×D311yescharacteristic, complemented
106C2×SD32—62no—
107C2×SD32C2×D312yescomplemented
108C2×D16C2×D311yescharacteristic
109#504—32no—
110#495C2×D312yescomplemented
111#473—32no—
112#472—32no—
113#471—62no—
114#455—62no—
115#454—62no—
116#443—31no—
117#443—32no—
118#368—31no—
119#364—31no—
120#362—32no—
121#361—32no—
122#359—32no—
123#350—32no—
124C4×C16C2×D311yescharacteristic, complemented
125C2×C3×Q8—22no—
126C2×C3×D4—22no—
127C2×#67—82no—
128#239—42no—
129#239—42no—
130D3×Q8—42no—
131D3×Q8—42no—
132#237—42no—
133#237—42no—
134D3×D4—42no—
135D3×D4—42no—
136#235—22no—
137#235—82no—
138#235C2×D412yes—
139C3×Q16—22no—
140C3×Q16—22no—
141C3×SD16—42no—
142C3×SD16—42no—
143C3×D8—22no—
144C3×D8—22no—
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8C2411yescharacteristic
147C3×C22⋊C4—82no—
148C3×C42C2×D411yescharacteristic
149#216—42no—
150#216—42no—
151#215—42no—
152#215—42no—
153#214—42no—
154#214—42no—
155#213—42no—
156#213—42no—
157#208—22no—
158#208C2×D412yes—
159Dic12D811yescharacteristic
160Dic12C2×D412yes—
161D24D811yescharacteristic
162D24C2×D412yes—
163#204—22no—
164#204D812yes—
165ΓL(1,𝔽25)—22no—
166ΓL(1,𝔽25)—42no—
167D3×C8—22no—
168D3×C8—42no—
169C3×C16—21no—
170C3×C16C2×D411yescharacteristic
171C3×C16C2×D411yescharacteristic
172C3⋊C16—22no—
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—31no—
184#135—32no—
185#135—62no—
186#135—62no—
187#135—122no—
188C2×Q16—62no—
189C2×Q16C22×D312yes—
190C2×SD16—22no—
191C2×SD16—122no—
192C2×D8—62no—
193C2×D8C22×D312yes—
194#131—32no—
195#131—62no—
196#124—22no—
197Q32—31no—
198Q32—41no—
199Q32—62no—
200SD32—22no—
201SD32—22no—
202SD32—32no—
203SD32—62no—
204SD32—62no—
205D16—31no—
206D16—41no—
207D16—62no—
208M32—32no—
209M32—62no—
210C2×C16—62no—
211C2×C16C22×D311yescharacteristic
212C2×C16C22×D311yescharacteristic
213#108—31no—
214#108—32no—
215#104—32no—
216#104—62no—
217#101—62no—
218#100—62no—
219C4×C8C22×D311yescharacteristic
220C23×C3—42no—
221C22×D3—82no—
222C3×Q8—42no—
223C3×Q8—42no—
224C3×D4—42no—
225C3×D4—42no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4C22×D411yescharacteristic
229#67—22no—
230#67—42no—
231#67—82no—
232#67—82no—
233C2×Dic3—82no—
234D12—22no—
235D12—82no—
236D12C2×D812yes—
237C4×D3—22no—
238C4×D3—42no—
239C4×D3—82no—
240C4×D3—82no—
241Dic6—22no—
242Dic6—82no—
243Dic6C2×D812yes—
244C3×C8C2×D811yescharacteristic
245C3×C8C2×D811yescharacteristic
246C3×C8C22×D411yescharacteristic
247C3×C8H(5,𝔽2)11yescharacteristic
248C3⋊C8—22no—
249C3⋊C8—42no—
250#41—32no—
251#41—62no—
252#41—122no—
253#41—122no—
254#41—122no—
255#41—122no—
256#41—242no—
257C2×Q8—22no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×D4—22no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—242no—
264Q16—22no—
265Q16—22no—
266Q16—31no—
267Q16—32no—
268Q16—122no—
269Q16—122no—
270SD16—32no—
271SD16—42no—
272SD16—42no—
273SD16—62no—
274SD16—122no—
275SD16—122no—
276SD16—122no—
277SD16—122no—
278D8—22no—
279D8—22no—
280D8—31no—
281D8—32no—
282D8—122no—
283D8—122no—
284M16—32no—
285M16—62no—
286M16—62no—
287M16—122no—
288C2×C8—62no—
289C2×C8—122no—
290C2×C8D3×D411yescharacteristic
291C2×C8C23×D311yescharacteristic
292C22⋊C4—82no—
293C42D3×D411yescharacteristic
294C16—21no—
295C16—62no—
296C16D3×D411yescharacteristic
297C16D3×D411yescharacteristic
298C22×C3—82no—
299C22×C3—162no—
300C22×C3#57611yescharacteristic
301C2×D3—22no—
302C2×D3—42no—
303C2×D3—322no—
304C3×C4—41no—
305C3×C4—82no—
306C3×C4D4211yescharacteristic
307C3×C4C22×D811yescharacteristic
308Dic3—22no—
309Dic3—42no—
310C23—42no—
311C23—242no—
312Q8—32no—
313Q8—42no—
314Q8—42no—
315Q8—62no—
316Q8—242no—
317D4—32no—
318D4—42no—
319D4—42no—
320D4—62no—
321D4—62no—
322D4—122no—
323D4—242no—
324D4—242no—
325D4—242no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—62no—
329C2×C4—122no—
330C2×C4—242no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4C2×D3×D411yescharacteristic
334C8—62no—
335C8—122no—
336C8D3×D811yescharacteristic
337C8D3×D811yescharacteristic
338C8C2×D3×D411yescharacteristic
339C8#100411yescharacteristic
340C2×C3—21no—
341C2×C3—162no—
342C2×C3D4×D811yescharacteristic
343D3—42no—
344D3—82no—
345C22—62no—
346C22—82no—
347C22—122no—
348C22—162no—
349C22—962no—
350C22#575711yescharacteristic
351C4—41no—
352C4—62no—
353C4—82no—
354C4—122no—
355C4#571711yescharacteristic
356C4C2×D3×D811yescharacteristic
357C3#3385711yescharacteristic, complemented, Sylow
358C2—21no—
359C2—122no—
360C2—162no—
361C2—242no—
362C2#8178911yescharacteristic
subgroupG/H
centerC2#81789
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2249C2
socleC2×C3D4×D8
hypercenter#2249D3
hypocenterC3#33857
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. #82024C2semidirect
2. #81835C2semidirect
3. C3×#2249C2semidirect
4. #71506C2semidirect
5. #71491C2semidirect
6. #71471C2semidirect
7. #71325C2semidirect
8. #69067C2semidirect
9. #69056C2semidirect
13. #5757C22semidirect
14. C2×C3×Q32C22semidirect
15. C2×C3×SD32C22semidirect
17. C3×#495C22semidirect
18. #5192C22semidirect
19. #5179C22semidirect
30. #4901C22semidirect
32. #4700C22semidirect
33. C3×C4×C16C22semidirect
34. #4565C22semidirect
35. #4563C22semidirect
36. #4510C22semidirect
37. #4508C22semidirect
38. #4506C22semidirect
39. #4463C22semidirect
42. #2249D3semidirect
105. C2×Q32C2×D3semidirect
107. C2×SD32C2×D3semidirect
110. #495C2×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
#820242
#818352
C3×#22492
#715062
#714912
#714712
#713252
#690672
#690562
#338573