Finite Group Databasebeta

← #10948647

Subgroups — #10948647

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3414C211yesmaximal, characteristic
2#83910C211yesmaximal, characteristic, complemented
3#83899C211yesmaximal, characteristic, complemented
4#72081C211yesmaximal, characteristic, complemented
5C2×#4566C211yesmaximal, characteristic, complemented
6#71521C214yesmaximal, complemented
7#71515C214yesmaximal, complemented
8C2×#4513C211yesmaximal, characteristic, complemented
9#71425C211yesmaximal, characteristic, complemented
10#33469—31nomaximal, Sylow
11C2×C3×#135C2211yescharacteristic
12C2×C3×#135C2211yescharacteristic
13#5827—21no—
14#5819—21no—
15C3×#510C412yescomplemented
16C3×#510C2212yes—
17C3×#509C412yescomplemented
18C3×#509C2212yes—
19C3×#508C414yescomplemented
20C2×C3×M32C2211yescharacteristic
21#5236—21no—
22#5232—21no—
23#5197C2214yes—
24#5196—22no—
25#5184C2212yescomplemented
26Dic3×SD16—22no—
27#5152C2212yescomplemented
28C2×#819C2211yescharacteristic
29C2×#817C2211yescharacteristic
30#5101C2211yescharacteristic
31C2×#812C2211yescharacteristic
32#4566C2214yes—
33#4565C2212yescomplemented
34#4563C2212yescomplemented
35#4513C2214yescomplemented
36#4512C2214yescomplemented
37#3414D311yescharacteristic
38#2957—31no—
39#2952—31no—
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42C2×#365—31no—
43#2148—31no—
44C2×#360—31no—
45#2110—31no—
46C2×C3×#41—21no—
47C2×C3×#41—21no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135—21no—
51C3×#135—21no—
52C3×#135C2×C412yes—
53C3×#135C2×C412yes—
54C3×#135C2×C412yescomplemented
55C3×#135C2312yes—
56C2×C3×Q16C2×C411yescharacteristic, complemented
57C2×C3×Q16C2311yescharacteristic
58C2×C3×SD16—21no—
59C2×C3×SD16—21no—
60C2×C3×D8C2×C411yescharacteristic, complemented
61C2×C3×D8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63#948—24no—
64#948—42no—
65Q8×Dic3—22no—
66D4×Dic3—22no—
67D4×Dic3—42no—
68#926—41no—
69#926D411yescharacteristic
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—41no—
72C2×#208—21no—
73C2×#208—42no—
74C2×#208D411yescharacteristic
75C3×Q32C2×C414yescomplemented
76C3×SD32C2×C414yescomplemented
77C3×SD32C2×C414yescomplemented
78C3×D16C2×C414yescomplemented
79C3×M32C2×C412yescomplemented
80C3×M32C2312yes—
81C2×C3×C16C2×C412yescomplemented
82#837—24no—
83#835—22no—
84#832—22no—
85#819—21no—
86#819C2312yes—
87#817—21no—
88#817C2312yes—
89#815C2312yes—
90C8×Dic3D412yes—
91#812—21no—
92#812C2312yes—
93C2×C3⋊C16—21no—
94C2×#135C2×D311yescharacteristic
95C2×#135C2×D311yescharacteristic
96#568—61no—
97#520—61no—
98#510Dic312yescomplemented
99#510C2×D312yes—
100#509Dic312yescomplemented
101#509C2×D312yes—
102#508Dic314yescomplemented
103C2×M32—31no—
104C2×M32C2×D311yescharacteristic
105#444—34no—
106#443—62no—
107#442—32no—
108#441—32no—
109C4×SD16—62no—
110C2×#108—31no—
111C2×#106—31no—
112#421—61no—
113#417—61no—
114#405—31no—
115#365—34no—
116#362—32no—
117#361—32no—
118#360—34no—
119#347—34no—
120C3×#41—22no—
121C3×#41—22no—
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—41no—
125C3×#41—41no—
126C3×#41—41no—
127C3×#41—41no—
128C2×C3×Q8—21no—
129C2×C3×Q8—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4C2×D411yescharacteristic
137C22×Dic3—42no—
138C22×Dic3—41no—
139C3×Q16—21no—
140C3×Q16—21no—
141C3×Q16C22×C412yes—
142C3×Q16C22×C412yes—
143C3×SD16—22no—
144C3×SD16—22no—
145C3×SD16—22no—
146C3×SD16—22no—
147C3×D8—21no—
148C3×D8—21no—
149C3×D8C22×C412yes—
150C3×D8C22×C412yes—
151C2×C3×C8C22×C411yescharacteristic
152C2×C3×C8C22×C411yescharacteristic
153C2×C3×C8C2×D411yescharacteristic
154C2×C3×C8#4111yescharacteristic
155C2×C3×C8#4111yescharacteristic
156C2×C3×C8C2411yescharacteristic
157#217—21no—
158#217—42no—
159C4⋊Dic3—21no—
160C4⋊Dic3—22no—
161C4⋊Dic3—42no—
162C4⋊Dic3—41no—
163C4×Dic3—42no—
164C4×Dic3C2×D412yes—
165#208—21no—
166#208—21no—
167#208—22no—
168#208—44no—
169#208—41no—
170C2×C3⋊C8—44no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8C2×D412yes—
173C3×C16C22×C414yes—
174C3⋊C16—22no—
175C2×#41—21no—
176C2×#41—21no—
177#135—21no—
178#135—21no—
179#135—21no—
180#135—21no—
181#135C2×Dic312yes—
182#135C2×Dic312yes—
183#135C2×Dic312yescomplemented
184#135C22×D312yes—
185C2×Q16C2×Dic311yescharacteristic, complemented
186C2×Q16C22×D311yescharacteristic
187C2×SD16—21no—
188C2×SD16—21no—
189C2×D8C2×Dic311yescharacteristic, complemented
190C2×D8C22×D311yescharacteristic
191#131—64no—
192#131—122no—
193C2×M16—31no—
194C2×M16—61no—
195C2×M16—122no—
196C22×C8C22×D311yescharacteristic
197C4×Q8—62no—
198C4×D4—62no—
199C4×D4—122no—
200#117—31no—
201#117—121no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—121no—
204Q32C2×Dic314yescomplemented
205SD32C2×Dic314yescomplemented
206SD32C2×Dic314yescomplemented
207D16C2×Dic314yescomplemented
208M32—32no—
209M32—61no—
210M32C2×Dic312yescomplemented
211M32C22×D312yes—
212C2×C16—61no—
213C2×C16C2×Dic312yescomplemented
214#108—32no—
215#108—61no—
216#106—32no—
217#106—61no—
218#104—64no—
219#103—62no—
220#102—62no—
221#97—32no—
222C4×C8—32no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C3×Q8—22no—
227C3×Q8—22no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×D4—22no—
231C3×D4—22no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—41no—
237C3×D4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4C22×D411yescharacteristic
250C2×Dic3—22no—
251C2×Dic3—42no—
252C2×Dic3—162no—
253C2×Dic3—161no—
254C3×C8C4×D411yescharacteristic
255C3×C8C4×D411yescharacteristic
256C3×C8C23×C411yescharacteristic
257C3×C8C2×#4111yescharacteristic
258C3⋊C8—22no—
259C3⋊C8—42no—
260#41—22no—
261#41—22no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—122no—
277C22×C4—121no—
278C22×C4D3×D411yescharacteristic
279Q16—21no—
280Q16—21no—
281Q16C22×Dic312yes—
282Q16C22×Dic312yes—
283SD16—22no—
284SD16—22no—
285SD16—22no—
286SD16—22no—
287D8—21no—
288D8—21no—
289D8C22×Dic312yes—
290D8C22×Dic312yes—
291M16—61no—
292M16—61no—
293M16—62no—
294M16—124no—
295M16—121no—
296C2×C8—32no—
297C2×C8—124no—
298C2×C8—121no—
299C2×C8D3×D411yescharacteristic
300C2×C8#23711yescharacteristic
301C2×C8#23711yescharacteristic
302C2×C8C22×Dic311yescharacteristic
303C2×C8C22×Dic311yescharacteristic
304C2×C8C23×D311yescharacteristic
305C4⋊C4—61no—
306C4⋊C4—62no—
307C4⋊C4—122no—
308C4⋊C4—121no—
309C22⋊C4—61no—
310C22⋊C4—122no—
311C42—32no—
312C42—122no—
313C16—62no—
314C16C22×Dic314yes—
315C22×C3—41no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—161no—
321C22×C3#44211yescharacteristic
322C22×C3#44211yescharacteristic
323C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
324C3×C4—42no—
325C3×C4—42no—
326C3×C4#44211yescharacteristic
327C3×C4#44211yescharacteristic
328C3×C4C2×C4×D411yescharacteristic
329C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
330Dic3—42no—
331Dic3—82no—
332C23—41no—
333C23—41no—
334C23#91111yescharacteristic
335Q8—22no—
336Q8—22no—
337Q8—41no—
338Q8—41no—
339D4—22no—
340D4—22no—
341D4—42no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—62no—
354C2×C4—122no—
355C2×C4—482no—
356C2×C4—481no—
357C2×C4#91111yescharacteristic
358C2×C4#91111yescharacteristic
359C2×C4#91111yescharacteristic
360C2×C4D4×Dic311yescharacteristic
361C2×C4D4×Dic311yescharacteristic
362C2×C4C2×D3×D411yescharacteristic
363C8—62no—
364C8—122no—
365C8D4×Dic311yescharacteristic
366C8D4×Dic311yescharacteristic
367C8C2×#23711yescharacteristic
368C8C23×Dic311yescharacteristic
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—82no—
373C2×C3—82no—
374C2×C3C2×#44211yescharacteristic
375C22—41no—
376C22—42no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—161no—
381C22#515211yescharacteristic
382C22#515211yescharacteristic
383C22C2×#91111yescharacteristic
384C4—42no—
385C4—42no—
386C4—122no—
387C4—242no—
388C4#515211yescharacteristic
389C4#515211yescharacteristic
390C4C2×#91111yescharacteristic
391C4C2×D4×Dic311yescharacteristic
392C3#3346911yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×#515211yescharacteristic
subgroupG/H
centerC2C2×#5152
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
Fitting, prime radicalC3×#3414C2
socleC2×C3C2×#442
cosocleC2×C3×C8C24
hypercenter#3414D3
hypocenterC3#33469
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
2. #83910C2semidirect
3. #83899C2semidirect
4. #72081C2semidirect
5. C2×#4566C2semidirect
6. #71521C2semidirect
7. #71515C2semidirect
8. C2×#4513C2semidirect
9. #71425C2semidirect
15. C3×#510C4semidirect
17. C3×#509C4semidirect
19. C3×#508C4semidirect
25. #5184C22semidirect
27. #5152C22semidirect
33. #4565C22semidirect
34. #4563C22semidirect
35. #4513C22semidirect
36. #4512C22semidirect
54. C3×#135C2×C4semidirect
56. C2×C3×Q16C2×C4semidirect
60. C2×C3×D8C2×C4semidirect
75. C3×Q32C2×C4semidirect
76. C3×SD32C2×C4semidirect
77. C3×SD32C2×C4semidirect
78. C3×D16C2×C4semidirect
79. C3×M32C2×C4semidirect
81. C2×C3×C16C2×C4semidirect
98. #510Dic3semidirect
100. #509Dic3semidirect
102. #508Dic3semidirect
183. #135C2×Dic3semidirect
185. C2×Q16C2×Dic3semidirect
189. C2×D8C2×Dic3semidirect
204. Q32C2×Dic3semidirect
205. SD32C2×Dic3semidirect
206. SD32C2×Dic3semidirect
207. D16C2×Dic3semidirect
210. M32C2×Dic3semidirect
213. C2×C16C2×Dic3semidirect
392. C3#33469semidirect

Maximal subgroups

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

subgroupindex
C3×#34142
#839102
#838992
#720812
C2×#45662
#715212
#715152
C2×#45132
#714252
#334693