Finite Group Databasebeta

← #10784426

Subgroups — #10784426

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3151C211yesmaximal, characteristic, complemented
2#83295C212yesmaximal, complemented
3#83163C212yesmaximal, complemented
4#78535C211yesmaximal, characteristic, complemented
5#71279C212yesmaximal, complemented
6#71277C211yesmaximal, characteristic, complemented
7#71276C211yesmaximal, characteristic, complemented
8#71262C211yesmaximal, characteristic, complemented
9#71259C212yesmaximal, complemented
10#71254C211yesmaximal, characteristic, complemented
11#69849C211yesmaximal, characteristic, complemented
12#19745—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)—22no—
14C3×#532C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×#932—24no—
17#5477C2212yes—
18C3×#496—22no—
19C3×#493C2211yescharacteristic
20C3×#493C2211yescharacteristic
21C3×#492C2211yescharacteristic
22C3×#492C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24#5135—22no—
25#5088C2212yescomplemented
26#5080C2212yescomplemented
27#5076C2212yes—
28#5076C2212yes—
29#5074—24no—
30#5073C2212yes—
31#5073C2212yes—
32#5073C2212yescomplemented
33#5070C2212yescomplemented
34#5039—24no—
35#5014—24no—
36C4×#208C2212yes—
37#4711—22no—
38#4695C2211yescharacteristic, complemented
39#4694C2212yescomplemented
40#4680C2211yescharacteristic, complemented
41#4455C2211yescharacteristic, complemented
42#4454C2212yescomplemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3184—31no—
45#3184—32no—
46#3151—32no—
47#3151D311yescharacteristic, complemented
48#1721—32no—
49#1721—31no—
50#1715—31no—
51#1715—32no—
52#1712—31no—
53#1705—31no—
54#1455—31no—
55C2×C3×#41—21no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C2×#235—22no—
59C3×AGL(1,ℤ/8)—48no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C2×C3×D8—22no—
63C2×C3×D8—22no—
64C2×C3×M16D412yes—
65C3×C4⋊Q8C2311yescharacteristic
66C3×#127—22no—
67C3×#127—21no—
68C3×#127C2311yescharacteristic
69C3×#127C2311yescharacteristic
70C3×#127C2311yescharacteristic
71C3×#124—21no—
72C3×#121—22no—
73C3×C4×D4—22no—
74C2×C3×C42C2311yescharacteristic
75C2×#214—24no—
76C2×#214—24no—
77#932—416no—
78C2×#213—24no—
79C2×#213—24no—
80C2×#213—24no—
81#930—22no—
82#930—22no—
83C2×#208D414yes—
84C2×D24—22no—
85#875—22no—
86#874C2312yes—
87C4×D12—22no—
88C4×D12—22no—
89#869C2312yes—
90C3×#97D412yes—
91C3×C4×C8C2311yescharacteristic
92C3×C4×C8C2311yescharacteristic
93#832—24no—
94#821—24no—
95#817—22no—
96#809—24no—
97#808—24no—
98#803D414yes—
99C4×C3⋊C8C2312yes—
100C4×C3⋊C8C2312yes—
101C2×AGL(1,ℤ/8)—22no—
102C2×AGL(1,ℤ/8)—64no—
103#532—32no—
104#532C2×D311yescharacteristic
105C2×#127C2×D311yescharacteristic
106#498—32no—
107#498—31no—
108#496—22no—
109#496—64no—
110#493—32no—
111#493—32no—
112#493—32no—
113#493—31no—
114#493C2×D311yescharacteristic
115#493C2×D311yescharacteristic
116#492—32no—
117#492—32no—
118#492C2×D311yescharacteristic
119#492C2×D311yescharacteristic
120#486—32no—
121#486—32no—
122#486—32no—
123#469—64no—
124#469—62no—
125#442—64no—
126#442—62no—
127C4×M16—32no—
128C4×M16C2×D311yescharacteristic
129#322—32no—
130#322—31no—
131C82—31no—
132C24×C3—41no—
133C3×#41—44no—
134C2×C3×Q8—21no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—48no—
142C2×C3×D4—44no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—42no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C2×#67—42no—
150#235—48no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×C4×D3—42no—
155C2×Dic6—22no—
156C3×SD16—44no—
157C3×SD16—44no—
158C3×D8—44no—
159C3×D8—44no—
160C3×M16—24no—
161C2×C3×C8C2×D412yes—
162C2×C3×C8C2×D412yes—
163C3×C4⋊C4—22no—
164C3×C22⋊C4—42no—
165C3×C42C2×D412yes—
166C3×C42C2×D411yescharacteristic
167C3×C42C2411yescharacteristic
168#214—48no—
169#214—48no—
170#213—48no—
171#213—48no—
172#213—48no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177#208—28no—
178C2×C3⋊C8C2×D414yes—
179C2×C3⋊C8C2×D414yes—
180D24—44no—
181C2×#41—21no—
182C2×#41—62no—
183C22×D4—22no—
184C22×D4—21no—
185AGL(1,ℤ/8)—48no—
186AGL(1,ℤ/8)—1216no—
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16—64no—
190C2×SD16—64no—
191C2×D8—22no—
192C2×D8—22no—
193C2×D8—64no—
194C2×D8—64no—
195C2×D8—64no—
196C2×D8—62no—
197C2×M16—34no—
198C2×M16#6712yes—
199C4⋊Q8—32no—
200C4⋊Q8C22×D311yescharacteristic
201#127—22no—
202#127—21no—
203#127—32no—
204#127C22×D311yescharacteristic
205#127C22×D311yescharacteristic
206#127C22×D311yescharacteristic
207#124—21no—
208#124—62no—
209#121—22no—
210#121—62no—
211#121—62no—
212C4×D4—22no—
213C4×D4—62no—
214C4×D4—62no—
215C2×C42C22×D311yescharacteristic
216#106—64no—
217#106—62no—
218#102—64no—
219#102—64no—
220#102—64no—
221#97—34no—
222#97#6712yes—
223C4×C8—32no—
224C4×C8—32no—
225C4×C8C22×D311yescharacteristic
226C4×C8C22×D311yescharacteristic
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—84no—
231C23×C3—82no—
232C23×C3#12711yescharacteristic
233C22×D3—42no—
234C3×Q8—42no—
235C3×D4—44no—
236C3×D4—44no—
237C3×D4—44no—
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—84no—
242C3×D4—82no—
243C3×D4—82no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4#12711yescharacteristic
249C2×C3×C4#12712yes—
250C2×C3×C4C22×D412yes—
251C2×C3×C4C22×D411yescharacteristic
252#67—84no—
253C2×Dic3—42no—
254D12—44no—
255D12—44no—
256D12—44no—
257C4×D3—84no—
258Dic6—44no—
259C3×C8—22no—
260C3×C8AGL(1,ℤ/8)14yescomplemented
261C3⋊C8—24no—
262C3⋊C8AGL(1,ℤ/8)18yescomplemented
263C24—41no—
264#41—44no—
265#41—128no—
266C2×Q8—21no—
267C2×Q8—62no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—122no—
283C22×C4—41no—
284C22×C4—122no—
285C22×C4D3×D412yes—
286C22×C4C2×#6711yescharacteristic
287SD16—44no—
288SD16—44no—
289SD16—128no—
290SD16—128no—
291D8—44no—
292D8—44no—
293D8—128no—
294D8—128no—
295D8—128no—
296D8—124no—
297M16—24no—
298M16—68no—
299C2×C8—34no—
300C2×C8—34no—
301C2×C8C2×#6712yes—
302C2×C8C2×#6712yes—
303C4⋊C4—22no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C22⋊C4—42no—
307C22⋊C4—122no—
308C22⋊C4—122no—
309C42D3×D412yes—
310C42C2×#6711yescharacteristic
311C42C23×D311yescharacteristic
312C22×C3—22no—
313C22×C3—21no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—164no—
321C22×C3C2×#12711yescharacteristic
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—82no—
325C3×C4—22no—
326C3×C4—21no—
327C3×C4—81no—
328C3×C4C2×AGL(1,ℤ/8)14yes—
329C3×C4C2×AGL(1,ℤ/8)12yes—
330Dic3—82no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—82no—
335C23—84no—
336C23—122no—
337C23#94011yescharacteristic
338Q8—42no—
339Q8—124no—
340D4—44no—
341D4—44no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—82no—
347D4—84no—
348D4—82no—
349D4—124no—
350D4—124no—
351D4—124no—
352D4—244no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—82no—
357C2×C4—122no—
358C2×C4—244no—
359C2×C4#94011yescharacteristic
360C2×C4#94012yes—
361C2×C4C2×D3×D412yes—
362C2×C4C22×#6711yescharacteristic
363C8—22no—
364C8—38no—
365C8—64no—
366C8#93214yescomplemented
367C2×C3—41no—
368C2×C3—161no—
369C2×C3—161no—
370C2×C3—161no—
371C2×C3#315111yescharacteristic
372C2×C3#315112yes—
373D3—162no—
374C22—21no—
375C22—22no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—164no—
383C22—242no—
384C22—242no—
385C22—242no—
386C22C2×#94011yescharacteristic
387C4—21no—
388C4—22no—
389C4—81no—
390C4—242no—
391C4D3×AGL(1,ℤ/8)14yes—
392C4C2×#93212yes—
393C3#1974511yescharacteristic, complemented, Sylow
394C2—41no—
395C2—161no—
396C2—161no—
397C2—161no—
398C2—482no—
399C2#8316311yescharacteristic
400C2#8393712yes—
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3151C2
socleC22×C3C2×#127
hypercenter#3151D3
hypocenterC3#19745
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. C3×#3151C2semidirect
2. #83295C2semidirect
3. #83163C2semidirect
4. #78535C2semidirect
5. #71279C2semidirect
6. #71277C2semidirect
7. #71276C2semidirect
8. #71262C2semidirect
9. #71259C2semidirect
10. #71254C2semidirect
11. #69849C2semidirect
25. #5088C22semidirect
26. #5080C22semidirect
32. #5073C22semidirect
33. #5070C22semidirect
38. #4695C22semidirect
39. #4694C22semidirect
40. #4680C22semidirect
41. #4455C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
47. #3151D3semidirect
260. C3×C8AGL(1,ℤ/8)semidirect
262. C3⋊C8AGL(1,ℤ/8)semidirect
366. C8#932semidirect
393. C3#19745semidirect

Maximal subgroups

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

subgroupindex
C3×#31512
#832952
#831632
#785352
#712792
#712772
#712762
#712622
#712592
#712542
#698492
#197453