Finite Group Databasebeta

← #10848591

Subgroups — #10848591

Order 768.

1448 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic, complemented
2C3×#3038C211yesmaximal, characteristic, complemented
3#83185C211yesmaximal, characteristic, complemented
4#83159C211yesmaximal, characteristic, complemented
5#72094C214yesmaximal, complemented
6C4×#837C211yesmaximal, characteristic, complemented
7#70521C211yesmaximal, characteristic, complemented
8#70514C211yesmaximal, characteristic, complemented
9#70323C214yesmaximal, complemented
10#25588—31nomaximal, Sylow
11C2×C3×AGL(1,ℤ/8)C2212yes—
12C2×C3×#127C2211yescharacteristic
13C3×C4×#41C2211yescharacteristic
14#5830C2212yes—
15C2×#940—24no—
16C2×#940C2214yes—
17#5585—22no—
18#5553—22no—
19#5550—24no—
20#5535—22no—
21#5521C2212yes—
22#5477—22no—
23C2×#874C2211yescharacteristic
24#5470C2211yescharacteristic
25C3×#460—22no—
26C3×#459—22no—
27C3×#423C2211yescharacteristic
28C3×#418C2212yes—
29C2×#837C2212yes—
30#5215C2214yes—
31C2×#823C2214yes—
32#5071—22no—
33#5070—22no—
34C42×Dic3C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4555C2214yes—
37#4532C2212yes—
38#3508—31no—
39#3138—31no—
40#3038D311yescharacteristic, complemented
41#2867—31no—
42#2002—34no—
43#1994—31no—
44#1965—31no—
45#1887—34no—
46C4×#104—31no—
47C2×C3×#41C2311yescharacteristic
48C22×C3×D4—22no—
49C22×C3×D4C2312yes—
50C22×#67—28no—
51C2×#235—24no—
52C22×D12D414yes—
53C22×D12C2312yes—
54C3×AGL(1,ℤ/8)—44no—
55C2×C3×SD16—22no—
56C2×C3×D8—22no—
57C2×C3×M16C2312yes—
58C3×#127—21no—
59C3×#127—21no—
60C3×#127—22no—
61C3×C4×Q8D412yescomplemented
62C3×C4×D4—22no—
63C3×C4×D4D412yescomplemented
64C3×#117—21no—
65C3×#117C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68#947—22no—
69#940—22no—
70#940—28no—
71#940—44no—
72#940—416no—
73#937—24no—
74#930—24no—
75#930—24no—
76#926C2312yes—
77#924—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3C2312yes—
80C2×#208C2312yes—
81#894—28no—
82#893—24no—
83#884—28no—
84D3×C22⋊C4—24no—
85#875—24no—
86#874—21no—
87#874—21no—
88#874—22no—
89C4×D12—24no—
90C4×D12—24no—
91#870D412yescomplemented
92#869D412yescomplemented
93C4×Dic6—22no—
94C3×C4⋊C8—22no—
95C3×#102—24no—
96#837—24no—
97#832—28no—
98#823—44no—
99#803—21no—
100C4×C3⋊C8—21no—
101C2×AGL(1,ℤ/8)C2×D312yes—
102#559—62no—
103#550—62no—
104#546—64no—
105#540—62no—
106#535—32no—
107#533—31no—
108#532—62no—
109C2×#127—34no—
110C2×#127—31no—
111C2×#127—64no—
112C2×#127C2×D311yescharacteristic
113#518—32no—
114C4×#41C2×D311yescharacteristic
115#490—62no—
116#486—62no—
117#460—22no—
118#459—22no—
119#423C2×D311yescharacteristic
120C2×#104—32no—
121#418—34no—
122#418C2×D312yes—
123C2×#100—34no—
124C4×M16—31no—
125C43—31no—
126#339—34no—
127#339—32no—
128C24×C3—22no—
129C23×D3C2×D414yes—
130C3×#41—22no—
131C3×#41—22no—
132C2×C3×Q8C2×D411yescharacteristic
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—48no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4C2×D411yescharacteristic
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—28no—
147C2×#67—416no—
148C2×#67—432no—
149C22×Dic3—22no—
150#235—88no—
151C2×D12—24no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—24no—
155C2×D12—24no—
156C2×D12—24no—
157C2×D12—48no—
158C2×C4×D3—28no—
159C2×Dic6—22no—
160C3×SD16—44no—
161C3×D8—44no—
162C3×M16—24no—
163C2×C3×C8—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4C2×D412yes—
166C3×C22⋊C4—21no—
167C3×C22⋊C4—22no—
168C3×C42—22no—
169C3×C42—21no—
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D412yes—
173#217—22no—
174(C2×D3)⋊C4—28no—
175(C2×D3)⋊C4—28no—
176C4⋊Dic3—22no—
177C4⋊Dic3C2×D414yes—
178Dic3⋊C4—24no—
179C4×Dic3—28no—
180C4×Dic3—22no—
181C4×Dic3—22no—
182C4×Dic3#4114yescomplemented
183#208—22no—
184#208—22no—
185C2×C3⋊C8—22no—
186C2×#41—64no—
187C2×#41C22×D311yescharacteristic
188C22×D4—22no—
189C22×D4—34no—
190C22×D4—32no—
191C22×D4—68no—
192C22×D4C22×D312yes—
193AGL(1,ℤ/8)—44no—
194C2×SD16—22no—
195C2×D8—22no—
196C2×M16—32no—
197C2×M16C22×D312yes—
198C4⋊Q8—32no—
199#127—21no—
200#127—21no—
201#127—22no—
202#127—68no—
203#127—62no—
204#127—61no—
205#127—61no—
206#127—62no—
207#127—124no—
208#127—1216no—
209#125—32no—
210#124—64no—
211#124—62no—
212#123—68no—
213C22⋊Q8—62no—
214#121—68no—
215#121—64no—
216#121—64no—
217#120—64no—
218C4×Q8—62no—
219C4×Q8#6712yescomplemented
220C4×D4—22no—
221C4×D4—64no—
222C4×D4—64no—
223C4×D4—64no—
224C4×D4#6712yescomplemented
225#117—21no—
226#117—32no—
227#117C22×D311yescharacteristic
228C2×C22⋊C4—64no—
229C2×C42—21no—
230C2×C42—32no—
231C2×C42—62no—
232C2×C42C22×D311yescharacteristic
233C4⋊C8—22no—
234#104—64no—
235#102—24no—
236#102—68no—
237#100—124no—
238#97—61no—
239C4×C8—61no—
240C23×C3—21no—
241C23×C3—48no—
242C23×C3—42no—
243C23×C3—44no—
244C23×C3C22×D411yescharacteristic
245C22×D3—24no—
246C22×D3—28no—
247C22×D3—48no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—44no—
256C3×D4—84no—
257C3×D4—84no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—44no—
263C2×C3×C4#12012yes—
264C2×C3×C4#12012yes—
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C2×#4112yes—
268C2×C3×C4H(5,𝔽2)12yes—
269#67—832no—
270#67—88no—
271C2×Dic3—28no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274D12—42no—
275D12—42no—
276D12—44no—
277D12—48no—
278D12—84no—
279C4×D3—88no—
280Dic6—44no—
281C3×C8—42no—
282C3⋊C8—42no—
283C24—22no—
284C24—34no—
285#41—22no—
286#41—22no—
287#41—248no—
288C2×Q8—62no—
289C2×Q8C2×#6711yescharacteristic
290C2×D4—21no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—48no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—68no—
299C2×D4—64no—
300C2×D4—64no—
301C2×D4—62no—
302C2×D4—64no—
303C2×D4—64no—
304C2×D4—62no—
305C2×D4—128no—
306C2×D4—1216no—
307C2×D4—1232no—
308C2×D4C2×#6711yescharacteristic
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—68no—
312C22×C4—62no—
313C22×C4D3×D412yes—
314C22×C4C23×D311yescharacteristic
315SD16—44no—
316D8—44no—
317M16—24no—
318M16—62no—
319M16—62no—
320C2×C8—22no—
321C2×C8—62no—
322C4⋊C4—22no—
323C4⋊C4—34no—
324C4⋊C4—62no—
325C4⋊C4—64no—
326C4⋊C4C2×#6712yes—
327C22⋊C4—21no—
328C22⋊C4—22no—
329C22⋊C4—68no—
330C22⋊C4—68no—
331C22⋊C4—62no—
332C42—22no—
333C42—21no—
334C42—34no—
335C42—68no—
336C42—62no—
337C42—62no—
338C42D3×D411yescharacteristic
339C42D3×D411yescharacteristic
340C42C2×#6712yes—
341C22×C3—22no—
342C22×C3—41no—
343C22×C3—42no—
344C22×C3—88no—
345C22×C3—82no—
346C22×C3—84no—
347C22×C3C2×#12011yescharacteristic
348C22×C3#54612yes—
349C2×D3—44no—
350C2×D3—48no—
351C2×D3—88no—
352C3×C4—22no—
353C3×C4—41no—
354C3×C4—42no—
355C3×C4#45314yes—
356C3×C4#53211yescharacteristic
357C3×C4#53511yescharacteristic
358C3×C4#55012yes—
359Dic3—28no—
360Dic3—44no—
361C23—21no—
362C23—48no—
363C23—42no—
364C23—44no—
365C23—64no—
366C23—68no—
367C23—128no—
368C23C2×D3×D411yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—124no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—44no—
378D4—84no—
379D4—84no—
380D4—128no—
381D4—122no—
382D4—122no—
383D4—124no—
384D4—244no—
385D4—2432no—
386D4—248no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—68no—
393C2×C4—64no—
394C2×C4—64no—
395C2×C4—248no—
396C2×C4#93712yes—
397C2×C4#93712yes—
398C2×C4C2×D3×D411yescharacteristic
399C2×C4#100412yes—
400C2×C4D3×#4112yes—
401C2×C4C22×#6711yescharacteristic
402C8—42no—
403C8—122no—
404C2×C3—22no—
405C2×C3—81no—
406C2×C3—162no—
407C2×C3#243111yescharacteristic
408C2×C3C2×#45312yes—
409D3—84no—
410C22—22no—
411C22—41no—
412C22—42no—
413C22—88no—
414C22—82no—
415C22—84no—
416C22—124no—
417C22—128no—
418C22—248no—
419C22#561212yes—
420C22C2×#93711yescharacteristic
421C4—22no—
422C4—41no—
423C4—42no—
424C4—68no—
425C4—124no—
426C4#519814yes—
427C4#559912yes—
428C4#580511yescharacteristic
429C4#583211yescharacteristic
430C3#2558811yescharacteristic, complemented, Sylow
431C2—22no—
432C2—81no—
433C2—162no—
434C2—244no—
435C2#8265211yescharacteristic
436C2C2×#519812yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3038C2
socleC22×C3C2×#120
hypercenter#3038D3
hypocenterC3#25588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #85848C2semidirect
2. C3×#3038C2semidirect
3. #83185C2semidirect
4. #83159C2semidirect
5. #72094C2semidirect
6. C4×#837C2semidirect
7. #70521C2semidirect
8. #70514C2semidirect
9. #70323C2semidirect
40. #3038D3semidirect
61. C3×C4×Q8D4semidirect
63. C3×C4×D4D4semidirect
91. #870D4semidirect
92. #869D4semidirect
182. C4×Dic3#41semidirect
219. C4×Q8#67semidirect
224. C4×D4#67semidirect
430. C3#25588semidirect

Maximal subgroups

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

subgroupindex
#858482
C3×#30382
#831852
#831592
#720942
C4×#8372
#705212
#705142
#703232
#255883