Finite Group Databasebeta

← #10848586

Subgroups — #10848586

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
1C3×#3508C211yesmaximal, characteristic, complemented
2#83336C211yesmaximal, characteristic, complemented
3#83185C211yesmaximal, characteristic, complemented
4#82526C211yesmaximal, characteristic, complemented
5#71856C214yesmaximal, complemented
6#71839C211yesmaximal, characteristic, complemented
7#71766C211yesmaximal, characteristic, complemented
8#71650C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic, complemented
10#25588—31nomaximal, Sylow
11C3×#559—22no—
12C3×#550—22no—
13C3×#546—24no—
14C3×#540—22no—
15C3×#535C2212yes—
16C3×#533C2211yescharacteristic
17C3×#532—22no—
18C2×C3×#127C2211yescharacteristic
19C3×#518C2212yes—
20C3×C4×#41C2211yescharacteristic
21C2×#949C2212yes—
22C2×#940—24no—
23C2×#940C2214yes—
24C2×#874C2211yescharacteristic
25C2×#837C2212yes—
26C2×#833C2214yes—
27#5088—22no—
28#5067—22no—
29#5027—22no—
30#5015—22no—
31#5001C2212yes—
32#5001C2214yes—
33C42×Dic3C2211yescharacteristic
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4532C2214yes—
37#4532C2212yes—
38#3508D311yescharacteristic, complemented
39#3138—31no—
40#3038—31no—
41#2867—31no—
42#2002—34no—
43#1994—31no—
44#1965—31no—
45#1887—34no—
46C4×#104—31no—
47C2×C3×#41—24no—
48C2×C3×#41C2311yescharacteristic
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C22×#67—28no—
52C22×D12—22no—
53C22×D12C2312yes—
54C3×C4⋊Q8D412yescomplemented
55C3×#127—22no—
56C3×#127—22no—
57C3×#127—21no—
58C3×#127—21no—
59C3×#125D412yescomplemented
60C3×#124—24no—
61C3×#124—22no—
62C3×#123—28no—
63C3×C22⋊Q8—22no—
64C3×#121—24no—
65C3×#121—28no—
66C3×#121—24no—
67C3×#120—24no—
68C3×C4×Q8—22no—
69C3×C4×Q8D412yescomplemented
70C3×C4×D4—24no—
71C3×C4×D4—22no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4D412yescomplemented
75C3×#117—21no—
76C3×#117C2311yescharacteristic
77C3×#117C2312yes—
78C2×C3×C22⋊C4—24no—
79C2×C3×C42—21no—
80C2×C3×C42C2311yescharacteristic
81#949—44no—
82C2×#215—22no—
83#940—28no—
84#940—44no—
85#940—416no—
86C2×#213—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3C2312yes—
89C2×#208C2312yes—
90C2×#208C2312yes—
91#874—22no—
92#874—21no—
93#874—21no—
94#837—24no—
95#833—44no—
96#809—24no—
97#809—28no—
98C4⋊(C3⋊C8)—22no—
99#803—21no—
100C4×C3⋊C8—21no—
101C2×AGL(1,ℤ/8)—32no—
102#559—22no—
103#550—22no—
104#546—24no—
105#540—22no—
106#535C2×D312yes—
107#533C2×D311yescharacteristic
108#532—22no—
109C2×#127—34no—
110C2×#127—31no—
111C2×#127—64no—
112C2×#127C2×D311yescharacteristic
113#518C2×D312yes—
114C4×#41C2×D311yescharacteristic
115#490—62no—
116#486—62no—
117#460—62no—
118#459—62no—
119#423—31no—
120C2×#104—32no—
121#418—34no—
122#418—32no—
123C2×#100—34no—
124C4×M16—31no—
125C43—31no—
126#339—32no—
127#339—34no—
128C24×C3C2×D414yes—
129C23×D3—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—88no—
133C2×C3×Q8—22no—
134C2×C3×Q8C2×D411yescharacteristic
135C2×C3×D4—21no—
136C2×C3×D4—28no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4—48no—
144C2×C3×D4C2×D411yescharacteristic
145C22×C3×C4—28no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2411yescharacteristic
150C2×#67—416no—
151C2×#67—432no—
152C22×Dic3—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—24no—
156C2×D12—24no—
157C2×D12—48no—
158C2×D12—42no—
159C2×D12—42no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4C2×D412yes—
164C3×C4⋊C4C2×D414yes—
165C3×C22⋊C4—28no—
166C3×C22⋊C4—28no—
167C3×C22⋊C4—21no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—22no—
170C3×C42—22no—
171C3×C42—21no—
172C3×C42—22no—
173C3×C42C2×D411yescharacteristic
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D412yes—
176#215—44no—
177#213—44no—
178C4×Dic3—28no—
179C4×Dic3—22no—
180C4×Dic3#4114yescomplemented
181#208—22no—
182#208—22no—
183#208—24no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×#41—24no—
187C2×#41C22×D311yescharacteristic
188C22×D4—32no—
189C22×D4—68no—
190C22×D4—62no—
191C22×D4#6714yes—
192C22×D4C22×D312yes—
193AGL(1,ℤ/8)—124no—
194C2×SD16—62no—
195C2×D8—62no—
196C2×M16—32no—
197C2×M16—32no—
198C4⋊Q8#6712yescomplemented
199#127—22no—
200#127—21no—
201#127—21no—
202#127—22no—
203#127—68no—
204#127—62no—
205#127—61no—
206#127—61no—
207#127—124no—
208#127—1216no—
209#125#6712yescomplemented
210#124—24no—
211#124—22no—
212#123—28no—
213C22⋊Q8—22no—
214#121—24no—
215#121—28no—
216#121—24no—
217#120—24no—
218C4×Q8—22no—
219C4×Q8#6712yescomplemented
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4#6712yescomplemented
225#117—21no—
226#117C22×D311yescharacteristic
227#117C22×D312yes—
228C2×C22⋊C4—24no—
229C2×C42—21no—
230C2×C42—32no—
231C2×C42—62no—
232C2×C42C22×D311yescharacteristic
233C4⋊C8—62no—
234#104—64no—
235#102—64no—
236#102—68no—
237#100—124no—
238#97—61no—
239C4×C8—61no—
240C23×C3—24no—
241C23×C3—21no—
242C23×C3—28no—
243C23×C3—48no—
244C23×C3C22×D411yescharacteristic
245C22×D3—48no—
246C22×D3—42no—
247C22×D3—44no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×Q8—44no—
251C3×D4—21no—
252C3×D4—21no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—44no—
257C3×D4—48no—
258C3×D4—84no—
259C3×D4—88no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—88no—
267C2×C3×C4#12012yes—
268C2×C3×C4#12012yes—
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4112yes—
272C2×C3×C4H(5,𝔽2)12yes—
273#67—832no—
274C2×Dic3—28no—
275C2×Dic3—24no—
276D12—42no—
277D12—42no—
278D12—44no—
279D12—84no—
280D12—84no—
281C3⋊C8—42no—
282C3⋊C8—42no—
283C24—62no—
284C24C2×#6714yes—
285#41—22no—
286#41—22no—
287#41—88no—
288C2×Q8—22no—
289C2×Q8C2×#6711yescharacteristic
290C2×D4—21no—
291C2×D4—28no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—48no—
299C2×D4—64no—
300C2×D4—64no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—128no—
304C2×D4—1216no—
305C2×D4—1232no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4C2×#6711yescharacteristic
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—62no—
313C22×C4C2×#6712yes—
314C22×C4C23×D311yescharacteristic
315SD16—124no—
316D8—124no—
317M16—62no—
318M16—62no—
319M16—64no—
320C2×C8—62no—
321C2×C8—62no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4C2×#6712yes—
326C4⋊C4C2×#6714yes—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—21no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C42—22no—
333C42—21no—
334C42—22no—
335C42—34no—
336C42—68no—
337C42—62no—
338C42C2×#6711yescharacteristic
339C42C2×#6711yescharacteristic
340C42C2×#6712yes—
341C22×C3—22no—
342C22×C3—41no—
343C22×C3—44no—
344C22×C3—42no—
345C22×C3—48no—
346C22×C3—88no—
347C22×C3C2×#12011yescharacteristic
348C22×C3#54612yes—
349C2×D3—88no—
350C2×D3—82no—
351C2×D3—84no—
352C3×C4—22no—
353C3×C4—41no—
354C3×C4—42no—
355C3×C4—44no—
356C3×C4#45314yes—
357C3×C4#53211yescharacteristic
358C3×C4#53511yescharacteristic
359C3×C4#55012yes—
360Dic3—28no—
361C23—24no—
362C23—21no—
363C23—28no—
364C23—48no—
365C23—128no—
366C23—124no—
367C23—122no—
368C23C22×#6711yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—44no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—48no—
376D4—42no—
377D4—42no—
378D4—44no—
379D4—84no—
380D4—88no—
381D4—122no—
382D4—122no—
383D4—124no—
384D4—244no—
385D4—2432no—
386D4—244no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—68no—
394C2×C4—64no—
395C2×C4—88no—
396C2×C4#95312yes—
397C2×C4#95312yes—
398C2×C4D3×#4112yes—
399C2×C4#100912yes—
400C2×C4C22×#6711yescharacteristic
401C2×C4C22×#6711yescharacteristic
402C8—122no—
403C8—122no—
404C2×C3—22no—
405C2×C3—81no—
406C2×C3—84no—
407C2×C3#243111yescharacteristic
408C2×C3C2×#45312yes—
409D3—162no—
410C22—22no—
411C22—41no—
412C22—44no—
413C22—42no—
414C22—48no—
415C22—88no—
416C22—248no—
417C22—242no—
418C22—244no—
419C22#582912yes—
420C22C2×#95311yescharacteristic
421C4—22no—
422C4—41no—
423C4—42no—
424C4—44no—
425C4—68no—
426C4#524114yes—
427C4#583011yescharacteristic
428C4#583211yescharacteristic
429C4#583412yes—
430C3#2558811yescharacteristic, complemented, Sylow
431C2—22no—
432C2—84no—
433C2—81no—
434C2—482no—
435C2#8409211yescharacteristic
436C2C2×#524112yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3508C2
socleC22×C3C2×#120
hypercenter#3508D3
hypocenterC3#25588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. C3×#3508C2semidirect
2. #83336C2semidirect
3. #83185C2semidirect
4. #82526C2semidirect
5. #71856C2semidirect
6. #71839C2semidirect
7. #71766C2semidirect
8. #71650C2semidirect
9. C4×#837C2semidirect
38. #3508D3semidirect
54. C3×C4⋊Q8D4semidirect
59. C3×#125D4semidirect
69. C3×C4×Q8D4semidirect
74. C3×C4×D4D4semidirect
180. C4×Dic3#41semidirect
198. C4⋊Q8#67semidirect
209. #125#67semidirect
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
C3×#35082
#833362
#831852
#825262
#718562
#718392
#717662
#716502
C4×#8372
#255883