Finite Group Databasebeta

← #10944072

Subgroups — #10944072

Order 768.

1646 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C2×#5217C212yesmaximal, complemented
3C2×#5015C212yesmaximal, complemented
4#82286C211yesmaximal, characteristic, complemented
5#71613C214yesmaximal, complemented
6#71605C212yesmaximal, complemented
7#71586C212yesmaximal, complemented
8#71583C211yesmaximal, characteristic, complemented
9#32911—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C3×#518—28no—
12C2×C3×C4×D4—22no—
13C2×C3×C4×D4C2212yes—
14C2×C3×C4×D4C2212yes—
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C22⋊C4—22no—
18C22×#213C2214yes—
19C2×#930C2212yes—
20C2×#874C2211yescharacteristic, complemented
21#5217—28no—
22C2×#832C2214yes—
23C2×#830C2212yes—
24#5015—28no—
25#4991—22no—
26C2×#809C2214yescomplemented
27#4960—22no—
28#4939C2212yescomplemented
29#4936C2211yescharacteristic, complemented
30C2×C4⋊(C3⋊C8)C2212yescomplemented
31#4529C2214yescomplemented
32#4524C2212yescomplemented
33C2×#518D311yescharacteristic, complemented
34C2×#459—32no—
35C2×#447—32no—
36#2619—31no—
37#1879—34no—
38#1875—32no—
39#1804—32no—
40#1798—31no—
41C25×C3—21no—
42C22×C3×D4—24no—
43C22×C3×D4—24no—
44C22×C3×D4—21no—
45C22×C3×D4—22no—
46C22×C3×D4D414yescomplemented
47C22×C3×D4C2312yes—
48C23×C3×C4—22no—
49C23×C3×C4—22no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—22no—
52C22×D12—22no—
53C22×D12C2312yes—
54C3×C4×D4—216no—
55C3×C4×D4—24no—
56C3×C4×D4—24no—
57C3×C4×D4—24no—
58C3×C4×D4D418yescomplemented
59C3×#117—24no—
60C3×#117—24no—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—216no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4D412yes—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#213—28no—
72C2×#213—28no—
73C2×#213—28no—
74#930—24no—
75#930—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×C4⋊Dic3C2312yes—
79C22×C3⋊C8C2314yes—
80#874—24no—
81#832—28no—
82C3⋊#95—22no—
83#830—24no—
84#809—28no—
85C4⋊(C3⋊C8)—24no—
86C23×D4C2×D311yescharacteristic
87C22×D8—34no—
88C2×#127—31no—
89C2×#121—32no—
90#518—28no—
91C2×C4×D4—22no—
92C2×C4×D4C2×D312yes—
93C2×C4×D4C2×D312yes—
94C2×#117C2×D311yescharacteristic
95C2×#117C2×D311yescharacteristic
96C22×C22⋊C4—22no—
97#459—68no—
98#447—68no—
99C2×C4⋊C8—32no—
100C2×#102—34no—
101C2×#102—34no—
102C2×C22⋊C8—32no—
103#394—62no—
104#392—62no—
105#390—32no—
106#382—31no—
107#340—34no—
108#336—32no—
109C24×C3—24no—
110C24×C3—24no—
111C24×C3—22no—
112C24×C3—24no—
113C24×C3—84no—
114C24×C3C2×D411yescharacteristic
115C23×D3—22no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—216no—
119C2×C3×D4—216no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4C2×D414yes—
125C2×C3×D4C2×D414yes—
126C2×C3×D4#4114yes—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—28no—
131C22×C3×C4—28no—
132C22×C3×C4—84no—
133C22×C3×C4—84no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4112yes—
139C22×C3×C4C2411yescharacteristic
140C2×#67—44no—
141C2×#67—84no—
142C22×Dic3—22no—
143C2×D12—24no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—24no—
147C2×D12—44no—
148C2×D12—84no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4C2×D414yes—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—28no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—28no—
156C3×C42—24no—
157C3×C42—22no—
158C3×C42C2×D414yes—
159#213—416no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—84no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C2×C3⋊C8—24no—
165C2×C3⋊C8—24no—
166C2×C3⋊C8—24no—
167C25—21no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—32no—
173C22×D4—62no—
174C22×D4—62no—
175C22×D4D1214yescomplemented
176C22×D4C22×D312yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4C22×D311yescharacteristic
180C2×D8—68no—
181C2×D8—68no—
182C2×D8—68no—
183C22×C8—34no—
184#127—64no—
185#121—64no—
186#121—64no—
187C4×D4—216no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4#6718yescomplemented
192#117—24no—
193#117—24no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4#6712yes—
196C2×C4⋊C4C22×D312yes—
197C2×C22⋊C4—216no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4#6712yes—
204C2×C42—21no—
205C2×C42C22×D311yescharacteristic
206C2×C42C22×D311yescharacteristic
207C4⋊C8—64no—
208#102—68no—
209#102—68no—
210C22⋊C8—64no—
211#95—62no—
212C23×C3—216no—
213C23×C3—21no—
214C23×C3—24no—
215C23×C3—24no—
216C23×C3—21no—
217C23×C3—24no—
218C23×C3—24no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3—88no—
222C23×C3—88no—
223C23×C3—88no—
224C23×C3#12112yes—
225C23×C3C22×D411yescharacteristic
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3—44no—
230C22×D3—44no—
231C3×D4—216no—
232C3×D4—24no—
233C3×D4—24no—
234C3×D4—24no—
235C3×D4#12118yes—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4—88no—
246C2×C3×C4—88no—
247C2×C3×C4#12014yes—
248C2×C3×C4#12112yes—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C22×D412yes—
251C2×C3×C4C2×#4112yes—
252C2×C3×C4C2×#4111yescharacteristic
253#67—88no—
254C2×Dic3—42no—
255C2×Dic3—44no—
256D12—44no—
257D12—44no—
258D12—88no—
259C3⋊C8—44no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—24no—
264C24—62no—
265C24—84no—
266C24C2×#6711yescharacteristic
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—216no—
270C2×D4—216no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—64no—
278C2×D4—64no—
279C2×D4—124no—
280C2×D4—124no—
281C2×D4—244no—
282C2×D4—244no—
283C2×D4C2×D1214yes—
284C2×D4#23514yes—
285C2×D4C2×#6714yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—62no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×D1212yes—
295C22×C4#23512yes—
296C22×C4C2×#6712yes—
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6712yes—
299C22×C4C23×D311yescharacteristic
300D8—1216no—
301C2×C8—64no—
302C2×C8—64no—
303C2×C8—64no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4C2×#6714yes—
309C22⋊C4—24no—
310C22⋊C4—28no—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—124no—
314C22⋊C4—244no—
315C42—24no—
316C42—22no—
317C42C2×#6714yes—
318C22×C3—22no—
319C22×C3—28no—
320C22×C3—24no—
321C22×C3—42no—
322C22×C3—48no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—48no—
327C22×C3—816no—
328C22×C3C2×#12011yescharacteristic
329C22×C3C2×#12111yescharacteristic
330C22×C3#52511yescharacteristic
331C22×C3#57614yes—
332C2×D3—82no—
333C2×D3—84no—
334C2×D3—84no—
335C2×D3—82no—
336C2×D3—82no—
337C3×C4—22no—
338C3×C4—24no—
339C3×C4—42no—
340C3×C4—44no—
341C3×C4C2×#12112yes—
342C3×C4#52512yes—
343Dic3—82no—
344C23—216no—
345C23—24no—
346C23—21no—
347C23—24no—
348C23—24no—
349C23—21no—
350C23—24no—
351C23—44no—
352C23—44no—
353C23—88no—
354C23—88no—
355C23—88no—
356C23—122no—
357C23—122no—
358C23—124no—
359C23—124no—
360C23—122no—
361C23#93012yes—
362C23C22×#6711yescharacteristic
363D4—216no—
364D4—24no—
365D4—24no—
366D4—24no—
367D4—124no—
368D4—124no—
369D4—248no—
370D4—248no—
371D4#93018yes—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—44no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—122no—
384C2×C4—124no—
385C2×C4#93012yes—
386C2×C4#95314yes—
387C2×C4C22×D1211yescharacteristic
388C2×C4C2×#23512yes—
389C2×C4C2×#23511yescharacteristic
390C2×C4C22×#6712yes—
391C8—124no—
392C2×C3—42no—
393C2×C3—48no—
394C2×C3#284711yescharacteristic
395C2×C3#300812yes—
396C2×C3#304112yes—
397C2×C3#312212yes—
398D3—162no—
399C22—22no—
400C22—28no—
401C22—24no—
402C22—48no—
403C22—42no—
404C22—48no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—816no—
409C22—242no—
410C22—242no—
411C22—242no—
412C22—244no—
413C22—244no—
414C22C2×#93011yescharacteristic
415C22#579111yescharacteristic
416C22#583514yes—
417C22C2×#95311yescharacteristic
418C4—22no—
419C4—24no—
420C4—42no—
421C4—44no—
422C4—242no—
423C4C2×#93012yes—
424C4#579112yes—
425C3#3291111yescharacteristic, complemented, Sylow
426C2—42no—
427C2—48no—
428C2—482no—
429C2#8254012yes—
430C2#8254212yes—
431C2#8403211yescharacteristic
432C2#8410112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#518C2
socleC23×C3C22×D4
hypercenterC2×#518D3
hypocenterC3#32911
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C2×#5217C2semidirect
3. C2×#5015C2semidirect
4. #82286C2semidirect
5. #71613C2semidirect
6. #71605C2semidirect
7. #71586C2semidirect
8. #71583C2semidirect
20. C2×#874C22semidirect
26. C2×#809C22semidirect
28. #4939C22semidirect
29. #4936C22semidirect
30. C2×C4⋊(C3⋊C8)C22semidirect
31. #4529C22semidirect
32. #4524C22semidirect
33. C2×#518D3semidirect
46. C22×C3×D4D4semidirect
58. C3×C4×D4D4semidirect
175. C22×D4D12semidirect
191. C4×D4#67semidirect
425. C3#32911semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×#52172
C2×#50152
#822862
#716132
#716052
#715862
#715832
#329113