Finite Group Databasebeta

← #10849217

Subgroups — #10849217

Order 768.

1346 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5553C211yesmaximal, characteristic
2C2×#5008C211yesmaximal, characteristic, complemented
3C2×#4694C211yesmaximal, characteristic, complemented
4#73721C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70177C214yesmaximal, complemented
7#69788C211yesmaximal, characteristic, complemented
8#69487C211yesmaximal, characteristic, complemented
9#68895C214yesmaximal, complemented
10#25649—31nomaximal, Sylow
11C22×#235—21no—
12C2×C3×C4×D4C2211yescharacteristic
13C2×#940C2212yes—
14C2×#930—21no—
15#5553—24no—
16#5553—24no—
17C2×#893C2212yes—
18C2×#884—22no—
19C2×#874C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#102C2212yes—
23C2×C3×C4×C8C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#832C2212yes—
26C2×#821C2214yes—
27#5008—24no—
28#4984C2214yes—
29C2×#809C2214yes—
30C2×#808C2212yes—
31#4935—21no—
32#4929—21no—
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4694—24no—
35#4656—22no—
36C3×#340C2212yes—
37#4550C2214yes—
38C2×#550—31no—
39C2×#486—31no—
40C2×#476—31no—
41#2567—31no—
42#2041—34no—
43#1984—31no—
44#1982—31no—
45#1864—34no—
46C4×#102D311yescharacteristic, complemented
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—21no—
49C22×#67—22no—
50C22×#67—22no—
51C2×#235—28no—
52C2×#235—44no—
53C2×#235—44no—
54C22×D12—21no—
55C22×D12—21no—
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C22×Dic6—21no—
59C22×Dic6D411yescharacteristic
60C22×C3×C8C2312yes—
61C3×C4×D4—22no—
62C3×C4×D4D414yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—21no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68#940—24no—
69#940—24no—
70#930—28no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4D412yes—
77C2×C4×Dic3C2312yes—
78C22×C3⋊C8C2312yes—
79#893—24no—
80#893—24no—
81#884—216no—
82#874—22no—
83#874D414yes—
84#869—22no—
85#869—22no—
86C4×Dic6—22no—
87C4×Dic6—22no—
88C3×#102—24no—
89C3×C4×C8—22no—
90C3×#95—21no—
91#832—24no—
92C3⋊#95—24no—
93#821—28no—
94#809—28no—
95#808—24no—
96C4⋊(C3⋊C8)—22no—
97C22×#41—61no—
98#550—64no—
99#550—64no—
100C2×C4⋊Q8—31no—
101C2×#127—32no—
102C2×#127—31no—
103C2×#121—62no—
104C2×#121—61no—
105C2×C4×Q8—31no—
106C2×C4×D4—32no—
107C2×C4×D4C2×D311yescharacteristic
108#486—64no—
109#476—64no—
110C2×#106—32no—
111C2×C4⋊C8—31no—
112C2×#102—32no—
113C2×#102—34no—
114C2×#102—34no—
115C2×#102C2×D312yes—
116C2×C4×C8C2×D311yescharacteristic
117#395—62no—
118#391—61no—
119#389—34no—
120#384—61no—
121C4×C4⋊C4C2×D311yescharacteristic
122#340C2×D312yes—
123#336—34no—
124C24×C3—21no—
125C23×D3—22no—
126C2×C3×D4—21no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4C2×D412yes—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C2×#67—28no—
139C2×#67—28no—
140C2×#67—44no—
141C2×#67—44no—
142C2×#67—84no—
143C22×Dic3—22no—
144C22×Dic3C2×D412yes—
145C22×Dic3C2×D412yes—
146#235—432no—
147C2×D12—24no—
148C2×D12—24no—
149C2×D12—24no—
150C2×D12—22no—
151C2×D12—24no—
152C2×D12—42no—
153C2×D12—42no—
154C2×D12—82no—
155C2×D12C2×D414yes—
156C2×C4×D3—28no—
157C2×C4×D3—44no—
158C2×C4×D3—44no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—42no—
162C2×Dic6—82no—
163C2×Dic6SD1614yes—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4C2×D412yes—
170C3×C4⋊C4#4114yescomplemented
171C3×C22⋊C4—24no—
172C3×C42—42no—
173C3×C42C2×D412yes—
174C3×C42C2×D412yes—
175(C2×D3)⋊C4—28no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—42no—
180C4⋊Dic3—84no—
181Dic3⋊C4—24no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—24no—
186C2×#41—68no—
187C2×#41—124no—
188C2×#41—124no—
189C22×Q8—31no—
190C22×Q8—61no—
191C22×D4—32no—
192C22×D4—62no—
193C22×D4—62no—
194C22×D4—61no—
195C22×D4—61no—
196C22×D4C22×D311yescharacteristic
197C23×C4—21no—
198C23×C4—62no—
199C22×C8—32no—
200C22×C8C22×D312yes—
201C4⋊Q8—62no—
202C4⋊Q8—62no—
203#127—34no—
204#127—64no—
205#127—64no—
206#127—62no—
207#121—616no—
208#121—68no—
209C4×Q8—62no—
210C4×Q8—62no—
211C4×D4—22no—
212C4×D4—64no—
213C4×D4—64no—
214C4×D4D1214yes—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4C22×D311yescharacteristic
221C2×C4⋊C4C22×D312yes—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—62no—
224C2×C42—21no—
225C2×C42—32no—
226C2×C42C22×D311yescharacteristic
227#106—64no—
228C4⋊C8—62no—
229#102—24no—
230#102—68no—
231#102—64no—
232#102—68no—
233C4×C8—22no—
234#95—21no—
235#95—64no—
236C23×C3—22no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3C22×D411yescharacteristic
242C22×D3—24no—
243C22×D3—44no—
244C22×D3—42no—
245C22×D3—42no—
246C22×D3—44no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—42no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—48no—
258C2×C3×C4#12012yes—
259C2×C3×C4#12012yes—
260C2×C3×C4C2×SD1612yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4112yes—
265C2×C3×C42−1+412yes—
266#67—416no—
267#67—88no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—24no—
272C2×Dic3—42no—
273C2×Dic3—44no—
274C2×Dic3C2×SD1614yes—
275D12—28no—
276D12—42no—
277D12—42no—
278D12—48no—
279D12—84no—
280C4×D3—416no—
281Dic6—28no—
282Dic6—84no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—21no—
286C24—62no—
287#41—1232no—
288C2×Q8—34no—
289C2×Q8—62no—
290C2×Q8—62no—
291C2×Q8—122no—
292C2×Q8—242no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—34no—
297C2×D4—68no—
298C2×D4—68no—
299C2×D4—64no—
300C2×D4—64no—
301C2×D4—62no—
302C2×D4—64no—
303C2×D4—64no—
304C2×D4—124no—
305C2×D4—124no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—242no—
309C2×D4—244no—
310C2×D4C2×D1212yes—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—68no—
319C22×C4—62no—
320C22×C4—124no—
321C22×C4—124no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—62no—
329C2×C8—64no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—64no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—122no—
337C4⋊C4—244no—
338C4⋊C4C2×D1212yes—
339C4⋊C4#23514yescomplemented
340C22⋊C4—24no—
341C22⋊C4—68no—
342C42—42no—
343C42—62no—
344C42—62no—
345C42C2×D1212yes—
346C42D3×D412yes—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3C2×#12011yescharacteristic
353C22×C3#54812yes—
354C22×C3C22×SD1611yescharacteristic
355C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
356C22×C3#57612yes—
357C2×D3—44no—
358C2×D3—44no—
359C2×D3—44no—
360C2×D3—84no—
361C2×D3—84no—
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—42no—
365C3×C4#45014yes—
366C3×C4#53211yescharacteristic
367C3×C4#53611yescharacteristic
368C3×C4#54712yes—
369Dic3—24no—
370Dic3—42no—
371Dic3—82no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—64no—
378C23—122no—
379C23—122no—
380C23—124no—
381C23—124no—
382C23C2×D3×D411yescharacteristic
383Q8—68no—
384Q8—244no—
385D4—22no—
386D4—22no—
387D4—42no—
388D4—68no—
389D4—1216no—
390D4—122no—
391D4—122no—
392D4—128no—
393D4—244no—
394D4—248no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—34no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—1216no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4#88212yes—
412C2×C4#88212yes—
413C2×C4#91112yes—
414C2×C4D3×SD1612yes—
415C2×C4C22×D1211yescharacteristic
416C2×C4C2×#23512yes—
417C2×C4C2×D3×D411yescharacteristic
418C2×C4#101012yes—
419C8—42no—
420C8—122no—
421C2×C3—82no—
422C2×C3#243211yescharacteristic
423C2×C3C2×#45011yescharacteristic
424C2×C3#300811yescharacteristic
425C2×C3#336212yes—
426C2×C3#336612yes—
427D3—84no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—124no—
434C22—124no—
435C22—124no—
436C22—244no—
437C22—244no—
438C22C2×#88211yescharacteristic
439C22#555212yes—
440C22C2×#91111yescharacteristic
441C22C2×D3×SD1611yescharacteristic
442C22#576912yes—
443C4—22no—
444C4—42no—
445C4—44no—
446C4—64no—
447C4—122no—
448C4—242no—
449C4#477614yes—
450C4#547711yescharacteristic
451C4#552611yescharacteristic
452C4#555112yes—
453C3#2564911yescharacteristic, complemented, Sylow
454C2—82no—
455C2—244no—
456C2#7348911yescharacteristic
457C2C2×#477611yescharacteristic
458C2#7883411yescharacteristic
459C2#7957012yes—
460C2#7962212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC2×#5553C2
hypercenterC4×#102D3
hypocenterC3#25649
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5008C2semidirect
3. C2×#4694C2semidirect
4. #73721C2semidirect
5. C3×C4×#102C2semidirect
6. #70177C2semidirect
7. #69788C2semidirect
8. #69487C2semidirect
9. #68895C2semidirect
46. C4×#102D3semidirect
170. C3×C4⋊C4#41semidirect
339. C4⋊C4#235semidirect
453. C3#25649semidirect

Maximal subgroups

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

subgroupindex
C2×#55532
C2×#50082
C2×#46942
#737212
C3×C4×#1022
#701772
#697882
#694872
#688952
#256493