Finite Group Databasebeta

← #10804549

Subgroups — #10804549

Order 768.

1354 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#4705C211yesmaximal, characteristic, complemented
3#74147C211yesmaximal, characteristic, complemented
4C3×#1960C211yesmaximal, characteristic, complemented
5#71786C211yesmaximal, characteristic, complemented
6#70267C214yesmaximal, complemented
7#70230C214yesmaximal, complemented
8#69823C211yesmaximal, characteristic, complemented
9#69553C211yesmaximal, characteristic, complemented
10#21812—31nomaximal, Sylow
11C22×#239C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×#947C2212yes—
14#5725—24no—
15#5725C418yescomplemented
16C2×#895—21no—
17C2×#895—21no—
18C2×#893—21no—
19C2×#893—21no—
20C2×#874C2211yescharacteristic
21C2×C42×D3C2211yescharacteristic
22C2×C3×#103C2214yes—
23C2×C3×#97C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#835C2214yes—
26C2×#821C2214yes—
27C2×(C2×D3)⋊C8C2212yes—
28C2×Dic3⋊C8C2212yes—
29#4987C2212yes—
30Dic3×C4⋊C4C2212yes—
31C2×#809C2214yes—
32#4936C2211yescharacteristic
33C2×#803C2211yescharacteristic
34#4705—24no—
35#4666—21no—
36#4664—21no—
37#4651—21no—
38#4648—21no—
39C2×#532—31no—
40C2×#432—31no—
41#2283—31no—
42#1983—34no—
43#1961—31no—
44#1961—31no—
45#1960—31no—
46#1960D311yescharacteristic, complemented
47#1788—34no—
48C22×C3×Q8C2312yes—
49C2×#239—24no—
50C2×#239—44no—
51C2×#239—44no—
52C2×#239C2×C418yes—
53C22×D12—22no—
54C22×D12—21no—
55C22×D12—21no—
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8C2×C414yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42C2311yescharacteristic
68#947—24no—
69#947C2×C418yes—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3—22no—
75C2×C4×Dic3D411yescharacteristic
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3C2311yescharacteristic
78C22×C3⋊C8C2312yes—
79#895—28no—
80#895—28no—
81#893—28no—
82#893—28no—
83#874—22no—
84#874C2×C414yes—
85C42×D3C2×C414yes—
86C42×D3D414yes—
87C3×#103—44no—
88C3×#97—22no—
89C3×#95—21no—
90C3×#95—21no—
91#835—44no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96#821—44no—
97(C2×D3)⋊C8—24no—
98Dic3⋊C8—24no—
99#809—44no—
100#803—22no—
101C22×#41—32no—
102#532—38no—
103#532—64no—
104C2×C4⋊Q8C2×D311yescharacteristic
105C2×#127—31no—
106C2×#124—32no—
107C2×#121—61no—
108C2×#121—61no—
109C2×C4×D4—61no—
110C2×C4×D4—61no—
111C22×C42—31no—
112#432—64no—
113C2×C4⋊C8—32no—
114C2×#103—34no—
115C2×#103C2×D314yes—
116C2×#102—34no—
117C2×#102—34no—
118C2×C22⋊C8—32no—
119C2×#97—31no—
120C2×#97C2×D311yescharacteristic
121#383—61no—
122#382—31no—
123#382C2×D311yescharacteristic
124#381—61no—
125#380—32no—
126C4×C4⋊C4—32no—
127C4×C22⋊C4—61no—
128#376—61no—
129C23×D3—22no—
130C23×D3C2×D411yescharacteristic
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8C22×C414yes—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D411yescharacteristic
143C22×Dic3C2×D411yescharacteristic
144#239—416no—
145#239—416no—
146C2×D12—24no—
147C2×D12—24no—
148C2×D12—28no—
149C2×D12—24no—
150C2×D12—24no—
151C2×D12—22no—
152C2×D12—44no—
153C2×D12—44no—
154C2×D12—44no—
155C2×D12—44no—
156C2×D12C22×C414yes—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—28no—
160C2×C4×D3—42no—
161C2×C4×D3—42no—
162C2×C4×D3—44no—
163C2×C4×D3—44no—
164C2×C4×D3C22⋊C414yes—
165C2×C4×D3C22⋊C414yes—
166C2×C4×D3C22×C412yes—
167C2×C4×D3C2×D412yes—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C42C22×C412yes—
175C3×C42C2×D412yes—
176(C2×D3)⋊C4—28no—
177(C2×D3)⋊C4—28no—
178(C2×D3)⋊C4—44no—
179C4⋊Dic3—24no—
180C4×Dic3—44no—
181C4×Dic3C22⋊C414yes—
182C4×Dic3C22⋊C414yes—
183C4×Dic3C22×C412yes—
184C4×Dic3C2×D412yes—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×#41—38no—
189C2×#41—64no—
190C2×#41—124no—
191C2×#41—124no—
192C22×Q8C22×D312yes—
193C22×D4—32no—
194C22×D4—62no—
195C22×D4—61no—
196C22×D4—61no—
197C23×C4—31no—
198C23×C4—31no—
199C23×C4—31no—
200C23×C4—62no—
201C22×C8—32no—
202C22×C8C22×D312yes—
203C4⋊Q8—22no—
204C4⋊Q8C4×D314yes—
205#127—34no—
206#127—62no—
207#124—38no—
208#124—64no—
209#121—68no—
210#121—68no—
211C4×D4—68no—
212C4×D4—68no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—32no—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—31no—
223C2×C42—34no—
224C2×C42—34no—
225C2×C42—62no—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—64no—
228#103—44no—
229#103—124no—
230#102—124no—
231#102—124no—
232C22⋊C8—64no—
233#97—22no—
234#97—62no—
235#95—21no—
236#95—21no—
237#95—62no—
238#95—62no—
239#95—61no—
240#95—61no—
241C23×C3C22×D411yescharacteristic
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—22no—
245C22×D3—22no—
246C22×D3—24no—
247C22×D3—42no—
248C22×D3—42no—
249C22×D3—44no—
250C22×D3—44no—
251C22×D3C2×C22⋊C412yes—
252C3×Q8—28no—
253C3×Q8—44no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4C2×C22⋊C412yes—
263C2×C3×C4C2×C22⋊C412yes—
264C2×C3×C4C23×C411yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4H(5,𝔽2)12yes—
267C2×C3×C4H(5,𝔽2)12yes—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3C2×C22⋊C412yes—
276C2×Dic3C2×C22⋊C412yes—
277C2×Dic3C2×C22⋊C412yes—
278D12—28no—
279D12—48no—
280D12—48no—
281D12—416no—
282D12—44no—
283C4×D3—24no—
284C4×D3—24no—
285C4×D3—44no—
286C4×D3—44no—
287C4×D3—416no—
288C3×C8—42no—
289C3⋊C8—42no—
290C24—31no—
291C24—62no—
292#41—1216no—
293#41—1216no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8C2×C4×D314yes—
298C2×D4—34no—
299C2×D4—68no—
300C2×D4—64no—
301C2×D4—64no—
302C2×D4—62no—
303C2×D4—64no—
304C2×D4—64no—
305C2×D4—124no—
306C2×D4—124no—
307C2×D4—124no—
308C2×D4—124no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—34no—
317C22×C4—34no—
318C22×C4—68no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—122no—
323C22×C4—122no—
324C22×C4—124no—
325C22×C4—124no—
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4C23×D311yescharacteristic
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—64no—
339C22⋊C4—68no—
340C22⋊C4—68no—
341C22⋊C4—124no—
342C42—34no—
343C42—34no—
344C42—32no—
345C42—32no—
346C42—124no—
347C42C2×C4×D312yes—
348C42D3×D412yes—
349C22×C3C22×C22⋊C411yescharacteristic
350C22×C3#53511yescharacteristic
351C22×C3#53511yescharacteristic
352C22×C3#57512yes—
353C22×C3#57512yes—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—44no—
360C2×D3—44no—
361C2×D3—44no—
362C2×D3—84no—
363C2×D3—84no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—44no—
367C3×C4—82no—
368C3×C4#51812yes—
369C3×C4#53512yes—
370Dic3—22no—
371Dic3—22no—
372Dic3—22no—
373Dic3—82no—
374C23—32no—
375C23—61no—
376C23—61no—
377C23—62no—
378C23—62no—
379C23—64no—
380C23—122no—
381C23—122no—
382C23—124no—
383C23—124no—
384C23C2×D3×D411yescharacteristic
385Q8—28no—
386Q8—44no—
387D4—68no—
388D4—128no—
389D4—128no—
390D4—1216no—
391D4—124no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—64no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4—1216no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4D3×C22⋊C412yes—
416C2×C4D3×C22⋊C412yes—
417C2×C4C22×C4×D311yescharacteristic
418C2×C4C2×D3×D411yescharacteristic
419C2×C4#100412yes—
420C2×C4#100912yes—
421C8—42no—
422C8—122no—
423C2×C3#230911yescharacteristic
424C2×C3#289611yescharacteristic
425C2×C3#289611yescharacteristic
426C2×C3#321912yes—
427C2×C3#321912yes—
428D3—42no—
429D3—84no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—122no—
434C22—122no—
435C22—124no—
436C22—124no—
437C22—124no—
438C22—244no—
439C22—244no—
440C22C2×D3×C22⋊C411yescharacteristic
441C22#567411yescharacteristic
442C22#571511yescharacteristic
443C22#577712yes—
444C22#577712yes—
445C4—22no—
446C4—22no—
447C4—44no—
448C4—62no—
449C4—62no—
450C4—62no—
451C4—82no—
452C4—242no—
453C4#554512yes—
454C4#560712yes—
455C3#2181211yescharacteristic, complemented, Sylow
456C2—122no—
457C2—244no—
458C2#7642811yescharacteristic
459C2#7895611yescharacteristic
460C2#7895611yescharacteristic
461C2#8174212yes—
462C2#8187712yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1960C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
cosocleC22×C3×C4C24
hypercenter#1960D3
hypocenterC3#21812
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
2. C2×#4705C2semidirect
3. #74147C2semidirect
4. C3×#1960C2semidirect
5. #71786C2semidirect
6. #70267C2semidirect
7. #70230C2semidirect
8. #69823C2semidirect
9. #69553C2semidirect
15. #5725C4semidirect
46. #1960D3semidirect
455. C3#21812semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#47052
#741472
C3×#19602
#717862
#702672
#702302
#698232
#695532
#218123