Finite Group Databasebeta

← #10801896

Subgroups — #10801896

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4867C211yesmaximal, characteristic
2C2×#4853C211yesmaximal, characteristic, complemented
3C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
4#75965C212yesmaximal
5C3×#1941C211yesmaximal, characteristic, complemented
6#70578C211yesmaximal, characteristic
7#70553C211yesmaximal, characteristic, complemented
8#69775C211yesmaximal, characteristic, complemented
9#69614C212yesmaximal, complemented
10#68884C212yesmaximal
11#68783C212yesmaximal, complemented
12#21539—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2211yescharacteristic
15C22×ΓL(1,𝔽25)C2211yescharacteristic
16C2×#897C2212yes—
17C2×#887C2212yes—
18C2×C42×D3C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×#389C2212yes—
23C2×#818C2211yescharacteristic
24C2×#816C2212yes—
25C2×#815C2211yescharacteristic
26C2×#815C2211yescharacteristic
27#4989—22no—
28#4987C2212yes—
29Dic3×C4⋊C4—22no—
30Dic3×C4⋊C4C2212yes—
31C2×#810C2212yes—
32C2×#810C2212yes—
33#4929C2212yes—
34C2×C4×C3⋊C8C2211yescharacteristic
35#4867C418yescomplemented
36#4853—24no—
37C4×ΓL(1,𝔽25)—24no—
38#4666—22no—
39#4654—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#470—31no—
44C2×#441—31no—
45C2×C4×M16—31no—
46C4×C22⋊Q8—32no—
47#1969—31no—
48#1941D311yescharacteristic, complemented
49#1933—32no—
50#1894—32no—
51#1809—31no—
52#1776—31no—
53#1763—32no—
54C22×C4×D3D412yes—
55C22×C4×D3C2311yescharacteristic
56C22×Dic6C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42C2311yescharacteristic
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67C2×Dic3⋊C4—22no—
68C2×C4×Dic3—22no—
69C2×C4×Dic3—22no—
70C2×C4×Dic3D412yes—
71C2×C4×Dic3C2311yescharacteristic
72C22×C3⋊C8C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74C2×Dic12—22no—
75C2×Dic12—22no—
76C2×Dic12C2×C414yescomplemented
77C2×ΓL(1,𝔽25)—22no—
78C2×ΓL(1,𝔽25)—22no—
79C2×ΓL(1,𝔽25)—22no—
80C2×ΓL(1,𝔽25)—22no—
81C2×ΓL(1,𝔽25)—22no—
82C2×ΓL(1,𝔽25)C2×C414yes—
83#897—24no—
84#897C2×C418yes—
85#887—24no—
86#887—24no—
87C42×D3—22no—
88C42×D3—22no—
89C3×#106—22no—
90C3×#106C2×C414yescomplemented
91C3×C4×C8—22no—
92C3×#95—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#818—22no—
98#816—24no—
99#815—22no—
100#815D414yes—
101#810—24no—
102#810C2×C418yescomplemented
103C4×C3⋊C8—22no—
104C22×Q16—31no—
105C22×M16—31no—
106C22×M16—31no—
107C2×C22⋊Q8—32no—
108C2×C4×Q8—32no—
109C22×C42—31no—
110#470—38no—
111#441—64no—
112C2×#107—31no—
113C2×#106C2×D311yescharacteristic
114C2×#106C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C4×M16—64no—
119C2×#97—31no—
120C2×#97—31no—
121C2×C4×C8—31no—
122C2×C4×C8C2×D311yescharacteristic
123#391—32no—
124#389C2×D312yes—
125#387—62no—
126#385—62no—
127#380—32no—
128C4×C4⋊C4—32no—
129C4×C4⋊C4—62no—
130C4×C22⋊C4—62no—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134C23×D3C2×D411yescharacteristic
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D412yes—
143C2×C4×D3—21no—
144C2×C4×D3—21no—
145C2×C4×D3—21no—
146C2×C4×D3—21no—
147C2×C4×D3—24no—
148C2×C4×D3—24no—
149C2×C4×D3—42no—
150C2×C4×D3C22×C412yes—
151C2×C4×D3#4112yes—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6—22no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6C22×C414yes—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8C22×C412yes—
161C2×C3×C8C2×D412yes—
162C2×C3×C8#4112yes—
163C2×C3×C8#4112yes—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4C22×C414yes—
169C3×C42—21no—
170C3×C42—21no—
171(C2×D3)⋊C4—28no—
172(C2×D3)⋊C4—44no—
173C4⋊Dic3—28no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176Dic3⋊C4—44no—
177C4×Dic3—24no—
178C4×Dic3—44no—
179C4×Dic3—44no—
180C4×Dic3C2×D412yes—
181C4×Dic3#4112yes—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8C22×C412yes—
187C2×C3⋊C8C2×D412yes—
188Dic12—28no—
189ΓL(1,𝔽25)—28no—
190ΓL(1,𝔽25)—28no—
191C22×Q8—32no—
192C23×C4—31no—
193C23×C4—32no—
194C2×Q16—34no—
195C2×Q16—62no—
196C2×Q16—62no—
197C2×M16—34no—
198C2×M16—62no—
199C2×M16—62no—
200C2×M16—62no—
201C2×M16—62no—
202C2×M16—62no—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8C22×D311yescharacteristic
206C22×C8C22×D311yescharacteristic
207C22⋊Q8—38no—
208C22⋊Q8—64no—
209C4×Q8—64no—
210C4×Q8—64no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4C22×D312yes—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C42—31no—
220C2×C42—32no—
221C2×C42—62no—
222C2×C42—62no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42C22×D311yescharacteristic
226#107—62no—
227#106—22no—
228#106C4×D314yescomplemented
229#103—38no—
230#103—64no—
231#103—64no—
232#97—34no—
233#97—62no—
234C4×C8—22no—
235C4×C8—62no—
236#95—22no—
237#95—62no—
238#95—62no—
239#95—62no—
240#95—62no—
241C23×C3C22×D411yescharacteristic
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—22no—
245C22×D3—22no—
246C22×D3C4×D412yes—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4#13714yes—
256C2×C3×C4C23×C411yescharacteristic
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—24no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3C4×D412yes—
273C2×Dic3#12114yes—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—48no—
279Dic6—24no—
280Dic6—24no—
281Dic6—42no—
282Dic6—42no—
283C3×C8—22no—
284C3×C8C4×D414yes—
285C3⋊C8—22no—
286C3⋊C8—22no—
287C24—31no—
288C2×Q8—34no—
289C2×Q8—62no—
290C2×Q8—62no—
291C2×Q8—62no—
292C2×Q8—62no—
293C2×Q8—62no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—31no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—64no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4—64no—
307C22×C4—62no—
308C22×C4—122no—
309C22×C4D3×D412yes—
310C22×C4C23×D311yescharacteristic
311Q16—68no—
312M16—68no—
313M16—68no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—32no—
317C2×C8—32no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8C2×C4×D312yes—
323C2×C8D3×D412yes—
324C2×C8#23912yes—
325C2×C8#23912yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—44no—
330C4⋊C4—68no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—124no—
334C4⋊C4C2×C4×D314yes—
335C22⋊C4—68no—
336C22⋊C4—124no—
337C42—21no—
338C42—21no—
339C42—32no—
340C42—32no—
341C42—64no—
342C42—124no—
343C42—124no—
344C22×C3C2×C4×D411yescharacteristic
345C22×C3C2×#12111yescharacteristic
346C22×C3#53211yescharacteristic
347C22×C3C2×#13712yes—
348C22×C3#57512yes—
349C2×D3—24no—
350C2×D3—22no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—42no—
354C3×C4—24no—
355C3×C4—44no—
356C3×C4—82no—
357C3×C4C2×C4×D411yescharacteristic
358C3×C4C4×#4111yescharacteristic
359C3×C4#52511yescharacteristic
360C3×C4#52911yescharacteristic
361Dic3—24no—
362Dic3—22no—
363Dic3—44no—
364Dic3—82no—
365C23—32no—
366C23—61no—
367C23—62no—
368C23—62no—
369C23—61no—
370C23C2×D3×D411yescharacteristic
371Q8—64no—
372Q8—64no—
373Q8—122no—
374Q8—122no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—32no—
379C2×C4—34no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—61no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—61no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—64no—
394C2×C4—128no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4#91514yes—
401C2×C4C22×C4×D311yescharacteristic
402C2×C4C2×D3×D411yescharacteristic
403C2×C4C2×#23911yescharacteristic
404C2×C4C2×#23911yescharacteristic
405C2×C4D3×#4111yescharacteristic
406C2×C4D3×#4111yescharacteristic
407C8—22no—
408C8—62no—
409C8—62no—
410C8#89314yes—
411C2×C3C4×#12111yescharacteristic
412C2×C3C4×#13712yes—
413C2×C3#305612yes—
414C2×C3#318311yescharacteristic
415C2×C3#318511yescharacteristic
416D3—42no—
417C22—64no—
418C22—62no—
419C22—122no—
420C22—121no—
421C22—121no—
422C22C4×D3×D411yescharacteristic
423C22D3×#12111yescharacteristic
424C22#572511yescharacteristic
425C22C2×#91512yes—
426C22#576812yes—
427C4—24no—
428C4—44no—
429C4—64no—
430C4—62no—
431C4—82no—
432C4—124no—
433C4—242no—
434C4C2×#89311yescharacteristic
435C4C4×#23911yescharacteristic
436C4#562711yescharacteristic
437C4#569111yescharacteristic
438C3#2153911yescharacteristic, complemented, Sylow
439C2—122no—
440C2#7596411yescharacteristic
441C2C4×#91512yes—
442C2#8065812yes—
443C2#8188911yescharacteristic
444C2#8191111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1941C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1941D3
hypocenterC3#21539
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. C2×#4853C2semidirect
5. C3×#1941C2semidirect
7. #70553C2semidirect
8. #69775C2semidirect
9. #69614C2semidirect
11. #68783C2semidirect
35. #4867C4semidirect
48. #1941D3semidirect
76. C2×Dic12C2×C4semidirect
90. C3×#106C2×C4semidirect
102. #810C2×C4semidirect
228. #106C4×D3semidirect
438. C3#21539semidirect

Maximal subgroups

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

subgroupindex
C2×#48672
C2×#48532
C2×C4×ΓL(1,𝔽25)2
#759652
C3×#19412
#705782
#705532
#697752
#696142
#688842
#687832
#215393