Finite Group Databasebeta

← #10801895

Subgroups — #10801895

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×C3×#441C211yesmaximal, characteristic, complemented
2C2×#5189C211yesmaximal, characteristic
3C4×#946C212yesmaximal
4C2×#4707C211yesmaximal, characteristic
5#72066C211yesmaximal, characteristic, complemented
6#71631C212yesmaximal, complemented
7#70578C211yesmaximal, characteristic
8#70559C211yesmaximal, characteristic, complemented
9#70271C212yesmaximal
10#70254C212yesmaximal, complemented
11#69814C211yesmaximal, characteristic, complemented
12#21539—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#946C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×C42×D3C2211yescharacteristic
19C3×#441—24no—
20C2×C3×#106C2211yescharacteristic
21C2×C3×#103C2212yes—
22C2×C3×#97C2211yescharacteristic
23#5229C2212yes—
24C2×#835C2212yes—
25C2×#835C2212yes—
26#5189C418yescomplemented
27C2×#818C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4989—22no—
32#4987C2212yes—
33#4978—22no—
34#4972C2212yes—
35C4×(C2×D3)⋊C4—22no—
36C4×C4⋊Dic3C2212yes—
37C4×Dic3⋊C4—22no—
38C2×#803C2211yescharacteristic
39#4707—24no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#470—31no—
44C2×#441D311yescharacteristic, complemented
45C2×C4×M16—31no—
46C4×C22⋊Q8—32no—
47#1969—31no—
48#1941—31no—
49#1933—32no—
50#1894—32no—
51#1809—31no—
52#1776—31no—
53#1763—32no—
54C22×C3×Q8C2312yes—
55C22×C4×D3D412yes—
56C22×C4×D3C2311yescharacteristic
57C2×C3×Q16—22no—
58C2×C3×Q16—22no—
59C2×C3×Q16C2×C414yescomplemented
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×C4×Q8—24no—
63C3×C4×Q8—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C42—22no—
67C2×C3×C42C2311yescharacteristic
68#946—24no—
69#946C2×C418yes—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3D412yes—
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×ΓL(1,𝔽25)—22no—
82C2×ΓL(1,𝔽25)—22no—
83C2×ΓL(1,𝔽25)—22no—
84C2×ΓL(1,𝔽25)—22no—
85C2×ΓL(1,𝔽25)—22no—
86C2×ΓL(1,𝔽25)C2×C414yes—
87C42×D3—22no—
88C42×D3—22no—
89C3×#106—22no—
90C3×#103—24no—
91C3×#97D414yes—
92#835—24no—
93#835C2×C418yescomplemented
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#818—22no—
100#817C2×C414yescomplemented
101C8×Dic3—22no—
102C8×Dic3—22no—
103#803—22no—
104C22×Q16C2×D311yescharacteristic
105C22×M16—31no—
106C22×M16—31no—
107C2×C22⋊Q8—32no—
108C2×C4×Q8C2×D312yes—
109C22×C42—31no—
110#470—38no—
111#441—24no—
112C2×#107—31no—
113C2×#106—31no—
114C2×#106C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103C2×D312yes—
118C4×M16—64no—
119C2×#97—31no—
120C2×#97C2×D311yescharacteristic
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#391—32no—
124#389—32no—
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#340—32no—
134C23×D3C2×D411yescharacteristic
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C22×C414yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D412yes—
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—24no—
154C2×C4×D3—24no—
155C2×C4×D3—42no—
156C2×C4×D3C22×C412yes—
157C2×C4×D3#4112yes—
158C3×Q16—28no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8C22×C412yes—
162C2×C3×C8C2×D412yes—
163C2×C3×C8C2×D412yes—
164C2×C3×C8#4112yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—44no—
168C3×C42—44no—
169C3×C42C2×D412yes—
170C3×C42#4112yes—
171(C2×D3)⋊C4—28no—
172(C2×D3)⋊C4—44no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3C22×C414yes—
177Dic3⋊C4—28no—
178Dic3⋊C4—44no—
179C4×Dic3—24no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—44no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8C22×C412yes—
188C2×C3⋊C8#4112yes—
189ΓL(1,𝔽25)—28no—
190ΓL(1,𝔽25)—28no—
191C22×Q8C22×D312yes—
192C23×C4—31no—
193C23×C4—32no—
194C2×Q16—22no—
195C2×Q16—22no—
196C2×Q16C4×D314yescomplemented
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—24no—
210C4×Q8—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C42—22no—
220C2×C42—32no—
221C2×C42—31no—
222C2×C42—62no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42C22×D311yescharacteristic
226#107—62no—
227#106—22no—
228#106—34no—
229#103—24no—
230#103—38no—
231#103—64no—
232#97—62no—
233#97#6714yes—
234C4×C8—62no—
235C4×C8—62no—
236#95—62no—
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—
247C3×Q8—24no—
248C3×Q8—24no—
249C3×Q8—42no—
250C3×Q8—42no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4#12114yes—
260C2×C3×C4C23×C411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×Dic3—22no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—22no—
270C2×Dic3—24no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3C4×D412yes—
277C2×Dic3#13714yes—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—48no—
283C3×C8—22no—
284C3×C8C4×D414yes—
285C3⋊C8—22no—
286C3⋊C8—22no—
287C24—31no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8C2×C4×D314yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—32no—
297C22×C4—31no—
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—28no—
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×C8#23512yes—
324C2×C8C2×#6712yes—
325C2×C8C2×#6712yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—34no—
329C4⋊C4—44no—
330C4⋊C4—68no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4—124no—
335C22⋊C4—68no—
336C22⋊C4—124no—
337C42—44no—
338C42—64no—
339C42—61no—
340C42—61no—
341C42—124no—
342C42#23712yes—
343C42C2×#6712yes—
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—62no—
367C23—62no—
368C23—61no—
369C23—61no—
370C23C2×D3×D411yescharacteristic
371Q8—24no—
372Q8—24no—
373Q8—42no—
374Q8—42no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—34no—
379C2×C4—32no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—61no—
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#93814yes—
401C2×C4C22×C4×D311yescharacteristic
402C2×C4C2×#23511yescharacteristic
403C2×C4C2×#23711yescharacteristic
404C2×C4D3×#4111yescharacteristic
405C2×C4D3×#4111yescharacteristic
406C2×C4C22×#6711yescharacteristic
407C8—22no—
408C8—62no—
409C8—62no—
410C8C4×#6714yes—
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
423C22#554811yescharacteristic
424C22D3×#13712yes—
425C22#577712yes—
426C22C2×#93811yescharacteristic
427C4—24no—
428C4—44no—
429C4—64no—
430C4—62no—
431C4—82no—
432C4—124no—
433C4—242no—
434C4C4×#23711yescharacteristic
435C4#555711yescharacteristic
436C4C2×C4×#6711yescharacteristic
437C4#582811yescharacteristic
438C3#2153911yescharacteristic, complemented, Sylow
439C2—122no—
440C2#7982611yescharacteristic
441C2#7982912yes—
442C2#7983811yescharacteristic
443C2C4×#93811yescharacteristic
444C2#8398512yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenterC2×#441D3
hypocenterC3#21539
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
1. C2×C3×#441C2semidirect
5. #72066C2semidirect
6. #71631C2semidirect
8. #70559C2semidirect
10. #70254C2semidirect
11. #69814C2semidirect
26. #5189C4semidirect
44. C2×#441D3semidirect
59. C2×C3×Q16C2×C4semidirect
93. #835C2×C4semidirect
100. #817C2×C4semidirect
196. C2×Q16C4×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×C3×#4412
C2×#51892
C4×#9462
C2×#47072
#720662
#716312
#705782
#705592
#702712
#702542
#698142
#215393