Finite Group Databasebeta

← #10950920

Subgroups — #10950920

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5188C211yesmaximal, characteristic, complemented
2D3×#466C211yesmaximal, characteristic
3C3×#2185C211yesmaximal, characteristic, complemented
4#72063C211yesmaximal, characteristic, complemented
5#71487C212yesmaximal, complemented
6#71485C212yesmaximal, complemented
7#71069C211yesmaximal, characteristic, complemented
8#69231C212yesmaximal, complemented
9#69228C212yesmaximal, complemented
10D3×C22⋊C16C211yesmaximal, characteristic, complemented
11#69096C211yesmaximal, characteristic, complemented
12#33753—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic, complemented
14C2×#946—21no—
15C2×D3×D8C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17D3×#121—21no—
18C2×C3×SD32C2212yescomplemented
19C3×#466C2211yescharacteristic
20C2×#835—21no—
21#5188—21no—
22#5188—21no—
23#5188C2212yes—
24#5188C2212yescomplemented
25C2×#828C2212yescomplemented
26#5157C2211yescharacteristic
27#5115C2211yescharacteristic
28C2×#818C2211yescharacteristic, complemented
29#5038—21no—
30C2×D3×C16C2212yescomplemented
31#4883C2211yescharacteristic
32D3×#107C2211yescharacteristic
33#4775—21no—
34D3×#102—21no—
35C3×#358C2212yescomplemented
36C3×C22⋊C16C2211yescharacteristic, complemented
37#4564C2212yescomplemented
38#4562C2212yescomplemented
39#4549C2211yescharacteristic, complemented
40#4509C2212yescomplemented
41(C2×D3)⋊C16C2212yes—
42#4505C2212yescomplemented
43#4490C2212yescomplemented
44C2×#467—31no—
45C2×#466—31no—
46#2235—32no—
47#2233—31no—
48#2233—31no—
49#2217—32no—
50#2217—32no—
51#2187—32no—
52#2185—31no—
53#2185D311yescharacteristic, complemented
54C2×C22⋊C16—31no—
55C22×C3×Q8—21no—
56C2×D3×D4—21no—
57C2×D3×D4—41no—
58C22×C4×D3D411yescharacteristic
59C2×C3×Q16—21no—
60C2×C3×Q16—42no—
61C2×C3×Q16C2312yes—
62C2×C3×D8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×#121—21no—
65#946—22no—
66#946—22no—
67#946—41no—
68#946—41no—
69#939—41no—
70#938—21no—
71C2×#213—21no—
72#930—21no—
73C2×(C2×D3)⋊C4—41no—
74C2×C4⋊Dic3—21no—
75C2×Dic3⋊C4—41no—
76C22×C3⋊C8D411yescharacteristic
77D3×D8—42no—
78C2×D24C2311yescharacteristic
79C2×D3×C8—42no—
80C2×D3×C8D411yescharacteristic
81C2×D3×C8D411yescharacteristic
82C2×D3×C8C2311yescharacteristic
83C2×D3×C8C2311yescharacteristic
84#895—21no—
85D3×C4⋊C4—21no—
86#884—41no—
87D3×C22⋊C4—41no—
88C3×SD32—42no—
89C2×C3×C16C2312yes—
90C3×#107C2311yescharacteristic
91C3×#102—21no—
92#835—22no—
93#835—22no—
94#832—21no—
95#828—42no—
96#821—21no—
97#818—21no—
98#818C2311yescharacteristic
99#818C2312yes—
100C2×C3⋊C16C2312yes—
101#809—21no—
102#807—21no—
103D3×C16—42no—
104C22×Q16C2×D311yescharacteristic, complemented
105C22×D8—31no—
106C23×C8—31no—
107C2×C22⋊Q8—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×SD32C2×D312yescomplemented
111C22×C16—32no—
112#467—32no—
113#467—32no—
114#467—61no—
115#467—61no—
116#466—31no—
117#466—31no—
118#466—31no—
119#466—61no—
120#466—61no—
121#466C2×D311yescharacteristic
122C2×#107—31no—
123C2×#107—31no—
124C2×#103—61no—
125C2×#102—61no—
126#367—32no—
127#367—32no—
128#358—32no—
129#358C2×D312yescomplemented
130#357—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic, complemented
135C23×D3—41no—
136C23×D3D811yescharacteristic
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×Q8—41no—
141C2×C3×D4—21no—
142C2×C3×D4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C22×Dic3—41no—
149C22×Dic3D811yescharacteristic
150D3×D4—42no—
151D3×D4—42no—
152D3×D4—84no—
153C2×D12—21no—
154C2×D12—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3—41no—
158C2×C4×D3D811yescharacteristic
159C2×C4×D3D811yescharacteristic
160C2×C4×D3C2×D411yescharacteristic
161C2×C4×D3C2×D411yescharacteristic
162C3×Q16—22no—
163C3×Q16—22no—
164C3×Q16—42no—
165C3×D8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2411yescharacteristic
169C3×C4⋊C4—21no—
170C3×C22⋊C4—41no—
171#217—41no—
172#213—42no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—42no—
182Dic3⋊C4—42no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8C2×D411yescharacteristic
185C2×C3⋊C8C2×D411yescharacteristic
186D24—22no—
187D3×C8—22no—
188D3×C8—22no—
189D3×C8—41no—
190D3×C8—41no—
191C3×C16—22no—
192C3⋊C16—22no—
193C22×Q8—21no—
194C22×D4—61no—
195C22×D4—121no—
196C23×C4—31no—
197C2×Q16—21no—
198C2×Q16—42no—
199C2×Q16C22×D312yes—
200C2×D8—31no—
201C2×D8—61no—
202C2×D8—122no—
203C2×D8C22×D311yescharacteristic
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8—122no—
210C22×C8C22×D311yescharacteristic
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C22⋊Q8—121no—
214C22⋊Q8—121no—
215#121—21no—
216#121—61no—
217#121—61no—
218#121—61no—
219#121—121no—
220#121—121no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225C2×C22⋊C4—121no—
226SD32—42no—
227SD32—122no—
228C2×C16—32no—
229C2×C16—122no—
230C2×C16C22×D312yes—
231#107—32no—
232#107—31no—
233#107—61no—
234#107—61no—
235#107C22×D311yescharacteristic
236#103—62no—
237#103—62no—
238#102—21no—
239#102—61no—
240#102—61no—
241#102—61no—
242C23×C3—41no—
243C23×C3C2×D811yescharacteristic
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—82no—
248C22×D3—82no—
249C22×D3—82no—
250C22×D3SD3214yescomplemented
251C22×D3C2×D811yescharacteristic
252C22×D3C2×D811yescharacteristic
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—82no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—82no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—82no—
264C2×C3×C4C2×D811yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266#67—82no—
267#67—82no—
268#67—82no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—82no—
273C2×Dic3C2×D811yescharacteristic
274C2×Dic3C2×D811yescharacteristic
275D12—41no—
276D12—41no—
277D12—82no—
278C4×D3—21no—
279C4×D3—21no—
280C4×D3—21no—
281C4×D3—21no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—82no—
285C3×C8—21no—
286C3×C8H(5,𝔽2)12yes—
287C3⋊C8—21no—
288C3⋊C8—21no—
289C24—31no—
290C24—121no—
291C2×Q8—22no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4—121no—
303C2×D4—121no—
304C2×D4—244no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—41no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4D3×D411yescharacteristic
316Q16—22no—
317Q16—22no—
318Q16—42no—
319D8—22no—
320D8—62no—
321D8—122no—
322C2×C8—22no—
323C2×C8—31no—
324C2×C8—31no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—121no—
328C2×C8—121no—
329C2×C8—121no—
330C2×C8D3×D411yescharacteristic
331C2×C8C23×D311yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—61no—
334C4⋊C4—62no—
335C4⋊C4—121no—
336C4⋊C4—122no—
337C4⋊C4—122no—
338C4⋊C4—121no—
339C22⋊C4—41no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—122no—
343C22⋊C4—122no—
344C22⋊C4—121no—
345C16—22no—
346C16—62no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3C2×SD3212yes—
353C22×C3C22×D811yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—44no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—164no—
363C2×D3C2×SD3212yes—
364C2×D3C2×SD3212yes—
365C3×C4—21no—
366C3×C4—81no—
367C3×C4—82no—
368C3×C4#53411yescharacteristic
369C3×C4#57511yescharacteristic
370Dic3—21no—
371Dic3—21no—
372Dic3—81no—
373Dic3—82no—
374C23—31no—
375C23—31no—
376C23—34no—
377C23—41no—
378C23—121no—
379C23—121no—
380C23—121no—
381C23—242no—
382C23—242no—
383C23—242no—
384C23D3×D811yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—82no—
388D4—41no—
389D4—41no—
390D4—82no—
391D4—121no—
392D4—121no—
393D4—242no—
394D4—242no—
395D4—242no—
396D4—242no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—82no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—242no—
414C2×C4—242no—
415C2×C4D3×D811yescharacteristic
416C2×C4C2×D3×D411yescharacteristic
417C8—21no—
418C8—61no—
419C8—61no—
420C8#100412yes—
421C2×C3—22no—
422C2×C3—161no—
423C2×C3#318611yescharacteristic
424C2×C3C22×SD3211yescharacteristic
425C2×C3#341411yescharacteristic
426D3—22no—
427D3—22no—
428D3—161no—
429C22—21no—
430C22—21no—
431C22—32no—
432C22—32no—
433C22—61no—
434C22—61no—
435C22—61no—
436C22—61no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—124no—
441C22—241no—
442C22—241no—
443C22—241no—
444C22—484no—
445C22D3×SD3212yes—
446C22C2×D3×D811yescharacteristic
447C4—21no—
448C4—61no—
449C4—61no—
450C4—81no—
451C4—82no—
452C4—241no—
453C4—242no—
454C4#558911yescharacteristic
455C4#577311yescharacteristic
456C3#3375311yescharacteristic, complemented, Sylow
457C2—22no—
458C2—62no—
459C2—62no—
460C2—161no—
461C2—481no—
462C2#8000611yescharacteristic
463C2C2×D3×SD3211yescharacteristic
464C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2185C2
socleC22×C3C22×D8
prime radicalD3×#466C2
hypercenter#2185D3
hypocenterC3#33753
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2185

Complements

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

NH ≅ G/Ntype
1. C2×#5188C2semidirect
3. C3×#2185C2semidirect
4. #72063C2semidirect
5. #71487C2semidirect
6. #71485C2semidirect
7. #71069C2semidirect
8. #69231C2semidirect
9. #69228C2semidirect
10. D3×C22⋊C16C2semidirect
11. #69096C2semidirect
13. C22×C3×Q16C22semidirect
18. C2×C3×SD32C22semidirect
24. #5188C22semidirect
25. C2×#828C22semidirect
28. C2×#818C22semidirect
30. C2×D3×C16C22semidirect
35. C3×#358C22semidirect
36. C3×C22⋊C16C22semidirect
37. #4564C22semidirect
38. #4562C22semidirect
39. #4549C22semidirect
40. #4509C22semidirect
42. #4505C22semidirect
43. #4490C22semidirect
53. #2185D3semidirect
104. C22×Q16C2×D3semidirect
110. C2×SD32C2×D3semidirect
129. #358C2×D3semidirect
134. C22⋊C16C2×D3semidirect
250. C22×D3SD32semidirect
456. C3#33753semidirect

Maximal subgroups

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

subgroupindex
C2×#51882
D3×#4662
C3×#21852
#720632
#714872
#714852
#710692
#692312
#692282
D3×C22⋊C162
#690962
#337533