Finite Group Databasebeta

← #80229

Subgroups — #80229

Order 384.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic
2C2×#918C211yesmaximal, characteristic, complemented
3C2×D3×SD16C211yesmaximal, characteristic, complemented
4#5628C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5239C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5169C211yesmaximal, characteristic, complemented
9#5046C211yesmaximal, characteristic, complemented
10#4808C211yesmaximal, characteristic, complemented
11#4804C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3009—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18#1009—24no—
19#1009—24no—
20C2×#239C2211yescharacteristic
21C2×#239C2211yescharacteristic
22C2×D3×Q8C2211yescharacteristic, complemented
23C2×D3×D4—21no—
24C2×D3×D4—21no—
25C2×D3×D4C2211yescharacteristic
26C2×D3×D4C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×D12—21no—
30C22×D12C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic, complemented
32C2×C3×SD16C2211yescharacteristic, complemented
33C3×C22⋊Q8C2211yescharacteristic, complemented
34#946C2211yescharacteristic, complemented
35C2×#216C2211yescharacteristic, complemented
36C2×#215C2211yescharacteristic, complemented
37C2×#215C2211yescharacteristic, complemented
38C2×#214C2211yescharacteristic, complemented
39#924C2211yescharacteristic, complemented
40#918—24no—
41D3×SD16—24no—
42C2×#204C2211yescharacteristic, complemented
43C2×#204C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45C2×D3×C8C2211yescharacteristic, complemented
46#897C2211yescharacteristic, complemented
47#894—21no—
48#892C2211yescharacteristic, complemented
49#888—21no—
50#885—21no—
51#883—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59#821C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61(C2×D3)⋊C8C2211yescharacteristic, complemented
62#809C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C2×H(5,𝔽2)—31no—
65C2×#137—31no—
66C22×SD16—31no—
67#536—31no—
68#452—31no—
69#452—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#450—31no—
73#450—31no—
74#450—31no—
75#450—31no—
76#418—31no—
77#416—31no—
78#408—31no—
79C23×D3—21no—
80C23×D3—21no—
81C23×D3—21no—
82C3×#41—22no—
83C3×#41—22no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×#67D411yescharacteristic
93C2×#67D411yescharacteristic
94#239—22no—
95#239—22no—
96#239—22no—
97#239D414yes—
98D3×Q8—22no—
99D3×Q8—22no—
100D3×D4—28no—
101D3×D4—28no—
102D3×D4—22no—
103D3×D4—22no—
104D3×D4—22no—
105D3×D4D414yes—
106#235—28no—
107#235—22no—
108#235—22no—
109C2×D12—28no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12D411yescharacteristic
116C2×D12C2311yescharacteristic
117C2×D12C2311yescharacteristic
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2311yescharacteristic
123C2×Dic6—21no—
124C2×Dic6D411yescharacteristic
125C2×Dic6C2311yescharacteristic
126C3×Q16—22no—
127C3×SD16—22no—
128C2×C3×C8C2311yescharacteristic
129C2×C3×C8C2311yescharacteristic
130C3×C4⋊C4—21no—
131C3×C4⋊C4C2311yescharacteristic
132C3×C22⋊C4—21no—
133#217—21no—
134#216—22no—
135#215—22no—
136#215—22no—
137#214—22no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3C2311yescharacteristic
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148#204—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
152H(5,𝔽2)—64no—
153H(5,𝔽2)—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—61no—
158C2×#41C2×D311yescharacteristic
159C22×Q8—31no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166#137—64no—
167C2×Q16—31no—
168C2×Q16C2×D311yescharacteristic, complemented
169C2×SD16—31no—
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16—31no—
173C2×SD16—31no—
174C2×SD16—64no—
175C2×SD16C2×D311yescharacteristic, complemented
176C2×M16—31no—
177C22×C8—31no—
178C4⋊Q8—61no—
179#124—61no—
180#123—61no—
181#123—61no—
182C22⋊Q8—31no—
183C22⋊Q8—31no—
184C22⋊Q8—31no—
185C22⋊Q8C2×D311yescharacteristic, complemented
186#117—31no—
187#103—31no—
188#103C2×D311yescharacteristic, complemented
189#102—31no—
190#102—31no—
191#102—31no—
192#102—31no—
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3C2×D411yescharacteristic
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—24no—
204C22×D3—24no—
205C22×D3—24no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3C2×D411yescharacteristic
213C22×D3C2×D411yescharacteristic
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8C2×D412yes—
218C3×D4—24no—
219C3×D4—21no—
220C3×D4C2×D412yes—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4C2×D411yescharacteristic
228C2×C3×C4C2411yescharacteristic
229#67—24no—
230#67—22no—
231#67—22no—
232#67—24no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3C2×D411yescharacteristic
236C2×Dic3C2×D411yescharacteristic
237D12—24no—
238D12—24no—
239D12—22no—
240D12—24no—
241D12—21no—
242D12—21no—
243D12C2×D412yes—
244D12C2×D412yes—
245C4×D3—24no—
246C4×D3—21no—
247C4×D3—21no—
248C4×D3—24no—
249C4×D3—42no—
250C4×D3C2×D412yes—
251C4×D3C2×D412yes—
252Dic6—22no—
253Dic6—21no—
254Dic6—21no—
255Dic6—42no—
256C3×C8—21no—
257C3×C8—21no—
258C3⋊C8—21no—
259C3⋊C8—21no—
260C24—61no—
261C24—61no—
262C24—61no—
263#41—22no—
264#41—22no—
265#41—34no—
266#41—62no—
267#41—62no—
268#41—68no—
269#41—62no—
270#41—62no—
271#41—62no—
272C2×Q8—31no—
273C2×Q8—31no—
274C2×Q8—61no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8C22×D311yescharacteristic
278C2×Q8C22×D311yescharacteristic
279C2×D4—21no—
280C2×D4—34no—
281C2×D4—31no—
282C2×D4—31no—
283C2×D4—31no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—68no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—62no—
295C2×D4—68no—
296C2×D4—68no—
297C2×D4—62no—
298C2×D4—61no—
299C2×D4C22×D311yescharacteristic
300C22×C4—21no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4C22×D311yescharacteristic
307Q16—22no—
308Q16—62no—
309SD16—22no—
310SD16—62no—
311SD16—62no—
312SD16—62no—
313SD16—62no—
314SD16—62no—
315M16—62no—
316C2×C8—31no—
317C2×C8—31no—
318C2×C8—62no—
319C2×C8C22×D311yescharacteristic
320C2×C8C22×D311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—31no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4C22×D311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C42—61no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3C22×D411yescharacteristic
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—21no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—84no—
352C2×D3—84no—
353C2×D3—84no—
354C2×D3#12012yes—
355C3×C4—21no—
356C3×C4—22no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4C22×D411yescharacteristic
360C3×C4C22×D411yescharacteristic
361Dic3—21no—
362Dic3—41no—
363Dic3—41no—
364Dic3#12012yes—
365C23—21no—
366C23—31no—
367C23—31no—
368C23—64no—
369C23—64no—
370C23—64no—
371C23—61no—
372C23—61no—
373C23—241no—
374C23—241no—
375C23—241no—
376C23—241no—
377C23—241no—
378C23—241no—
379C23D3×D411yescharacteristic
380Q8—21no—
381Q8—21no—
382Q8—21no—
383Q8—62no—
384Q8—61no—
385Q8—61no—
386Q8—122no—
387Q8D3×D412yes—
388D4—24no—
389D4—21no—
390D4—32no—
391D4—32no—
392D4—62no—
393D4—64no—
394D4—64no—
395D4—64no—
396D4—62no—
397D4—64no—
398D4—62no—
399D4—64no—
400D4—61no—
401D4—61no—
402D4D3×D412yes—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—31no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—31no—
413C2×C4—64no—
414C2×C4—64no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—122no—
420C2×C4D3×D411yescharacteristic
421C2×C4C23×D311yescharacteristic
422C8—21no—
423C8—21no—
424C8—61no—
425C8—61no—
426C2×C3—41no—
427C2×C3—42no—
428C2×C3C2×#12011yescharacteristic
429C2×C3#57711yescharacteristic
430C2×C3#57711yescharacteristic
431D3—22no—
432D3—41no—
433D3—42no—
434D3—42no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—32no—
440C22—41no—
441C22—41no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—62no—
447C22—62no—
448C22—61no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—244no—
454C22—244no—
455C22—244no—
456C22C2×D3×D411yescharacteristic
457C4—21no—
458C4—22no—
459C4—32no—
460C4—41no—
461C4—41no—
462C4—61no—
463C4—121no—
464C4—121no—
465C4C2×D3×D411yescharacteristic
466C4C2×D3×D411yescharacteristic
467C3#300911yescharacteristic, complemented, Sylow
468C2—41no—
469C2—42no—
470C2—62no—
471C2—121no—
472C2—122no—
473C2—122no—
474C2D3×#12011yescharacteristic
475C2#577011yescharacteristic
476C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radicalC2×#1009C2
hypercenter#452D3
hypocenterC3#3009
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#918C2semidirect
3. C2×D3×SD16C2semidirect
4. #5628C2semidirect
5. C3×#452C2semidirect
6. #5239C2semidirect
7. #5187C2semidirect
8. #5169C2semidirect
9. #5046C2semidirect
10. #4808C2semidirect
11. #4804C2semidirect
12. #4776C2semidirect
13. #4770C2semidirect
14. #4731C2semidirect
15. #4727C2semidirect
22. C2×D3×Q8C22semidirect
31. C2×C3×Q16C22semidirect
32. C2×C3×SD16C22semidirect
33. C3×C22⋊Q8C22semidirect
34. #946C22semidirect
35. C2×#216C22semidirect
36. C2×#215C22semidirect
37. C2×#215C22semidirect
38. C2×#214C22semidirect
39. #924C22semidirect
42. C2×#204C22semidirect
43. C2×#204C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
45. C2×D3×C8C22semidirect
46. #897C22semidirect
48. #892C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. #821C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. (C2×D3)⋊C8C22semidirect
62. #809C22semidirect
63. #809C22semidirect
71. #452D3semidirect
168. C2×Q16C2×D3semidirect
175. C2×SD16C2×D3semidirect
185. C22⋊Q8C2×D3semidirect
188. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
467. C3#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C2×#9182
C2×D3×SD162
#56282
C3×#4522
#52392
#51872
#51692
#50462
#48082
#48042
#47762
#47702
#47312
#47272
#30093