Finite Group Databasebeta

← #80005

Subgroups — #80005

Order 384.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1004C211yesmaximal, characteristic, complemented
2C2×#912C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4#5608C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5218C211yesmaximal, characteristic, complemented
7#5158C211yesmaximal, characteristic, complemented
8#5156C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4783C211yesmaximal, characteristic, complemented
11#4776C211yesmaximal, characteristic, complemented
12#4771C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3012—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C22×#67—21no—
20#1004—24no—
21#1004—24no—
22C2×#237—21no—
23C2×#237C2211yescharacteristic
24C2×#237C2211yescharacteristic
25C2×#237C2211yescharacteristic, complemented
26C2×D3×D4—21no—
27C2×D3×D4C2211yescharacteristic
28C2×D3×D4C2211yescharacteristic
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic
31C2×C3×D8C2211yescharacteristic, complemented
32C2×C3×D8C2211yescharacteristic, complemented
33C3×#121C2211yescharacteristic, complemented
34#939—21no—
35#938—21no—
36#938C2211yescharacteristic, complemented
37C2×#214C2211yescharacteristic, complemented
38C2×#214C2211yescharacteristic, complemented
39C2×#214C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic, complemented
41#924C2211yescharacteristic, complemented
42#912—24no—
43#911—24no—
44C2×Dic12C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47C2×D3×C8C2211yescharacteristic, complemented
48#897C2211yescharacteristic, complemented
49D3×C4⋊C4C2211yescharacteristic, complemented
50#884—21no—
51#878—21no—
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#832C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59(C2×D3)⋊C8C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62#810C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C2×H(5,𝔽2)—31no—
65C2×AGL(1,ℤ/8)—31no—
66C2×#135—31no—
67#537—31no—
68#452—31no—
69#452—31no—
70#450—31no—
71#450—31no—
72#449—31no—
73#449—31no—
74#447—31no—
75#447D311yescharacteristic, complemented
76#417—31no—
77C2×#102—31no—
78#408—31no—
79C24×C3—21no—
80C23×D3—21no—
81C23×D3—21no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4C2311yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C2×#67—28no—
91C2×#67—28no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C2×#67—21no—
98C2×#67—21no—
99C2×#67D411yescharacteristic
100C2×#67D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104#237—28no—
105#237—22no—
106#237—22no—
107#237—22no—
108#237—22no—
109#237—22no—
110#237D414yescomplemented
111D3×D4—28no—
112D3×D4—22no—
113D3×D4—22no—
114D3×D4—22no—
115D3×D4D414yescomplemented
116#235—22no—
117#235—22no—
118#235—22no—
119#235—22no—
120C2×D12D411yescharacteristic
121C2×D12C2311yescharacteristic
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×Dic6D411yescharacteristic
127C2×Dic6C2311yescharacteristic
128C2×Dic6C2311yescharacteristic
129C3×D8—22no—
130C3×D8—22no—
131C2×C3×C8C2311yescharacteristic
132C2×C3×C8C2311yescharacteristic
133C3×C4⋊C4C2311yescharacteristic
134C3×C22⋊C4—21no—
135#217—21no—
136#217—21no—
137#214—22no—
138#214—22no—
139#214—22no—
140#213—22no—
141(C2×D3)⋊C4—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3C2311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148Dic12—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—31no—
158C2×#41—31no—
159C2×#41—61no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—61no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4C2×D311yescharacteristic
166AGL(1,ℤ/8)—64no—
167#135—64no—
168C2×Q16—31no—
169C2×SD16—31no—
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16—31no—
173C2×D8—31no—
174C2×D8C2×D311yescharacteristic, complemented
175C2×D8C2×D311yescharacteristic, complemented
176C2×M16—31no—
177C22×C8—31no—
178C22⋊Q8—31no—
179C22⋊Q8—31no—
180C22⋊Q8—61no—
181#121—31no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121C2×D311yescharacteristic, complemented
186C2×C4⋊C4—31no—
187#103—31no—
188#103—31no—
189#102—31no—
190#102—31no—
191#102—31no—
192#102—31no—
193#102C2×D311yescharacteristic, complemented
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—24no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3C2×D411yescharacteristic
206C22×D3—21no—
207C22×D3—24no—
208C22×D3—24no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C22×D3C2×D411yescharacteristic
214C22×D3C2×D411yescharacteristic
215C3×D4—24no—
216C3×D4—21no—
217C3×D4—21no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—81no—
221C3×D4C2×D412yes—
222C3×D4C2×D412yes—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4C2×D411yescharacteristic
227C2×C3×C4C2411yescharacteristic
228#67—24no—
229#67—24no—
230#67—24no—
231#67—22no—
232#67—24no—
233#67—22no—
234#67—24no—
235#67—42no—
236#67—81no—
237C2×Dic3—24no—
238C2×Dic3—21no—
239C2×Dic3—24no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—42no—
243C2×Dic3C2×D411yescharacteristic
244C2×Dic3C2×D411yescharacteristic
245D12—22no—
246D12—21no—
247D12C2×D412yes—
248C4×D3—24no—
249C4×D3—21no—
250C4×D3—21no—
251C4×D3—81no—
252C4×D3C2×D412yes—
253C4×D3C2×D412yes—
254Dic6—21no—
255Dic6—22no—
256Dic6—21no—
257Dic6—21no—
258Dic6C2×D412yes—
259C3×C8—21no—
260C3×C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C24—21no—
264C24—61no—
265C24—61no—
266#41—34no—
267#41—62no—
268#41—62no—
269#41—62no—
270#41—62no—
271#41—62no—
272#41—62no—
273#41—62no—
274#41—62no—
275#41—68no—
276#41—62no—
277C2×Q8—31no—
278C2×Q8—31no—
279C2×Q8—31no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—34no—
285C2×D4—31no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4—62no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—62no—
293C2×D4—68no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—68no—
297C2×D4—68no—
298C2×D4—61no—
299C2×D4—62no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4C22×D311yescharacteristic
303C2×D4C22×D311yescharacteristic
304C2×D4C22×D311yescharacteristic
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4C22×D311yescharacteristic
313Q16—62no—
314SD16—62no—
315SD16—62no—
316SD16—62no—
317SD16—62no—
318D8—22no—
319D8—22no—
320D8—62no—
321M16—62no—
322C2×C8—31no—
323C2×C8—31no—
324C2×C8—62no—
325C2×C8C22×D311yescharacteristic
326C2×C8C22×D311yescharacteristic
327C4⋊C4—31no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4C22×D311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—84no—
349C22×C3C22×D411yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—22no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—41no—
357C2×D3—41no—
358C2×D3—84no—
359C2×D3—84no—
360C2×D3#12012yes—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4C22×D411yescharacteristic
364C3×C4C22×D411yescharacteristic
365Dic3—21no—
366Dic3—22no—
367Dic3—41no—
368Dic3—41no—
369Dic3#12012yes—
370C23—24no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—31no—
375C23—31no—
376C23—64no—
377C23—64no—
378C23—61no—
379C23—81no—
380C23—81no—
381C23—241no—
382C23—241no—
383C23—241no—
384C23—241no—
385C23D3×D411yescharacteristic
386Q8—32no—
387Q8—62no—
388Q8—61no—
389Q8—61no—
390Q8—61no—
391D4—24no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—32no—
397D4—62no—
398D4—62no—
399D4—64no—
400D4—62no—
401D4—64no—
402D4—64no—
403D4—64no—
404D4—64no—
405D4—61no—
406D4—81no—
407D4—122no—
408D4—241no—
409D4D3×D412yes—
410D4D3×D412yes—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—31no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—64no—
419C2×C4—64no—
420C2×C4—64no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4D3×D411yescharacteristic
429C2×C4C23×D311yescharacteristic
430C8—21no—
431C8—21no—
432C8—61no—
433C8—61no—
434C2×C3—42no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—81no—
438C2×C3C2×#12011yescharacteristic
439C2×C3#57611yescharacteristic
440C2×C3#57711yescharacteristic
441D3—22no—
442D3—41no—
443D3—42no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—32no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—62no—
462C22—61no—
463C22—84no—
464C22—121no—
465C22—121no—
466C22—244no—
467C22—244no—
468C22C2×D3×D411yescharacteristic
469C4—21no—
470C4—32no—
471C4—41no—
472C4—61no—
473C4—62no—
474C4—121no—
475C4—121no—
476C4C2×D3×D411yescharacteristic
477C4C2×D3×D411yescharacteristic
478C3#301211yescharacteristic, complemented, Sylow
479C2—41no—
480C2—42no—
481C2—42no—
482C2—62no—
483C2—81no—
484C2—121no—
485C2—122no—
486C2D3×#12011yescharacteristic
487C2#575711yescharacteristic
488C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3012
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C2×#912C2semidirect
3. C2×#911C2semidirect
4. #5608C2semidirect
5. C3×#447C2semidirect
6. #5218C2semidirect
7. #5158C2semidirect
8. #5156C2semidirect
9. #5037C2semidirect
10. #4783C2semidirect
11. #4776C2semidirect
12. #4771C2semidirect
13. D3×#102C2semidirect
14. #4733C2semidirect
15. #4727C2semidirect
25. C2×#237C22semidirect
31. C2×C3×D8C22semidirect
32. C2×C3×D8C22semidirect
33. C3×#121C22semidirect
36. #938C22semidirect
37. C2×#214C22semidirect
38. C2×#214C22semidirect
39. C2×#214C22semidirect
40. C2×#213C22semidirect
41. #924C22semidirect
44. C2×Dic12C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. C2×D3×C8C22semidirect
48. #897C22semidirect
49. D3×C4⋊C4C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #832C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. (C2×D3)⋊C8C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #809C22semidirect
75. #447D3semidirect
110. #237D4semidirect
115. D3×D4D4semidirect
174. C2×D8C2×D3semidirect
175. C2×D8C2×D3semidirect
185. #121C2×D3semidirect
193. #102C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
478. C3#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C2×#9122
C2×#9112
#56082
C3×#4472
#52182
#51582
#51562
#50372
#47832
#47762
#47712
D3×#1022
#47332
#47272
#30123