Finite Group Databasebeta

← #84012

Subgroups — #84012

Order 384.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#577C211yesmaximal, characteristic, complemented
2#5838C211yesmaximal, characteristic, complemented
3#5835C211yesmaximal, characteristic, complemented
4#5770C211yesmaximal, characteristic, complemented
5#5245C211yesmaximal, characteristic, complemented
6#5242C211yesmaximal, characteristic, complemented
7#5204C211yesmaximal, characteristic, complemented
8#5202C211yesmaximal, characteristic, complemented
9#5201C211yesmaximal, characteristic, complemented
10#5199C211yesmaximal, characteristic, complemented
11#5196C211yesmaximal, characteristic, complemented
12#5177C211yesmaximal, characteristic, complemented
13#5168C211yesmaximal, characteristic, complemented
14#5144C211yesmaximal, characteristic, complemented
15#5141C211yesmaximal, characteristic, complemented
16#3290—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic, complemented
18C3×H(5,𝔽2)C2211yescharacteristic
19#1010C2211yescharacteristic, complemented
20#1009C2211yescharacteristic
21C3×#137—21no—
22C3×#137C2211yescharacteristic
23C3×AGL(1,ℤ/8)—21no—
24C3×AGL(1,ℤ/8)C2211yescharacteristic, complemented
25C3×#135—21no—
26C3×#135C2211yescharacteristic
27C2×C3×SD16—21no—
28C2×C3×SD16C2211yescharacteristic, complemented
29C3×#131C2211yescharacteristic, complemented
30#951—21no—
31#951C2211yescharacteristic
32#951C2211yescharacteristic, complemented
33#950—21no—
34#950—21no—
35#950C2211yescharacteristic
36#949—21no—
37#949C2211yescharacteristic
38#949C2211yescharacteristic, complemented
39#948C2211yescharacteristic
40#948C2211yescharacteristic, complemented
41#947C2211yescharacteristic, complemented
42C2×#216—21no—
43C2×#216C2211yescharacteristic, complemented
44#942—21no—
45#942C2211yescharacteristic, complemented
46#936C2211yescharacteristic, complemented
47#932—21no—
48#932C2211yescharacteristic
49C2×#213—21no—
50C2×#213C2211yescharacteristic
51#918—21no—
52#916—21no—
53D3×SD16—21no—
54#911—21no—
55C2×#204C2211yescharacteristic, complemented
56#901C2211yescharacteristic, complemented
57#837C2211yescharacteristic
58#837C2211yescharacteristic, complemented
59#837C2211yescharacteristic, complemented
60#836C2211yescharacteristic, complemented
61#833C2211yescharacteristic, complemented
62#825C2211yescharacteristic, complemented
63#824C2211yescharacteristic, complemented
64#823C2211yescharacteristic, complemented
65#822C2211yescharacteristic, complemented
66#819C2211yescharacteristic, complemented
67C8×Dic3C2211yescharacteristic, complemented
68#578—31no—
69#577—31no—
70#577D311yescharacteristic, complemented
71#576—31no—
72#495—31no—
73#473—31no—
74#472—31no—
75#471—31no—
76#471—31no—
77#455—31no—
78#455—31no—
79#454—31no—
80#454—31no—
81#443—31no—
82#443—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41—21no—
86C3×#41—21no—
87C3×#41—41no—
88C3×#41C2311yescharacteristic
89C3×#41C2311yescharacteristic
90C3×#41C2311yescharacteristic
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C2×C3×Q8C2311yescharacteristic
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—41no—
97C2×C3×D4C2311yescharacteristic
98#239—21no—
99D3×Q8—21no—
100#237—21no—
101#237—41no—
102D3×D4—21no—
103D3×D4—41no—
104#235—21no—
105#235—21no—
106#235C2311yescharacteristic
107C2×D12—21no—
108C2×D12C2311yescharacteristic
109C2×Dic6—21no—
110C2×Dic6C2311yescharacteristic
111C3×Q16—21no—
112C3×Q16D411yescharacteristic
113C3×SD16—21no—
114C3×SD16—21no—
115C3×SD16—21no—
116C3×SD16—21no—
117C3×SD16D411yescharacteristic
118C3×SD16D411yescharacteristic
119C3×D8—21no—
120C3×D8D411yescharacteristic
121C3×M16—21no—
122C3×M16C2311yescharacteristic
123C2×C3×C8—21no—
124C2×C3×C8C2311yescharacteristic
125#217—41no—
126#216—21no—
127#216—21no—
128#216—21no—
129#216—21no—
130#216—21no—
131#216D411yescharacteristic
132#215—21no—
133#215—21no—
134#215—21no—
135#215D411yescharacteristic
136#214—21no—
137#214—21no—
138#214—21no—
139#214D411yescharacteristic
140#213—21no—
141#213—21no—
142#213—21no—
143#213—21no—
144#213—21no—
145#213D411yescharacteristic
146(C2×D3)⋊C4—41no—
147C4×Dic3C2311yescharacteristic
148#208—21no—
149#208—21no—
150#208C2311yescharacteristic
151#208C2311yescharacteristic
152#208C2311yescharacteristic
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8C2311yescharacteristic
156#204—21no—
157#204—21no—
158ΓL(1,𝔽25)—21no—
159D3×C8—21no—
1602−1+4—31no—
1612−1+4C2×D311yescharacteristic, complemented
162H(5,𝔽2)—31no—
163H(5,𝔽2)C2×D311yescharacteristic
164#137—21no—
165#137—31no—
166#137—31no—
167#137—31no—
168#137—61no—
169#137—61no—
170#137—61no—
171#137C2×D311yescharacteristic
172AGL(1,ℤ/8)—21no—
173AGL(1,ℤ/8)—31no—
174AGL(1,ℤ/8)—31no—
175AGL(1,ℤ/8)—31no—
176AGL(1,ℤ/8)—61no—
177AGL(1,ℤ/8)—61no—
178AGL(1,ℤ/8)—61no—
179AGL(1,ℤ/8)C2×D311yescharacteristic, complemented
180#135—21no—
181#135—31no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135C2×D311yescharacteristic
186C2×Q16—31no—
187C2×Q16—61no—
188C2×SD16—21no—
189C2×SD16—31no—
190C2×SD16—61no—
191C2×SD16C2×D311yescharacteristic, complemented
192C2×D8—31no—
193C2×D8—61no—
194#131—31no—
195#131—31no—
196#131—31no—
197#131C2×D311yescharacteristic, complemented
198#124—31no—
199#124—31no—
200#108—31no—
201#108—31no—
202#104—31no—
203#104—31no—
204#104—31no—
205#104—31no—
206#101—31no—
207#101—31no—
208#100—31no—
209#100—31no—
210C4×C8—31no—
211C23×C3—21no—
212C23×C3—41no—
213C22×D3—21no—
214C22×D3—41no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—41no—
218C3×Q8C2×D411yescharacteristic
219C3×Q8C2×D411yescharacteristic
220C3×Q8C2×D411yescharacteristic
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—21no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4C2×D411yescharacteristic
230C3×D4C2×D411yescharacteristic
231C3×D4C2×D411yescharacteristic
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4C2411yescharacteristic
240#67—21no—
241#67—41no—
242#67—41no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—41no—
246D12—21no—
247D12—21no—
248D12—41no—
249D12C2×D411yescharacteristic
250C4×D3—21no—
251C4×D3—41no—
252C4×D3—41no—
253Dic6—21no—
254Dic6—21no—
255Dic6—41no—
256Dic6C2×D411yescharacteristic
257C3×C8—21no—
258C3×C8C2×D411yescharacteristic
259C3×C8C2×D411yescharacteristic
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8C2×D411yescharacteristic
264C3⋊C8C2×D411yescharacteristic
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—31no—
270#41—41no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—121no—
276#41C22×D311yescharacteristic
277#41C22×D311yescharacteristic
278#41C22×D311yescharacteristic
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—31no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8C22×D311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—31no—
288C2×D4—41no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C2×D4C22×D311yescharacteristic
293Q16—21no—
294Q16—31no—
295Q16—61no—
296Q16—61no—
297Q16—61no—
298Q16—61no—
299Q16—61no—
300Q16#6711yescharacteristic
301SD16—21no—
302SD16—21no—
303SD16—21no—
304SD16—21no—
305SD16—31no—
306SD16—31no—
307SD16—61no—
308SD16—61no—
309SD16—61no—
310SD16—61no—
311SD16—61no—
312SD16—61no—
313SD16—61no—
314SD16—61no—
315SD16#6711yescharacteristic
316SD16#6711yescharacteristic
317D8—21no—
318D8—31no—
319D8—61no—
320D8—61no—
321D8—61no—
322D8—61no—
323D8—61no—
324D8#6711yescharacteristic
325M16—21no—
326M16—31no—
327M16—31no—
328M16—31no—
329M16—61no—
330M16—61no—
331M16—61no—
332M16C22×D311yescharacteristic
333C2×C8—21no—
334C2×C8—31no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8C22×D311yescharacteristic
339C22⋊C4—121no—
340C22⋊C4—121no—
341C42—31no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—81no—
347C22×C3—161no—
348C22×C3H(5,𝔽2)11yescharacteristic
349C2×D3—21no—
350C2×D3—41no—
351C2×D3—81no—
352C2×D3—161no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—41no—
357C3×C4C22×D411yescharacteristic
358C3×C4C22×D411yescharacteristic
359Dic3—21no—
360Dic3—41no—
361Dic3—41no—
362C23—21no—
363C23—41no—
364C23—61no—
365C23—121no—
366Q8—21no—
367Q8—21no—
368Q8—31no—
369Q8—41no—
370Q8—61no—
371Q8—61no—
372Q8—121no—
373Q8D3×D411yescharacteristic
374Q8C2×#6711yescharacteristic
375Q8C2×#6711yescharacteristic
376D4—21no—
377D4—21no—
378D4—21no—
379D4—21no—
380D4—21no—
381D4—31no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—61no—
386D4—61no—
387D4—61no—
388D4—121no—
389D4—121no—
390D4—121no—
391D4D3×D411yescharacteristic
392D4C2×#6711yescharacteristic
393D4C2×#6711yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4C23×D311yescharacteristic
408C8—21no—
409C8—31no—
410C8—31no—
411C8—61no—
412C8—61no—
413C8—61no—
414C8C2×#6711yescharacteristic
415C8C2×#6711yescharacteristic
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—81no—
421C2×C3D4211yescharacteristic
422D3—41no—
423D3—81no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—61no—
429C22—81no—
430C22—121no—
431C22—161no—
432C22—241no—
433C22—481no—
434C22#100411yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—61no—
440C4—121no—
441C4—121no—
442C4C2×D3×D411yescharacteristic
443C4C22×#6711yescharacteristic
444C3#329011yescharacteristic, complemented, Sylow
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—81no—
450C2—121no—
451C2—241no—
452C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#577C2
socleC2×C3D42
hypercenter#577D3
hypocenterC3#3290
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. C3×#577C2semidirect
2. #5838C2semidirect
3. #5835C2semidirect
4. #5770C2semidirect
5. #5245C2semidirect
6. #5242C2semidirect
7. #5204C2semidirect
8. #5202C2semidirect
9. #5201C2semidirect
10. #5199C2semidirect
11. #5196C2semidirect
12. #5177C2semidirect
13. #5168C2semidirect
14. #5144C2semidirect
15. #5141C2semidirect
17. C3×2−1+4C22semidirect
19. #1010C22semidirect
24. C3×AGL(1,ℤ/8)C22semidirect
28. C2×C3×SD16C22semidirect
29. C3×#131C22semidirect
32. #951C22semidirect
38. #949C22semidirect
40. #948C22semidirect
41. #947C22semidirect
43. C2×#216C22semidirect
45. #942C22semidirect
46. #936C22semidirect
55. C2×#204C22semidirect
56. #901C22semidirect
58. #837C22semidirect
59. #837C22semidirect
60. #836C22semidirect
61. #833C22semidirect
62. #825C22semidirect
63. #824C22semidirect
64. #823C22semidirect
65. #822C22semidirect
66. #819C22semidirect
67. C8×Dic3C22semidirect
70. #577D3semidirect
161. 2−1+4C2×D3semidirect
179. AGL(1,ℤ/8)C2×D3semidirect
191. C2×SD16C2×D3semidirect
197. #131C2×D3semidirect
444. C3#3290semidirect

Maximal subgroups

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

subgroupindex
C3×#5772
#58382
#58352
#57702
#52452
#52422
#52042
#52022
#52012
#51992
#51962
#51772
#51682
#51442
#51412
#32903