Finite Group Databasebeta

← #84019

Subgroups — #84019

Order 384.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#578C211yesmaximal, characteristic, complemented
2#5838C211yesmaximal, characteristic, complemented
3#5837C211yesmaximal, characteristic, complemented
4#5770C211yesmaximal, characteristic, complemented
5#5246C211yesmaximal, characteristic, complemented
6#5245C211yesmaximal, characteristic, complemented
7#5205C211yesmaximal, characteristic, complemented
8#5204C211yesmaximal, characteristic, complemented
9#5203C211yesmaximal, characteristic, complemented
10#5202C211yesmaximal, characteristic, complemented
11#5197C211yesmaximal, characteristic, complemented
12#5192C211yesmaximal, characteristic, complemented
13#5190C211yesmaximal, characteristic, complemented
14#5145C211yesmaximal, characteristic, complemented
15#5141C211yesmaximal, characteristic, complemented
16#3289—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic
18C3×2−1+4C2211yescharacteristic, complemented
19#1010C2211yescharacteristic
20#1009C2211yescharacteristic
21C3×#137—21no—
22C3×#137—21no—
23C3×#137C2211yescharacteristic
24C3×#137C2211yescharacteristic, complemented
25C3×#135—21no—
26C3×#135C2211yescharacteristic
27C2×C3×Q16—21no—
28C2×C3×Q16C2211yescharacteristic, 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
39#948C2211yescharacteristic
40#948C2211yescharacteristic, complemented
41#947C2211yescharacteristic
42#944C2211yescharacteristic, complemented
43C2×#216—21no—
44C2×#216C2211yescharacteristic, complemented
45#942—21no—
46#942—21no—
47#942C2211yescharacteristic
48#942C2211yescharacteristic
49C2×#215—21no—
50C2×#215C2211yescharacteristic
51#918—21no—
52#916—21no—
53D3×SD16—21no—
54#911—21no—
55C2×#204C2211yescharacteristic
56#901C2211yescharacteristic
57#837C2211yescharacteristic
58#837C2211yescharacteristic, complemented
59#837C2211yescharacteristic, complemented
60#836C2211yescharacteristic, complemented
61#836C2211yescharacteristic, complemented
62#825C2211yescharacteristic
63#824C2211yescharacteristic, complemented
64#823C2211yescharacteristic
65#822C2211yescharacteristic, complemented
66#819C2211yescharacteristic, complemented
67#815C2211yescharacteristic, complemented
68#578—31no—
69#578D311yescharacteristic, complemented
70#577—31no—
71#577—31no—
72#497—31no—
73#473—31no—
74#473—31no—
75#471—31no—
76#471—31no—
77#456—31no—
78#456—31no—
79#455—31no—
80#454—31no—
81#444—31no—
82#444—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41—21no—
86C3×#41—21no—
87C3×#41—41no—
88C3×#41—41no—
89C3×#41C2311yescharacteristic
90C3×#41C2311yescharacteristic
91C3×#41C2311yescharacteristic
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8C2311yescharacteristic
97C2×C3×Q8C2311yescharacteristic
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×Q16—21no—
113C3×Q16—21no—
114C3×Q16—21no—
115C3×Q16D411yescharacteristic
116C3×SD16—21no—
117C3×SD16—21no—
118C3×SD16D411yescharacteristic
119C3×SD16D411yescharacteristic
120C3×D8D411yescharacteristic
121C3×M16—21no—
122C3×M16C2311yescharacteristic
123C2×C3×C8—21no—
124C2×C3×C8C2311yescharacteristic
125#216—21no—
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#215—21no—
136#215—21no—
137#215—21no—
138#215D411yescharacteristic
139#214—21no—
140#214—21no—
141#214D411yescharacteristic
142#213—21no—
143#213—21no—
144#213D411yescharacteristic
145(C2×D3)⋊C4—41no—
146Dic3⋊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
1622−1+4C2×D311yescharacteristic, complemented
163H(5,𝔽2)—31no—
164#137—21no—
165#137—21no—
166#137—31no—
167#137—31no—
168#137—31no—
169#137—31no—
170#137—61no—
171#137—61no—
172#137—61no—
173#137—61no—
174#137C2×D311yescharacteristic
175#137C2×D311yescharacteristic, complemented
176AGL(1,ℤ/8)—31no—
177AGL(1,ℤ/8)—31no—
178AGL(1,ℤ/8)—61no—
179AGL(1,ℤ/8)—61no—
180#135—21no—
181#135—31no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135C2×D311yescharacteristic
186C2×Q16—21no—
187C2×Q16—31no—
188C2×Q16—61no—
189C2×Q16C2×D311yescharacteristic, complemented
190C2×SD16—31no—
191C2×SD16—31no—
192C2×SD16—61no—
193C2×SD16—61no—
194#131—31no—
195#131—31no—
196#131—31no—
197#131C2×D311yescharacteristic, complemented
198C4⋊Q8—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#101—31no—
209#100—31no—
210#97—31no—
211C22×D3—21no—
212C22×D3—41no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—41no—
218C3×Q8—41no—
219C3×Q8C2×D411yescharacteristic
220C3×Q8C2×D411yescharacteristic
221C3×Q8C2×D411yescharacteristic
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4C2×D411yescharacteristic
228C3×D4C2×D411yescharacteristic
229C3×D4C2×D411yescharacteristic
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
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—41no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—61no—
276#41—121no—
277#41C22×D311yescharacteristic
278#41C22×D311yescharacteristic
279#41C22×D311yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—31no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8C22×D311yescharacteristic
288C2×Q8C22×D311yescharacteristic
289C2×D4—31no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—121no—
293Q16—21no—
294Q16—21no—
295Q16—21no—
296Q16—21no—
297Q16—31no—
298Q16—61no—
299Q16—61no—
300Q16—61no—
301Q16—61no—
302Q16—61no—
303Q16—61no—
304Q16#6711yescharacteristic
305SD16—21no—
306SD16—21no—
307SD16—31no—
308SD16—31no—
309SD16—61no—
310SD16—61no—
311SD16—61no—
312SD16—61no—
313SD16—61no—
314SD16—61no—
315SD16—61no—
316SD16—61no—
317SD16—61no—
318SD16—61no—
319SD16#6711yescharacteristic
320SD16#6711yescharacteristic
321D8—31no—
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
339C4⋊C4—121no—
340C22⋊C4—121no—
341C42—31no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3H(5,𝔽2)11yescharacteristic
346C2×D3—21no—
347C2×D3—41no—
348C2×D3—81no—
349C2×D3—161no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4C22×D411yescharacteristic
356C3×C4C22×D411yescharacteristic
357Dic3—21no—
358Dic3—41no—
359Dic3—41no—
360C23—61no—
361C23—121no—
362Q8—21no—
363Q8—21no—
364Q8—21no—
365Q8—21no—
366Q8—31no—
367Q8—41no—
368Q8—41no—
369Q8—61no—
370Q8—61no—
371Q8—121no—
372Q8D3×D411yescharacteristic
373Q8C2×#6711yescharacteristic
374Q8C2×#6711yescharacteristic
375D4—21no—
376D4—21no—
377D4—21no—
378D4—31no—
379D4—41no—
380D4—41no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—121no—
385D4—121no—
386D4—121no—
387D4D3×D411yescharacteristic
388D4C2×#6711yescharacteristic
389D4C2×#6711yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4C23×D311yescharacteristic
406C8—21no—
407C8—31no—
408C8—31no—
409C8—61no—
410C8—61no—
411C8—61no—
412C8C2×#6711yescharacteristic
413C8C2×#6711yescharacteristic
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3D4211yescharacteristic
419D3—41no—
420D3—81no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—61no—
425C22—121no—
426C22—241no—
427C22—481no—
428C22#100411yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—61no—
435C4—121no—
436C4—121no—
437C4C2×D3×D411yescharacteristic
438C4C22×#6711yescharacteristic
439C3#328911yescharacteristic, complemented, Sylow
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—121no—
445C2—241no—
446C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#578C2
socleC2×C3D42
hypercenter#578D3
hypocenterC3#3289
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. C3×#578C2semidirect
2. #5838C2semidirect
3. #5837C2semidirect
4. #5770C2semidirect
5. #5246C2semidirect
6. #5245C2semidirect
7. #5205C2semidirect
8. #5204C2semidirect
9. #5203C2semidirect
10. #5202C2semidirect
11. #5197C2semidirect
12. #5192C2semidirect
13. #5190C2semidirect
14. #5145C2semidirect
15. #5141C2semidirect
18. C3×2−1+4C22semidirect
24. C3×#137C22semidirect
28. C2×C3×Q16C22semidirect
29. C3×#131C22semidirect
32. #951C22semidirect
40. #948C22semidirect
42. #944C22semidirect
44. C2×#216C22semidirect
58. #837C22semidirect
59. #837C22semidirect
60. #836C22semidirect
61. #836C22semidirect
63. #824C22semidirect
65. #822C22semidirect
66. #819C22semidirect
67. #815C22semidirect
69. #578D3semidirect
162. 2−1+4C2×D3semidirect
175. #137C2×D3semidirect
189. C2×Q16C2×D3semidirect
197. #131C2×D3semidirect
439. C3#3289semidirect

Maximal subgroups

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

subgroupindex
C3×#5782
#58382
#58372
#57702
#52462
#52452
#52052
#52042
#52032
#52022
#51972
#51922
#51902
#51452
#51412
#32893