Finite Group Databasebeta

← #80301

Subgroups — #80301

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
1#5838C211yesmaximal, characteristic, complemented
2#5837C211yesmaximal, characteristic, complemented
3#5779C211yesmaximal, characteristic, complemented
4#5778C211yesmaximal, characteristic, complemented
5C3×#456C211yesmaximal, characteristic, complemented
6#5204C211yesmaximal, characteristic, complemented
7#5203C211yesmaximal, characteristic, complemented
8#5095C211yesmaximal, characteristic, complemented
9#5086C211yesmaximal, characteristic, complemented
10#4827C211yesmaximal, characteristic, complemented
11#4825C211yesmaximal, characteristic, complemented
12#4823C211yesmaximal, characteristic, complemented
13#4823C211yesmaximal, characteristic, complemented
14#4755C211yesmaximal, characteristic, complemented
15#4754C211yesmaximal, characteristic, complemented
16#3289—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic
18#1010C2211yescharacteristic, complemented
19#1009C2211yescharacteristic
20#1007C2211yescharacteristic
21C3×#137C2211yescharacteristic
22C3×#137C2211yescharacteristic, complemented
23C3×C4⋊Q8C2211yescharacteristic, complemented
24#951—21no—
25#951C2211yescharacteristic
26#950—21no—
27#950—21no—
28#950C2211yescharacteristic
29#950C2211yescharacteristic
30#949—21no—
31#949C2211yescharacteristic
32#948C2211yescharacteristic
33#948C2211yescharacteristic, complemented
34C2×#216—21no—
35C2×#216C2211yescharacteristic, complemented
36C2×#216C2211yescharacteristic, complemented
37#942—21no—
38#942—21no—
39#942C2211yescharacteristic
40#942C2211yescharacteristic, complemented
41C2×#215—21no—
42C2×#215C2211yescharacteristic
43C2×#215C2211yescharacteristic
44#919—21no—
45#918—21no—
46#918—21no—
47D3×Q16—21no—
48#916—21no—
49#915—21no—
50#914—21no—
51D3×SD16—21no—
52#909C2211yescharacteristic
53#908C2211yescharacteristic
54#907C2211yescharacteristic
55#907C2211yescharacteristic, complemented
56#875C2211yescharacteristic
57C3×#104C2211yescharacteristic
58C3×#104C2211yescharacteristic, complemented
59C3×#101C2211yescharacteristic, complemented
60#836C2211yescharacteristic, complemented
61#824C2211yescharacteristic, complemented
62#823C2211yescharacteristic
63#822C2211yescharacteristic, complemented
64#822C2211yescharacteristic, complemented
65#805C2211yescharacteristic
66#805C2211yescharacteristic, complemented
67#803C2211yescharacteristic, complemented
68#578—31no—
69#578—31no—
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#456D311yescharacteristic, complemented
79#455—31no—
80#454—31no—
81#444—31no—
82#444—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41—41no—
86C3×#41C2311yescharacteristic
87C3×#41C2311yescharacteristic
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C2×C3×Q8C2311yescharacteristic
91C2×C3×Q8C2311yescharacteristic
92#239—21no—
93#239—21no—
94#239—21no—
95D3×Q8—21no—
96D3×Q8—21no—
97D3×Q8—21no—
98#237—21no—
99#237—41no—
100D3×D4—21no—
101D3×D4—41no—
102#235—21no—
103#235—21no—
104#235—41no—
105#235C2311yescharacteristic
106#235C2311yescharacteristic
107C2×D12—21no—
108C2×D12C2311yescharacteristic
109C2×Dic6—21no—
110C2×Dic6C2311yescharacteristic
111C3×Q16—21no—
112C3×Q16—21no—
113C3×SD16—21no—
114C3×SD16—21no—
115C3×M16C2311yescharacteristic
116C3×M16C2311yescharacteristic
117C3×C4⋊C4—41no—
118C3×C42C2311yescharacteristic
119#216—21no—
120#216—21no—
121#216—21no—
122#216—21no—
123#216—21no—
124#216—21no—
125#216—21no—
126#216D411yescharacteristic
127#216D411yescharacteristic
128#215—21no—
129#215—21no—
130#215—21no—
131#215—21no—
132#215—21no—
133#215—21no—
134#215—21no—
135#215D411yescharacteristic
136#215D411yescharacteristic
137#214—21no—
138#214D411yescharacteristic
139#214D411yescharacteristic
140#213—21no—
141#213D411yescharacteristic
142#213D411yescharacteristic
143(C2×D3)⋊C4—41no—
144#208—21no—
145#208—21no—
146#208C2311yescharacteristic
147#208C2311yescharacteristic
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8C2311yescharacteristic
151C2×C3⋊C8C2311yescharacteristic
152Dic12—21no—
153D24—21no—
154#204—21no—
155#204—21no—
156ΓL(1,𝔽25)—21no—
157ΓL(1,𝔽25)—21no—
158D3×C8—21no—
159D3×C8—21no—
1602−1+4—31no—
1612−1+4—31no—
1622−1+4C2×D311yescharacteristic
163H(5,𝔽2)—31no—
164#137—31no—
165#137—31no—
166#137—31no—
167#137—31no—
168#137—61no—
169#137—61no—
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—31no—
181#135—31no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135—61no—
186C2×Q16—31no—
187C2×Q16—31no—
188C2×Q16—61no—
189C2×Q16—61no—
190C2×SD16—31no—
191C2×SD16—31no—
192C2×SD16—61no—
193C2×SD16—61no—
194#131—31no—
195#131—31no—
196#131—31no—
197#131—31no—
198C4⋊Q8C2×D311yescharacteristic, complemented
199#124—31no—
200#108—31no—
201#108—31no—
202#104—31no—
203#104—31no—
204#104C2×D311yescharacteristic
205#104C2×D311yescharacteristic, complemented
206#101—31no—
207#101—31no—
208#101C2×D311yescharacteristic, complemented
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×Q8C2×D411yescharacteristic
219C3×Q8C2×D411yescharacteristic
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—41no—
223C3×D4C2×D411yescharacteristic
224C3×D4C2×D411yescharacteristic
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4C2411yescharacteristic
233#67—21no—
234#67—21no—
235#67—41no—
236#67—41no—
237C2×Dic3—21no—
238C2×Dic3—41no—
239D12—21no—
240D12—21no—
241D12—41no—
242D12—41no—
243D12C2×D411yescharacteristic
244D12C2×D411yescharacteristic
245C4×D3—21no—
246C4×D3—21no—
247C4×D3—41no—
248C4×D3—41no—
249C4×D3—41no—
250C4×D3—41no—
251Dic6—21no—
252Dic6—21no—
253Dic6—41no—
254Dic6—41no—
255Dic6C2×D411yescharacteristic
256Dic6C2×D411yescharacteristic
257C3×C8—21no—
258C3×C8—21no—
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8C2×D411yescharacteristic
262C3⋊C8C2×D411yescharacteristic
263C3⋊C8C2×D411yescharacteristic
264C3⋊C8C2×D411yescharacteristic
265#41—21no—
266#41—21no—
267#41—31no—
268#41—31no—
269#41—41no—
270#41—61no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—61no—
276#41—121no—
277#41—121no—
278#41C22×D311yescharacteristic
279#41C22×D311yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—31no—
283C2×Q8—61no—
284C2×Q8—61no—
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—31no—
296Q16—31no—
297Q16—61no—
298Q16—61no—
299Q16—61no—
300Q16—61no—
301Q16—61no—
302Q16—61no—
303Q16—61no—
304Q16—61no—
305SD16—21no—
306SD16—21no—
307SD16—31no—
308SD16—31no—
309SD16—31no—
310SD16—31no—
311SD16—61no—
312SD16—61no—
313SD16—61no—
314SD16—61no—
315SD16—61no—
316SD16—61no—
317SD16—61no—
318SD16—61no—
319SD16—61no—
320SD16—61no—
321D8—31no—
322D8—31no—
323D8—61no—
324D8—61no—
325M16—31no—
326M16—31no—
327M16—61no—
328M16—61no—
329M16—61no—
330M16—61no—
331M16C22×D311yescharacteristic
332M16C22×D311yescharacteristic
333C2×C8—31no—
334C2×C8—31no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C4⋊C4—41no—
340C22⋊C4—121no—
341C42C22×D311yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3H(5,𝔽2)11yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—41no—
348C2×D3—81no—
349C2×D3—161no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4C22×D411yescharacteristic
356C3×C4C22×D411yescharacteristic
357Dic3—21no—
358Dic3—21no—
359Dic3—41no—
360C23—61no—
361C23—121no—
362Q8—21no—
363Q8—21no—
364Q8—21no—
365Q8—21no—
366Q8—31no—
367Q8—31no—
368Q8—41no—
369Q8—61no—
370Q8—61no—
371Q8—121no—
372Q8—121no—
373Q8D3×D411yescharacteristic
374Q8D3×D411yescharacteristic
375D4—21no—
376D4—21no—
377D4—31no—
378D4—31no—
379D4—41no—
380D4—61no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—121no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4D3×D411yescharacteristic
389D4D3×D411yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4C23×D311yescharacteristic
406C8—21no—
407C8—21no—
408C8—31no—
409C8—31no—
410C8—31no—
411C8—31no—
412C8—61no—
413C8—61no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3D4211yescharacteristic
418D3—41no—
419D3—41no—
420D3—81no—
421C22—21no—
422C22—21no—
423C22—61no—
424C22—61no—
425C22—121no—
426C22—241no—
427C22—481no—
428C22#100411yescharacteristic
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—61no—
435C4—61no—
436C4—121no—
437C4C2×D3×D411yescharacteristic
438C4C2×D3×D411yescharacteristic
439C3#328911yescharacteristic, complemented, Sylow
440C2—21no—
441C2—41no—
442C2—41no—
443C2—121no—
444C2—121no—
445C2—241no—
446C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#456C2
socleC2×C3D42
hypercenter#456D3
hypocenterC3#3289
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #5838C2semidirect
2. #5837C2semidirect
3. #5779C2semidirect
4. #5778C2semidirect
5. C3×#456C2semidirect
6. #5204C2semidirect
7. #5203C2semidirect
8. #5095C2semidirect
9. #5086C2semidirect
10. #4827C2semidirect
11. #4825C2semidirect
12. #4823C2semidirect
13. #4823C2semidirect
14. #4755C2semidirect
15. #4754C2semidirect
18. #1010C22semidirect
22. C3×#137C22semidirect
23. C3×C4⋊Q8C22semidirect
33. #948C22semidirect
35. C2×#216C22semidirect
36. C2×#216C22semidirect
40. #942C22semidirect
55. #907C22semidirect
58. C3×#104C22semidirect
59. C3×#101C22semidirect
60. #836C22semidirect
61. #824C22semidirect
63. #822C22semidirect
64. #822C22semidirect
66. #805C22semidirect
67. #803C22semidirect
78. #456D3semidirect
175. #137C2×D3semidirect
198. C4⋊Q8C2×D3semidirect
205. #104C2×D3semidirect
208. #101C2×D3semidirect
439. C3#3289semidirect

Maximal subgroups

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

subgroupindex
#58382
#58372
#57792
#57782
C3×#4562
#52042
#52032
#50952
#50862
#48272
#48252
#48232
#47552
#47542
#32893