Finite Group Databasebeta

← #80929

Subgroups — #80929

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5777C211yesmaximal, characteristic, complemented
2#5773C211yesmaximal, characteristic, complemented
3C2×#916C211yesmaximal, characteristic, complemented
4D3×#131C211yesmaximal, characteristic, complemented
5C3×#471C211yesmaximal, characteristic, complemented
6#5204C211yesmaximal, characteristic, complemented
7#5199C211yesmaximal, characteristic, complemented
8#5177C211yesmaximal, characteristic, complemented
9#5141C211yesmaximal, characteristic, complemented
10#4898C211yesmaximal, characteristic, complemented
11D3×#108C211yesmaximal, characteristic, complemented
12#4755C211yesmaximal, characteristic, complemented
13#4751C211yesmaximal, characteristic, complemented
14#4749C211yesmaximal, characteristic, complemented
15#4745C211yesmaximal, characteristic, complemented
16#3067—31nomaximal, Sylow
17D3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#237C2211yescharacteristic
20C3×#137C2211yescharacteristic
21C3×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C3×#131C2211yescharacteristic
24#951C2211yescharacteristic
25#949C2211yescharacteristic
26#948C2211yescharacteristic
27C2×#216C2211yescharacteristic, complemented
28#942C2211yescharacteristic
29#932C2211yescharacteristic
30C2×#213C2211yescharacteristic, complemented
31#919—21no—
32#918—21no—
33#916—21no—
34#916—21no—
35#916—21no—
36#916—21no—
37#916—21no—
38#916—21no—
39#915—21no—
40#914—21no—
41#912—21no—
42#911—21no—
43#909C2211yescharacteristic
44#908C2211yescharacteristic
45#907—21no—
46#907C2211yescharacteristic
47D3×M16—21no—
48D3×M16C2211yescharacteristic
49D3×M16C2211yescharacteristic
50D3×M16C2211yescharacteristic
51C2×#204C2211yescharacteristic, complemented
52#901—21no—
53#901C2211yescharacteristic
54C2×D3×C8—21no—
55C2×D3×C8C2211yescharacteristic
56C3×#108C2211yescharacteristic, complemented
57C3×#101C2211yescharacteristic, complemented
58C3×#100C2211yescharacteristic, complemented
59#836C2211yescharacteristic, complemented
60#833C2211yescharacteristic, complemented
61#824C2211yescharacteristic, complemented
62#824C2211yescharacteristic, complemented
63#823C2211yescharacteristic, complemented
64#823C2211yescharacteristic, complemented
65#822C2211yescharacteristic, complemented
66#822C2211yescharacteristic, complemented
67#819C2211yescharacteristic, complemented
68#575—31no—
69#575—31no—
70C2×#135—31no—
71C2×#131—31no—
72#473—31no—
73#473—31no—
74#472—31no—
75#472—31no—
76#471—31no—
77#471—31no—
78#471—31no—
79#471D311yescharacteristic, complemented
80C2×#108—31no—
81#413—31no—
82#413—31no—
83C3×#41C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C2×#67—21no—
87C22×Dic3—21no—
88#239—21no—
89#239—21no—
90#239—21no—
91#239—21no—
92#239D411yescharacteristic
93D3×Q8D411yescharacteristic
94#237—21no—
95#237—21no—
96#237—21no—
97#237—21no—
98#237—21no—
99#237D411yescharacteristic
100D3×D4—21no—
101D3×D4D411yescharacteristic
102#235—21no—
103#235C2311yescharacteristic
104C2×D12—21no—
105C2×D12C2311yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3C2311yescharacteristic
109C2×Dic6C2311yescharacteristic
110C3×Q16—21no—
111C3×SD16—21no—
112C3×SD16—21no—
113C3×SD16—21no—
114C3×SD16—21no—
115C3×D8—21no—
116C3×M16—21no—
117C3×M16C2311yescharacteristic
118C3×M16C2311yescharacteristic
119C3×M16C2311yescharacteristic
120C2×C3×C8—21no—
121C2×C3×C8C2311yescharacteristic
122#216—21no—
123#216—21no—
124#216—21no—
125#216—21no—
126#215—21no—
127#215—21no—
128#214—21no—
129#214—21no—
130#213—21no—
131#213—21no—
132#213—21no—
133#213—21no—
134#208—21no—
135#208C2311yescharacteristic
136#208C2311yescharacteristic
137#208C2311yescharacteristic
138C2×C3⋊C8—21no—
139C2×C3⋊C8C2311yescharacteristic
140Dic12—21no—
141D24—21no—
142#204—21no—
143#204—21no—
144#204—21no—
145#204—21no—
146ΓL(1,𝔽25)—21no—
147ΓL(1,𝔽25)—21no—
148ΓL(1,𝔽25)—21no—
149ΓL(1,𝔽25)—21no—
150ΓL(1,𝔽25)—21no—
151D3×C8—21no—
152D3×C8—21no—
153D3×C8—21no—
154D3×C8—21no—
155D3×C8—21no—
156D3×C8D411yescharacteristic
157D3×C8D411yescharacteristic
158D3×C8D411yescharacteristic
159D3×C8D411yescharacteristic
160C2×#41—31no—
161C2×#41—31no—
162C2×#41—31no—
163#137—31no—
164#137—31no—
165#137—31no—
166#137—61no—
167#137—61no—
168#137C2×D311yescharacteristic
169AGL(1,ℤ/8)—31no—
170AGL(1,ℤ/8)—31no—
171AGL(1,ℤ/8)—31no—
172AGL(1,ℤ/8)—61no—
173AGL(1,ℤ/8)—61no—
174AGL(1,ℤ/8)C2×D311yescharacteristic
175#135—61no—
176#135—61no—
177#135—61no—
178#135—61no—
179#135—61no—
180#135—61no—
181#135—61no—
182#135—61no—
183C2×Q16—31no—
184C2×SD16—31no—
185C2×SD16C2×D311yescharacteristic, complemented
186C2×D8—31no—
187#131—31no—
188#131—31no—
189#131—31no—
190#131—61no—
191#131—61no—
192#131C2×D311yescharacteristic
193C2×M16—31no—
194C2×M16—31no—
195C2×M16—31no—
196C2×M16—61no—
197C22×C8—31no—
198C22×C8—61no—
199#108—31no—
200#108—31no—
201#108—31no—
202#108C2×D311yescharacteristic, complemented
203#101—31no—
204#101—31no—
205#101—31no—
206#101C2×D311yescharacteristic, complemented
207#100—31no—
208#100—31no—
209#100—31no—
210#100C2×D311yescharacteristic, complemented
211C23×C3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3#4111yescharacteristic
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8C2×D411yescharacteristic
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—21no—
221C3×D4C2×D411yescharacteristic
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4C2411yescharacteristic
225#67—21no—
226#67—21no—
227#67—21no—
228#67—41no—
229#67—41no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3#4111yescharacteristic
235D12—21no—
236D12—21no—
237D12—21no—
238D12—41no—
239D12—41no—
240D12C2×D411yescharacteristic
241C4×D3—21no—
242C4×D3—21no—
243C4×D3—21no—
244C4×D3—41no—
245C4×D3—41no—
246C4×D3C2×D411yescharacteristic
247C4×D3C2×D411yescharacteristic
248C4×D3C2×D411yescharacteristic
249C4×D3C2×D411yescharacteristic
250Dic6—21no—
251Dic6—21no—
252Dic6—21no—
253Dic6C2×D411yescharacteristic
254C3×C8—21no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8C2×D411yescharacteristic
258C3×C8C2×D411yescharacteristic
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8C2×D411yescharacteristic
263C3⋊C8C2×D411yescharacteristic
264#41—31no—
265#41—31no—
266#41—31no—
267#41—61no—
268#41—61no—
269#41—61no—
270#41—61no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—61no—
276#41—61no—
277#41C22×D311yescharacteristic
278C2×Q8—31no—
279C2×Q8—31no—
280C2×Q8C22×D311yescharacteristic
281C2×D4—31no—
282C2×D4—31no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4C22×D311yescharacteristic
287C22×C4—31no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291Q16—21no—
292Q16—61no—
293Q16—61no—
294Q16—61no—
295Q16—61no—
296Q16—61no—
297SD16—21no—
298SD16—21no—
299SD16—21no—
300SD16—21no—
301SD16—61no—
302SD16—61no—
303SD16—61no—
304SD16—61no—
305SD16—61no—
306SD16—61no—
307SD16—61no—
308SD16—61no—
309D8—21no—
310D8—61no—
311D8—61no—
312D8—61no—
313D8—61no—
314D8—61no—
315M16—21no—
316M16—31no—
317M16—31no—
318M16—31no—
319M16—61no—
320M16—61no—
321M16—61no—
322M16—61no—
323M16—61no—
324M16—61no—
325M16C22×D311yescharacteristic
326M16C22×D311yescharacteristic
327M16C22×D311yescharacteristic
328C2×C8—21no—
329C2×C8—31no—
330C2×C8—31no—
331C2×C8—31no—
332C2×C8—31no—
333C2×C8—31no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8C22×D311yescharacteristic
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—81no—
344C22×C3C2×#4111yescharacteristic
345C2×D3—21no—
346C2×D3—41no—
347C2×D3—42no—
348C2×D3—41no—
349C2×D3—81no—
350C2×D3#12111yescharacteristic
351C2×D3#12111yescharacteristic
352C3×C4—21no—
353C3×C4—41no—
354C3×C4C22×D411yescharacteristic
355C3×C4C22×D411yescharacteristic
356Dic3—21no—
357Dic3—41no—
358Dic3#12111yescharacteristic
359Dic3#12111yescharacteristic
360C23—21no—
361C23—31no—
362C23—61no—
363C23—61no—
364Q8—21no—
365Q8—21no—
366Q8—31no—
367Q8—61no—
368Q8—61no—
369Q8—61no—
370Q8D3×D411yescharacteristic
371D4—21no—
372D4—21no—
373D4—21no—
374D4—31no—
375D4—61no—
376D4—61no—
377D4—61no—
378D4—61no—
379D4—61no—
380D4—61no—
381D4—121no—
382D4—121no—
383D4—121no—
384D4—121no—
385D4D3×D411yescharacteristic
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4C23×D311yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8—31no—
407C8—31no—
408C8—61no—
409C8—61no—
410C8—61no—
411C8D3×D411yescharacteristic
412C8D3×D411yescharacteristic
413C2×C3—21no—
414C2×C3—41no—
415C2×C3—81no—
416C2×C3C2×#12111yescharacteristic
417D3—21no—
418D3—21no—
419D3—41no—
420D3—81no—
421C22—21no—
422C22—31no—
423C22—31no—
424C22—41no—
425C22—61no—
426C22—81no—
427C22—121no—
428C22—122no—
429C22—121no—
430C22—241no—
431C22D3×#4111yescharacteristic
432C4—21no—
433C4—31no—
434C4—31no—
435C4—41no—
436C4—61no—
437C4—121no—
438C4C2×D3×D411yescharacteristic
439C4C2×D3×D411yescharacteristic
440C3#306711yescharacteristic, complemented, Sylow
441C2—21no—
442C2—41no—
443C2—61no—
444C2—61no—
445C2—81no—
446C2—121no—
447C2—241no—
448C2D3×#12111yescharacteristic
subgroupG/H
centerC2D3×#121
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#471C2
socleC2×C3C2×#121
hypercenter#471D3
hypocenterC3#3067
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #5777C2semidirect
2. #5773C2semidirect
3. C2×#916C2semidirect
4. D3×#131C2semidirect
5. C3×#471C2semidirect
6. #5204C2semidirect
7. #5199C2semidirect
8. #5177C2semidirect
9. #5141C2semidirect
10. #4898C2semidirect
11. D3×#108C2semidirect
12. #4755C2semidirect
13. #4751C2semidirect
14. #4749C2semidirect
15. #4745C2semidirect
22. C2×C3×SD16C22semidirect
27. C2×#216C22semidirect
30. C2×#213C22semidirect
51. C2×#204C22semidirect
56. C3×#108C22semidirect
57. C3×#101C22semidirect
58. C3×#100C22semidirect
59. #836C22semidirect
60. #833C22semidirect
61. #824C22semidirect
62. #824C22semidirect
63. #823C22semidirect
64. #823C22semidirect
65. #822C22semidirect
66. #822C22semidirect
67. #819C22semidirect
79. #471D3semidirect
185. C2×SD16C2×D3semidirect
202. #108C2×D3semidirect
206. #101C2×D3semidirect
210. #100C2×D3semidirect
440. C3#3067semidirect

Maximal subgroups

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

subgroupindex
#57772
#57732
C2×#9162
D3×#1312
C3×#4712
#52042
#51992
#51772
#51412
#48982
D3×#1082
#47552
#47512
#47492
#47452
#30673