Finite Group Databasebeta

← #84004

Subgroups — #84004

Order 384.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C211yesmaximal, characteristic, complemented
2C3×#575C211yesmaximal, characteristic, complemented
3#5832C211yesmaximal, characteristic, complemented
4#5822C211yesmaximal, characteristic, complemented
5C2×#837C211yesmaximal, characteristic, complemented
6#5205C212yesmaximal, complemented
7#5202C212yesmaximal, complemented
8#5201C212yesmaximal, complemented
9#5198C212yesmaximal, complemented
10C2×#825C211yesmaximal, characteristic, complemented
11#5132C211yesmaximal, characteristic, complemented
12#3018—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×C3×#41C2211yescharacteristic, complemented
15#1010—21no—
16#1010C2212yes—
17#1009—21no—
18#1009C2212yes—
19D3×#41—22no—
20D3×#41—22no—
21#1007—21no—
22#1004—21no—
23#1004—22no—
24C2×#235—22no—
25C2×#235C2211yescharacteristic
26C2×#235C2211yescharacteristic
27C3×#137—21no—
28C3×#137C2212yescomplemented
29C3×AGL(1,ℤ/8)—21no—
30C3×AGL(1,ℤ/8)C2212yescomplemented
31C3×#135—22no—
32C2×C3×M16C2211yescharacteristic, complemented
33#951—21no—
34#950—22no—
35#949—21no—
36#947C2211yescharacteristic, complemented
37#944C2211yescharacteristic, complemented
38#942C2212yescomplemented
39#940C2211yescharacteristic, complemented
40#939—22no—
41#936C2211yescharacteristic, complemented
42#932C2212yescomplemented
43C4×#67—22no—
44C2×C4×Dic3C2211yescharacteristic, complemented
45C2×#208C2211yescharacteristic, complemented
46#837C2212yescomplemented
47#837C2212yescomplemented
48#825C2212yescomplemented
49#825C2212yescomplemented
50#824C2212yescomplemented
51#823C2212yescomplemented
52#585—31no—
53#575—31no—
54#575D311yescharacteristic, complemented
55#532—31no—
56#456—32no—
57#455—32no—
58#454—32no—
59#453—32no—
60C2×#104—31no—
61C2×#104—31no—
62#413—31no—
63C3×#41—21no—
64C3×#41—21no—
65C3×#41—21no—
66C3×#41—21no—
67C3×#41—21no—
68C3×#41—21no—
69C3×#41D412yescomplemented
70C3×#41C2312yes—
71C2×C3×Q8D411yescharacteristic, complemented
72C2×C3×Q8C2311yescharacteristic
73C2×C3×D4—21no—
74C2×C3×D4—21no—
75C2×C3×D4D411yescharacteristic, complemented
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4C2311yescharacteristic
80C2×#67—24no—
81C2×#67—21no—
82C2×#67—22no—
83C2×#67D412yes—
84C22×Dic3—21no—
85#239—22no—
86#239—22no—
87D3×Q8—22no—
88D3×Q8—22no—
89#237—24no—
90#237—22no—
91#237—22no—
92#237—24no—
93D3×D4—24no—
94D3×D4—22no—
95D3×D4—24no—
96D3×D4—22no—
97#235—24no—
98#235—24no—
99#235—21no—
100#235—21no—
101#235—21no—
102#235—21no—
103#235—21no—
104#235—21no—
105#235—22no—
106#235—22no—
107#235D412yescomplemented
108#235C2312yes—
109C2×D12—22no—
110C2×D12D411yescharacteristic, complemented
111C2×D12C2311yescharacteristic
112C2×C4×D3—24no—
113C2×C4×D3—21no—
114C2×C4×D3D412yes—
115C2×Dic6—22no—
116C2×Dic6D411yescharacteristic, complemented
117C2×Dic6C2311yescharacteristic
118C3×Q16—22no—
119C3×SD16—22no—
120C3×SD16—22no—
121C3×D8—22no—
122C3×M16—21no—
123C3×M16C2312yes—
124C2×C3×C8—21no—
125#217—22no—
126#216—22no—
127#215—22no—
128#214—22no—
129#213—22no—
130(C2×D3)⋊C4—22no—
131Dic3⋊C4—22no—
132C4×Dic3—21no—
133C4×Dic3C2311yescharacteristic
134C4×Dic3C2311yescharacteristic
135#208—21no—
136#208C2312yes—
137C2×C3⋊C8—21no—
1382−1+4—32no—
1392−1+4—61no—
1402−1+4—61no—
141H(5,𝔽2)—32no—
142H(5,𝔽2)—62no—
143H(5,𝔽2)—61no—
144H(5,𝔽2)—61no—
145C2×#41—31no—
146C2×#41—31no—
147C2×#41—62no—
148C2×#41—62no—
149C2×#41—62no—
150C2×#41C2×D311yescharacteristic
151C2×#41C2×D311yescharacteristic, complemented
152#137—21no—
153#137—32no—
154#137—61no—
155#137C2×D312yescomplemented
156AGL(1,ℤ/8)—21no—
157AGL(1,ℤ/8)—32no—
158AGL(1,ℤ/8)—61no—
159AGL(1,ℤ/8)C2×D312yescomplemented
160#135—22no—
161#135—62no—
162C2×M16—31no—
163C2×M16C2×D311yescharacteristic, complemented
164C4⋊Q8—31no—
165#127—31no—
166#124—31no—
167#124—31no—
168#121—62no—
169C4×D4—62no—
170C2×C42—31no—
171#104—32no—
172#104—32no—
173#104—32no—
174#104—32no—
175#101—32no—
176#100—32no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3C2×D411yescharacteristic
180C22×D3—24no—
181C22×D3—21no—
182C22×D3C2×D412yes—
183C3×Q8—21no—
184C3×Q8—21no—
185C3×Q8—21no—
186C3×Q8C2×D412yes—
187C3×D4—22no—
188C3×D4—22no—
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—41no—
193C3×D4—41no—
194C3×D4C2×D412yes—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—24no—
208#67—24no—
209#67—22no—
210#67—22no—
211#67—22no—
212#67—22no—
213#67—44no—
214C2×Dic3—24no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—42no—
219C2×Dic3C2×D412yes—
220D12—24no—
221D12—21no—
222D12—21no—
223D12—21no—
224D12C2×D412yes—
225C4×D3—24no—
226C4×D3—24no—
227C4×D3—22no—
228C4×D3—22no—
229C4×D3—22no—
230C4×D3—22no—
231Dic6—24no—
232Dic6—21no—
233Dic6—21no—
234Dic6—21no—
235Dic6C2×D412yes—
236C3×C8—22no—
237C3⋊C8—22no—
238#41—21no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—32no—
245#41—32no—
246#41—62no—
247#41—62no—
248#41—64no—
249#41—64no—
250#41—64no—
251#41—64no—
252#41—62no—
253#41—62no—
254#41—62no—
255#41—62no—
256#41—61no—
257#41—61no—
258#41—61no—
259#41—61no—
260#41—61no—
261#41—61no—
262#41#6712yescomplemented
263#41C22×D312yes—
264C2×Q8—31no—
265C2×Q8—31no—
266C2×Q8—62no—
267C2×Q8—62no—
268C2×Q8—62no—
269C2×Q8#6711yescharacteristic, complemented
270C2×Q8C22×D311yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—32no—
274C2×D4—31no—
275C2×D4—31no—
276C2×D4—62no—
277C2×D4—64no—
278C2×D4—64no—
279C2×D4—62no—
280C2×D4—64no—
281C2×D4—61no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4#6711yescharacteristic, complemented
285C2×D4C22×D311yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—32no—
289C22×C4—64no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4C22×D311yescharacteristic
293Q16—22no—
294Q16—62no—
295SD16—22no—
296SD16—22no—
297SD16—62no—
298SD16—62no—
299D8—22no—
300D8—62no—
301M16—21no—
302M16—32no—
303M16—61no—
304M16C22×D312yes—
305C2×C8—21no—
306C2×C8—61no—
307C4⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C42—31no—
311C42—31no—
312C42—61no—
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—42no—
316C22×C3—41no—
317C22×C3—81no—
318C22×C3#12011yescharacteristic
319C22×C3#12011yescharacteristic
320C22×C3C22×D411yescharacteristic
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—42no—
324C2×D3—81no—
325C2×D3—84no—
326C3×C4—22no—
327C3×C4—41no—
328C3×C4#12011yescharacteristic
329C3×C4#12011yescharacteristic
330C3×C4C22×D411yescharacteristic
331C3×C4C22×D411yescharacteristic
332Dic3—22no—
333Dic3—22no—
334Dic3—22no—
335Dic3—41no—
336C23—21no—
337C23—21no—
338C23—32no—
339C23—64no—
340C23—61no—
341C23D3×D411yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—32no—
346Q8—64no—
347Q8—61no—
348Q8—61no—
349Q8—61no—
350Q8C2×#6712yes—
351D4—22no—
352D4—22no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—32no—
357D4—41no—
358D4—41no—
359D4—64no—
360D4—64no—
361D4—64no—
362D4—62no—
363D4—62no—
364D4—62no—
365D4—62no—
366D4—61no—
367D4—61no—
368D4—61no—
369D4—124no—
370D4C2×#6712yes—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—32no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—64no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—122no—
389C2×C4D3×D411yescharacteristic
390C2×C4D3×D411yescharacteristic
391C2×C4D3×D411yescharacteristic
392C2×C4C2×#6711yescharacteristic
393C2×C4C2×#6711yescharacteristic
394C2×C4C23×D311yescharacteristic
395C8—22no—
396C8—62no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—42no—
401C2×C3—81no—
402C2×C3C2×#12011yescharacteristic
403D3—42no—
404D3—42no—
405C22—22no—
406C22—41no—
407C22—42no—
408C22—41no—
409C22—62no—
410C22—62no—
411C22—81no—
412C22—122no—
413C22—241no—
414C22—244no—
415C22#93711yescharacteristic
416C22#93711yescharacteristic
417C22C2×D3×D411yescharacteristic
418C4—22no—
419C4—41no—
420C4—62no—
421C4—62no—
422C4—62no—
423C4—121no—
424C4#93711yescharacteristic
425C4#93711yescharacteristic
426C4C2×D3×D411yescharacteristic
427C4C22×#6711yescharacteristic
428C3#301811yescharacteristic, complemented, Sylow
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—81no—
434C2—122no—
435C2—122no—
436C2C2×#93711yescharacteristic
subgroupG/H
centerC4#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#575C2
socleC2×C3C2×#120
hypercenter#575D3
hypocenterC3#3018
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C3×#575C2semidirect
3. #5832C2semidirect
4. #5822C2semidirect
5. C2×#837C2semidirect
6. #5205C2semidirect
7. #5202C2semidirect
8. #5201C2semidirect
9. #5198C2semidirect
10. C2×#825C2semidirect
11. #5132C2semidirect
14. C2×C3×#41C22semidirect
28. C3×#137C22semidirect
30. C3×AGL(1,ℤ/8)C22semidirect
32. C2×C3×M16C22semidirect
36. #947C22semidirect
37. #944C22semidirect
38. #942C22semidirect
39. #940C22semidirect
41. #936C22semidirect
42. #932C22semidirect
44. C2×C4×Dic3C22semidirect
45. C2×#208C22semidirect
46. #837C22semidirect
47. #837C22semidirect
48. #825C22semidirect
49. #825C22semidirect
50. #824C22semidirect
51. #823C22semidirect
54. #575D3semidirect
69. C3×#41D4semidirect
71. C2×C3×Q8D4semidirect
75. C2×C3×D4D4semidirect
107. #235D4semidirect
110. C2×D12D4semidirect
116. C2×Dic6D4semidirect
151. C2×#41C2×D3semidirect
155. #137C2×D3semidirect
159. AGL(1,ℤ/8)C2×D3semidirect
163. C2×M16C2×D3semidirect
262. #41#67semidirect
269. C2×Q8#67semidirect
284. C2×D4#67semidirect
428. C3#3018semidirect

Maximal subgroups

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

subgroupindex
#59642
C3×#5752
#58322
#58222
C2×#8372
#52052
#52022
#52012
#51982
C2×#8252
#51322
#30183