Finite Group Databasebeta

← #78992

Subgroups — #78992

Order 384.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C211yesmaximal, characteristic
2#5745C211yesmaximal, characteristic, complemented
3#5522C211yesmaximal, characteristic, complemented
4C3×#421C211yesmaximal, characteristic, complemented
5C2×#837C211yesmaximal, characteristic, complemented
6C2×#825C211yesmaximal, characteristic, complemented
7#5005C211yesmaximal, characteristic, complemented
8#4821C214yesmaximal, complemented
9D3×#104C214yesmaximal, complemented
10#2901—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1010—21no—
13#1010C412yescomplemented
14#1009—21no—
15#1009C412yescomplemented
16D3×#41—22no—
17D3×#41C2214yes—
18#1007C412yescomplemented
19#1004—22no—
20#1004C412yescomplemented
21C2×#235—22no—
22C2×#235C2211yescharacteristic
23C2×#235C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic, complemented
25C3×#117C2211yescharacteristic, complemented
26C2×C4×Dic3C2211yescharacteristic, complemented
27C2×#208C2211yescharacteristic, complemented
28#907—22no—
29D3×M16C2214yescomplemented
30#901—22no—
31#893—21no—
32#887—21no—
33#67⋊C4—22no—
34#872C2212yescomplemented
35C42×D3C2212yescomplemented
36C3×#104C2214yescomplemented
37#837C2212yescomplemented
38#837C2212yescomplemented
39#825C2212yescomplemented
40#825C2212yescomplemented
41#805C2214yescomplemented
42#585—31no—
43#568—31no—
44C4×#41—31no—
45#421—31no—
46#421—34no—
47#421D311yescharacteristic, complemented
48C2×#104—31no—
49C2×#104—31no—
50C2×#104—34no—
51C3×#41—21no—
52C3×#41—21no—
53C3×#41C2×C412yescomplemented
54C3×#41C2312yes—
55C2×C3×Q8C2×C411yescharacteristic, complemented
56C2×C3×D4—21no—
57C2×C3×D4C2×C411yescharacteristic, complemented
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61C2×#67—24no—
62C2×#67D412yes—
63C22×Dic3—21no—
64#239—22no—
65#239C2×C414yescomplemented
66D3×Q8—22no—
67D3×Q8C2×C414yescomplemented
68#237—24no—
69#237—24no—
70#237—22no—
71#237C2×C414yescomplemented
72D3×D4—24no—
73D3×D4—24no—
74D3×D4—22no—
75D3×D4C2×C414yescomplemented
76#235—21no—
77#235—21no—
78#235—24no—
79#235—22no—
80#235—24no—
81#235—22no—
82#235C2×C412yes—
83#235C2×C412yes—
84#235C2×C412yescomplemented
85#235D412yes—
86#235D412yes—
87#235C2312yes—
88C2×D12—22no—
89C2×D12C2×C411yescharacteristic, complemented
90C2×D12D411yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—24no—
93C2×C4×D3—22no—
94C2×C4×D3C2312yes—
95C2×Dic6—22no—
96C2×Dic6C2×C411yescharacteristic, complemented
97C2×Dic6D411yescharacteristic
98C3×M16—21no—
99C3×M16C2312yes—
100C2×C3×C8—21no—
101C3×C4⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C42C2312yes—
104(C2×D3)⋊C4—22no—
105Dic3⋊C4—22no—
106C4×Dic3—21no—
107C4×Dic3—22no—
108C4×Dic3C2311yescharacteristic
109C4×Dic3C2311yescharacteristic
110#208—21no—
111#208C2312yes—
112C2×C3⋊C8—21no—
113ΓL(1,𝔽25)—24no—
114D3×C8—24no—
1152−1+4—32no—
1162−1+4—32no—
1172−1+4—61no—
118H(5,𝔽2)—32no—
119H(5,𝔽2)—32no—
120H(5,𝔽2)—62no—
121H(5,𝔽2)—61no—
122C2×#41—31no—
123C2×#41—34no—
124C2×#41—31no—
125C2×#41—62no—
126C2×#41—62no—
127C2×#41C2×D311yescharacteristic
128#131—62no—
129#131—62no—
130C2×M16—31no—
131C2×M16—34no—
132C2×M16C2×D311yescharacteristic, complemented
133C4×Q8—61no—
134C4×D4—62no—
135C4×D4—61no—
136#117—32no—
137#117C2×D311yescharacteristic, complemented
138C2×C42—31no—
139C2×C42—32no—
140#104—32no—
141#104—32no—
142#104—32no—
143#104—32no—
144#104—34no—
145#104C2×D314yescomplemented
146C23×C3—21no—
147C23×C3C2×D411yescharacteristic
148C22×D3—21no—
149C22×D3—24no—
150C22×D3C2×D412yes—
151C3×Q8—21no—
152C3×Q8C22×C412yes—
153C3×D4—22no—
154C3×D4—22no—
155C3×D4—21no—
156C3×D4C22×C412yes—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4C22×C411yescharacteristic
162C2×C3×C4C22×C411yescharacteristic
163C2×C3×C4C2×D411yescharacteristic
164C2×C3×C4C2×D411yescharacteristic
165C2×C3×C4C2×D411yescharacteristic
166C2×C3×C4C2411yescharacteristic
167#67—22no—
168#67—22no—
169#67—24no—
170#67—24no—
171#67C22⋊C414yes—
172#67C22⋊C414yes—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—24no—
176C2×Dic3—21no—
177C2×Dic3—42no—
178C2×Dic3C2×D412yes—
179D12—24no—
180D12—21no—
181D12C22⋊C412yes—
182D12C22⋊C412yes—
183D12C22×C412yes—
184C4×D3—24no—
185C4×D3—22no—
186C4×D3—22no—
187C4×D3—24no—
188C4×D3—44no—
189C4×D3C22×C414yes—
190C4×D3C2×D414yes—
191Dic6—21no—
192Dic6—24no—
193Dic6C22⋊C412yes—
194Dic6C22⋊C412yes—
195Dic6C22×C412yes—
196C3×C8—22no—
197C3⋊C8—22no—
198#41—21no—
199#41—21no—
200#41—32no—
201#41—32no—
202#41—32no—
203#41—32no—
204#41—34no—
205#41—34no—
206#41—32no—
207#41—32no—
208#41—64no—
209#41—64no—
210#41—62no—
211#41—62no—
212#41—64no—
213#41—64no—
214#41—62no—
215#41—62no—
216#41—61no—
217#41—61no—
218#41C4×D312yescomplemented
219#41C22×D312yes—
220C2×Q8—31no—
221C2×Q8—34no—
222C2×Q8—31no—
223C2×Q8—62no—
224C2×Q8—62no—
225C2×Q8C4×D311yescharacteristic, complemented
226C2×D4—21no—
227C2×D4—32no—
228C2×D4—34no—
229C2×D4—31no—
230C2×D4—31no—
231C2×D4—62no—
232C2×D4—64no—
233C2×D4—62no—
234C2×D4—64no—
235C2×D4—64no—
236C2×D4—61no—
237C2×D4C4×D311yescharacteristic, complemented
238C22×C4—21no—
239C22×C4—32no—
240C22×C4—64no—
241C22×C4—62no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4C22×D311yescharacteristic
245M16—21no—
246M16—32no—
247M16—64no—
248M16—61no—
249M16C22×D312yes—
250C2×C8—21no—
251C2×C8—64no—
252C2×C8—61no—
253C4⋊C4—21no—
254C4⋊C4—62no—
255C22⋊C4—21no—
256C22⋊C4—62no—
257C42—31no—
258C42—31no—
259C42—62no—
260C42—61no—
261C42C22×D312yes—
262C22×C3—22no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3C2×C22⋊C411yescharacteristic
266C22×C3C2×C22⋊C411yescharacteristic
267C22×C3C22×D411yescharacteristic
268C2×D3—22no—
269C2×D3—42no—
270C2×D3—84no—
271C2×D3—81no—
272C2×D3C2×C22⋊C414yes—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4C2×C22⋊C411yescharacteristic
276C3×C4C2×C22⋊C411yescharacteristic
277C3×C4C23×C411yescharacteristic
278C3×C4C22×D411yescharacteristic
279Dic3—22no—
280Dic3—22no—
281Dic3—41no—
282Dic3C2×C22⋊C414yes—
283C23—21no—
284C23—32no—
285C23—64no—
286C23—61no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—32no—
290Q8—32no—
291Q8—32no—
292Q8—64no—
293Q8—61no—
294Q8C2×C4×D312yes—
295D4—22no—
296D4—22no—
297D4—21no—
298D4—34no—
299D4—34no—
300D4—32no—
301D4—32no—
302D4—32no—
303D4—64no—
304D4—64no—
305D4—64no—
306D4—62no—
307D4—62no—
308D4—61no—
309D4C2×C4×D312yes—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—32no—
315C2×C4—34no—
316C2×C4—34no—
317C2×C4—64no—
318C2×C4—64no—
319C2×C4—64no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—61no—
325C2×C4—124no—
326C2×C4—122no—
327C2×C4C2×C4×D311yescharacteristic
328C2×C4C2×C4×D311yescharacteristic
329C2×C4D3×D411yescharacteristic
330C2×C4D3×D411yescharacteristic
331C2×C4D3×D411yescharacteristic
332C2×C4C23×D311yescharacteristic
333C8—22no—
334C8—62no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—42no—
339C2×C3C22×C22⋊C411yescharacteristic
340D3—24no—
341D3—42no—
342C22—22no—
343C22—34no—
344C22—41no—
345C22—62no—
346C22—81no—
347C22—122no—
348C22—241no—
349C22—244no—
350C22D3×C22⋊C411yescharacteristic
351C22D3×C22⋊C411yescharacteristic
352C22C2×D3×D411yescharacteristic
353C4—22no—
354C4—34no—
355C4—42no—
356C4—62no—
357C4—62no—
358C4—121no—
359C4D3×C22⋊C411yescharacteristic
360C4D3×C22⋊C411yescharacteristic
361C4C22×C4×D311yescharacteristic
362C4C2×D3×D411yescharacteristic
363C3#290111yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—64no—
369C2—122no—
370C2C2×D3×C22⋊C411yescharacteristic
subgroupG/H
centerC4C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#421C2
socleC2×C3C22×C22⋊C4
prime radical#5964C2
cosocleC2×C3×C4C24
hypercenter#421D3
hypocenterC3#2901
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #5745C2semidirect
3. #5522C2semidirect
4. C3×#421C2semidirect
5. C2×#837C2semidirect
6. C2×#825C2semidirect
7. #5005C2semidirect
8. #4821C2semidirect
9. D3×#104C2semidirect
13. #1010C4semidirect
15. #1009C4semidirect
18. #1007C4semidirect
20. #1004C4semidirect
24. C2×C3×M16C22semidirect
25. C3×#117C22semidirect
26. C2×C4×Dic3C22semidirect
27. C2×#208C22semidirect
29. D3×M16C22semidirect
34. #872C22semidirect
35. C42×D3C22semidirect
36. C3×#104C22semidirect
37. #837C22semidirect
38. #837C22semidirect
39. #825C22semidirect
40. #825C22semidirect
41. #805C22semidirect
47. #421D3semidirect
53. C3×#41C2×C4semidirect
55. C2×C3×Q8C2×C4semidirect
57. C2×C3×D4C2×C4semidirect
65. #239C2×C4semidirect
67. D3×Q8C2×C4semidirect
71. #237C2×C4semidirect
75. D3×D4C2×C4semidirect
84. #235C2×C4semidirect
89. C2×D12C2×C4semidirect
96. C2×Dic6C2×C4semidirect
132. C2×M16C2×D3semidirect
137. #117C2×D3semidirect
145. #104C2×D3semidirect
218. #41C4×D3semidirect
225. C2×Q8C4×D3semidirect
237. C2×D4C4×D3semidirect
363. C3#2901semidirect

Maximal subgroups

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

subgroupindex
#59642
#57452
#55222
C3×#4212
C2×#8372
C2×#8252
#50052
#48212
D3×#1042
#29013