Finite Group Databasebeta

← #13161322

Subgroups — #13161322

Order 1728.

896 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88862C211yesmaximal, characteristic, complemented
2C32×#942C211yesmaximal, characteristic, complemented
3#11723367C211yesmaximal, characteristic
4#11723350C211yesmaximal, characteristic, complemented
5C3×#63663C211yesmaximal, characteristic, complemented
6#11722059C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592902—31nomaximal
9#10591643—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88862—21no—
14#88862—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C32×#216—21no—
19C32×#215—21no—
20C32×#208C2211yescharacteristic
21#88795—21no—
22#88790—21no—
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×#137—31no—
26#64158—31no—
27#64157—31no—
28C2×#3755—31no—
29C2×#3755—34no—
30C2×#3755—64no—
31C3×#942D314yescomplemented
32#63825—34no—
33#63815—34no—
34#63663D311yescharacteristic, complemented
35#63650—31no—
36#63649—31no—
37C3×#836—34no—
38#63578—34no—
39#63559—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C32×#67—21no—
48C32×D12D411yescharacteristic
49C32×C4×D3—21no—
50C32×Dic6D411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3×C3⋊(C3⋊C8)—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5203—94no—
57#4754—91no—
58C32×#41—31no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C3×#235C2×D314yes—
66C32×Q16—61no—
67C32×SD16—61no—
68C32×M16—31no—
69#3755—61no—
70#3755—61no—
71#3755—61no—
72#3755—64no—
73#3755—64no—
74#3755—68no—
75#3755—68no—
76#3753—61no—
77#3748—61no—
78#3748C2×D311yescharacteristic
79#3748C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—64no—
84C3×#216—24no—
85C3×#215—24no—
86C3×#208—34no—
87C3×#208—34no—
88C3×#208C2×D314yes—
89#3717—64no—
90#3714—64no—
91#3711—64no—
92D3×C3⋊C8—64no—
93C22×C33#4111yescharacteristic
94C2×C3⋊(C3⋊D3)—41no—
95C2×C3⋊(C3⋊D3)—81no—
96C2×C32×D3—21no—
97C33×C4—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C32×Dic3—21no—
101C3×#137—34no—
102#949—94no—
103#948—94no—
104#942C3⋊D311yescharacteristic, complemented
105C2×#215—94no—
106C2×#215—91no—
107C2×#215—184no—
108#908—91no—
109#907—91no—
110C3×#101—91no—
111#836—34no—
112#823—94no—
113#823—91no—
114#822—94no—
115#822—91no—
116C22×C3⋊D3—61no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C32×Q8—28no—
120C32×Q8—24no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—24no—
124C32×Q8—31no—
125C32×Q8—44no—
126C32×D4—31no—
127C32×D4—61no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141C3×#67—24no—
142C3×D12#6714yes—
143C3×C4×D3—24no—
144C3×Dic6#6714yes—
145C32×C8—61no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—31no—
149C3⋊(C3⋊C8)—31no—
150C3⋊(C3⋊C8)—34no—
151C3⋊(C3⋊C8)—34no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—64no—
154C3×C3⋊C8—24no—
155C3×C3⋊C8—64no—
156C3×C3⋊C8—64no—
157#471—271noSylow
158C2×C33—21no—
159C2×C33#12111yescharacteristic
160C3⋊(C3⋊D3)—81no—
161C32×D3—41no—
162C3×#41—34no—
163C2×C3×Q8—24no—
164C2×C3×Q8D3214yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166#235C2×C3⋊D311yescharacteristic
167C2×D12—94no—
168C2×D12—91no—
169C2×D12—184no—
170C3×Q16—64no—
171C3×SD16—64no—
172C3×M16—34no—
173C3×M16—91no—
174C3×M16—91no—
175#216—21no—
176#215—21no—
177#215—184no—
178#215—184no—
179#215—184no—
180#215—181no—
181#215—181no—
182#215—188no—
183#215—188no—
184#213—184no—
185#208—34no—
186#208—34no—
187#208—184no—
188#208C2×C3⋊D311yescharacteristic
189C2×C3⋊C8—94no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—184no—
192C2×C3⋊C8—184no—
193D24—181no—
194#204—181no—
195ΓL(1,𝔽25)—181no—
196D3×C8—181no—
197C22×C32—24no—
198C22×C32—61no—
199C22×C32#23511yescharacteristic
200C22×C32#23714yes—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—248no—
205C2×C3⋊D3—244no—
206C2×C3⋊D3—241no—
207C2×C3×D3—24no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—41no—
212C32×C4—44no—
213C32×C4—61no—
214C32×C4D3×D414yes—
215C32×C4D3×D411yescharacteristic
216C32×C4D3×D411yescharacteristic
217C32×C4C2×#6714yes—
218C3×Dic3—24no—
219#137—31no—
220AGL(1,ℤ/8)—271no—
221C2×SD16—271no—
222#131—271no—
223#108—271no—
224#101—91no—
225#100—271no—
226C33#47111yescharacteristic, complemented, Sylow
227C22×D3—184no—
228C22×D3—181no—
229C22×D3—364no—
230C3×Q8—28no—
231C3×Q8—24no—
232C3×Q8—24no—
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—34no—
236C3×Q8—44no—
237C3×D4—34no—
238C3×D4—64no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4—64no—
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246#67—21no—
247D12—188no—
248D12—184no—
249D12—184no—
250D12—181no—
251D12—181no—
252D12—364no—
253D12#63811yescharacteristic
254C4×D3—21no—
255Dic6#63811yescharacteristic
256C3×C8—64no—
257C3×C8—181no—
258C3×C8—181no—
259C3⋊C8—21no—
260C3⋊C8—64no—
261C3⋊C8—64no—
262C3⋊C8—94no—
263C3⋊C8—94no—
264C3⋊C8—91no—
265C3⋊C8—91no—
266C3⋊C8—184no—
267C3⋊C8—184no—
268C2×C32—28no—
269C2×C32—24no—
270C2×C32—21no—
271C2×C32—24no—
272C2×C32—121no—
273C2×C32#88411yescharacteristic
274C2×C32#93914yes—
275C3⋊D3—248no—
276C3⋊D3—244no—
277C3⋊D3—241no—
278C3×D3—44no—
279#41—31no—
280C2×Q8D3×C3⋊D311yescharacteristic
281C2×D4—271no—
282Q16—61no—
283SD16—61no—
284SD16—541no—
285SD16—541no—
286SD16—541no—
287D8—541no—
288M16—31no—
289M16—91no—
290M16—91no—
291M16—541no—
292C2×C8—271no—
293C2×C8—541no—
294C22×C3—24no—
295C22×C3—64no—
296C22×C3#380414yes—
297C22×C3#383011yescharacteristic
298C2×D3—21no—
299C2×D3—368no—
300C2×D3—361no—
301C2×D3—364no—
302C2×D3—728no—
303C2×D3—721no—
304C2×D3—724no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—48no—
308C3×C4—41no—
309C3×C4—44no—
310C3×C4—64no—
311C3×C4D3×#6714yes—
312C3×C4#381114yes—
313C3×C4D4×C3⋊D311yescharacteristic
314C3×C4C2×#63811yescharacteristic
315Dic3—21no—
316C32—24no—
317C32#475411yescharacteristic, complemented
318C32#520314yescomplemented
319C23—541no—
320Q8—21no—
321Q8—21no—
322Q8—31no—
323D4—31no—
324D4—61no—
325D4—541no—
326D4—541no—
327C2×C4—21no—
328C2×C4—61no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—61no—
331C8—181no—
332C8—181no—
333C8—271no—
334C8—271no—
335C2×C3—28no—
336C2×C3—21no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—124no—
340C2×C3#6397114yes—
341C2×C3#6414811yescharacteristic
342D3—41no—
343D3—728no—
344D3—721no—
345D3—724no—
346C22—61no—
347C22—1081no—
348C22—2161no—
349C22#8902811yescharacteristic
350C4—41no—
351C4—61no—
352C4D3×#63811yescharacteristic
353C4#8903911yescharacteristic
354C3—24no—
355C3#1059164314yescomplemented
356C3#1059290211yescharacteristic, complemented
357C2—21no—
358C2—121no—
359C2—2161no—
360C2#1172361611yescharacteristic
subgroupG/H
centerC2#11723616
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radical#11723367C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C32×#942C2semidirect
4. #11723350C2semidirect
5. C3×#63663C2semidirect
6. #11722059C2semidirect
7. #11722027C2semidirect
31. C3×#942D3semidirect
34. #63663D3semidirect
104. #942C3⋊D3semidirect
226. C33#471semidirect
317. C32#4754semidirect
318. C32#5203semidirect
355. C3#10591643semidirect
356. C3#10592902semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C32×#9422
#117233672
#117233502
C3×#636632
#117220592
#117220272
#105929023
#105916433