Finite Group Databasebeta

← #13161381

Subgroups — #13161381

Order 1728.

896 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 144 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
1#11723990C211yesmaximal, characteristic
2C3×#64157C211yesmaximal, characteristic, complemented
3C32×#951C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722028C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592902—31nomaximal
9#10591680—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88862—21no—
14#88860—21no—
15#88855C2211yescharacteristic
16C3×#3748—21no—
17C3×#3748C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)—21no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×#216—21no—
21C32×#214—21no—
22C32×#208C2211yescharacteristic
23#88795—21no—
24#88795—21no—
25C32×#137—31no—
26#64158—31no—
27#64158—34no—
28#64158—64no—
29#64157D311yescharacteristic, complemented
30C2×#3755—31no—
31C3×#951D314yescomplemented
32C3×#948—34no—
33C2×#3717—34no—
34#63663—31no—
35#63650—31no—
36#63649—31no—
37#63577—34no—
38#63560—34no—
39#63559—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C33×Q8D411yescharacteristic
42C33×D4—21no—
43C33×D4D411yescharacteristic
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)D411yescharacteristic
54C3×C3⋊(C3⋊C8)D411yescharacteristic
55C32×C3⋊C8—21no—
56#5203—94no—
57#5141—91no—
58C32×#41—24no—
59C32×#41C2×D311yescharacteristic
60C32×#41C2×D314yes—
61C2×C32×Q8—31no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—61no—
67C32×SD16—61no—
68C32×M16—31no—
69#3755—68no—
70#3755—61no—
71#3755—64no—
72#3755—61no—
73#3755—61no—
74#3753—68no—
75#3753—61no—
76#3753—64no—
77#3748—21no—
78#3748—31no—
79#3748—34no—
80#3748—64no—
81#3748C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—21no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×#216—24no—
85C3×#214—24no—
86C3×#208—34no—
87C3×#208—64no—
88C3×#208C2×D314yes—
89C2×C3×C3⋊C8—34no—
90C2×C3×C3⋊C8—64no—
91#3717—64no—
92#3717—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C3⋊(C3⋊D3)—41no—
96C2×C3⋊(C3⋊D3)—81no—
97C33×C4—21no—
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C32×Dic3—41no—
101C3×#137—34no—
102C3×#131—91no—
103#951C3⋊D311yescharacteristic, complemented
104#949—94no—
105#949—91no—
106#949—184no—
107#948—34no—
108C2×#215—94no—
109C2×#204—91no—
110#836—94no—
111#824—91no—
112#823—94no—
113#823—91no—
114#822—94no—
115#819—91no—
116C22×C3⋊D3—61no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C32×Q8—24no—
120C32×Q8—61no—
121C32×Q8—61no—
122C32×Q8D1211yescharacteristic
123C32×Q8#6714yes—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—24no—
127C32×D4—44no—
128C32×D4D1211yescharacteristic
129C32×D4#6714yes—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C2×C3×Dic3—24no—
144C3×Dic6—24no—
145C3×Dic6—24no—
146C32×C8—61no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—61no—
150C3⋊(C3⋊C8)—64no—
151C3⋊(C3⋊C8)#6711yescharacteristic
152C3⋊(C3⋊C8)#6711yescharacteristic
153C3×C3⋊C8—24no—
154C3×C3⋊C8—34no—
155C3×C3⋊C8—34no—
156C3×C3⋊C8—64no—
157#471—271noSylow
158C2×C33—21no—
159C2×C33—41no—
160C2×C33#12111yescharacteristic
161C3⋊(C3⋊D3)—81no—
162C3×#41—24no—
163C3×#41D3214yes—
164C3×#41C2×C3⋊D311yescharacteristic
165C2×C3×Q8—34no—
166C2×D12—94no—
167C2×D12—91no—
168C2×D12—184no—
169C2×Dic6C2×C3⋊D311yescharacteristic
170C3×Q16—64no—
171C3×SD16—64no—
172C3×M16—34no—
173C3×M16—91no—
174C3×M16—181no—
175C2×C3×C8—91no—
176C2×C3×C8—181no—
177#216—21no—
178#215—184no—
179#215—184no—
180#215—188no—
181#215—184no—
182#215—181no—
183#214—21no—
184#213—188no—
185#213—184no—
186#213—181no—
187#208—34no—
188#208—64no—
189#208—94no—
190#208—91no—
191#208—184no—
192#208C2×C3⋊D311yescharacteristic
193C2×C3⋊C8—34no—
194C2×C3⋊C8—64no—
195#204—181no—
196#204—181no—
197C22×C32—21no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32#23511yescharacteristic
202C22×C32#23714yes—
203C2×C3⋊D3—128no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—248no—
207C2×C3⋊D3—244no—
208C2×C3⋊D3—241no—
209C32×C4—21no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—44no—
214C32×C4—121no—
215C32×C4C2×D1211yescharacteristic
216C32×C4D3×D414yes—
217C32×C4C2×#6714yes—
218C32×C4C2×#6711yescharacteristic
219C3×Dic3—44no—
220#137—31no—
221AGL(1,ℤ/8)—271no—
222C2×SD16—271no—
223#131—91no—
224#108—271no—
225#101—271no—
226#100—271no—
227C33#47111yescharacteristic, complemented, Sylow
228C22×D3—184no—
229C22×D3—181no—
230C22×D3—364no—
231C3×Q8—24no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×Q8#62614yes—
235C3×Q8#63811yescharacteristic
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—24no—
239C3×D4—44no—
240C3×D4#62614yes—
241C3×D4#63811yescharacteristic
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—44no—
246C2×C3×C4—64no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—21no—
250D12—188no—
251D12—184no—
252D12—184no—
253D12—181no—
254D12—181no—
255D12—364no—
256Dic6—21no—
257Dic6—21no—
258C3×C8—64no—
259C3×C8—91no—
260C3×C8—91no—
261C3×C8—181no—
262C3⋊C8—21no—
263C3⋊C8—34no—
264C3⋊C8—34no—
265C3⋊C8—64no—
266C3⋊C8—188no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—44no—
273C2×C32—41no—
274C2×C32—44no—
275C2×C32—84no—
276C2×C32#93011yescharacteristic
277C2×C32#93914yes—
278C3⋊D3—248no—
279C3⋊D3—241no—
280C3⋊D3—244no—
281#41D3×C3⋊D311yescharacteristic
282C2×Q8—31no—
283C2×D4—271no—
284Q16—61no—
285SD16—61no—
286SD16—541no—
287SD16—541no—
288SD16—541no—
289D8—541no—
290M16—31no—
291M16—91no—
292M16—181no—
293M16—271no—
294C2×C8—91no—
295C2×C8—181no—
296C22×C3—21no—
297C22×C3—24no—
298C22×C3—24no—
299C22×C3—44no—
300C22×C3#380414yes—
301C22×C3#383011yescharacteristic
302C2×D3—368no—
303C2×D3—361no—
304C2×D3—364no—
305C2×D3—728no—
306C2×D3—721no—
307C2×D3—724no—
308C3×C4—21no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—44no—
313C3×C4—124no—
314C3×C4C2×#62614yes—
315C3×C4D3×#6714yes—
316C3×C4D4×C3⋊D311yescharacteristic
317C3×C4C2×#63811yescharacteristic
318Dic3—41no—
319C32—24no—
320C32#514111yescharacteristic, complemented
321C32#520314yescomplemented
322C23—541no—
323Q8—61no—
324Q8—61no—
325Q8#629111yescharacteristic
326D4—21no—
327D4—541no—
328D4—541no—
329D4#629111yescharacteristic
330C2×C4—21no—
331C2×C4—61no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—61no—
334C8—91no—
335C8—91no—
336C8—181no—
337C8—541no—
338C2×C3—21no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—44no—
342C2×C3—41no—
343C2×C3—44no—
344C2×C3—84no—
345C2×C3#6397814yes—
346C2×C3#6414811yescharacteristic
347D3—728no—
348D3—721no—
349D3—724no—
350C22—21no—
351C22—1081no—
352C22—2161no—
353C22#8902811yescharacteristic
354C4—21no—
355C4—121no—
356C4C2×#629111yescharacteristic
357C4C3⋊D3×#6711yescharacteristic
358C3—24no—
359C3#1059168014yescomplemented
360C3#1059290211yescharacteristic, complemented
361C2—21no—
362C2—41no—
363C2—2161no—
364C2#1172363511yescharacteristic
subgroupG/H
centerC2#11723635
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#121
prime radical#11723990C2
hypercenter#41D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#64157C2semidirect
3. C32×#951C2semidirect
4. C2×#88795C2semidirect
5. #11722060C2semidirect
6. #11722028C2semidirect
7. #11722027C2semidirect
29. #64157D3semidirect
31. C3×#951D3semidirect
103. #951C3⋊D3semidirect
227. C33#471semidirect
320. C32#5141semidirect
321. C32#5203semidirect
359. C3#10591680semidirect
360. C3#10592902semidirect

Maximal subgroups

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

subgroupindex
#117239902
C3×#641572
C32×#9512
C2×#887952
#117220602
#117220282
#117220272
#105929023
#105916803