Finite Group Databasebeta

← #13161383

Subgroups — #13161383

Order 1728.

896 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 152 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×#64160C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5#11722058C211yesmaximal, characteristic, complemented
6#11722029C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592840—31nomaximal
9#10591680—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88862—21no—
14#88860—21no—
15#88855C2211yescharacteristic
16C3×#3756—21no—
17C3×#3754—21no—
18C3×#3748C2211yescharacteristic
19C32×#208—21no—
20C32×#208C2211yescharacteristic
21C2×C32×C3⋊C8—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88797—21no—
24#88797—21no—
25C32×#131—31no—
26#64160D311yescharacteristic, complemented
27#64158—31no—
28#64158—34no—
29#64158—64no—
30C2×#3744—31no—
31C3×#951—34no—
32C3×#948D314yescomplemented
33C2×#3717—34no—
34#63651—31no—
35#63650—31no—
36#63646—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×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#5203—91no—
57#5141—94no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—34no—
66C32×M16—31no—
67C32×M16—61no—
68C2×C32×C8—31no—
69C2×C32×C8—61no—
70#3756—21no—
71#3755—68no—
72#3755—64no—
73#3755—61no—
74#3754—21no—
75#3753—68no—
76#3753—64no—
77#3753—61no—
78#3748—31no—
79#3748—34no—
80#3748—64no—
81#3748C2×D311yescharacteristic
82#3744—61no—
83#3744—61no—
84C3×#216—64no—
85C3×#214—64no—
86C3×#208—24no—
87C3×#208—34no—
88C3×#208C2×D314yes—
89C2×C3×C3⋊C8—24no—
90C2×C3×C3⋊C8C2×D314yes—
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
100C3×C3⋊Dic3—41no—
101C3×#137—91no—
102C3×#131—34no—
103#951—34no—
104#949—94no—
105#949—91no—
106#949—184no—
107#948C3⋊D311yescharacteristic, complemented
108C2×#215—91no—
109C2×#204—94no—
110#836—91no—
111#824—94no—
112#823—94no—
113#823—91no—
114#822—91no—
115#819—94no—
116C22×C3⋊D3—61no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C32×Q8—24no—
120C32×Q8D1214yes—
121C32×Q8#6711yescharacteristic
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4D1214yes—
127C32×D4#6711yescharacteristic
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—64no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C32×C8—31no—
147C32×C8—31no—
148C32×C8—61no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—24no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8#6714yes—
156C3×C3⋊C8#6714yes—
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—91no—
166C2×D12—94no—
167C2×D12—91no—
168C2×D12—184no—
169C2×Dic6—34no—
170C3×Q16—181no—
171C3×SD16—181no—
172C3×M16—34no—
173C3×M16—64no—
174C3×M16—91no—
175C2×C3×C8—34no—
176C2×C3×C8—64no—
177#216—64no—
178#215—188no—
179#215—184no—
180#215—181no—
181#215—181no—
182#215—181no—
183#214—64no—
184#213—188no—
185#213—184no—
186#213—181no—
187#208—21no—
188#208—34no—
189#208—94no—
190#208—91no—
191#208—184no—
192#208C2×C3⋊D311yescharacteristic
193C2×C3⋊C8—21no—
194C2×C3⋊C8C2×C3⋊D311yescharacteristic
195#204—184no—
196#204—184no—
197C22×C32—21no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32#23514yes—
202C22×C32#23711yescharacteristic
203C2×C3⋊D3—128no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—121no—
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×C4C2×D1214yes—
215C32×C4D3×D411yescharacteristic
216C32×C4C2×#6714yes—
217C32×C4C2×#6711yescharacteristic
218C3⋊Dic3—41no—
219C3×Dic3—124no—
220#137—91no—
221AGL(1,ℤ/8)—271no—
222C2×SD16—271no—
223#131—31no—
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—181no—
233C3×Q8—181no—
234C3×Q8#62614yes—
235C3×Q8C4⋊(C3⋊D3)11yescharacteristic
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—24no—
239C3×D4—44no—
240C3×D4#62614yes—
241C3×D4C4⋊(C3⋊D3)11yescharacteristic
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—44no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—64no—
250D12—188no—
251D12—184no—
252D12—184no—
253D12—181no—
254D12—181no—
255D12—364no—
256Dic6—64no—
257Dic6—64no—
258C3×C8—34no—
259C3×C8—34no—
260C3×C8—64no—
261C3×C8—181no—
262C3⋊C8—21no—
263C3⋊C8—64no—
264C3⋊C8—188no—
265C3⋊C8—184no—
266C3⋊C8—181no—
267C3⋊C8#63811yescharacteristic
268C3⋊C8#63811yescharacteristic
269C2×C32—21no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—44no—
273C2×C32—41no—
274C2×C32—44no—
275C2×C32—84no—
276C2×C32#93014yes—
277C2×C32#93911yescharacteristic
278C3⋊D3—248no—
279C3⋊D3—244no—
280C3⋊D3—241no—
281#41D3×C3⋊D311yescharacteristic
282C2×Q8—91no—
283C2×D4—271no—
284Q16—181no—
285SD16—181no—
286SD16—541no—
287SD16—541no—
288SD16—541no—
289D8—541no—
290M16—31no—
291M16—61no—
292M16—91no—
293M16—271no—
294C2×C8—31no—
295C2×C8—61no—
296C22×C3—21no—
297C22×C3—24no—
298C22×C3—24no—
299C22×C3—44no—
300C22×C3#380414yes—
301C22×C3#382811yescharacteristic
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—361no—
314C3×C4C2×#62614yes—
315C3×C4D3×#6714yes—
316C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
317C3×C4C2×#63811yescharacteristic
318Dic3—124no—
319C32—24no—
320C32#514114yescomplemented
321C32#520311yescharacteristic, complemented
322C23—541no—
323Q8—181no—
324Q8—181no—
325Q8#629211yescharacteristic
326D4—21no—
327D4—541no—
328D4—541no—
329D4#629211yescharacteristic
330C2×C4—21no—
331C2×C4—181no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—31no—
334C8—31no—
335C8—61no—
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#6413911yescharacteristic
347D3—728no—
348D3—721no—
349D3—724no—
350C22—21no—
351C22—1081no—
352C22—2161no—
353C22#8903111yescharacteristic
354C4—21no—
355C4—361no—
356C4C2×#629211yescharacteristic
357C4D3×#63811yescharacteristic
358C3—24no—
359C3#1059168014yescomplemented
360C3#1059284011yescharacteristic, complemented
361C2—21no—
362C2—41no—
363C2—2161no—
364C2#1172363211yescharacteristic
subgroupG/H
centerC2#11723632
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×#64160C2semidirect
3. C32×#948C2semidirect
4. C2×#88797C2semidirect
5. #11722058C2semidirect
6. #11722029C2semidirect
7. #11722027C2semidirect
26. #64160D3semidirect
32. C3×#948D3semidirect
107. #948C3⋊D3semidirect
227. C33#471semidirect
320. C32#5141semidirect
321. C32#5203semidirect
359. C3#10591680semidirect
360. C3#10592840semidirect

Maximal subgroups

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

subgroupindex
#117239902
C3×#641602
C32×#9482
C2×#887972
#117220582
#117220292
#117220272
#105928403
#105916803