Finite Group Databasebeta

← #13160015

Subgroups — #13160015

Order 1728.

872 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723942C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723381C211yesmaximal, characteristic, complemented
4#11723378C211yesmaximal, characteristic
5C3×#63646C211yesmaximal, characteristic, complemented
6#11722028C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592876—31nomaximal
9#10590928—34nomaximal
10#89076C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14#88855C2211yescharacteristic
15#88850—21no—
16C8×C3⋊(C3⋊D3)—21no—
17C3×#3748C2211yescharacteristic
18C3×#3748C2211yescharacteristic
19C32×#204—21no—
20C32×#204—21no—
21#88799—21no—
22#88795—21no—
23#88794—21no—
24#88790—21no—
25C2×C32×SD16—31no—
26#64151—31no—
27#64141—31no—
28#64110—31no—
29#64110—34no—
30#64110—64no—
31C2×C3×#204D314yescomplemented
32#63833—34no—
33#63830—34no—
34#63663—31no—
35#63660—31no—
36#63646D311yescharacteristic, complemented
37C3×#819—34no—
38#63560—34no—
39#63558—34no—
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42#6312—21no—
43C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45#6308D411yescharacteristic
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C33×C8D411yescharacteristic
52C33×C8D411yescharacteristic
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5177—94no—
57#4898—91no—
58C2×C32×Q8—31no—
59C2×C32×D4—31no—
60#3828—31no—
61#3828—34no—
62#3828—64no—
63C2×C3×D12C2×D314yes—
64C2×C3×Dic6C2×D314yes—
65C32×SD16—61no—
66C32×SD16—61no—
67C2×C32×C8—24no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C32×C8C2×D314yes—
70#3756—61no—
71#3755—61no—
72#3754—61no—
73#3753—61no—
74#3748—31no—
75#3748—34no—
76#3748—64no—
77#3748C2×D311yescharacteristic
78#3748C2×D311yescharacteristic
79#3743—68no—
80#3743—61no—
81#3743—64no—
82C8×C3⋊D3—68no—
83C8×C3⋊D3—61no—
84C8×C3⋊D3—64no—
85C3×#208—34no—
86C3×#208—34no—
87C3×#204—24no—
88C3×#204—24no—
89#3719—64no—
90#3717—64no—
91#3716—64no—
92#3714—64no—
93C22×C33#4111yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C32×D3—41no—
96C2×C32×D3—81no—
97C33×C4C2×D411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—21no—
100C32×Dic3—41no—
101C2×C3×SD16—34no—
102#942—94no—
103#932—94no—
104#909—91no—
105#908—91no—
106C2×#204C3⋊D311yescharacteristic, complemented
107#901—94no—
108#901—91no—
109#901—184no—
110C3×#108—91no—
111#836—94no—
112#833—94no—
113#824—91no—
114#823—91no—
115#819—34no—
116C23×C32—61no—
117C22×C3×D3—24no—
118C32×Q8—61no—
119C32×Q8—61no—
120C32×D4—61no—
121C32×D4—61no—
122C2×C32×C4—24no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126#638—68no—
127#638—64no—
128#638—61no—
129C4⋊(C3⋊D3)—31no—
130C4⋊(C3⋊D3)—34no—
131C4⋊(C3⋊D3)—64no—
132C4×C3⋊D3—68no—
133C4×C3⋊D3—61no—
134C4×C3⋊D3—64no—
135#634—31no—
136#634—34no—
137#634—64no—
138C2×C3×Dic3—24no—
139C3×D12—24no—
140C3×D12—24no—
141C3×Dic6—24no—
142C3×Dic6—24no—
143C32×C8—24no—
144C32×C8—24no—
145C32×C8D1211yescharacteristic
146C32×C8D1211yescharacteristic
147C32×C8#6714yes—
148C32×C8#6714yes—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—21no—
151C3⋊(C3⋊C8)—68no—
152C3⋊(C3⋊C8)—61no—
153C3⋊(C3⋊C8)—64no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8—64no—
156#471—271noSylow
157C2×C33—21no—
158C2×C33#12111yescharacteristic
159C3⋊(C3⋊D3)—41no—
160C32×D3—81no—
161C2×C3×Q8—34no—
162C2×C3×D4—34no—
163#235—94no—
164#235—91no—
165#235—184no—
166C2×D12C2×C3⋊D311yescharacteristic
167C2×Dic6C2×C3⋊D311yescharacteristic
168C3×SD16—64no—
169C3×SD16—64no—
170C3×M16—91no—
171C3×M16—91no—
172C2×C3×C8—24no—
173C2×C3×C8D3214yes—
174C2×C3×C8C2×C3⋊D311yescharacteristic
175#216—184no—
176#215—184no—
177#214—184no—
178#213—184no—
179#208—34no—
180#208—34no—
181#208—94no—
182#208—91no—
183#208—184no—
184Dic12—181no—
185D24—181no—
186#204—21no—
187#204—21no—
188#204—181no—
189#204—181no—
190ΓL(1,𝔽25)—188no—
191ΓL(1,𝔽25)—184no—
192ΓL(1,𝔽25)—181no—
193D3×C8—188no—
194D3×C8—184no—
195D3×C8—181no—
196C22×C32—24no—
197C22×C32—121no—
198C22×C32—241no—
199C22×C32#23714yes—
200C22×C32#23911yescharacteristic
201C2×C3⋊D3—68no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—64no—
204C2×C3×D3—44no—
205C2×C3×D3—84no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—121no—
209C32×C4C2×D1211yescharacteristic
210C32×C4D3×D414yes—
211C32×C4D3×D411yescharacteristic
212C32×C4C2×#6714yes—
213C3⋊Dic3—68no—
214C3⋊Dic3—61no—
215C3⋊Dic3—64no—
216C3×Dic3—44no—
217#137—271no—
218AGL(1,ℤ/8)—271no—
219C2×SD16—31no—
220#131—271no—
221#108—91no—
222#101—271no—
223#100—271no—
224C33#47111yescharacteristic, complemented, Sylow
225C23×C3—64no—
226C22×D3—21no—
227C3×Q8—64no—
228C3×Q8—64no—
229C3×D4—64no—
230C3×D4—64no—
231C2×C3×C4—24no—
232C2×C3×C4—64no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235#67—188no—
236#67—184no—
237#67—181no—
238C2×Dic3—21no—
239D12—21no—
240D12—21no—
241D12—94no—
242D12—91no—
243D12—184no—
244C4×D3—188no—
245C4×D3—184no—
246C4×D3—181no—
247Dic6—21no—
248Dic6—21no—
249Dic6—94no—
250Dic6—91no—
251Dic6—184no—
252C3×C8—24no—
253C3×C8—24no—
254C3×C8—181no—
255C3×C8—181no—
256C3×C8#62614yes—
257C3×C8#62614yes—
258C3×C8#63811yescharacteristic
259C3×C8#63811yescharacteristic
260C3⋊C8—64no—
261C3⋊C8—64no—
262C3⋊C8—188no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—241no—
270C2×C32#89511yescharacteristic
271C2×C32#93814yes—
272C3⋊D3—128no—
273C3⋊D3—121no—
274C3⋊D3—124no—
275C3×D3—84no—
276#41—271no—
277C2×Q8—31no—
278C2×D4—31no—
279Q16—541no—
280SD16—61no—
281SD16—61no—
282SD16—541no—
283SD16—541no—
284D8—541no—
285M16—91no—
286M16—91no—
287M16—271no—
288M16—541no—
289C2×C8—541no—
290C2×C8D3×C3⋊D311yescharacteristic
291C22×C3—24no—
292C22×C3—124no—
293C22×C3—244no—
294C22×C3#379614yes—
295C22×C3#383011yescharacteristic
296C2×D3—41no—
297C2×D3—81no—
298C2×D3—188no—
299C2×D3—181no—
300C2×D3—184no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—124no—
304C3×C4#380214yes—
305C3×C4C2×#62614yes—
306C3×C4D4×C3⋊D311yescharacteristic
307C3×C4C2×#63811yescharacteristic
308Dic3—41no—
309Dic3—188no—
310Dic3—181no—
311Dic3—184no—
312C32—24no—
313C32#489811yescharacteristic, complemented
314C32#517714yescomplemented
315C23—61no—
316Q8—61no—
317Q8—61no—
318Q8—271no—
319D4—61no—
320D4—61no—
321D4—271no—
322D4—541no—
323C2×C4—61no—
324C2×C4—541no—
325C2×C4C2×D3×C3⋊D311yescharacteristic
326C8—181no—
327C8—181no—
328C8—541no—
329C8#629111yescharacteristic
330C8#629111yescharacteristic
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—244no—
336C2×C3#6391614yes—
337C2×C3#6414711yescharacteristic
338D3—81no—
339D3—368no—
340D3—361no—
341D3—364no—
342C22—121no—
343C22—241no—
344C22—541no—
345C22#8901511yescharacteristic
346C4—121no—
347C4—541no—
348C4#8902611yescharacteristic
349C4C2×#629111yescharacteristic
350C3—24no—
351C3#1059092814yescomplemented
352C3#1059287611yescharacteristic, complemented
353C2—21no—
354C2—241no—
355C2—1081no—
356C2#1172353111yescharacteristic
subgroupG/H
centerC2#11723531
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33#121
prime radical#11723378C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11723942C2semidirect
2. C2×C32×#204C2semidirect
3. #11723381C2semidirect
5. C3×#63646C2semidirect
6. #11722028C2semidirect
7. #11722025C2semidirect
31. C2×C3×#204D3semidirect
36. #63646D3semidirect
106. C2×#204C3⋊D3semidirect
224. C33#471semidirect
313. C32#4898semidirect
314. C32#5177semidirect
351. C3#10590928semidirect
352. C3#10592876semidirect

Maximal subgroups

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

subgroupindex
#117239422
C2×C32×#2042
#117233812
#117233782
C3×#636462
#117220282
#117220252
#105928763
#105909283