Finite Group Databasebeta

← #13161331

Subgroups — #13161331

Order 1728.

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

normal subgroups only

This group has 840 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×#88863C211yesmaximal, characteristic, complemented
2C32×#942C211yesmaximal, characteristic
3#11723379C211yesmaximal, characteristic
4#11723350C211yesmaximal, characteristic
5C3×#63663C211yesmaximal, characteristic, complemented
6#11722059C211yesmaximal, characteristic, complemented
7#11722030C211yesmaximal, characteristic, complemented
8#10592903—31nomaximal
9#10591644—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88863—21no—
14#88863—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C32×#216—21no—
19C32×#215—21no—
20C32×#208C2211yescharacteristic
21#88799—21no—
22#88794—21no—
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×#137—31no—
26#64160—31no—
27#64157—31no—
28C2×#3756—31no—
29C2×#3756—34no—
30C2×#3756—64no—
31C3×#942D314yes—
32#63831—34no—
33#63815—34no—
34#63663D311yescharacteristic, complemented
35#63651—31no—
36#63649—31no—
37C3×#836—34no—
38#63578—34no—
39#63561—34no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308—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#5204—94no—
57#4755—91no—
58C32×#41—31no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62C2×#634—31no—
63C2×#634—34no—
64C2×#634—64no—
65C3×#235C2×D314yes—
66C32×Q16—61no—
67C32×SD16—61no—
68C32×M16—31no—
69#3756—61no—
70#3756—61no—
71#3756—61no—
72#3756—64no—
73#3756—64no—
74#3756—68no—
75#3756—68no—
76#3754—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#3719—64no—
90#3716—64no—
91#3711—64no—
92D3×C3⋊C8—64no—
93C22×C33#4111yescharacteristic
94C2×C32×D3—21no—
95C33×C4—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C32×Dic3—21no—
100C3×#137—34no—
101#951—94no—
102#948—94no—
103C2×#216—94no—
104C2×#216—91no—
105C2×#216—184no—
106#942C3⋊D311yescharacteristic
107#909—91no—
108#907—91no—
109C3×#101—91no—
110#836—34no—
111#824—94no—
112#824—91no—
113#822—94no—
114#822—91no—
115C32×Q8—28no—
116C32×Q8—24no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—24no—
120C32×Q8—31no—
121C32×Q8—44no—
122C32×D4—31no—
123C32×D4—61no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134#634—68no—
135#634—61no—
136#634—64no—
137#634—61no—
138#634—64no—
139#634—124no—
140C3×#67—24no—
141C3×D12#6714yes—
142C3×C4×D3—24no—
143C3×Dic6#6714yes—
144C32×C8—61no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—31no—
148C3⋊(C3⋊C8)—31no—
149C3⋊(C3⋊C8)—34no—
150C3⋊(C3⋊C8)—34no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—64no—
153C3×C3⋊C8—24no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8—64no—
156#473—271noSylow
157C2×C33—21no—
158C2×C33#12111yescharacteristic
159C32×D3—41no—
160C3×#41—34no—
161C2×C3×Q8—24no—
162C2×C3×Q8D3214yes—
163C2×C3×Q8C2×C3⋊D311yescharacteristic
164#235C2×C3⋊D311yescharacteristic
165C2×Dic6—94no—
166C2×Dic6—91no—
167C2×Dic6—184no—
168C3×Q16—64no—
169C3×SD16—64no—
170C3×M16—34no—
171C3×M16—91no—
172C3×M16—91no—
173#216—21no—
174#216—184no—
175#216—184no—
176#216—184no—
177#216—181no—
178#216—181no—
179#216—188no—
180#216—188no—
181#215—21no—
182#214—184no—
183#208—34no—
184#208—34no—
185#208—184no—
186#208C2×C3⋊D311yescharacteristic
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190C2×C3⋊C8—184no—
191Dic12—181no—
192#204—181no—
193ΓL(1,𝔽25)—181no—
194D3×C8—181no—
195C22×C32—24no—
196C22×C32—61no—
197C22×C32#23511yescharacteristic
198C22×C32#23714yes—
199C2×C3×D3—24no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—48no—
203C32×C4—41no—
204C32×C4—44no—
205C32×C4—61no—
206C32×C4D3×D414yes—
207C32×C4D3×D411yescharacteristic
208C32×C4D3×D411yescharacteristic
209C32×C4C2×#6714yes—
210C3⋊Dic3—128no—
211C3⋊Dic3—121no—
212C3⋊Dic3—124no—
213C3×Dic3—24no—
214#137—31no—
215#137—271no—
216C2×Q16—271no—
217#131—271no—
218#108—271no—
219#101—91no—
220#101—271no—
221C33#47311yescharacteristic, complemented, Sylow
222C3×Q8—28no—
223C3×Q8—24no—
224C3×Q8—24no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—34no—
228C3×Q8—44no—
229C3×D4—34no—
230C3×D4—64no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—44no—
235C2×C3×C4—64no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—21no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—364no—
242D12#63811yescharacteristic
243C4×D3—21no—
244Dic6—188no—
245Dic6—184no—
246Dic6—184no—
247Dic6—181no—
248Dic6—181no—
249Dic6—364no—
250Dic6#63811yescharacteristic
251C3×C8—64no—
252C3×C8—181no—
253C3×C8—181no—
254C3⋊C8—21no—
255C3⋊C8—64no—
256C3⋊C8—64no—
257C3⋊C8—94no—
258C3⋊C8—94no—
259C3⋊C8—91no—
260C3⋊C8—91no—
261C3⋊C8—184no—
262C3⋊C8—184no—
263C2×C32—28no—
264C2×C32—24no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—121no—
268C2×C32#88411yescharacteristic
269C2×C32#93914yes—
270C3×D3—44no—
271#41—31no—
272C2×Q8—271no—
273C2×Q8D3×C3⋊D311yescharacteristic
274Q16—61no—
275Q16—541no—
276Q16—541no—
277Q16—541no—
278SD16—61no—
279SD16—541no—
280M16—31no—
281M16—91no—
282M16—91no—
283M16—541no—
284C2×C8—271no—
285C2×C8—541no—
286C22×C3—24no—
287C22×C3—64no—
288C22×C3#380414yes—
289C22×C3#383011yescharacteristic
290C2×D3—21no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—48no—
294C3×C4—41no—
295C3×C4—44no—
296C3×C4—64no—
297C3×C4D3×#6714yes—
298C3×C4#381114yes—
299C3×C4D4×C3⋊D311yescharacteristic
300C3×C4C2×#63811yescharacteristic
301Dic3—21no—
302Dic3—368no—
303Dic3—361no—
304Dic3—364no—
305C32—24no—
306C32#475511yescharacteristic, complemented
307C32#520414yescomplemented
308Q8—21no—
309Q8—21no—
310Q8—31no—
311Q8—541no—
312Q8—541no—
313D4—31no—
314D4—61no—
315C2×C4—21no—
316C2×C4—61no—
317C2×C4—541no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—181no—
321C8—181no—
322C8—271no—
323C8—271no—
324C2×C3—28no—
325C2×C3—21no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—124no—
329C2×C3#6397114yes—
330C2×C3#6414811yescharacteristic
331D3—41no—
332C22—61no—
333C22#8902811yescharacteristic
334C4—41no—
335C4—61no—
336C4—1081no—
337C4D3×#63811yescharacteristic
338C4#8903911yescharacteristic
339C3—24no—
340C3#1059164414yescomplemented
341C3#1059290311yescharacteristic, complemented
342C2—21no—
343C2—121no—
344C2#1172361611yescharacteristic
subgroupG/H
centerC2#11723616
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radicalC32×#235C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#473
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×#88863C2semidirect
5. C3×#63663C2semidirect
6. #11722059C2semidirect
7. #11722030C2semidirect
34. #63663D3semidirect
221. C33#473semidirect
306. C32#4755semidirect
307. C32#5204semidirect
340. C3#10591644semidirect
341. C3#10592903semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C32×#9422
#117233792
#117233502
C3×#636632
#117220592
#117220302
#105929033
#105916443