Finite Group Databasebeta

← #13161761

Subgroups — #13161761

Order 1728.

558 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#64116C211yesmaximal, characteristic
2C2×#88812C211yesmaximal, characteristic, complemented
3#11723663C212yesmaximal
4C3×#63651C212yesmaximal, complemented
5#11722081C211yesmaximal, characteristic, complemented
6#10590999—32nomaximal
7#10590993—31nomaximal
8C3×#3828C2211yescharacteristic
9C2×C3×#634C2212yes—
10C33×M16C2211yescharacteristic
11C3×#3748C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C3×#3743—21no—
14C3×C8×C3⋊D3—21no—
15#88812—22no—
16#88812—21no—
17#88812—21no—
18#88811—22no—
19#64116D311yescharacteristic
20C3×#907—32no—
21C3×#907—61no—
22C2×#3719—32no—
23C2×#3718—31no—
24#63833—32no—
25#63831—32no—
26#63827—32no—
27#63651D312yescomplemented
28C3×#824—32no—
29C3×#824—32no—
30C3×#824—62no—
31#63578—31no—
32#63577—32no—
33C2×C33×C4C2311yescharacteristic
34C3×#638—21no—
35C2×C3×C3⋊Dic3—22no—
36C3×C4⋊(C3⋊D3)D411yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×#634—22no—
39C3×#634—22no—
40C3×#634D411yescharacteristic
41C33×C8—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C3×C3⋊(C3⋊C8)D411yescharacteristic
45#5143—91no—
46#4755—92no—
47#3828C2×D311yescharacteristic
48C2×#634C2×D312yes—
49C3×#235—32no—
50C3×#235—61no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C32×M16—21no—
55C32×M16—22no—
56C32×M16—41no—
57C32×M16C2×D312yes—
58C32×M16C2×D311yescharacteristic
59#3748C2×D312yes—
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61#3743—21no—
62C8×C3⋊D3—21no—
63C3×#208—32no—
64C3×#208—32no—
65C3×#208—62no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68C3×ΓL(1,𝔽25)—62no—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71C3×D3×C8—62no—
72#3719—62no—
73#3719—62no—
74#3719—62no—
75#3719—62no—
76#3718—62no—
77#3718—61no—
78#3718—61no—
79#3717—62no—
80#3716—62no—
81#3715—62no—
82C22×C33#4111yescharacteristic
83C2×C3×C3⋊D3—21no—
84C33×C4C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C3×C3⋊Dic3—21no—
87C3×C3⋊Dic3—42no—
88C3×#131—91no—
89#951—92no—
90C2×#216—92no—
91#942—92no—
92#909—92no—
93#907—32no—
94#907—61no—
95C2×Dic12—91no—
96C3×#101—92no—
97#824—32no—
98#824—32no—
99#824—62no—
100#822—92no—
101#819—91no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—41no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107#638—21no—
108C2×C3⋊Dic3—22no—
109C4⋊(C3⋊D3)D1211yescharacteristic
110C4×C3⋊D3—21no—
111#634—22no—
112#634—22no—
113#634D1211yescharacteristic
114C3×#67—62no—
115C3×#67—62no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C3×D12—32no—
120C3×D12—61no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—32no—
124C3×Dic6—61no—
125C3×Dic6—62no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—62no—
129C3×Dic6—122no—
130C3×Dic6—122no—
131C32×C8—21no—
132C32×C8—22no—
133C32×C8—22no—
134C32×C8—42no—
135C32×C8—42no—
136C3⋊(C3⋊C8)—22no—
137C3⋊(C3⋊C8)#6711yescharacteristic
138C3⋊(C3⋊C8)#6711yescharacteristic
139C3×C3⋊C8—32no—
140C3×C3⋊C8—32no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—122no—
146#473—271noSylow
147C2×C33—21no—
148C2×C33#12111yescharacteristic
149C3×C3⋊D3—41no—
150C3×#41—91no—
151C2×C3×Q8—92no—
152#235—32no—
153#235—61no—
154C2×Dic6—32no—
155C2×Dic6—32no—
156C2×Dic6—62no—
157C3×M16—21no—
158C3×M16—22no—
159C3×M16—41no—
160C3×M16—92no—
161C3×M16—181no—
162C3×M16D3211yescharacteristic
163C3×M16D3212yes—
164C2×C3×C8—91no—
165C2×C3×C8—181no—
166#216—182no—
167#216—182no—
168#216—182no—
169#216—182no—
170#215—182no—
171#214—182no—
172#208—32no—
173#208—32no—
174#208—62no—
175C2×C3⋊C8—32no—
176C2×C3⋊C8—61no—
177Dic12—181no—
178Dic12—181no—
179Dic12—182no—
180#204—182no—
181ΓL(1,𝔽25)—62no—
182ΓL(1,𝔽25)—62no—
183D3×C8—62no—
184D3×C8—62no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32#23512yes—
189C22×C32#23511yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C32×C4—22no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4C2×D1211yescharacteristic
200C32×C4D3×D412yes—
201C32×C4D3×D412yes—
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3—42no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—124no—
208C3×Dic3—122no—
209C3×Dic3—122no—
210#137—272no—
211C2×Q16—271no—
212#131—91no—
213#108—271no—
214#101—92no—
215C33#47311yescharacteristic, complemented, Sylow
216C3×Q8—91no—
217C3×Q8—182no—
218C3×Q8—182no—
219C3×D4—91no—
220C3×D4—181no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—181no—
225C2×C3×C4—182no—
226C2×C3×C4C2×D3212yes—
227C2×C3×C4C2×D3211yescharacteristic
228#67—62no—
229#67—62no—
230C2×Dic3—64no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233D12—32no—
234D12—61no—
235C4×D3—62no—
236C4×D3—62no—
237Dic6—32no—
238Dic6—61no—
239Dic6—62no—
240Dic6—62no—
241Dic6—62no—
242Dic6—62no—
243Dic6—122no—
244Dic6—122no—
245C3×C8—22no—
246C3×C8—22no—
247C3×C8—21no—
248C3×C8—42no—
249C3×C8—42no—
250C3×C8—91no—
251C3×C8—91no—
252C3×C8—182no—
253C3⋊C8—32no—
254C3⋊C8—32no—
255C3⋊C8—61no—
256C3⋊C8—61no—
257C3⋊C8—62no—
258C3⋊C8—62no—
259C3⋊C8—122no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—81no—
268C2×C32#88412yes—
269C2×C32#93011yescharacteristic
270C3⋊D3—41no—
271C3×D3—122no—
272C3×D3—122no—
273#41—91no—
274C2×Q8—92no—
275Q16—541no—
276Q16—541no—
277Q16—542no—
278SD16—542no—
279M16—92no—
280M16—181no—
281M16#111311yescharacteristic
282C2×C8—91no—
283C2×C8—181no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—41no—
287C22×C3—181no—
288C22×C3#379812yes—
289C22×C3#379811yescharacteristic
290C2×D3—62no—
291C2×D3—62no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—181no—
299C3×C4—362no—
300C3×C4D3×D1212yes—
301C3×C4#380211yescharacteristic
302C3×C4D3×#6712yes—
303C3×C4#381111yescharacteristic
304Dic3—62no—
305Dic3—62no—
306Dic3—124no—
307Dic3—122no—
308Dic3—122no—
309C32—21no—
310C32—22no—
311C32—41no—
312C32#475512yescomplemented
313C32#514311yescharacteristic, complemented
314Q8—91no—
315Q8—182no—
316Q8—182no—
317D4—91no—
318D4—181no—
319C2×C4—181no—
320C2×C4—182no—
321C2×C4C2×#111311yescharacteristic
322C8—21no—
323C8—91no—
324C8—91no—
325C8—182no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—361no—
335C2×C3#6390612yes—
336C2×C3#6391211yescharacteristic
337D3—122no—
338D3—122no—
339C22—181no—
340C22#8904211yescharacteristic
341C4—181no—
342C4—362no—
343C4C4⋊#111311yescharacteristic
344C4#8904911yescharacteristic
345C3—21no—
346C3—22no—
347C3—41no—
348C3#1059099311yescharacteristic, complemented
349C3#1059099912yescomplemented
350C2—21no—
351C2—361no—
352C2#1172369511yescharacteristic
subgroupG/H
centerC2#11723695
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33#121
prime radicalC3×#3828C22
hypercenterM16#1113
hypocenterC33#473
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×#88812C2semidirect
4. C3×#63651C2semidirect
5. #11722081C2semidirect
27. #63651D3semidirect
215. C33#473semidirect
312. C32#4755semidirect
313. C32#5143semidirect
348. C3#10590993semidirect
349. C3#10590999semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
C2×#888122
#117236632
C3×#636512
#117220812
#105909993
#105909933