Finite Group Databasebeta

← #13161757

Subgroups — #13161757

Order 1728.

572 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 115 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×M16×C3⋊D3C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic
4C3×#63649C212yesmaximal, complemented
5#11722081C212yesmaximal, complemented
6#10590972—31nomaximal
7D3×#822—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×M16C2211yescharacteristic
10C3×#3748C2211yescharacteristic
11C3×#3748C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C3×#3743—21no—
14C3×C8×C3⋊D3—21no—
15#88809—21no—
16#88808—21no—
17#88808C412yes—
18#88808C412yes—
19M16×C3⋊D3D311yescharacteristic
20C3×D3×M16—32no—
21C3×D3×M16—61no—
22#63818—31no—
23C2×#3710—31no—
24D3×#208—32no—
25C2×D3×C3⋊C8—32no—
26#63649D312yescomplemented
27C3×#822—32no—
28C3×#822—32no—
29C3×#822—62no—
30#63578—32no—
31#63577—32no—
32#63577—32no—
33C22×C3×C3⋊D3Q811yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3Q811yescharacteristic
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C33×C8—21no—
41C3×C3⋊(C3⋊C8)—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)C2×C414yes—
44C2×#822—91no—
45D3×#108—92no—
46C2×C4×C3⋊D3C2×D311yescharacteristic
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—61no—
49C32×M16—21no—
50C32×M16—22no—
51C32×M16—41no—
52C32×M16C2×D312yes—
53C32×M16C2×D311yescharacteristic
54#3748C2×D312yes—
55#3748C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D312yes—
57#3743—21no—
58C8×C3⋊D3—21no—
59C3×#208—32no—
60C3×#208—32no—
61C3×#208—32no—
62C3×#208—61no—
63C3×#208—62no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67C3×ΓL(1,𝔽25)—62no—
68C3×ΓL(1,𝔽25)—62no—
69C3×D3×C8—62no—
70C3×D3×C8—62no—
71#3712—61no—
72#3711—62no—
73#3710—32no—
74#3710—32no—
75#3710—61no—
76D3×C3⋊C8—34no—
77D3×C3⋊C8—34no—
78D3×C3⋊C8—62no—
79C22×C33C2×Q811yescharacteristic
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3C4⋊C411yescharacteristic
82C2×C3×C3⋊D3C4⋊C411yescharacteristic
83C33×C4C22×C411yescharacteristic
84C33×C4C2×D411yescharacteristic
85C3×C3⋊Dic3C4⋊C411yescharacteristic
86C3×C3⋊Dic3C4⋊C411yescharacteristic
87C2×C3×M16—91no—
88C2×#208—91no—
89C22×C3⋊C8—91no—
90D3×M16—32no—
91D3×M16—61no—
92D3×M16—92no—
93C2×D3×C8—92no—
94C3×#108—92no—
95#822—32no—
96#822—32no—
97#822—62no—
98#822—92no—
99#822—92no—
100#819—92no—
101C22×C3⋊D3Dic611yescharacteristic
102C22×C3×D3—32no—
103C22×C3×D3—61no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3Dic611yescharacteristic
110C4×C3⋊D3C4×D311yescharacteristic
111C4×C3⋊D3C4×D311yescharacteristic
112C4×C3⋊D3#6711yescharacteristic
113C4×C3⋊D3#6711yescharacteristic
114C2×C3×Dic3—32no—
115C2×C3×Dic3—61no—
116C3×C4×D3—32no—
117C3×C4×D3—32no—
118C3×C4×D3—32no—
119C3×C4×D3—32no—
120C3×C4×D3—61no—
121C3×C4×D3—61no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C32×C8—21no—
125C32×C8—22no—
126C32×C8—22no—
127C32×C8—42no—
128C32×C8—42no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)—22no—
131C3⋊(C3⋊C8)C4×D314yes—
132C3×C3⋊C8—34no—
133C3×C3⋊C8—34no—
134C3×C3⋊C8—64no—
135C3×C3⋊C8—62no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—122no—
140C2×#108—271noSylow
141C2×C33—21no—
142C2×C33C2×C4⋊C411yescharacteristic
143C3×C3⋊D3—21no—
144C3×C3⋊D3#10812yescomplemented
145C22×C3×C4—91no—
146C2×C4×D3—32no—
147C2×C4×D3—61no—
148C3×M16—21no—
149C3×M16—22no—
150C3×M16—41no—
151C3×M16—91no—
152C3×M16—92no—
153C3×M16—181no—
154C3×M16D3211yescharacteristic
155C3×M16D3212yes—
156C2×C3×C8—92no—
157C2×C3×C8—181no—
158#208—32no—
159#208—32no—
160#208—32no—
161#208—61no—
162#208—62no—
163#208—181no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—62no—
167C2×C3⋊C8—92no—
168C2×C3⋊C8—92no—
169C2×C3⋊C8—181no—
170ΓL(1,𝔽25)—62no—
171ΓL(1,𝔽25)—62no—
172ΓL(1,𝔽25)—182no—
173D3×C8—62no—
174D3×C8—62no—
175D3×C8—94no—
176D3×C8—94no—
177D3×C8—182no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32C2×Dic611yescharacteristic
182C22×C32D3×Q812yes—
183C2×C3⋊D3—22no—
184C2×C3⋊D3Dic3⋊C411yescharacteristic
185C2×C3⋊D3Dic3⋊C411yescharacteristic
186C2×C3×D3—32no—
187C2×C3×D3—32no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—64no—
191C2×C3×D3—122no—
192C32×C4—22no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4C2×C4×D312yes—
199C32×C4C2×C4×D311yescharacteristic
200C32×C4D3×D412yes—
201C32×C4C2×#6711yescharacteristic
202C3⋊Dic3Dic3⋊C411yescharacteristic
203C3⋊Dic3Dic3⋊C411yescharacteristic
204C3×Dic3—32no—
205C3×Dic3—32no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C2×M16—91no—
209C2×M16—271no—
210C22×C8—271no—
211#108—92no—
212#108—272no—
213C33C2×#10811yescharacteristic, complemented, Sylow
214C23×C3—91no—
215C22×D3—32no—
216C22×D3—61no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—32no—
228C2×Dic3—61no—
229C4×D3—32no—
230C4×D3—32no—
231C4×D3—32no—
232C4×D3—32no—
233C4×D3—61no—
234C4×D3—61no—
235C4×D3—61no—
236C4×D3—61no—
237C3×C8—22no—
238C3×C8—22no—
239C3×C8—21no—
240C3×C8—42no—
241C3×C8—42no—
242C3×C8—94no—
243C3×C8—182no—
244C3×C8—181no—
245C3⋊C8—34no—
246C3⋊C8—34no—
247C3⋊C8—64no—
248C3⋊C8—62no—
249C3⋊C8—62no—
250C3⋊C8—62no—
251C3⋊C8—62no—
252C3⋊C8—122no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—81no—
261C2×C32D3×C4⋊C412yes—
262C2×C32C2×Dic3⋊C411yescharacteristic
263C3⋊D3—21no—
264C3⋊D3#82212yescomplemented
265C3×D3—34no—
266C3×D3—62no—
267C3×D3—62no—
268C3×D3—121no—
269C22×C4—91no—
270M16—92no—
271M16—91no—
272M16—181no—
273M16—541no—
274M16#111311yescharacteristic
275C2×C8—92no—
276C2×C8—181no—
277C2×C8—272no—
278C2×C8—272no—
279C2×C8—541no—
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—91no—
284C22×C3—91no—
285C22×C3—182no—
286C22×C3D3×Dic612yes—
287C22×C3#379711yescharacteristic
288C2×D3—32no—
289C2×D3—32no—
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—64no—
293C2×D3—122no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—91no—
301C3×C4—91no—
302C3×C4C4×D3212yes—
303C3×C4C4×D3211yescharacteristic
304C3×C4D3×#6712yes—
305C3×C4#381111yescharacteristic
306Dic3—32no—
307Dic3—32no—
308Dic3—61no—
309Dic3—61no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32D3×#10812yescomplemented
314C32C2×#82211yescharacteristic, complemented
315C23—91no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4C2×#111311yescharacteristic
322C8—21no—
323C8—94no—
324C8—181no—
325C8—182no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—92no—
335C2×C3—181no—
336C2×C3D3×Dic3⋊C412yes—
337C2×C3#6388311yescharacteristic
338D3—34no—
339D3—62no—
340D3—62no—
341D3—121no—
342C22—91no—
343C22—91no—
344C22—182no—
345C22#8904011yescharacteristic
346C4—91no—
347C4—91no—
348C4C4×#111311yescharacteristic
349C4#8904911yescharacteristic
350C3—21no—
351C3—22no—
352C3—41no—
353C3D3×#82212yescomplemented
354C3#1059097211yescharacteristic, complemented
355C2—21no—
356C2—92no—
357C2—181no—
358C2#1172368211yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. C3×#63649C2semidirect
5. #11722081C2semidirect
26. #63649D3semidirect
144. C3×C3⋊D3#108semidirect
213. C33C2×#108semidirect
264. C3⋊D3#822semidirect
313. C32D3×#108semidirect
314. C32C2×#822semidirect
353. C3D3×#822semidirect
354. C3#10590972semidirect

Maximal subgroups

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

subgroupindex
C3×M16×C3⋊D32
#117236562
C2×#888082
C3×#636492
#117220812
#105909723
D3×#8223