Finite Group Databasebeta

← #13162024

Subgroups — #13162024

Order 1728.

854 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 131 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2#11723656C212yesmaximal
3C3×#63663C212yesmaximal, complemented
4#11722074C212yesmaximal, complemented
5#10591639—31nomaximal
6D3×#836—32nomaximal
7C2×C33×Q8C2211yescharacteristic
8C3×Q8×C3⋊D3—22no—
9C3×Q8×C3⋊D3—22no—
10C2×C3×C4×C3⋊D3C412yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C412yes—
13C2×C3×#634C2211yescharacteristic
14C3×#3748C2214yes—
15#88809—22no—
16#88808—22no—
17C2×Q8×C3⋊D3D311yescharacteristic
18C2×C3×D3×Q8—32no—
19C2×C3×D3×Q8—61no—
20#63818—32no—
21D3×#208—34no—
22#63663D312yescomplemented
23C3×#836—32no—
24C3×#836—32no—
25C3×#836—62no—
26#63561—32no—
27#63560—34no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C33×Q8—22no—
30C2×C33×C4C2×C412yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3C2×C412yes—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3D412yes—
37C3×C4×C3⋊D3D412yes—
38C3×#634—22no—
39C3×#634—24no—
40C3×C3⋊(C3⋊C8)—24no—
41C2×#836—91no—
42D3×#101—92no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C2×D311yescharacteristic
48Q8×C3⋊D3—22no—
49Q8×C3⋊D3—22no—
50C2×C4×C3⋊D3Dic312yes—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634Dic312yes—
53C2×#634C2×D311yescharacteristic
54C3×D3×Q8—64no—
55C3×D3×Q8—64no—
56C3×D3×Q8—64no—
57C3×D3×Q8—64no—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—62no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—34no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—62no—
65C2×C3×Dic6—61no—
66#3748C2×D314yes—
67C3×#208—34no—
68C3×#208—34no—
69C3×#208—64no—
70#3712—62no—
71#3711—64no—
72#3710—62no—
73D3×C3⋊C8—64no—
74C22×C33C22×C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C22⋊C411yescharacteristic
78C33×C4—22no—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3C22⋊C412yes—
82C22×C3×Q8—91no—
83C2×D3×Q8—32no—
84C2×D3×Q8—61no—
85C2×#208—92no—
86D3×M16—94no—
87C3×#101—92no—
88#836—32no—
89#836—32no—
90#836—62no—
91#836—92no—
92#824—94no—
93C22×C3⋊D3C2×Dic311yescharacteristic
94C22×C3×D3—32no—
95C22×C3×D3—61no—
96C32×Q8—22no—
97C32×Q8—24no—
98C32×Q8—24no—
99C32×Q8—44no—
100C32×Q8—44no—
101C2×C32×C4—24no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—42no—
106C2×C32×C4—41no—
107C2×C32×C4C4×D314yes—
108C2×C32×C4C2×Dic312yes—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3C2×Dic312yes—
112C2×C3⋊Dic3C2×Dic311yescharacteristic
113C4×C3⋊D3—22no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3#6712yes—
116C4×C3⋊D3#6712yes—
117#634—22no—
118#634—24no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C3×C4×D3—34no—
124C3×C4×D3—34no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—64no—
128C3×C4×D3—64no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131C3×Dic6—68no—
132C3×Dic6—68no—
133C3×Dic6—64no—
134C3×Dic6—64no—
135C3⋊(C3⋊C8)—24no—
136C3×C3⋊C8—64no—
137C3×C3⋊C8—64no—
138C3×C3⋊C8—124no—
139C2×#101—271noSylow
140C2×C33—21no—
141C2×C33C2×C22⋊C411yescharacteristic
142C3×C3⋊D3—21no—
143C3×C3⋊D3#10112yescomplemented
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—41no—
147C2×C3×Q8—92no—
148C2×C3×Q8—91no—
149C2×C3×Q8—182no—
150C2×C3×Q8—182no—
151C2×C3×Q8D3212yes—
152C2×C3×Q8D3211yescharacteristic
153C22×C3×C4—91no—
154C22×C3×C4—92no—
155D3×Q8—64no—
156D3×Q8—64no—
157D3×Q8—64no—
158D3×Q8—64no—
159C2×C4×D3—32no—
160C2×C4×D3—34no—
161C2×C4×D3—62no—
162C2×C4×D3—61no—
163C2×Dic6—34no—
164C2×Dic6—32no—
165C2×Dic6—62no—
166C2×Dic6—61no—
167C3×M16—94no—
168#208—34no—
169#208—34no—
170#208—64no—
171#208—182no—
172C2×C3⋊C8—182no—
173ΓL(1,𝔽25)—184no—
174D3×C8—184no—
175C22×C32—22no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32C2×C4×D312yes—
179C22×C32C22×Dic311yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3⋊D3#21711yescharacteristic
182C2×C3⋊D3#21711yescharacteristic
183C2×C3×D3—32no—
184C2×C3×D3—32no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—22no—
194C32×C4—42no—
195C32×C4—44no—
196C32×C4—44no—
197C32×C4D3×D414yes—
198C32×C4C2×#6712yes—
199C3⋊Dic3—22no—
200C3⋊Dic3#21712yes—
201C3×Dic3—34no—
202C3×Dic3—62no—
203C3×Dic3—64no—
204C3×Dic3—64no—
205C22×Q8—91no—
206C2×M16—272no—
207#101—92no—
208#101—272no—
209C33C2×#10111yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—32no—
212C22×D3—61no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—22no—
216C3×Q8—44no—
217C3×Q8—44no—
218C3×Q8—184no—
219C3×Q8—182no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—91no—
227C2×C3×C4—92no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4D3×Dic314yes—
233C2×C3×C4#62412yes—
234C2×C3×C4C2×D3212yes—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—32no—
237C2×Dic3—34no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C4×D3—34no—
241C4×D3—34no—
242C4×D3—62no—
243C4×D3—62no—
244C4×D3—64no—
245C4×D3—64no—
246C4×D3—64no—
247C4×D3—64no—
248Dic6—68no—
249Dic6—68no—
250Dic6—64no—
251Dic6—64no—
252C3×C8—184no—
253C3⋊C8—64no—
254C3⋊C8—64no—
255C3⋊C8—124no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32—42no—
264C2×C32D3×C22⋊C412yes—
265C2×C32C2×#21711yescharacteristic
266C3⋊D3—21no—
267C3⋊D3#83612yescomplemented
268C3×D3—34no—
269C3×D3—62no—
270C3×D3—62no—
271C3×D3—62no—
272C2×Q8—92no—
273C2×Q8—91no—
274C2×Q8—182no—
275C2×Q8—182no—
276C2×Q8#111311yescharacteristic
277C22×C4—91no—
278C22×C4—92no—
279M16—94no—
280M16—542no—
281C2×C8—542no—
282C22×C3—21no—
283C22×C3—22no—
284C22×C3—41no—
285C22×C3—91no—
286C22×C3—91no—
287C22×C3—182no—
288C22×C3C2×D3×Dic312yes—
289C22×C3C2×#62411yescharacteristic
290C2×D3—32no—
291C2×D3—32no—
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—64no—
295C2×D3—64no—
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—22no—
299C3×C4—24no—
300C3×C4—22no—
301C3×C4—44no—
302C3×C4—44no—
303C3×C4—42no—
304C3×C4—92no—
305C3×C4—182no—
306C3×C4D3×#6714yes—
307C3×C4#381112yes—
308Dic3—34no—
309Dic3—62no—
310Dic3—64no—
311Dic3—64no—
312C32—22no—
313C32—21no—
314C32—41no—
315C32D3×#10112yescomplemented
316C32C2×#83611yescharacteristic, complemented
317C23—91no—
318Q8—22no—
319Q8—184no—
320Q8—182no—
321C2×C4—91no—
322C2×C4—92no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4#629412yes—
328C2×C4C2×#111311yescharacteristic
329C8—184no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—22no—
335C2×C3—42no—
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—92no—
339C2×C3—181no—
340C2×C3D3×#21712yes—
341C2×C3#6397411yescharacteristic
342D3—34no—
343D3—62no—
344D3—62no—
345D3—62no—
346C22—91no—
347C22—91no—
348C22—182no—
349C22C2×#629411yescharacteristic
350C4—22no—
351C4—92no—
352C4—182no—
353C4#8904912yes—
354C3—21no—
355C3—22no—
356C3—41no—
357C3D3×#83612yescomplemented
358C3#1059163911yescharacteristic, complemented
359C2—21no—
360C2—92no—
361C2—181no—
362C2#1172372311yescharacteristic
subgroupG/H
centerC2#11723723
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63663C2semidirect
4. #11722074C2semidirect
22. #63663D3semidirect
143. C3×C3⋊D3#101semidirect
209. C33C2×#101semidirect
267. C3⋊D3#836semidirect
315. C32D3×#101semidirect
316. C32C2×#836semidirect
357. C3D3×#836semidirect
358. C3#10591639semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117236562
C3×#636632
#117220742
#105916393
D3×#8363