Finite Group Databasebeta

← #13161824

Subgroups — #13161824

Order 1728.

702 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 147 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×C4×C3⋊Dic3C211yesmaximal, characteristic
2C3×#63657C212yesmaximal
3#11722078C212yesmaximal, complemented
4#11722071C212yesmaximal, complemented
5#10591066—31nomaximal
6#10591063—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C22×C3×C3⋊Dic3C411yescharacteristic
10C3×C4×C3⋊Dic3C411yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2214yes—
15C2×C4×C3⋊Dic3D311yescharacteristic
16C2×C3×C4×Dic3—32no—
17C2×C3×C4×Dic3—61no—
18#63657D312yes—
19C3×#830—32no—
20C3×#830—32no—
21C3×#830—62no—
22#63572—32no—
23Dic3⋊(C3⋊C8)—34no—
24#63555—32no—
25#63554—34no—
26C23×C33C2×C411yescharacteristic
27C2×C33×C4—22no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C3⋊(C3⋊C8)—24no—
37#4926—91no—
38#4719—92no—
39C22×C32×C4—21no—
40C22×C32×C4—22no—
41C22×C32×C4—41no—
42C22×C32×C4C2×D312yes—
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3Dic311yescharacteristic
45C22×C3⋊Dic3Dic311yescharacteristic
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—61no—
50C4×C3⋊Dic3Dic311yescharacteristic
51C4×C3⋊Dic3Dic311yescharacteristic
52C4×C3⋊Dic3C2×D311yescharacteristic
53C4×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)C2×D314yes—
55C3×C4×Dic3—32no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—34no—
64C2×C3×C3⋊C8—34no—
65C2×C3×C3⋊C8—64no—
66C22×C33—21no—
67C22×C33—21no—
68C22×C33M1612yes—
69C22×C33C22×C411yescharacteristic
70C33×C4—21no—
71C33×C4#4112yes—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3M1612yes—
76C2×C3×C42—91no—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—61no—
79C3×C22⋊C8—92no—
80#830—32no—
81#830—32no—
82#830—62no—
83#815—94no—
84Dic3⋊C8—94no—
85C4⋊(C3⋊C8)—92no—
86#803—92no—
87C23×C32—22no—
88C23×C32—21no—
89C23×C32—41no—
90C23×C32C4×D312yes—
91C23×C32C2×Dic311yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—44no—
102C2×C32×C4—44no—
103C2×C32×C4C4×D312yes—
104C2×C32×C4C2×Dic311yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C2×C3⋊Dic3C2×Dic311yescharacteristic
111C2×C3⋊Dic3C2×Dic311yescharacteristic
112C2×C3⋊Dic3C2×Dic311yescharacteristic
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C3⋊(C3⋊C8)—24no—
126C3×C3⋊C8—64no—
127C3×C3⋊C8—64no—
128C3×C3⋊C8—124no—
129#432—271noSylow
130C2×C33—22no—
131C2×C33#11711yescharacteristic
132C2×C33C2×M1611yescharacteristic
133C2×C33C2×M1611yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—41no—
137C22×C3×C4—91no—
138C22×C3×C4—91no—
139C22×C3×C4D3212yes—
140C22×C3×C4D3211yescharacteristic
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—61no—
144C22×Dic3—61no—
145C2×C3×C8—94no—
146C3×C42—91no—
147C3×C42—91no—
148C3×C42—91no—
149C3×C42—91no—
150C4×Dic3—32no—
151C4×Dic3—32no—
152C4×Dic3—32no—
153C4×Dic3—32no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8—34no—
160C2×C3⋊C8—64no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—24no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—42no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—42no—
177C22×C32ΓL(1,𝔽25)14yes—
178C22×C32#20812yes—
179C22×C32C2×C4×D312yes—
180C22×C32C22×Dic311yescharacteristic
181C32×C4—21no—
182C32×C4—22no—
183C32×C4—22no—
184C32×C4—24no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—42no—
189C32×C4#23512yes—
190C32×C4#23714yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3#20812yes—
195C3×Dic3—34no—
196C3×Dic3—62no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C2×C42—91no—
204C4⋊C8—272no—
205C22⋊C8—92no—
206#97—272no—
207C33#43211yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3D3×Dic312yes—
212C23×C3#62411yescharacteristic
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—44no—
221C2×C3×C4—44no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4D3×Dic312yes—
231C2×C3×C4#62411yescharacteristic
232C2×C3×C4C2×D3212yes—
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×Dic3—64no—
246C3×C8—184no—
247C3⋊C8—64no—
248C3⋊C8—64no—
249C3⋊C8—124no—
250C2×C32—22no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—44no—
263C2×C32—44no—
264C2×C32#87712yes—
265C2×C32C2×ΓL(1,𝔽25)12yes—
266C2×C32D3×M1612yes—
267C2×C32C2×#20811yescharacteristic
268C2×C32C2×#20811yescharacteristic
269C2×C32#92611yescharacteristic
270C22×C4—91no—
271C22×C4—91no—
272C22×C4#111311yescharacteristic
273C2×C8—94no—
274C42—91no—
275C42—91no—
276C42—91no—
277C42—91no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—41no—
293C22×C3—42no—
294C22×C3#371114yes—
295C22×C3#371212yes—
296C22×C3C2×D3×Dic312yes—
297C22×C3C2×#62411yescharacteristic
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—24no—
302C3×C4—22no—
303C3×C4—42no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—92no—
307C3×C4—181no—
308C3×C4—181no—
309C3×C4—181no—
310C3×C4#380414yes—
311C3×C4#380512yes—
312Dic3—34no—
313Dic3—62no—
314Dic3—62no—
315Dic3—62no—
316Dic3—62no—
317Dic3—62no—
318Dic3—62no—
319Dic3—62no—
320C32—22no—
321C32—21no—
322C32—41no—
323C32#471912yescomplemented
324C32#492611yescharacteristic, complemented
325C23#629411yescharacteristic
326C2×C4—22no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4#629411yescharacteristic
334C2×C4C2×#111311yescharacteristic
335C8—184no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3—22no—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—44no—
346C2×C3—44no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3D3×#20812yes—
351C2×C3#6381811yescharacteristic
352C2×C3C2×#371112yes—
353C2×C3C2×#371211yescharacteristic
354C2×C3#6395512yes—
355C2×C3#6395711yescharacteristic
356C22—21no—
357C22—21no—
358C22#8880912yes—
359C22C2×#629411yescharacteristic
360C4—21no—
361C4—92no—
362C4—181no—
363C4—181no—
364C4—181no—
365C4#8904612yes—
366C3—21no—
367C3—22no—
368C3—41no—
369C3#1059106312yescomplemented
370C3#1059106611yescharacteristic, complemented
371C2—22no—
372C2#1172365611yescharacteristic
373C2C2×#8880911yescharacteristic
374C2#1172371511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722078C2semidirect
4. #11722071C2semidirect
207. C33#432semidirect
323. C32#4719semidirect
324. C32#4926semidirect
369. C3#10591063semidirect
370. C3#10591066semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C3×#636572
#117220782
#117220712
#105910663
#105910633