Finite Group Databasebeta

← #13155595

Subgroups — #13155595

Order 1728.

654 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 134 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×C4×#625C211yesmaximal, characteristic
2C2×C3×#3711C212yesmaximal, complemented
3#11722078C211yesmaximal, characteristic, complemented
4(C2×D3)⋊(C3⋊(C3⋊C8))C212yesmaximal, complemented
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10591156—32nomaximal
7#10590733—31nomaximal
8C2×C32×C4×D3C2212yes—
9C2×C3×#625C411yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C3×#3724C411yescharacteristic
15C3×(C2×D3)⋊Dic3C412yes—
16C3×#3711—24no—
17C3×C4×#67—32no—
18C2×C3×#208—32no—
19C2×C3×ΓL(1,𝔽25)—32no—
20C4×#625D311yescharacteristic
21C2×#3711D312yescomplemented
22#63657—32no—
23C8×C3⋊Dic3—31no—
24#63572—31no—
25Dic3⋊(C3⋊C8)—32no—
26(C2×D3)⋊(C3⋊C8)—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28(C2×D3)⋊(C3⋊C8)—62no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C32×D3C2×C412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C412yes—
36C32×C4×D3—22no—
37C32×C4×D3—22no—
38C3×#625—22no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40C3×C3⋊(C3⋊C8)—22no—
41C32×C3⋊C8D412yescomplemented
42#5128—92no—
43#5012—91no—
44C22×C32×C4—32no—
45C2×C3×#67—32no—
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3C2×D312yes—
48C2×C3×C4×D3C2×D312yes—
49C2×#625Dic311yescharacteristic
50C2×C32×C8—31no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—31no—
54C2×C3⋊(C3⋊C8)—61no—
55C2×C3⋊(C3⋊C8)—62no—
56C2×C3⋊(C3⋊C8)—121no—
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×#217—32no—
59C3×(C2×D3)⋊C4—32no—
60C3×Dic3⋊C4—32no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C3×#208—64no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—62no—
68C2×C3×C3⋊C8C2×D312yes—
69C3×ΓL(1,𝔽25)—64no—
70#3724Dic311yescharacteristic
71(C2×D3)⋊Dic3Dic312yes—
72#3711—24no—
73C22×C33C22×C411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—22no—
76C2×C32×D3—22no—
77C33×C4C2×D411yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3M1612yescomplemented
81C32×Dic3—22no—
82C2×C3×M16—92no—
83C3×C4×D4—91no—
84C4×#67—32no—
85C2×#208—32no—
86C2×ΓL(1,𝔽25)—32no—
87#830—92no—
88#830—92no—
89#830—182no—
90(C2×D3)⋊C8—92no—
91Dic3⋊C8—92no—
92C8×Dic3—92no—
93C8×Dic3—181no—
94C4⋊(C3⋊C8)—91no—
95C4×C3⋊C8—91no—
96C23×C32—32no—
97C22×C3×D3—22no—
98C22×C3×D3C4×D312yes—
99C22×C3×D3C2×Dic312yes—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—32no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4—62no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3C2×Dic311yescharacteristic
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C3×#67—64no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3C4×D312yes—
117C2×C3×Dic3C2×Dic312yes—
118C3×C4×D3—22no—
119C3×C4×D3—22no—
120C3×C4×D3—22no—
121C3×C4×D3—22no—
122C3×C4×D3—42no—
123C3×C4×D3—42no—
124#625—22no—
125C32×C8—32no—
126C3⋊(C3⋊C8)—22no—
127C3⋊(C3⋊C8)—61no—
128C3⋊(C3⋊C8)—62no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—122no—
131C3⋊(C3⋊C8)—122no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—122no—
136C3×C3⋊C8#6714yescomplemented
137#436—271noSylow
138C2×C33C4×D411yescharacteristic
139C2×C33C2×M1611yescharacteristic
140C2×C33#13111yescharacteristic
141C32×D3—42no—
142C2×C3×D4—91no—
143C22×C3×C4—32no—
144C22×C3×C4—32no—
145C22×C3×C4—62no—
146C2×#67—32no—
147C2×C4×D3D3212yes—
148C3×M16—184no—
149C2×C3×C8—32no—
150C2×C3×C8—61no—
151C2×C3×C8—92no—
152C3×C4⋊C4—91no—
153C3×C22⋊C4—92no—
154C3×C42—91no—
155#217—32no—
156(C2×D3)⋊C4—32no—
157Dic3⋊C4—32no—
158C4×Dic3—32no—
159C4×Dic3—61no—
160#208—64no—
161C2×C3⋊C8—32no—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—92no—
166C2×C3⋊C8—181no—
167C2×C3⋊C8—182no—
168C2×C3⋊C8—361no—
169C2×C3⋊C8D3211yescharacteristic
170ΓL(1,𝔽25)—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32—62no—
175C22×C32—62no—
176C22×C32—62no—
177C22×C32C2×C4×D312yes—
178C22×C32C22×Dic311yescharacteristic
179C2×C3×D3—24no—
180C2×C3×D3—22no—
181C2×C3×D3—22no—
182C2×C3×D3—22no—
183C2×C3×D3—22no—
184C2×C3×D3—22no—
185C2×C3×D3—22no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—22no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—62no—
195C32×C4#23512yes—
196C32×C4D3×D411yescharacteristic
197C32×C4#23711yescharacteristic
198C32×C4C2×#6712yes—
199C3⋊Dic3—21no—
200C3⋊Dic3#20812yescomplemented
201C3×Dic3—22no—
202C3×Dic3—22no—
203C3×Dic3—34no—
204C3×Dic3—42no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C2×M16—92no—
209C4×D4—91no—
210C4⋊C8—271no—
211C22⋊C8—272no—
212C4×C8—271no—
213C33#43611yescharacteristic, complemented, Sylow
214C23×C3—32no—
215C23×C3—32no—
216C23×C3—62no—
217C22×D3D3×Dic312yes—
218C3×D4—182no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—122no—
232C2×C3×C4—122no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×C3×C4C2×D3212yes—
235#67—64no—
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3D3×Dic312yes—
241C4×D3—22no—
242C4×D3—22no—
243C3×C8—34no—
244C3×C8—62no—
245C3×C8—182no—
246C3⋊C8—62no—
247C3⋊C8—62no—
248C3⋊C8—122no—
249C3⋊C8—182no—
250C3⋊C8—181no—
251C3⋊C8—182no—
252C3⋊C8—362no—
253C3⋊C8—362no—
254C3⋊C8#62512yescomplemented
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—122no—
265C2×C32D3×M1612yes—
266C2×C32#90712yes—
267C2×C32C2×#20811yescharacteristic
268C2×C32C4×#6712yes—
269C2×C32D4×Dic311yescharacteristic
270C2×C32#94811yescharacteristic
271C3×D3—44no—
272C3×D3—42no—
273C3×D3—42no—
274C2×D4—91no—
275C22×C4—32no—
276M16—184no—
277C2×C8—31no—
278C2×C8—92no—
279C2×C8—271no—
280C4⋊C4—91no—
281C22⋊C4—92no—
282C42—91no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—41no—
286C22×C3—64no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3C4×D3211yescharacteristic
296C22×C3C2×D3×Dic312yes—
297C2×D3—22no—
298C2×D3—22no—
299C2×D3—22no—
300C3×C4—22no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—62no—
307C3×C4—62no—
308C3×C4—92no—
309C3×C4—122no—
310C3×C4—181no—
311C3×C4#379811yescharacteristic
312C3×C4#380412yes—
313C3×C4C2×#62511yescharacteristic
314C3×C4D3×#6712yes—
315Dic3—22no—
316Dic3—34no—
317Dic3—62no—
318Dic3—62no—
319Dic3—62no—
320C32—21no—
321C32—22no—
322C32—41no—
323C32#501211yescharacteristic, complemented
324C32#512812yescomplemented
325C23—32no—
326D4—182no—
327C2×C4—32no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4D3311yescharacteristic
333C8—32no—
334C8—182no—
335C8—541no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—124no—
346C2×C3—122no—
347C2×C3—122no—
348C2×C3#6379111yescharacteristic
349C2×C3#6380511yescharacteristic
350C2×C3D3×#20812yes—
351C2×C3#6381512yes—
352C2×C3C4×#62511yescharacteristic
353C2×C3Dic3×#6712yes—
354D3—42no—
355C22—62no—
356C22—62no—
357C22—62no—
358C22D32×Dic311yescharacteristic
359C4—62no—
360C4—92no—
361C4—181no—
362C4D3×#62511yescharacteristic
363C4#8895911yescharacteristic
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059073311yescharacteristic, complemented
368C3#1059115612yescomplemented
369C2—122no—
370C2#1172281611yescharacteristic
371C2#1172282311yescharacteristic
372C2Dic3×#62511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×C3×#625C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3711C2semidirect
3. #11722078C2semidirect
4. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
21. C2×#3711D3semidirect
41. C32×C3⋊C8D4semidirect
80. C3×C3⋊Dic3M16semidirect
136. C3×C3⋊C8#67semidirect
200. C3⋊Dic3#208semidirect
213. C33#436semidirect
254. C3⋊C8#625semidirect
323. C32#5012semidirect
324. C32#5128semidirect
367. C3#10590733semidirect
368. C3#10591156semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6252
C2×C3×#37112
#117220782
(C2×D3)⋊(C3⋊(C3⋊C8))2
C3⋊C8×C3⋊Dic32
#105911563
#105907333