Finite Group Databasebeta

← #13156588

Subgroups — #13156588

Order 1728.

866 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 124 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×#88810C211yesmaximal, characteristic, complemented
2C2×#88789C212yesmaximal, complemented
3C3×#63911C212yesmaximal, complemented
4C2×C3×#3713C211yesmaximal, characteristic, complemented
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10591482—32nomaximal
7#10590352—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×D12C2212yes—
10C2×C3×#626—24no—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×C4×Dic3C2211yescharacteristic
14#88810—22no—
15#88789—24no—
16C3×#3713—22no—
17C2×#3753—32no—
18C3×#940—32no—
19C2×C3×#213—32no—
20C3×#874—32no—
21#63911D312yescomplemented
22C2×#3714—32no—
23C2×#3713—32no—
24C2×#3713—32no—
25C2×#3713—31no—
26C2×#3713—62no—
27C2×#3713D311yescharacteristic, complemented
28C4×C3⋊(C3⋊C8)—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3—22no—
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—22no—
36C2×C32×Dic3D412yes—
37C32×D12—22no—
38C32×D12—22no—
39C3×#626—48no—
40C3⋊(C3⋊(C3⋊C8))D412yescomplemented
41C3×C3⋊(C3⋊C8)D412yescomplemented
42#5151—91no—
43#5073—92no—
44C2×C32×D4—32no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×C3×#67—64no—
47C2×C3×D12—22no—
48C2×C3×D12—32no—
49C2×C3×D12—32no—
50C2×C3×D12—62no—
51C2×C3×D12—64no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C2×D312yes—
54C2×#626—24no—
55C32×C42—31no—
56#3753—64no—
57C2×C3⋊(C3⋊C8)—32no—
58C2×C3⋊(C3⋊C8)—31no—
59C2×C3⋊(C3⋊C8)—61no—
60C2×C3⋊(C3⋊C8)—62no—
61C2×C3⋊(C3⋊C8)—121no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#213—64no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3C2×D312yes—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68#3714—64no—
69#3713—22no—
70#3713—64no—
71#3713—64no—
72#3713—68no—
73#3713—62no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—44no—
77C2×C32×D3—42no—
78C2×C32×D3—44no—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C32×Dic3D814yescomplemented
82C2×C3×D8—91no—
83C3×#127—92no—
84#940—32no—
85C2×#213—32no—
86C2×#213—92no—
87C2×#213—92no—
88C2×#213—92no—
89C2×#213—92no—
90C2×#213—182no—
91C2×D24—91no—
92#874—32no—
93C8×Dic3—91no—
94C4×C3⋊C8—92no—
95C4×C3⋊C8—181no—
96C23×C32—62no—
97C22×C3⋊D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C22×C3×D3—42no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—122no—
104C32×D4—62no—
105C32×D4—62no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—32no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C22×D311yescharacteristic
112C4⋊(C3⋊D3)—22no—
113C4⋊(C3⋊D3)—22no—
114C3×#67—128no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3#6714yes—
117C3×D12—24no—
118C3×D12—22no—
119C3×D12—22no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—42no—
123C3×D12—64no—
124C3×D12—62no—
125C3×D12—62no—
126C3×D12—62no—
127C3×D12—62no—
128C3×D12—122no—
129C3×D12—128no—
130#626—48no—
131C3⋊(C3⋊C8)—32no—
132C3⋊(C3⋊C8)—34no—
133C3⋊(C3⋊C8)—64no—
134C3⋊(C3⋊C8)—62no—
135C3⋊(C3⋊C8)—122no—
136C3⋊(C3⋊C8)#6712yescomplemented
137C3×C3⋊C8—34no—
138C3×C3⋊C8—62no—
139#493—271noSylow
140C2×C33#12711yescharacteristic
141C2×C33C2×D812yes—
142C3×C3⋊D3—82no—
143C32×D3—82no—
144C2×C3×D4—32no—
145C2×C3×D4—32no—
146C2×C3×D4—62no—
147C2×C3×D4—92no—
148C2×C3×D4—184no—
149C2×#67—64no—
150C2×D12—32no—
151C2×D12—32no—
152C2×D12—62no—
153C2×D12—64no—
154C2×D12D3212yes—
155C3×D8—182no—
156C2×C3×C8—91no—
157C3×C42—32no—
158C3×C42—61no—
159#213—64no—
160#213—184no—
161#213—184no—
162#213—184no—
163#213—184no—
164#213—188no—
165C4×Dic3D3211yescharacteristic
166C2×C3⋊C8—32no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—92no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—181no—
171C2×C3⋊C8—182no—
172C2×C3⋊C8—361no—
173D24—182no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—41no—
177C22×C32—122no—
178C22×C32—124no—
179C22×C32D3×D411yescharacteristic
180C22×C32C2×#6712yes—
181C2×C3⋊D3—42no—
182C2×C3⋊D3—44no—
183C2×C3×D3—44no—
184C2×C3×D3—44no—
185C2×C3×D3—44no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—84no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C2×C3×D3—244no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—34no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4D3×D412yes—
203C32×C4D3×D412yes—
204C32×C4D3×D411yescharacteristic
205C32×C4C2×#6711yescharacteristic
206C3×Dic3—24no—
207C3×Dic3#21318yescomplemented
208C2×D8—91no—
209C2×D8—271no—
210C2×D8—272no—
211#127—92no—
212C4×C8—271no—
213C33#49311yescharacteristic, complemented, Sylow
214C23×C3—62no—
215C23×C3—62no—
216C23×C3—122no—
217C23×C3—182no—
218C22×D3—22no—
219C22×D3—62no—
220C22×D3—62no—
221C22×D3—122no—
222C3×D4—64no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—122no—
228C3×D4—182no—
229C3×D4—182no—
230C3×D4—368no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—34no—
234C2×C3×C4—41no—
235C2×C3×C4—62no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C2×D3212yes—
238#67—128no—
239C2×Dic3#62512yes—
240D12—22no—
241D12—22no—
242D12—64no—
243D12—62no—
244D12—62no—
245D12—62no—
246D12—62no—
247D12—122no—
248D12—128no—
249C3×C8—92no—
250C3⋊C8—34no—
251C3⋊C8—62no—
252C3⋊C8—92no—
253C3⋊C8—94no—
254C3⋊C8—182no—
255C3⋊C8—184no—
256C3⋊C8—362no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—24no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32—242no—
264C2×C32D3×D812yes—
265C2×C32C2×#21314yes—
266C2×C32#94012yes—
267C2×C32#94011yescharacteristic
268C3⋊D3—82no—
269C3×D3—84no—
270C3×D3—82no—
271C3×D3—82no—
272C3×D3—244no—
273C3×D3—242no—
274C3×D3—242no—
275C2×D4—32no—
276C2×D4—92no—
277C2×D4—184no—
278D8—182no—
279D8—542no—
280D8—544no—
281C2×C8—91no—
282C2×C8—271no—
283C42—31no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—41no—
287C22×C3—124no—
288C22×C3—124no—
289C22×C3—122no—
290C22×C3—124no—
291C22×C3—122no—
292C22×C3—244no—
293C22×C3—364no—
294C22×C3—362no—
295C22×C3C2×#62511yescharacteristic
296C22×C3D3×#6712yes—
297C2×D3—44no—
298C2×D3—42no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—122no—
302C2×D3—124no—
303C2×D3—122no—
304C2×D3—244no—
305C3×C4—21no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—38no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—64no—
313C3×C4#380211yescharacteristic
314C3×C4#380211yescharacteristic
315C3×C4D3×#6712yes—
316C3×C4#381112yes—
317Dic3#371314yescomplemented
318C32—21no—
319C32—22no—
320C32—41no—
321C32#507312yescomplemented
322C32#515111yescharacteristic, complemented
323C23—62no—
324C23—182no—
325D4—62no—
326D4—62no—
327D4—182no—
328D4—182no—
329D4—368no—
330C2×C4—32no—
331C2×C4D3311yescharacteristic
332C8—92no—
333C8—272no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—24no—
338C2×C3—41no—
339C2×C3—42no—
340C2×C3—244no—
341C2×C3—242no—
342C2×C3—242no—
343C2×C3—722no—
344C2×C3C2×#371312yes—
345C2×C3#6391411yescharacteristic
346C2×C3D3×#21314yes—
347C2×C3#6397912yes—
348D3—82no—
349D3—244no—
350D3—242no—
351D3—242no—
352C22—122no—
353C22—124no—
354C22—362no—
355C22—364no—
356C22D3×#62511yescharacteristic
357C4—34no—
358C4#8895311yescharacteristic
359C4#8895411yescharacteristic
360C3—22no—
361C3—21no—
362C3—41no—
363C3#1059035211yescharacteristic, complemented
364C3#1059148212yescomplemented
365C2—242no—
366C2—722no—
367C2D3×#371312yes—
368C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#493
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88810C2semidirect
2. C2×#88789C2semidirect
3. C3×#63911C2semidirect
4. C2×C3×#3713C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
21. #63911D3semidirect
27. C2×#3713D3semidirect
40. C3⋊(C3⋊(C3⋊C8))D4semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
81. C32×Dic3D8semidirect
136. C3⋊(C3⋊C8)#67semidirect
207. C3×Dic3#213semidirect
213. C33#493semidirect
317. Dic3#3713semidirect
321. C32#5073semidirect
322. C32#5151semidirect
363. C3#10590352semidirect
364. C3#10591482semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
C2×#887892
C3×#639112
C2×C3×#37132
Dic3×C3⋊(C3⋊C8)2
#105914823
#105903523