Finite Group Databasebeta

← #13173396

Subgroups — #13173396

Order 1728.

960 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#88989C212yesmaximal, complemented
3#11723112C212yesmaximal, complemented
4#11723103C212yesmaximal, complemented
5#10593803—31nomaximal
6C23×C3×C3⋊D3C411yescharacteristic
7C22×C33×C4C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C3×C4×C3⋊D3—24no—
10C2×C3×C4×C3⋊D3C411yescharacteristic
11C2×C3×C4×C3⋊D3C411yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3⋊#622C2214yescomplemented
15#88989—24no—
16C22×C4×C3⋊D3D311yescharacteristic
17C22×C3×C4×D3—62no—
18C2×#3788—32no—
19C22⋊#622—32no—
20#63778—32no—
21C23×C33C2×C411yescharacteristic
22C22×C3×C3⋊D3—22no—
23C22×C3×C3⋊D3—22no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4—22no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3D414yes—
37C3⋊#622—24no—
38#5207—91nomaximal
39C23×C3⋊D3Dic311yescharacteristic
40C23×C3×D3—62no—
41C22×C32×C4—22no—
42C22×C32×C4—42no—
43C22×C32×C4Dic311yescharacteristic
44C22×C3⋊Dic3C2×D311yescharacteristic
45C2×C4×C3⋊D3—24no—
46C2×C4×C3⋊D3Dic311yescharacteristic
47C2×C4×C3⋊D3Dic311yescharacteristic
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C22×C3×Dic3—62no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—62no—
53C2×C3×C4×D3—62no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—616no—
56C2×#622—34no—
57#3788—64no—
58C22×C33—21no—
59C22×C33—21no—
60C22×C33M1612yescomplemented
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—28no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C2×C3×C3⋊D3—21no—
66C2×C3×C3⋊D3—21no—
67C2×C3×C3⋊D3C22⋊C412yes—
68C2×C3×C3⋊D3M1612yescomplemented
69C33×C4—21no—
70C33×C4C22⋊C412yes—
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3C2×D412yes—
73C23×C3×C4—91no—
74C22×C4×D3—62no—
75C2×#208—92no—
76#830—92no—
77#830—92no—
78C23×C32—22no—
79C23×C32—42no—
80C23×C32C2×Dic311yescharacteristic
81C22×C3⋊D3—22no—
82C22×C3⋊D3—22no—
83C22×C3⋊D3—22no—
84C22×C3⋊D3C2×Dic311yescharacteristic
85C22×C3⋊D3C2×Dic311yescharacteristic
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—68no—
89C22×C3×D3—68no—
90C22×C3×D3—68no—
91C2×C32×C4—22no—
92C2×C32×C4—28no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—48no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4C2×Dic311yescharacteristic
99C2×C32×C4C2×Dic311yescharacteristic
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3C2×Dic311yescharacteristic
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4×C3⋊D3—22no—
104C4×C3⋊D3—22no—
105C4×C3⋊D3—22no—
106C4×C3⋊D3#6714yes—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—68no—
110C3×C4×D3—68no—
111C3×C4×D3—68no—
112C3×C4×D3—68no—
113C3×C4×D3—68no—
114#622—64no—
115#407—271noSylow
116C2×C33—22no—
117C2×C33C2×C22⋊C411yescharacteristic
118C2×C33C2×M1611yescharacteristic
119C2×C33C2×M1611yescharacteristic
120C3×C3⋊D3—22no—
121C3×C3⋊D3—22no—
122C24×C3—91no—
123C23×D3—62no—
124C22×C3×C4—22no—
125C22×C3×C4—42no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C22×C3×C4—91no—
130C22×C3×C4—91no—
131C22×C3×C4—184no—
132C22×C3×C4#15711yescharacteristic
133C22×Dic3—62no—
134C2×C4×D3—62no—
135C2×C4×D3—62no—
136C2×C4×D3—62no—
137C2×C4×D3—62no—
138C2×C4×D3—616no—
139#208—184no—
140C2×C3⋊C8—94no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—24no—
144C22×C32—24no—
145C22×C32—22no—
146C22×C32—24no—
147C22×C32—44no—
148C22×C32—44no—
149C22×C32—44no—
150C22×C32—42no—
151C22×C32#20812yescomplemented
152C22×C32C22×Dic311yescharacteristic
153C2×C3⋊D3—28no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3—21no—
158C2×C3⋊D3#20812yescomplemented
159C2×C3⋊D3#21712yes—
160C2×C3×D3—64no—
161C2×C3×D3—64no—
162C2×C3×D3—64no—
163C2×C3×D3—632no—
164C2×C3×D3—64no—
165C2×C3×D3—64no—
166C2×C3×D3—64no—
167C32×C4—21no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—44no—
171C32×C4—44no—
172C32×C4#21712yes—
173C3⋊Dic3—21no—
174C3⋊Dic3C2×#6712yes—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C23×C4—91no—
178C2×M16—272no—
179C22⋊C8—272no—
180C22⋊C8—272no—
181C33#40711yescharacteristic, complemented, Sylow
182C23×C3—22no—
183C23×C3—42no—
184C23×C3—91no—
185C23×C3—91no—
186C23×C3—182no—
187C23×C3—182no—
188C23×C3—182no—
189C23×C3C2×#15711yescharacteristic
190C22×D3—62no—
191C22×D3—62no—
192C22×D3—68no—
193C22×D3—68no—
194C22×D3—68no—
195C2×C3×C4—28no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—48no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—91no—
203C2×C3×C4—94no—
204C2×C3×C4—91no—
205C2×C3×C4—182no—
206C2×C3×C4—182no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4C2×#15711yescharacteristic
210C2×C3×C4C2×#15711yescharacteristic
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—68no—
214C4×D3—68no—
215C4×D3—68no—
216C4×D3—68no—
217C4×D3—68no—
218C3⋊C8—184no—
219C2×C32—22no—
220C2×C32—28no—
221C2×C32—22no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—48no—
225C2×C32—42no—
226C2×C32—42no—
227C2×C32—42no—
228C2×C32C2×#20811yescharacteristic
229C2×C32C2×#20811yescharacteristic
230C2×C32C2×#21711yescharacteristic
231C3⋊D3—22no—
232C3⋊D3—22no—
233C3×D3—68no—
234C3×D3—68no—
235C24—91no—
236C22×C4—91no—
237C22×C4—91no—
238C22×C4—91no—
239C22×C4—91no—
240C22×C4—91no—
241C22×C4—184no—
242C22×C4C3⋊#15711yescharacteristic
243M16—544no—
244C2×C8—274no—
245C22×C3—24no—
246C22×C3—24no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—22no—
250C22×C3—24no—
251C22×C3—44no—
252C22×C3—44no—
253C22×C3—42no—
254C22×C3—44no—
255C22×C3—92no—
256C22×C3—92no—
257C22×C3—188no—
258C22×C3—181no—
259C22×C3—181no—
260C22×C3—181no—
261C22×C3—181no—
262C22×C3#378812yescomplemented
263C22×C3C22×#15711yescharacteristic
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—64no—
267C2×D3—632no—
268C2×D3—64no—
269C2×D3—64no—
270C2×D3—64no—
271C3×C4—24no—
272C3×C4—21no—
273C3×C4—24no—
274C3×C4—44no—
275C3×C4—44no—
276C3×C4—92no—
277C3×C4—181no—
278C3×C4C22⋊#15712yes—
279Dic3—64no—
280Dic3—64no—
281C32—22no—
282C32—42no—
283C32#520711yescharacteristic, complemented
284C23—91no—
285C23—91no—
286C23—182no—
287C23—182no—
288C23—182no—
289C23C2×C3⋊#15711yescharacteristic
290C2×C4—22no—
291C2×C4—91no—
292C2×C4—94no—
293C2×C4—91no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—182no—
298C2×C4C2×C3⋊#15711yescharacteristic
299C2×C4C2×C3⋊#15711yescharacteristic
300C8—544no—
301C2×C3—28no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—48no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—42no—
310C2×C3—182no—
311C2×C3—182no—
312C2×C3C2×#378811yescharacteristic
313C2×C3#6428311yescharacteristic
314C2×C3C2×C22⋊#15711yescharacteristic
315D3—68no—
316D3—68no—
317C22—21no—
318C22—21no—
319C22—92no—
320C22—92no—
321C22—188no—
322C22—181no—
323C22—181no—
324C22—181no—
325C22—181no—
326C22#8898912yescomplemented
327C22C22×C3⋊#15711yescharacteristic
328C4—21no—
329C4—92no—
330C4—181no—
331C4#8899412yes—
332C3—22no—
333C3—42no—
334C3#1059380311yescharacteristic, complemented
335C2—22no—
336C2—182no—
337C2—182no—
338C2C2×#8898911yescharacteristic
339C2#1172451111yescharacteristic
340C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. #11723112C2semidirect
4. #11723103C2semidirect
14. C2×C3⋊#622C22semidirect
60. C22×C33M16semidirect
68. C2×C3×C3⋊D3M16semidirect
151. C22×C32#208semidirect
158. C2×C3⋊D3#208semidirect
181. C33#407semidirect
262. C22×C3#3788semidirect
283. C32#5207semidirect
326. C22#88989semidirect
334. C3#10593803semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×#889892
#117231122
#117231032
#105938033
#52079