Finite Group Databasebeta

← #13140092

Subgroups — #13140092

Order 1728.

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

normal subgroups only

This group has 615 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×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#88448C212yesmaximal
3#11720574C211yesmaximal, characteristic
4#63658⋊C3C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591360—31nomaximal
7#10591206—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4×(C3⋊Dic3)⋊C3C414yescomplemented
14#88448—24no—
15C2×C4×C3⋊Dic3D311yescharacteristic
16C2×C3×C4×Dic3—31no—
17C2×#3724—32no—
18C2×Dic3⋊Dic3—32no—
19#63658D311yescharacteristic
20C3×C3⋊#95—31no—
21C3×C3⋊#95—31no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—61no—
24#63579—32no—
25#63579—32no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×#1084—22no—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3C2×C412yes—
37C2×(C3⋊Dic3)⋊C3C2×C412yes—
38C2×(C3⋊Dic3)⋊C3D412yes—
39C2×(C3⋊Dic3)⋊C3Q812yes—
40#4936—91no—
41#4650—91no—
42#4650—91no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D311yescharacteristic
48C22×C3⋊Dic3C2×D311yescharacteristic
49C22×C3×Dic3—31no—
50C22×C3×Dic3—31no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—62no—
55C4×C3⋊Dic3Dic314yescomplemented
56C3×C4×Dic3—34no—
57#3724—64no—
58Dic3⋊Dic3—64no—
59C22×H(3,𝔽3)C22×C411yescharacteristic
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)#4112yes—
63C22×H(3,𝔽3)#4112yes—
64C4×H(3,𝔽3)—22no—
65#1084—42no—
66(C3⋊Dic3)⋊C3—22no—
67(C3⋊Dic3)⋊C3—42no—
68(C3⋊Dic3)⋊C3C4⋊C414yes—
69C2×C3×C42—91no—
70C2×C4⋊Dic3—92no—
71C2×Dic3⋊C4—92no—
72C2×Dic3⋊C4—92no—
73C2×C4×Dic3—31no—
74C2×C4×Dic3—91no—
75C3×#95—91no—
76C3×#95—91no—
77C3⋊#95—31no—
78C3⋊#95—31no—
79C3⋊#95—31no—
80C3⋊#95—61no—
81C3⋊#95—91no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C3⋊#95—92no—
85C23×C32—21no—
86C23×C32C22×D311yescharacteristic
87C23×C32C22×D311yescharacteristic
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—42no—
92C2×C32×C4C4×D312yes—
93C2×C32×C4C2×Dic312yes—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3Dic612yes—
99C2×C3⋊Dic3D1212yes—
100C2×C3⋊Dic3C2×Dic312yes—
101C2×C3⋊Dic3C2×Dic312yes—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—122no—
118C2×C3×Dic3—122no—
119#382—271noSylow
120C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
121C2×H(3,𝔽3)#11711yescharacteristic
122C2×H(3,𝔽3)#11711yescharacteristic
123C2×H(3,𝔽3)#12412yes—
124C2×H(3,𝔽3)#12512yes—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—61no—
128C22×C3×C4—91no—
129C22×C3×C4—91no—
130C22×C3×C4—92no—
131C22×C3×C4—92no—
132C22×C3×C4D3211yescharacteristic
133C22×Dic3—31no—
134C22×Dic3—31no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—62no—
139C22×Dic3—91no—
140C22×Dic3—91no—
141C22×Dic3—92no—
142C3×C42—94no—
143C4⋊Dic3—184no—
144Dic3⋊C4—184no—
145Dic3⋊C4—184no—
146C4×Dic3—34no—
147C4×Dic3—94no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—22no—
153C22×C32C2×Dic611yescharacteristic
154C22×C32C2×C4×D311yescharacteristic
155C22×C32C2×D1211yescharacteristic
156C22×C32#23512yes—
157C22×C32#23512yes—
158C22×C32#23512yes—
159C22×C32D3×D411yescharacteristic
160C22×C32#23712yes—
161C22×C32D3×Q811yescharacteristic
162C22×C32C22×Dic311yescharacteristic
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C3⋊Dic3—22no—
167C3⋊Dic3—42no—
168C3⋊Dic3C4⋊Dic314yes—
169C3×Dic3—34no—
170C3×Dic3—62no—
171C3×Dic3—124no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C3×Dic3—122no—
175C2×C4⋊C4—272no—
176C2×C42—91no—
177#95—91no—
178#95—91no—
179#95—272no—
180H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3C2×D3211yescharacteristic
185C2×C3×C4—22no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—92no—
194C2×C3×C4—92no—
195C2×C3×C4—122no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4—182no—
202C2×C3×C4—182no—
203C2×C3×C4D3×Dic312yes—
204C2×Dic3—32no—
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—32no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—64no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—92no—
220C2×Dic3—92no—
221C2×Dic3—92no—
222C2×Dic3—92no—
223C2×Dic3—122no—
224C2×Dic3—122no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—22no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32#87012yes—
235C2×C32#87211yescharacteristic
236C2×C32#87512yes—
237C2×C32#87711yescharacteristic
238C2×C32#88512yes—
239C2×C32#88912yes—
240C2×C32D3×C4⋊C411yescharacteristic
241C2×C32C2×C4⋊Dic311yescharacteristic
242C2×C32#92611yescharacteristic
243C2×C32#92611yescharacteristic
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—92no—
247C22×C4—92no—
248C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C4⋊C4—544no—
250C42—94no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—32no—
260C22×C3—32no—
261C22×C3—61no—
262C22×C3—62no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—62no—
266C22×C3D3×Dic611yescharacteristic
267C22×C3#379812yes—
268C22×C3D3×D1211yescharacteristic
269C22×C3C2×D3×Dic311yescharacteristic
270C22×C3#380412yes—
271C3×C4—22no—
272C3×C4—62no—
273C3×C4—62no—
274C3×C4—94no—
275C3×C4—122no—
276C3×C4—182no—
277C3×C4—362no—
278C3×C4—362no—
279Dic3—34no—
280Dic3—62no—
281Dic3—94no—
282Dic3—122no—
283Dic3—122no—
284Dic3—124no—
285Dic3—122no—
286Dic3—182no—
287Dic3—362no—
288C32—21no—
289C32#465011yescharacteristic, complemented
290C32#493611yescharacteristic, complemented
291C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C2×C4—22no—
293C2×C4—92no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4((C3⋊D3)⋊C3)⋊C412yes—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—32no—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—32no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—61no—
316C2×C3—62no—
317C2×C3—62no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3#6385212yes—
321C2×C3#6385511yescharacteristic
322C2×C3#6386712yes—
323C2×C3D3×C4⋊Dic311yescharacteristic
324C2×C3#6395511yescharacteristic
325C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
326C22#8865012yes—
327C22#8865411yescharacteristic
328C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
329C22#8867112yes—
330C4—22no—
331C4—94no—
332C4—182no—
333C4—362no—
334C4—362no—
335C3—31no—
336C3—31no—
337C3—61no—
338C3#1059120611yescharacteristic
339C2#1172109112yes—
340C2#1172110312yes—
341C2#1172110511yescharacteristic
342C2#1172112911yescharacteristic
343C2#1172129611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#382
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C4×(C3⋊Dic3)⋊C3C4semidirect
55. C4×C3⋊Dic3Dic3semidirect
180. H(3,𝔽3)#382semidirect
289. C32#4650semidirect
290. C32#4936semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊Dic3)⋊C32
C2×#884482
#117205742
#63658⋊C32
#117203342
#105913603
#105912063