Finite Group Databasebeta

← #13140006

Subgroups — #13140006

Order 1728.

561 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 119 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⋊#1084C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#11720334C212yesmaximal
5#11720334C212yesmaximal
6#10591227—31nomaximal
7#10591225—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊#1084—22no—
14#3750⋊C3—22no—
15C2×#3750D311yescharacteristic
16C2×C3×C4⋊Dic3—31no—
17C2×C3×C4⋊Dic3—31no—
18C2×C3×C4⋊Dic3—61no—
19C2×C3×Dic3⋊C4—31no—
20#63658D311yescharacteristic
21C3×C3⋊#95—31no—
22#63579—32no—
23#63579—32no—
24#63579—32no—
25#63579—32no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)D412yes—
29C2×#1084—22no—
30C2×#1084—24no—
31C2×(C3⋊Dic3)⋊C3—21no—
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3Q812yes—
39#4986—91no—
40#4985—91no—
41#4659—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3C2×D311yescharacteristic
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D311yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—62no—
54#3750—22no—
55C3×C4⋊Dic3—64no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3—62no—
58C3×Dic3⋊C4—62no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C22×H(3,𝔽3)C2×Q811yescharacteristic
61C22×H(3,𝔽3)#4111yescharacteristic
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—41no—
68(C3⋊Dic3)⋊C3—42no—
69C2×C3×C4⋊C4—91no—
70C2×C3×C4⋊C4—91no—
71C2×C4⋊Dic3—31no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—61no—
74C2×Dic3⋊C4—31no—
75C2×Dic3⋊C4—91no—
76C3×#95—91no—
77C3⋊#95—31no—
78C3⋊#95—91no—
79C3⋊#95—92no—
80C3⋊#95—92no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C3⋊#95—92no—
85C23×C32—21no—
86C23×C32C22×D311yescharacteristic
87C23×C32C22×D311yescharacteristic
88C2×C32×C4—24no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4D1212yes—
93C2×C32×C4#6712yes—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3Dic612yes—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—124no—
116#400—271noSylow
117C2×H(3,𝔽3)C22⋊Q811yescharacteristic
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)#12411yescharacteristic
120C2×H(3,𝔽3)#12512yes—
121C2×H(3,𝔽3)#12612yes—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—92no—
127C22×C3×C4—91no—
128C22×C3×C4—92no—
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—31no—
131C22×Dic3—32no—
132C22×Dic3—32no—
133C22×Dic3—32no—
134C22×Dic3—31no—
135C22×Dic3—62no—
136C22×Dic3—91no—
137C22×Dic3—92no—
138C22×Dic3—91no—
139C3×C4⋊C4—182no—
140C3×C4⋊C4—182no—
141C4⋊Dic3—62no—
142C4⋊Dic3—62no—
143C4⋊Dic3—64no—
144Dic3⋊C4—62no—
145Dic3⋊C4—182no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32C2×Dic611yescharacteristic
152C22×C32C2×D1211yescharacteristic
153C22×C32#23511yescharacteristic
154C22×C32#23512yes—
155C22×C32#23711yescharacteristic
156C22×C32#23712yes—
157C22×C32#23712yes—
158C22×C32D3×Q811yescharacteristic
159C22×C32#23912yes—
160C22×C32C2×#6711yescharacteristic
161C32×C4—24no—
162C32×C4—22no—
163C32×C4—22no—
164C3⋊Dic3—22no—
165C3⋊Dic3—42no—
166C3⋊Dic3—41no—
167C3×Dic3—62no—
168C3×Dic3—124no—
169C3×Dic3—122no—
170C3×Dic3—121no—
171C3×Dic3—122no—
172C3×Dic3—122no—
173C2×C4⋊C4—91no—
174C2×C4⋊C4—91no—
175#95—91no—
176#95—272no—
177#95—272no—
178H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
179C23×C3—31no—
180C23×C3—31no—
181C23×C3—61no—
182C23×C3C2×D3211yescharacteristic
183C2×C3×C4—22no—
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—64no—
189C2×C3×C4—62no—
190C2×C3×C4—92no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—181no—
194C2×C3×C4—182no—
195C2×C3×C4—182no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4—184no—
199C2×C3×C4—182no—
200C2×C3×C4#62612yes—
201C2×Dic3—32no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×Dic3—64no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—61no—
214C2×Dic3—92no—
215C2×Dic3—124no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—182no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—182no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32#87511yescharacteristic
229C2×C32#87912yes—
230C2×C32#88611yescharacteristic
231C2×C32#88912yes—
232C2×C32#89012yes—
233C2×C32#89711yescharacteristic
234C2×C32#89812yes—
235C2×C32#92411yescharacteristic
236C2×C32#93511yescharacteristic
237C2×C32#94711yescharacteristic
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—92no—
241C22×C4—91no—
242C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4⋊C4—182no—
244C4⋊C4—182no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3D3×Dic611yescharacteristic
261C22×C3#379511yescharacteristic
262C22×C3#379912yes—
263C22×C3#380512yes—
264C22×C3C2×#62611yescharacteristic
265C3×C4—22no—
266C3×C4—62no—
267C3×C4—62no—
268C3×C4—64no—
269C3×C4—182no—
270C3×C4—361no—
271C3×C4—362no—
272C3×C4—362no—
273Dic3—62no—
274Dic3—122no—
275Dic3—122no—
276Dic3—124no—
277Dic3—122no—
278Dic3—121no—
279Dic3—182no—
280Dic3—361no—
281Dic3—362no—
282C32—21no—
283C32#465911yescharacteristic, complemented
284C32#498611yescharacteristic, complemented
285C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C2×C4—22no—
287C2×C4—92no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—182no—
291C2×C4—184no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—181no—
297C2×C4#620012yes—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—62no—
310C2×C3—61no—
311C2×C3—62no—
312C2×C3—61no—
313C2×C3#6384512yes—
314C2×C3#6385711yescharacteristic
315C2×C3#6386112yes—
316C2×C3#6386411yescharacteristic
317C2×C3#6396211yescharacteristic
318C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
319C22#8864811yescharacteristic
320C22#8865212yes—
321C22#8867212yes—
322C22C2×#620011yescharacteristic
323C4—22no—
324C4—182no—
325C4—361no—
326C4—362no—
327C4—362no—
328C3—31no—
329C3—31no—
330C3—61no—
331C3#1059122611yescharacteristic
332C2#1172108512yes—
333C2#1172110711yescharacteristic
334C2#1172111112yes—
335C2#1172111411yescharacteristic
336C2#1172130711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
178. H(3,𝔽3)#400semidirect
283. C32#4659semidirect
284. C32#4986semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3750⋊C32
#63658⋊C32
#117203342
#105912273
#105912253