Finite Group Databasebeta

← #13139954

Subgroups — #13139954

Order 1728.

456 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 85 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
2#64133⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720309C212yesmaximal, complemented
5#11720306C212yesmaximal, complemented
6#10591135—31nomaximal
7#10591132—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#634⋊C3C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084C2212yes—
13#3757⋊C3—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64133D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×C4⋊Dic3—61no—
21C3×#924—31no—
22#63657D311yescharacteristic
23C3×#830—31no—
24#63576—32no—
25#63575—32no—
26#63567—32no—
27#63565—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—22no—
33C2×#1084—42no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—21no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5044—91no—
39#4970—91no—
40#4722—91no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C2×#634C2×D311yescharacteristic
45C22×C3×Dic3—62no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C2×C3×Dic6—31no—
49#3757—21no—
50C4⋊(C3⋊Dic3)C2×D311yescharacteristic
51#3750—21no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×#217—61no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)Q1612yes—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)#4112yes—
69#1084—42no—
70(C3⋊Dic3)⋊C3—41no—
71(C3⋊Dic3)⋊C3—41no—
72C3×C22⋊Q8—91no—
73C2×C3×C4⋊C4—91no—
74C2×C4⋊Dic3—31no—
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—61no—
77#924—31no—
78#924—91no—
79C3×C22⋊C8—91no—
80#835—92no—
81#830—31no—
82#830—91no—
83#818—92no—
84#816—92no—
85#810—92no—
86#807—92no—
87#807—92no—
88C23×C32—21no—
89C23×C32D1211yescharacteristic
90C23×C32#6711yescharacteristic
91C2×C32×C4—24no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4D1211yescharacteristic
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102#634—22no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—124no—
109C2×C3×Dic3—122no—
110C2×C3×Dic3—122no—
111C3×Dic6—62no—
112C3⋊(C3⋊C8)—22no—
113C3×C3⋊C8—62no—
114#484—271noSylow
115C2×H(3,𝔽3)—22no—
116C2×H(3,𝔽3)#12311yescharacteristic
117C2×H(3,𝔽3)C2×Q1611yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
119C2×C3×Q8—91no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—61no—
123C22×C3×C4—181no—
124C22×C3×C4D3211yescharacteristic
125C22×Dic3—61no—
126C22×Dic3—61no—
127C22×Dic3—62no—
128C2×Dic6—31no—
129C2×Dic6—91no—
130C2×C3×C8—92no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—92no—
133C3×C4⋊C4—181no—
134C3×C4⋊C4—181no—
135C3×C22⋊C4—181no—
136#217—61no—
137#217—181no—
138C4⋊Dic3—32no—
139C4⋊Dic3—32no—
140C4⋊Dic3—31no—
141C4⋊Dic3—62no—
142C4⋊Dic3—61no—
143C4⋊Dic3—61no—
144C4⋊Dic3—62no—
145C4⋊Dic3—91no—
146Dic3⋊C4—61no—
147Dic3⋊C4—181no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—92no—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32Dic1212yes—
159C22×C32#21612yes—
160C22×C32C2×D1211yescharacteristic
161C22×C32C2×#6711yescharacteristic
162C32×C4—22no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4—22no—
166C32×C4#23712yes—
167C32×C4#23712yes—
168C3⋊Dic3—41no—
169C3⋊Dic3—41no—
170C3×Dic3—124no—
171C3×Dic3—121no—
172C3×Dic3—121no—
173C3×Dic3—122no—
174C3×Dic3—122no—
175C22⋊Q8—91no—
176C2×C4⋊C4—91no—
177#107—272no—
178#103—272no—
179C22⋊C8—91no—
180H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3#62611yescharacteristic
185C3×Q8—182no—
186C2×C3×C4—22no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—64no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4—182no—
199C2×C3×C4—362no—
200C2×C3×C4#62611yescharacteristic
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—64no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—122no—
208C2×Dic3—122no—
209C2×Dic3—124no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212Dic6—62no—
213Dic6—182no—
214C3×C8—182no—
215C3⋊C8—62no—
216C3⋊C8—182no—
217C2×C32—24no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32#88611yescharacteristic
224C2×C32C2×Dic1211yescharacteristic
225C2×C32#90811yescharacteristic
226C2×C32#93511yescharacteristic
227C2×C32C2×#21611yescharacteristic
228C2×C32#94911yescharacteristic
229C2×Q8—91no—
230C22×C4—181no—
231C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
232C2×C8—92no—
233C4⋊C4—92no—
234C4⋊C4—91no—
235C4⋊C4—181no—
236C4⋊C4—181no—
237C22⋊C4—181no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—61no—
251C22×C3—62no—
252C22×C3#371912yes—
253C22×C3C2×#62611yescharacteristic
254C3×C4—21no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—62no—
260C3×C4—62no—
261C3×C4—361no—
262C3×C4—361no—
263C3×C4—362no—
264C3×C4#380512yes—
265Dic3—122no—
266Dic3—122no—
267Dic3—124no—
268Dic3—121no—
269Dic3—121no—
270Dic3—361no—
271Dic3—361no—
272C32—21no—
273C32#472211yescharacteristic, complemented
274C32#504411yescharacteristic, complemented
275C23#620011yescharacteristic
276Q8—182no—
277C2×C4—22no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—182no—
281C2×C4—362no—
282C2×C4#620011yescharacteristic
283C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C8—182no—
285C2×C3—22no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—62no—
293C2×C3—62no—
294C2×C3—64no—
295C2×C3—61no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3#6382511yescharacteristic
299C2×C3C2×#371911yescharacteristic
300C2×C3#6396211yescharacteristic
301C22—21no—
302C22—21no—
303C22#8843912yes—
304C22C2×#620011yescharacteristic
305C4—21no—
306C4—361no—
307C4—362no—
308C4—361no—
309C4#8867212yes—
310C3—31no—
311C3—31no—
312C3—61no—
313C3#1059110311yescharacteristic
314C2—22no—
315C2#1172106811yescharacteristic
316C2C2×#8843911yescharacteristic
317C2#1172130711yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720309C2semidirect
5. #11720306C2semidirect
180. H(3,𝔽3)#484semidirect
273. C32#4722semidirect
274. C32#5044semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64133⋊C32
#63657⋊C32
#117203092
#117203062
#105911353
#105911323