Finite Group Databasebeta

← #13139971

Subgroups — #13139971

Order 1728.

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

normal subgroups only

This group has 414 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⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720315C212yesmaximal, complemented
5#11720313C212yesmaximal, complemented
6#10591135—31nomaximal
7#10591103—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×#634⋊C3C2211yescharacteristic
11C2×#6180C2212yes—
12#3757⋊C3—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic
18#64133D311yescharacteristic
19C2×C3×C4⋊Dic3—31no—
20C3×#924—31no—
21C3×#830—31no—
22C3×#830—31no—
23C3×#830—61no—
24#63575—32no—
25#63575—32no—
26#63567—32no—
27#63566—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×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—42no—
36#634⋊C3—22no—
37#6180—22no—
38#5044—91no—
39#4970—91no—
40#4722—91no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—21no—
45C2×#634C2×D311yescharacteristic
46C22×C3×Dic3—61no—
47C2×C3×Dic6—31no—
48#3757—21no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)C2×D312yes—
51C4⋊(C3⋊Dic3)C2×D311yescharacteristic
52#3750—21no—
53C3×#217—61no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—61no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)Q1612yes—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)#4112yes—
67(C3⋊Dic3)⋊C3—41no—
68(C3⋊Dic3)⋊C3—41no—
69(C3⋊Dic3)⋊C3—42no—
70C3×C22⋊Q8—91no—
71C2×C3×C4⋊C4—91no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—91no—
74#924—31no—
75#924—91no—
76C3×C22⋊C8—91no—
77#835—92no—
78#830—31no—
79#830—31no—
80#830—61no—
81#818—92no—
82#816—92no—
83#810—92no—
84#807—92no—
85#807—92no—
86C23×C32—21no—
87C23×C32#6711yescharacteristic
88C23×C32#6711yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—42no—
94C2×C32×C4#6711yescharacteristic
95C2×C32×C4#6711yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—42no—
102#634—22no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—122no—
107C3×Dic6—62no—
108C3×C3⋊C8—62no—
109C3×C3⋊C8—62no—
110C3×C3⋊C8—122no—
111#484—271noSylow
112C2×H(3,𝔽3)—22no—
113C2×H(3,𝔽3)#12311yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
116C2×C3×Q8—91no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—61no—
120C22×C3×C4—181no—
121C22×C3×C4D3211yescharacteristic
122C22×Dic3—61no—
123C22×Dic3—181no—
124C2×Dic6—31no—
125C2×Dic6—91no—
126C2×C3×C8—92no—
127C3×C4⋊C4—91no—
128C3×C4⋊C4—92no—
129C3×C4⋊C4—181no—
130C3×C4⋊C4—181no—
131C3×C22⋊C4—181no—
132#217—61no—
133#217—181no—
134C4⋊Dic3—32no—
135C4⋊Dic3—31no—
136C4⋊Dic3—61no—
137C4⋊Dic3—91no—
138C4⋊Dic3—92no—
139C4⋊Dic3—181no—
140Dic3⋊C4—61no—
141Dic3⋊C4—181no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—62no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32#21612yes—
154C22×C32#21612yes—
155C22×C32C2×#6711yescharacteristic
156C22×C32C2×#6711yescharacteristic
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—22no—
160C32×C4—41no—
161C32×C4#23512yes—
162C32×C4#23712yes—
163C3⋊Dic3—42no—
164C3⋊Dic3—41no—
165C3⋊Dic3—41no—
166C3×Dic3—121no—
167C3×Dic3—121no—
168C3×Dic3—122no—
169C22⋊Q8—91no—
170C2×C4⋊C4—91no—
171#107—272no—
172#103—272no—
173C22⋊C8—91no—
174H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—61no—
178C23×C3#62511yescharacteristic
179C3×Q8—182no—
180C2×C3×C4—22no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—122no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—182no—
193C2×C3×C4—362no—
194C2×C3×C4#62511yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—62no—
199C2×Dic3—122no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—182no—
203C2×Dic3—362no—
204Dic6—62no—
205Dic6—182no—
206C3×C8—182no—
207C3⋊C8—62no—
208C3⋊C8—62no—
209C3⋊C8—122no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—42no—
216C2×C32#92911yescharacteristic
217C2×C32#93211yescharacteristic
218C2×C32#93511yescharacteristic
219C2×C32C2×#21611yescharacteristic
220C2×C32C2×#21611yescharacteristic
221C2×C32#94911yescharacteristic
222C2×Q8—91no—
223C22×C4—181no—
224C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C2×C8—92no—
226C4⋊C4—92no—
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C22⋊C4—181no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—62no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3#371812yes—
246C22×C3C2×#62511yescharacteristic
247C3×C4—21no—
248C3×C4—32no—
249C3×C4—32no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—62no—
253C3×C4—121no—
254C3×C4—362no—
255C3×C4—361no—
256C3×C4—361no—
257C3×C4#380412yes—
258Dic3—122no—
259Dic3—121no—
260Dic3—121no—
261Dic3—361no—
262Dic3—362no—
263Dic3—361no—
264C32—21no—
265C32#497011yescharacteristic, complemented
266C32#504411yescharacteristic, complemented
267C23#619811yescharacteristic
268Q8—182no—
269C2×C4—22no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C4—362no—
274C2×C4#619811yescharacteristic
275C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C8—182no—
277C2×C3—22no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—62no—
285C2×C3—62no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—122no—
290C2×C3#6382311yescharacteristic
291C2×C3C2×#371811yescharacteristic
292C2×C3#6396111yescharacteristic
293C22—21no—
294C22—21no—
295C22#8843312yes—
296C22C2×#619811yescharacteristic
297C4—21no—
298C4—361no—
299C4—362no—
300C4—361no—
301C4#8867112yes—
302C3—31no—
303C3—31no—
304C3—61no—
305C3#1059113211yescharacteristic
306C2—22no—
307C2#1172106111yescharacteristic
308C2C2×#8843311yescharacteristic
309C2#1172130611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
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. #11720315C2semidirect
5. #11720313C2semidirect
174. H(3,𝔽3)#484semidirect
265. C32#4970semidirect
266. C32#5044semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64133⋊C32
#117205732
#117203152
#117203132
#105911353
#105911033