Finite Group Databasebeta

← #13139976

Subgroups — #13139976

Order 1728.

481 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 123 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×#88538C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
4#11720315C212yesmaximal, complemented
5#11720314C212yesmaximal, complemented
6#10591131—31nomaximal
7#10591130—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10#88538C414yescomplemented
11C2×#6180C2212yes—
12#3757⋊C3—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3—21no—
18#64135D311yescharacteristic
19C2×C4⋊(C3⋊Dic3)D311yescharacteristic
20C3×#926—31no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×#208—31no—
23C2×C3×#208—31no—
24C2×C3×#208—61no—
25#63575—32no—
26#63575—32no—
27#63573—32no—
28#63573—32no—
29C23×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)Q812yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—42no—
38#6180C2×C414yescomplemented
39#5121—91no—
40#4995—91no—
41#4964—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—21no—
46C22×C3×Dic3—61no—
47#3757—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51C4⋊(C3⋊Dic3)C2×D311yescharacteristic
52C4×C3⋊Dic3—21no—
53C3×#217—61no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—61no—
58C3×C4×Dic3—61no—
59C3×#208—34no—
60C3×#208—34no—
61C3×#208—64no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)—21no—
67C22×H(3,𝔽3)C4⋊C412yes—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)C4⋊C412yes—
70C4×H(3,𝔽3)C22×C411yescharacteristic
71C4×H(3,𝔽3)C2×Q811yescharacteristic
72(C3⋊Dic3)⋊C3—41no—
73(C3⋊Dic3)⋊C3—42no—
74(C3⋊Dic3)⋊C3—41no—
75C2×C3×M16—91no—
76C3×#117—91no—
77C2×C3×C4⋊C4—91no—
78#926—31no—
79#926—91no—
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—91no—
82C2×#208—31no—
83C2×#208—31no—
84C2×#208—61no—
85#818—92no—
86#817—92no—
87#808—92no—
88#808—92no—
89#807—92no—
90#807—92no—
91C23×C32—21no—
92C23×C32#6711yescharacteristic
93C23×C32#6711yescharacteristic
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4Dic612yes—
99C2×C32×C4Dic612yes—
100C2×C32×C4C4×D312yes—
101C2×C32×C4C4×D312yes—
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—42no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C3×C3⋊C8—34no—
115C3×C3⋊C8—34no—
116C3×C3⋊C8—64no—
117#428—271noSylow
118C2×H(3,𝔽3)—22no—
119C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
120C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
121C2×H(3,𝔽3)#13711yescharacteristic
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4—181no—
126C22×C3×C4D3211yescharacteristic
127C22×Dic3—61no—
128C22×Dic3—181no—
129C3×M16—94no—
130C2×C3×C8—92no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—92no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—181no—
135C3×C22⋊C4—181no—
136C3×C42—181no—
137#217—61no—
138#217—181no—
139C4⋊Dic3—31no—
140C4⋊Dic3—32no—
141C4⋊Dic3—31no—
142C4⋊Dic3—61no—
143C4⋊Dic3—91no—
144C4⋊Dic3—92no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147C4×Dic3—61no—
148C4×Dic3—181no—
149#208—34no—
150#208—34no—
151#208—64no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—62no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32Dic3⋊C412yes—
164C22×C32Dic3⋊C412yes—
165C22×C32C2×#6711yescharacteristic
166C22×C32C2×#6711yescharacteristic
167C32×C4—22no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4Dic3⋊C412yes—
171C32×C4Dic3⋊C412yes—
172C32×C4C2×Dic611yescharacteristic
173C32×C4C2×Dic611yescharacteristic
174C32×C4C2×C4×D311yescharacteristic
175C32×C4C2×C4×D311yescharacteristic
176C3⋊Dic3—42no—
177C3⋊Dic3—41no—
178C3⋊Dic3—41no—
179C3×Dic3—121no—
180C3×Dic3—121no—
181C3×Dic3—122no—
182C2×M16—91no—
183#117—91no—
184C2×C4⋊C4—91no—
185#107—272no—
186#106—272no—
187H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—61no—
191C23×C3#62511yescharacteristic
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4—182no—
207C2×C3×C4—362no—
208C2×C3×C4#62412yes—
209C2×C3×C4#62511yescharacteristic
210C2×C3×C4#62712yes—
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—62no—
215C2×Dic3—122no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C2×Dic3—182no—
219C2×Dic3—362no—
220C3×C8—94no—
221C3⋊C8—34no—
222C3⋊C8—34no—
223C3⋊C8—64no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—42no—
230C2×C32C2×Dic3⋊C411yescharacteristic
231C2×C32C2×Dic3⋊C411yescharacteristic
232C2×C32#93211yescharacteristic
233C2×C32#94211yescharacteristic
234C2×C32#94911yescharacteristic
235C2×C32#95111yescharacteristic
236C22×C4—181no—
237C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
238M16—94no—
239C2×C8—92no—
240C4⋊C4—91no—
241C4⋊C4—92no—
242C4⋊C4—91no—
243C4⋊C4—181no—
244C22⋊C4—181no—
245C42—181no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3#372412yes—
261C22×C3C2×#62511yescharacteristic
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—32no—
267C3×C4—32no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—62no—
271C3×C4—362no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4#372412yes—
275C3×C4C2×#62411yescharacteristic
276C3×C4C2×#62711yescharacteristic
277Dic3—122no—
278Dic3—121no—
279Dic3—121no—
280Dic3—362no—
281Dic3—361no—
282Dic3—361no—
283C32—21no—
284C32#496411yescharacteristic, complemented
285C32#499511yescharacteristic, complemented
286C23#619811yescharacteristic
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—362no—
291C2×C4#619612yes—
292C2×C4#619811yescharacteristic
293C2×C4#619912yes—
294C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C8—94no—
296C2×C3—22no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—62no—
304C2×C3—62no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—122no—
309C2×C3#6382311yescharacteristic
310C2×C3#6382711yescharacteristic
311C2×C3C2×#372411yescharacteristic
312C22—21no—
313C22—21no—
314C22#8844912yes—
315C22C2×#619811yescharacteristic
316C4—361no—
317C4—362no—
318C4—361no—
319C4#8844912yes—
320C4C2×#619611yescharacteristic
321C4C2×#619911yescharacteristic
322C3—31no—
323C3—31no—
324C3—61no—
325C3#1059113911yescharacteristic
326C2—22no—
327C2#1172105811yescharacteristic
328C2#1172106111yescharacteristic
329C2C2×#8844911yescharacteristic
subgroupG/H
centerC22C2×#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#428
LCSG ▷ 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. #11720314C2semidirect
10. #88538C4semidirect
38. #6180C2×C4semidirect
187. H(3,𝔽3)#428semidirect
284. C32#4964semidirect
285. C32#4995semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64135⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117203152
#117203142
#105911313
#105911303