Finite Group Databasebeta

← #13139936

Subgroups — #13139936

Order 1728.

505 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 131 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#64135⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#11720307C212yesmaximal, complemented
5#11720306C212yesmaximal, complemented
6#10591139—31nomaximal
7#10591130—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C4⋊#1084—21no—
11C4⋊#1084C2212yes—
12#3757⋊C3—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3—21no—
16#3748⋊C3C414yescomplemented
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#64135D311yescharacteristic
19C2×#3748D311yescharacteristic
20C3×#926—31no—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×C4⋊Dic3—31no—
23C2×C3×C4⋊Dic3—61no—
24C2×C3×#208—31no—
25#63576—32no—
26#63575—32no—
27#63574—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×#1084—22no—
35C2×#1084—42no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—21no—
38(C3⋊(C3⋊C8))⋊C3C2×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—62no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48#3757—21no—
49C4⋊(C3⋊Dic3)C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D311yescharacteristic
51C4×C3⋊Dic3—21no—
52#3748Dic314yescomplemented
53C2×C3⋊(C3⋊C8)C2×D312yes—
54C3×#217—61no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—61no—
63C3×C4×Dic3—61no—
64C3×#208—34no—
65C2×C3×C3⋊C8—32no—
66C22×H(3,𝔽3)—21no—
67C22×H(3,𝔽3)—21no—
68C22×H(3,𝔽3)C4⋊C412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C4⋊C412yes—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×Q811yescharacteristic
73#1084—42no—
74(C3⋊Dic3)⋊C3—41no—
75(C3⋊Dic3)⋊C3—41no—
76C2×C3×M16—91no—
77C3×#117—91no—
78C2×C3×C4⋊C4—91no—
79#926—31no—
80#926—91no—
81C2×C4⋊Dic3—31no—
82C2×C4⋊Dic3—31no—
83C2×C4⋊Dic3—61no—
84C2×#208—31no—
85C2×#208—91no—
86#818—92no—
87#817—92no—
88#808—92no—
89#808—92no—
90#807—92no—
91#807—92no—
92C23×C32—21no—
93C23×C32D1211yescharacteristic
94C23×C32#6711yescharacteristic
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4Dic612yes—
100C2×C32×C4Dic612yes—
101C2×C32×C4C4×D312yes—
102C2×C32×C4D1211yescharacteristic
103C2×C32×C4C2×Dic312yes—
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—124no—
115C2×C3×Dic3—122no—
116C2×C3×Dic3—122no—
117C3⋊(C3⋊C8)C2×Dic314yescomplemented
118C3×C3⋊C8—34no—
119#428—271noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
122C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
123C2×H(3,𝔽3)#13711yescharacteristic
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—61no—
127C22×C3×C4—181no—
128C22×C3×C4D3211yescharacteristic
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—62no—
132C3×M16—94no—
133C2×C3×C8—92no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—92no—
137C3×C4⋊C4—181no—
138C3×C22⋊C4—181no—
139C3×C42—181no—
140#217—61no—
141#217—181no—
142C4⋊Dic3—32no—
143C4⋊Dic3—32no—
144C4⋊Dic3—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—62no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—62no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4×Dic3—61no—
153C4×Dic3—181no—
154#208—34no—
155#208—94no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—92no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32Dic3⋊C412yes—
167C22×C32C4⋊Dic312yes—
168C22×C32C2×D1211yescharacteristic
169C22×C32C2×#6711yescharacteristic
170C32×C4—22no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4Dic3⋊C412yes—
174C32×C4C4⋊Dic312yes—
175C32×C4C2×Dic611yescharacteristic
176C32×C4C2×Dic611yescharacteristic
177C32×C4C2×C4×D311yescharacteristic
178C32×C4C22×Dic311yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3—41no—
181C3×Dic3—124no—
182C3×Dic3—121no—
183C3×Dic3—122no—
184C3×Dic3—121no—
185C3×Dic3—122no—
186C2×M16—91no—
187#117—91no—
188C2×C4⋊C4—91no—
189#107—272no—
190#106—272no—
191H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—61no—
195C23×C3#62611yescharacteristic
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—182no—
211C2×C3×C4—362no—
212C2×C3×C4D3×Dic312yes—
213C2×C3×C4#62611yescharacteristic
214C2×C3×C4#62712yes—
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—64no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—122no—
222C2×Dic3—122no—
223C2×Dic3—124no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C3×C8—94no—
227C3⋊C8—34no—
228C3⋊C8—94no—
229C2×C32—24no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32#90811yescharacteristic
236C2×C32#90911yescharacteristic
237C2×C32C2×Dic3⋊C411yescharacteristic
238C2×C32C2×C4⋊Dic311yescharacteristic
239C2×C32#94911yescharacteristic
240C2×C32#95111yescharacteristic
241C22×C4—181no—
242C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
243M16—94no—
244C2×C8—92no—
245C4⋊C4—91no—
246C4⋊C4—92no—
247C4⋊C4—91no—
248C4⋊C4—181no—
249C22⋊C4—181no—
250C42—181no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—32no—
256C22×C3—32no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3—61no—
264C22×C3—62no—
265C22×C3Dic3⋊Dic312yes—
266C22×C3C2×#62611yescharacteristic
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—32no—
272C3×C4—32no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—62no—
276C3×C4—361no—
277C3×C4—361no—
278C3×C4—362no—
279C3×C4Dic3⋊Dic312yes—
280C3×C4C2×D3×Dic311yescharacteristic
281C3×C4C2×#62711yescharacteristic
282Dic3—122no—
283Dic3—122no—
284Dic3—124no—
285Dic3—121no—
286Dic3—121no—
287Dic3—361no—
288Dic3—361no—
289C32—21no—
290C32#499511yescharacteristic, complemented
291C32#512111yescharacteristic, complemented
292C23#620011yescharacteristic
293C2×C4—181no—
294C2×C4—181no—
295C2×C4—182no—
296C2×C4—362no—
297C2×C4#619612yes—
298C2×C4((C3⋊D3)⋊C3)⋊C412yes—
299C2×C4#620011yescharacteristic
300C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C8—94no—
302C2×C3—22no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—62no—
310C2×C3—62no—
311C2×C3—64no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3#6382511yescharacteristic
316C2×C3#6383111yescharacteristic
317C2×C3C2×Dic3⋊Dic311yescharacteristic
318C22—21no—
319C22—21no—
320C22#8844812yes—
321C22C2×#620011yescharacteristic
322C4—362no—
323C4—361no—
324C4—361no—
325C4#8844812yes—
326C4C2×#619611yescharacteristic
327C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
328C3—31no—
329C3—31no—
330C3—61no—
331C3#1059113111yescharacteristic
332C2—22no—
333C2#1172106811yescharacteristic
334C2#1172107411yescharacteristic
335C2C2×#8844811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
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. #11720307C2semidirect
5. #11720306C2semidirect
16. #3748⋊C3C4semidirect
38. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
52. #3748Dic3semidirect
117. C3⋊(C3⋊C8)C2×Dic3semidirect
191. H(3,𝔽3)#428semidirect
290. C32#4995semidirect
291. C32#5121semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64135⋊C32
C2×#3748⋊C32
#117203072
#117203062
#105911393
#105911303