Finite Group Databasebeta

← #13138041

Subgroups — #13138041

Order 1728.

659 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88651C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63637⋊C3C212yesmaximal, complemented
5#11720309C212yesmaximal, complemented
6D3×#810—32nomaximal
7C2×C4×#272—21no—
8C2×C4×#272C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×#6199—21no—
11C2×#6196—21no—
12#88651—22no—
13#88651C414yescomplemented
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88434—22no—
19C2×D3×Dic6—32no—
20C3×D3×C4⋊C4—32no—
21C3×D3×C4⋊C4—61no—
22C2×D3×C3⋊C8—32no—
23#63637D312yescomplemented
24C3×#810—32no—
25#63567—32no—
26#63565—32no—
27C22×#272D411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31C2×#1084D411yescharacteristic
32C4×#272—42no—
33C4×#272C2×C412yes—
34C4×#272D412yes—
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—22no—
37#634⋊C3C2×C414yescomplemented
38#6199—24no—
39#6196—24no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41C2×#810—91no—
42D3×#103—92no—
43C2×#634C2×D312yes—
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—62no—
48C2×C3×Dic6—32no—
49C2×#627—62no—
50C2×D3×Dic3—62no—
51D3×Dic6—38no—
52D3×Dic6—64no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D312yes—
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—61no—
58C3×Dic3⋊C4—62no—
59C3×Dic3⋊C4—62no—
60C2×C3×C3⋊C8—32no—
61D3×C3⋊C8—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×#272C22⋊C412yes—
64C2×#272SD1612yes—
65C2×#272Q1612yes—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C22×C411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69#1084—41no—
70#1084C22⋊C412yes—
71(C3⋊Dic3)⋊C3—24no—
72C2×D3×Q8—92no—
73C22×Dic6—91no—
74C2×C3×C4⋊C4—91no—
75C22×C3⋊C8—91no—
76C2×D3×C8—92no—
77D3×C4⋊C4—32no—
78D3×C4⋊C4—61no—
79C3×#103—92no—
80#835—92no—
81#816—92no—
82#810—32no—
83#810—92no—
84#810—92no—
85C22×C3×D3—32no—
86C22×C3×D3—61no—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—21no—
90C2×C32×C4C22×D312yes—
91C2×C3⋊Dic3—22no—
92#634—22no—
93#634C4×D314yescomplemented
94C2×C3×Dic3—32no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×C4×D3—34no—
100C3×C4×D3—34no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—124no—
104C3×C4×D3—124no—
105C3×Dic6—34no—
106C3×Dic6—62no—
107#627—68no—
108D3×Dic3—68no—
109C3⋊(C3⋊C8)—22no—
110C3×C3⋊C8—62no—
111C2×#103—271noSylow
112C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
113C2×H(3,𝔽3)C2×SD1611yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115#272#10314yescomplemented
116C2×C3×Q8—92no—
117C22×C3×C4—91no—
118C22×C3×C4—181no—
119C22×Dic3—181no—
120D3×Q8—98no—
121D3×Q8—184no—
122C2×C4×D3—32no—
123C2×C4×D3—61no—
124C2×C4×D3—62no—
125C2×C4×D3—62no—
126C2×C4×D3—182no—
127C2×Dic6—32no—
128C2×Dic6—92no—
129C2×Dic6—94no—
130C2×Dic6—182no—
131C2×Dic6—182no—
132C2×Dic6—181no—
133C2×C3×C8—92no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4D3211yescharacteristic
139C4⋊Dic3—32no—
140C4⋊Dic3—61no—
141Dic3⋊C4—62no—
142Dic3⋊C4—62no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—92no—
145C2×C3⋊C8—182no—
146D3×C8—184no—
147C22×C32—21no—
148C22×C32D3×D412yes—
149C2×C3×D3—34no—
150C2×C3×D3—34no—
151C2×C3×D3—34no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—42no—
158C32×C4—42no—
159C32×C4C2×C4×D312yes—
160C32×C4D3×D412yes—
161C3⋊Dic3—24no—
162C3×Dic3—34no—
163C3×Dic3—62no—
164C3×Dic3—64no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C22×Q8—271no—
168C22×C8—271no—
169C2×C4⋊C4—91no—
170#103—92no—
171#103—272no—
172H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
173C23×C3—91no—
174C22×D3—32no—
175C22×D3—61no—
176C3×Q8—94no—
177C3×Q8—182no—
178C2×C3×C4—21no—
179C2×C3×C4—32no—
180C2×C3×C4—62no—
181C2×C3×C4—62no—
182C2×C3×C4—61no—
183C2×C3×C4—91no—
184C2×C3×C4—92no—
185C2×C3×C4—92no—
186C2×C3×C4—182no—
187C2×C3×C4—181no—
188C2×C3×C4—362no—
189C2×C3×C4C2×D3211yescharacteristic
190C2×Dic3—32no—
191C2×Dic3—61no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—182no—
196C2×Dic3—184no—
197C4×D3—34no—
198C4×D3—34no—
199C4×D3—62no—
200C4×D3—62no—
201C4×D3—124no—
202C4×D3—124no—
203C4×D3—188no—
204Dic6—34no—
205Dic6—62no—
206Dic6—94no—
207Dic6—182no—
208Dic6—188no—
209Dic6—184no—
210C3×C8—182no—
211C3⋊C8—62no—
212C3⋊C8—182no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32D3×C22⋊C412yes—
217C2×C32D3×SD1612yes—
218C2×C32D3×Q1612yes—
219C3×D3—38no—
220C3×D3—64no—
221C2×Q8—92no—
222C2×Q8—274no—
223C2×Q8—541no—
224C2×Q8—542no—
225C22×C4—91no—
226C22×C4—181no—
227C22×C4—541no—
228C2×C8—92no—
229C2×C8—542no—
230C4⋊C4—91no—
231C4⋊C4—181no—
232C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C22×C3—32no—
234C22×C3—61no—
235C22×C3—92no—
236C22×C3—92no—
237C22×C3—92no—
238C22×C3#381111yescharacteristic
239C2×D3—34no—
240C2×D3—34no—
241C2×D3—34no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C3×C4—32no—
246C3×C4—32no—
247C3×C4—41no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—92no—
251C3×C4—122no—
252C3×C4—122no—
253C3×C4—184no—
254C3×C4—361no—
255C3×C4C4×D3211yescharacteristic
256C3×C4#380211yescharacteristic
257Dic3—34no—
258Dic3—62no—
259Dic3—64no—
260Dic3—122no—
261Dic3—122no—
262Dic3—184no—
263C32—21no—
264C32D3×#10312yescomplemented
265C23—91no—
266Q8—94no—
267Q8—182no—
268Q8—544no—
269C2×C4—21no—
270C2×C4—91no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—182no—
274C2×C4—181no—
275C2×C4—362no—
276C2×C4—544no—
277C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
278C8—182no—
279C2×C3—32no—
280C2×C3—32no—
281C2×C3—32no—
282C2×C3—61no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—94no—
286C2×C3#6392111yescharacteristic
287C2×C3#6393011yescharacteristic
288C2×C3#6394411yescharacteristic
289D3—38no—
290D3—64no—
291C22—92no—
292C22—92no—
293C22—92no—
294C22#8867711yescharacteristic
295C4—41no—
296C4—92no—
297C4—184no—
298C4—361no—
299C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C4#8865511yescharacteristic
301C3—32no—
302C3—61no—
303C3#1059060111yescharacteristic
304C2—94no—
305C2#1172116111yescharacteristic
306C2#1172118811yescharacteristic
307C2#1172122611yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63637⋊C3C2semidirect
5. #11720309C2semidirect
13. #88651C4semidirect
23. #63637D3semidirect
37. #634⋊C3C2×C4semidirect
93. #634C4×D3semidirect
115. #272#103semidirect
172. H(3,𝔽3)C2×#103semidirect
264. C32D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C4⋊C4×#2722
C2×#884342
#63637⋊C32
#117203092
D3×#8103