Finite Group Databasebeta

← #13138779

Subgroups — #13138779

Order 1728.

499 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 127 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721105C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic
4C8×#1084C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720303C211yesmaximal, characteristic, complemented
8#10590700—31nomaximal
9#10590692—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C8×(C3⋊D3)⋊C3C414yescomplemented
18#88450C412yes—
19#88448C412yes—
20#88430C414yescomplemented
21C2×C8×C3⋊D3D311yescharacteristic
22C2×C3×D3×C8—31no—
23#63874—31no—
24#63855—31no—
25C2×#3712—31no—
26C2×#3711—31no—
27C8×C3⋊Dic3D311yescharacteristic, complemented
28C3×C8×Dic3—31no—
29C3×C8×Dic3—31no—
30C3×C8×Dic3—31no—
31C3×C8×Dic3—61no—
32#63555—31no—
33#63554—31no—
34#63554—31no—
35#63554—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C412yes—
39C2×(C3⋊Dic3)⋊C3C2×C412yes—
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C4×(C3⋊D3)⋊C3C2×C412yes—
42C4×(C3⋊D3)⋊C3C2×C412yes—
43C8×H(3,𝔽3)C2×C412yes—
44#6180C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3C2×C412yes—
46(C3⋊(C3⋊C8))⋊C3C2×C412yes—
47#5101—91no—
48#4689—91no—
49#4689—91no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—31no—
52C2×C32×C8—21no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C2×D311yescharacteristic
55C4×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
58C8×C3⋊D3Dic314yescomplemented
59C3×C4×Dic3—31no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—61no—
68C3×D3×C8—34no—
69#3724—32no—
70Dic3⋊Dic3—32no—
71#3722—32no—
72(C2×D3)⋊Dic3—32no—
73#3712—34no—
74#3711—34no—
75C22×H(3,𝔽3)C22×C411yescharacteristic
76C2×(C3⋊D3)⋊C3—22no—
77C2×(C3⋊D3)⋊C3C4212yes—
78C4×H(3,𝔽3)C22×C411yescharacteristic
79C4×H(3,𝔽3)C22×C411yescharacteristic
80#1084—22no—
81(C3⋊Dic3)⋊C3—22no—
82(C3⋊Dic3)⋊C3C4212yes—
83C22×C3×C8—91no—
84#926—91no—
85C2×#208—91no—
86C2×ΓL(1,𝔽25)—91no—
87C2×ΓL(1,𝔽25)—91no—
88C2×D3×C8—31no—
89C2×D3×C8—91no—
90#872—91no—
91#872—91no—
92C3×C4×C8—91no—
93C3×C4×C8—91no—
94#815—91no—
95#815—91no—
96#815—91no—
97#815—91no—
98C8×Dic3—31no—
99C8×Dic3—31no—
100C8×Dic3—31no—
101C8×Dic3—61no—
102C8×Dic3—91no—
103#803—91no—
104#803—91no—
105C22×C3⋊D3C4×D311yescharacteristic
106C22×C3×D3—31no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3C4×D312yes—
111C2×C3⋊Dic3C4×D311yescharacteristic
112C4×C3⋊D3C4×D312yes—
113C4×C3⋊D3C2×Dic312yes—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—62no—
119C3×C4×D3—32no—
120C3×C4×D3—32no—
121C32×C8—22no—
122C32×C8C4×D312yes—
123C32×C8C2×Dic312yes—
124C3⋊(C3⋊C8)C4×D312yes—
125C3⋊(C3⋊C8)C2×Dic312yes—
126C3×C3⋊C8—32no—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—32no—
129C3×C3⋊C8—32no—
130C3×C3⋊C8—62no—
131#405—271noSylow
132C2×H(3,𝔽3)C2×C4211yescharacteristic
133C2×H(3,𝔽3)#13112yes—
134(C3⋊D3)⋊C3—22no—
135C22×C3×C4—91no—
136C2×C4×D3—31no—
137C2×C4×D3—91no—
138C2×C3×C8—31no—
139C2×C3×C8—31no—
140C2×C3×C8—61no—
141C2×C3×C8—94no—
142C2×C3×C8—91no—
143C2×C3×C8—91no—
144C2×C3×C8—91no—
145C2×C3×C8D3211yescharacteristic
146C3×C42—91no—
147C3×C42—91no—
148#217—92no—
149(C2×D3)⋊C4—92no—
150(C2×D3)⋊C4—92no—
151C4⋊Dic3—92no—
152Dic3⋊C4—92no—
153Dic3⋊C4—92no—
154C4×Dic3—31no—
155C4×Dic3—31no—
156C4×Dic3—31no—
157C4×Dic3—61no—
158C4×Dic3—91no—
159#208—94no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—91no—
167ΓL(1,𝔽25)—94no—
168ΓL(1,𝔽25)—94no—
169D3×C8—34no—
170D3×C8—94no—
171C22×C32—21no—
172C22×C32C2×C4×D311yescharacteristic
173C22×C32C2×C4×D311yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3C4×Dic312yes—
176C2×C3×D3—32no—
177C2×C3×D3—62no—
178C32×C4—21no—
179C32×C4—21no—
180C32×C4C2×C4×D311yescharacteristic
181C32×C4C2×C4×D311yescharacteristic
182C32×C4C2×C4×D311yescharacteristic
183C32×C4C22×Dic311yescharacteristic
184C3⋊Dic3—22no—
185C3⋊Dic3C4×Dic312yes—
186C3×Dic3—32no—
187C3×Dic3—64no—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C2×M16—271no—
192C22×C8—91no—
193#117—271no—
194#97—271no—
195#97—271no—
196C4×C8—91no—
197C4×C8—91no—
198H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—31no—
201C22×D3—91no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—61no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4—91no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—31no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—62no—
216C2×Dic3—91no—
217C2×Dic3—92no—
218C4×D3—32no—
219C4×D3—32no—
220C4×D3—92no—
221C4×D3—92no—
222C3×C8—32no—
223C3×C8—32no—
224C3×C8—62no—
225C3×C8—92no—
226C3×C8—92no—
227C3×C8—92no—
228C3×C8D3×Dic312yes—
229C3⋊C8—32no—
230C3⋊C8—32no—
231C3⋊C8—32no—
232C3⋊C8—32no—
233C3⋊C8—62no—
234C3⋊C8—92no—
235C3⋊C8—92no—
236C2×C32—22no—
237C2×C32—21no—
238C2×C32C42×D311yescharacteristic
239C2×C32#90112yes—
240C2×C32#90112yes—
241C2×C32C2×C4×Dic311yescharacteristic
242C3⋊D3—22no—
243C3×D3—62no—
244C22×C4—91no—
245M16—274no—
246C2×C8—91no—
247C2×C8—91no—
248C2×C8—91no—
249C2×C8—94no—
250C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
251C4⋊C4—272no—
252C22⋊C4—272no—
253C42—91no—
254C42—91no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—92no—
259C22×C3—182no—
260C22×C3C4×D3211yescharacteristic
261C2×D3—32no—
262C2×D3—62no—
263C2×D3—92no—
264C2×D3—182no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—92no—
272C3×C4—182no—
273C3×C4—182no—
274C3×C4C4×D3211yescharacteristic
275C3×C4C2×D3×Dic311yescharacteristic
276Dic3—32no—
277Dic3—62no—
278Dic3—62no—
279Dic3—64no—
280Dic3—62no—
281Dic3—92no—
282Dic3—182no—
283C32—21no—
284C32#468911yescharacteristic, complemented
285C32#510111yescharacteristic, complemented
286C23—91no—
287C2×C4—91no—
288C2×C4—92no—
289C2×C4—92no—
290C2×C4—92no—
291C2×C4—92no—
292C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
293C8—92no—
294C8—92no—
295C8—92no—
296C8((C3⋊D3)⋊C3)⋊C412yes—
297C2×C3—32no—
298C2×C3—32no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—62no—
302C2×C3—61no—
303C2×C3—182no—
304C2×C3#6380312yes—
305C2×C3C4×D3×Dic311yescharacteristic
306D3—62no—
307D3—182no—
308C22—92no—
309C22—182no—
310C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4—92no—
312C4—182no—
313C4—182no—
314C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
316C3—31no—
317C3—31no—
318C3—61no—
319C3#1059069211yescharacteristic
320C2—182no—
321C2#1172084012yes—
322C2C4×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C8×#1084C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720303C2semidirect
17. C8×(C3⋊D3)⋊C3C4semidirect
20. #88430C4semidirect
27. C8×C3⋊Dic3D3semidirect
58. C8×C3⋊D3Dic3semidirect
198. H(3,𝔽3)#405semidirect
284. C32#4689semidirect
285. C32#5101semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
#117211052
C2×#884302
C8×#10842
C8×(C3⋊Dic3)⋊C32
#117203122
#117203032
#105907003
#105906923