Finite Group Databasebeta

← #13165167

Subgroups — #13165167

Order 1728.

428 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C4×D3×#157C211yesmaximal, characteristic
2#11722818C212yesmaximal
3#11722786C211yesmaximal, characteristic, complemented
4#11722763C212yesmaximal, complemented
5C3×#63732C211yesmaximal, characteristic, complemented
6#10593328—31nomaximal
7C2×D3×#157C411yescharacteristic
8C2×D3×#157C411yescharacteristic
9C4×D3×C3⋊D3C2211yescharacteristic
10C4×C3⋊#157C2211yescharacteristic
11C3×C4×#157C2211yescharacteristic
12Dic3×#157C411yescharacteristic
13Dic3×#157C411yescharacteristic
14#88808C2212yes—
15#88788—22no—
16C3×#3711—22no—
17C3×#3710C2212yes—
18C2×C4×#157—31no—
19#63818—32no—
20D3×ΓL(1,𝔽25)—62no—
21#63732—31no—
22#63732—32no—
23#63732D311yescharacteristic, complemented
24C2×D3×C3⋊D3C2×C411yescharacteristic
25C2×C3⋊#157C2×C411yescharacteristic
26C2×C3⋊#157C2×C411yescharacteristic
27C2×C3×#157C2×C411yescharacteristic
28C2×C3×#157C2×C411yescharacteristic
29D3×#157—22no—
30D3×#157—22no—
31C4×C3⋊(C3⋊D3)D411yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C32×C4×D3D411yescharacteristic
34#6289Q811yescharacteristic
35Dic3×C3⋊D3C2×C411yescharacteristic
36D3×C3⋊Dic3Q811yescharacteristic
37C3×C3⋊(C3⋊C8)—22no—
38C32×C3⋊C8—22no—
39#4837—91nomaximal
40C22×#157—31no—
41C22×#157—31no—
42C2×C4×C3⋊D3—31no—
43C4×D32—62no—
44C4×#157—31no—
45C4×#157—31no—
46C4×#157—31no—
47C4×#157C2×D311yescharacteristic
48#3743—62no—
49C3×#208—62no—
50C3×ΓL(1,𝔽25)—62no—
51C3×D3×C8—62no—
52#3712—62no—
53#3712—62no—
54#3712—122no—
55#3711—22no—
56#3710—32no—
57#3710C2×D312yes—
58D3×C3⋊C8—62no—
59C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C2×C32×D3C4⋊C411yescharacteristic
62D3×C3⋊D3—21no—
63D3×C3⋊D3—21no—
64C3⋊#157—21no—
65C3⋊#157M1612yescomplemented
66C3×#157—21no—
67C3×#157M1612yescomplemented
68C33×C4C2×D411yescharacteristic
69C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
70C3×C3⋊Dic3C2×Q811yescharacteristic
71C32×Dic3C4⋊C411yescharacteristic
72D3×M16—182no—
73C2×ΓL(1,𝔽25)—92no—
74C42×D3—91no—
75C3×C4⋊C8—91no—
76Dic3⋊C8—92no—
77C4⋊(C3⋊C8)—91no—
78C22×C3⋊D3—31no—
79C2×D32—62no—
80C2×#157—31no—
81C2×#157—31no—
82C2×#157—62no—
83C2×#157—62no—
84C2×#157C4×D311yescharacteristic
85C2×#157C4×D311yescharacteristic
86C2×C32×C4—31no—
87C2×C3⋊Dic3—31no—
88C4×C3⋊D3—31no—
89C4×C3⋊D3—31no—
90C4×C3⋊D3—31no—
91C4×C3⋊D3—62no—
92C4×C3⋊D3—122no—
93C4×C3⋊D3C22×D311yescharacteristic
94C3×C4×D3—22no—
95C3×C4×D3—62no—
96#624—62no—
97D3×Dic3—62no—
98D3×Dic3—62no—
99C32×C8—62no—
100C3⋊(C3⋊C8)—22no—
101C3×C3⋊C8—22no—
102C3×C3⋊C8—22no—
103C3×C3⋊C8—42no—
104C3×C3⋊C8—62no—
105C3×C3⋊C8—62no—
106C3×C3⋊C8—122no—
107#423—271noSylow
108C2×C33C2×C4⋊C411yescharacteristic
109C3⋊(C3⋊D3)—21no—
110C3×C3⋊D3C2×M1611yescharacteristic
111C3×C3⋊D3C2×M1611yescharacteristic
112C32×D3—21no—
113C2×C4×D3—91no—
114C2×C4×D3—91no—
115C2×C4×D3—91no—
116C2×C4×D3—182no—
117C3×M16—182no—
118C2×C3×C8—92no—
119C3×C42—91no—
120C4×Dic3—91no—
121C4×Dic3—91no—
122C4×Dic3—91no—
123#208—62no—
124C2×C3⋊C8—92no—
125ΓL(1,𝔽25)—62no—
126ΓL(1,𝔽25)—182no—
127ΓL(1,𝔽25)—182no—
128ΓL(1,𝔽25)—182no—
129ΓL(1,𝔽25)—362no—
130D3×C8—62no—
131D3×C8—182no—
132C22×C32—31no—
133C2×C3⋊D3—31no—
134C2×C3⋊D3—62no—
135C2×C3⋊D3—61no—
136C2×C3⋊D3—61no—
137C2×C3⋊D3—122no—
138C2×C3⋊D3C2×C4×D311yescharacteristic
139C2×C3×D3—22no—
140C2×C3×D3—62no—
141D32—122no—
142D32—122no—
143#157—21no—
144#157—32no—
145#157—61no—
146#157ΓL(1,𝔽25)12yescomplemented
147C32×C4—22no—
148C32×C4—31no—
149C32×C4—42no—
150C32×C4D3×D411yescharacteristic
151C3⋊Dic3—31no—
152C3⋊Dic3—62no—
153C3⋊Dic3—122no—
154C3⋊Dic3D3×Q811yescharacteristic
155C3×Dic3—22no—
156C3×Dic3—62no—
157C2×M16—272no—
158C2×C42—271no—
159C4⋊C8—91no—
160C4⋊C8—271no—
161C4⋊C8—272no—
162C33#42311yescharacteristic, complemented, Sylow
163C22×D3—91no—
164C22×D3—182no—
165C2×C3×C4—62no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×Dic3—91no—
170C2×Dic3—91no—
171C2×Dic3—91no—
172C2×Dic3—182no—
173C4×D3—62no—
174C4×D3—91no—
175C4×D3—91no—
176C4×D3—91no—
177C4×D3—182no—
178C4×D3—182no—
179C4×D3—182no—
180C4×D3—182no—
181C4×D3—182no—
182C4×D3—362no—
183C4×D3GO+(4,𝔽2)11yescharacteristic
184C3×C8—62no—
185C3×C8—62no—
186C3×C8—122no—
187C3×C8—182no—
188C3⋊C8—22no—
189C3⋊C8—62no—
190C3⋊C8—62no—
191C3⋊C8—122no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—61no—
195C2×C32D3×C4⋊C411yescharacteristic
196C3⋊D3—61no—
197C3⋊D3—122no—
198C3⋊D3—242no—
199C3⋊D3C2×ΓL(1,𝔽25)11yescharacteristic
200C3⋊D3D3×M1611yescharacteristic
201C3×D3—42no—
202C3×D3—62no—
203C3×D3—62no—
204C22×C4—271no—
205C22×C4—271no—
206C22×C4—271no—
207M16—182no—
208M16—542no—
209C2×C8—92no—
210C2×C8—272no—
211C42—91no—
212C42—271no—
213C42—271no—
214C42—271no—
215C22×C3—62no—
216C22×C3—91no—
217C2×D3—62no—
218C2×D3—91no—
219C2×D3—182no—
220C2×D3—181no—
221C2×D3—181no—
222C2×D3—362no—
223C2×D3—362no—
224C2×D3—362no—
225C2×D3#377311yescharacteristic
226C3×C4—22no—
227C3×C4—42no—
228C3×C4—62no—
229C3×C4—92no—
230C3×C4—91no—
231C3×C4—181no—
232C3×C4C2×GO+(4,𝔽2)11yescharacteristic
233Dic3—62no—
234Dic3—92no—
235Dic3—91no—
236Dic3—182no—
237Dic3—181no—
238Dic3—362no—
239Dic3#377311yescharacteristic
240C32—22no—
241C32—42no—
242C32#483711yescharacteristic, complemented
243C23—271no—
244C2×C4—31no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—271no—
249C2×C4—271no—
250C2×C4—271no—
251C2×C4—271no—
252C2×C4—271no—
253C2×C4—271no—
254C2×C4—542no—
255C2×C4—542no—
256C8—62no—
257C8—182no—
258C2×C3—22no—
259C2×C3—42no—
260C2×C3—91no—
261C2×C3—91no—
262C2×C3—122no—
263C2×C3C2×#377311yescharacteristic
264D3—21no—
265D3—62no—
266D3—62no—
267D3—181no—
268D3—362no—
269D3—722no—
270C22—31no—
271C22—91no—
272C22—271no—
273C22—541no—
274C22—541no—
275C4—31no—
276C4—92no—
277C4—91no—
278C4—181no—
279C4—272no—
280C4—271no—
281C4—541no—
282C4D3×GO+(4,𝔽2)11yescharacteristic
283C3—22no—
284C3—42no—
285C3#1059332811yescharacteristic, complemented
286C2—61no—
287C2—91no—
288C2—91no—
289C2—541no—
290C2D3×#377311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
FittingC33×C4C2×D4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypocenterC33#423
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
3. #11722786C2semidirect
4. #11722763C2semidirect
5. C3×#63732C2semidirect
23. #63732D3semidirect
65. C3⋊#157M16semidirect
67. C3×#157M16semidirect
146. #157ΓL(1,𝔽25)semidirect
162. C33#423semidirect
242. C32#4837semidirect
285. C3#10593328semidirect

Maximal subgroups

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

subgroupindex
C4×D3×#1572
#117228182
#117227862
#117227632
C3×#637322
#105933283
#48379