Finite Group Databasebeta

← #13166264

Subgroups — #13166264

Order 1728.

380 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#11724255C211yesmaximal, characteristic
2D3×#3788C211yesmaximal, characteristic
3C3⋊D3×ΓL(1,𝔽25)C211yesmaximal, characteristic
4#11723095C211yesmaximal, characteristic, complemented
5C3×#63771C211yesmaximal, characteristic, complemented
6#11722717C211yesmaximal, characteristic, complemented
7#11722716C211yesmaximal, characteristic, complemented
8#10593776—31nomaximal
9C4×D3×C3⋊D3C2211yescharacteristic
10#88989C2211yescharacteristic
11#88989C2211yescharacteristic
12C3×#3788C2211yescharacteristic
13C3×#3788C2211yescharacteristic
14#88926—21no—
15#88925—21no—
16#88924—21no—
17D3×#622—21no—
18#88850C411yescharacteristic
19C3×C8×C3⋊D3C2211yescharacteristic
20C32×ΓL(1,𝔽25)C411yescharacteristic
21#88788C411yescharacteristic
22#88786C411yescharacteristic
23C3⋊D3×C3⋊C8C2211yescharacteristic
24#64283—31no—
25C2×#3788—31no—
26M16×C3⋊D3—31no—
27D3×ΓL(1,𝔽25)—62no—
28#63771—31no—
29#63771D311yescharacteristic, complemented
30#63770—31no—
31#63770—31no—
32C2×D3×C3⋊D3Q811yescharacteristic
33C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C32×C4×D3C2×C411yescharacteristic
36#6289D411yescharacteristic
37Dic3×C3⋊D3Q811yescharacteristic
38D3×C3⋊Dic3D411yescharacteristic
39C3⋊#622—21no—
40C3⋊#622—21no—
41C3×#622—21no—
42C3×#622—21no—
43C33×C8C2×C411yescharacteristic
44C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
45C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
46C32×C3⋊C8C2×C411yescharacteristic
47#4893—91nomaximal
48C2×C4×C3⋊D3—31no—
49C4×D32—62no—
50#3792—61no—
51C2×#622—61no—
52#3788—31no—
53#3788—31no—
54#3788—61no—
55#3788C2×D311yescharacteristic
56#3788C2×D311yescharacteristic
57#3787—61no—
58C32×M16—31no—
59#3748—31no—
60#3743—31no—
61#3743—31no—
62#3743—62no—
63#3743—122no—
64C8×C3⋊D3—31no—
65C8×C3⋊D3C2×D311yescharacteristic
66C3×ΓL(1,𝔽25)—22no—
67C3×D3×C8—62no—
68#3712—62no—
69#3711—62no—
70D3×C3⋊C8—62no—
71C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
72C2×C3×C3⋊D3C2×Q811yescharacteristic
73C2×C32×D3C4⋊C411yescharacteristic
74D3×C3⋊D3—41no—
75C33×C4C22×C411yescharacteristic
76C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
77C3×C3⋊Dic3C2×D411yescharacteristic
78C32×Dic3C4⋊C411yescharacteristic
79D3×M16—91no—
80D3×M16—91no—
81D3×M16—182no—
82C2×ΓL(1,𝔽25)—91no—
83C3×#108—91no—
84#822—91no—
85#822—91no—
86#819—91no—
87C22×C3⋊D3—31no—
88C2×D32—62no—
89C2×C32×C4—31no—
90C2×C3⋊Dic3—31no—
91C4×C3⋊D3—31no—
92C4×C3⋊D3—31no—
93C4×C3⋊D3—31no—
94C4×C3⋊D3—62no—
95C4×C3⋊D3—122no—
96C4×C3⋊D3C22×D311yescharacteristic
97C3×C4×D3—22no—
98C3×C4×D3—62no—
99#624—62no—
100D3×Dic3—62no—
101D3×Dic3—62no—
102#622—21no—
103#622—21no—
104#622—61no—
105#622—61no—
106C32×C8—22no—
107C32×C8—31no—
108C32×C8—42no—
109C32×C8C4×D311yescharacteristic
110C3⋊(C3⋊C8)—31no—
111C3⋊(C3⋊C8)—62no—
112C3⋊(C3⋊C8)—122no—
113C3⋊(C3⋊C8)C4×D311yescharacteristic
114C3×C3⋊C8—22no—
115C3×C3⋊C8—62no—
116#430—271noSylow
117C2×C33C2×C4⋊C411yescharacteristic
118C3⋊(C3⋊D3)—21no—
119C3×C3⋊D3—21no—
120C32×D3—21no—
121C2×C4×D3—91no—
122C2×C4×D3—182no—
123C3×M16—62no—
124C3×M16—91no—
125C3×M16—91no—
126C2×C3×C8—91no—
127#208—91no—
128#208—91no—
129#208—182no—
130C2×C3⋊C8—91no—
131ΓL(1,𝔽25)—91no—
132ΓL(1,𝔽25)—91no—
133ΓL(1,𝔽25)—91no—
134ΓL(1,𝔽25)—182no—
135ΓL(1,𝔽25)—182no—
136ΓL(1,𝔽25)—181no—
137ΓL(1,𝔽25)—181no—
138ΓL(1,𝔽25)—362no—
139ΓL(1,𝔽25)#15711yescharacteristic
140D3×C8—62no—
141D3×C8—182no—
142D3×C8—181no—
143D3×C8—181no—
144C22×C32—31no—
145C2×C3⋊D3—31no—
146C2×C3⋊D3—62no—
147C2×C3⋊D3—122no—
148C2×C3⋊D3—121no—
149C2×C3⋊D3D3×Q811yescharacteristic
150C2×C3×D3—22no—
151C2×C3×D3—62no—
152D32—124no—
153C32×C4—22no—
154C32×C4—31no—
155C32×C4—42no—
156C32×C4C2×C4×D311yescharacteristic
157C3⋊Dic3—31no—
158C3⋊Dic3—62no—
159C3⋊Dic3—122no—
160C3⋊Dic3D3×D411yescharacteristic
161C3×Dic3—22no—
162C3×Dic3—62no—
163C2×M16—271no—
164C2×M16—271no—
165C2×M16—271no—
166#108—91no—
167#108—271no—
168#108—271no—
169#108—271no—
170C33#43011yescharacteristic, complemented, Sylow
171C22×D3—91no—
172C22×D3—182no—
173C2×C3×C4—62no—
174C2×C3×C4—91no—
175C2×Dic3—91no—
176C2×Dic3—182no—
177C4×D3—62no—
178C4×D3—91no—
179C4×D3—91no—
180C4×D3—91no—
181C4×D3—182no—
182C4×D3—182no—
183C4×D3—182no—
184C4×D3—362no—
185C4×D3C2×#15711yescharacteristic
186C3×C8—22no—
187C3×C8—42no—
188C3×C8—62no—
189C3×C8—91no—
190C3×C8—181no—
191C3×C8—181no—
192C3×C8C2×#15711yescharacteristic
193C3⋊C8—62no—
194C3⋊C8—91no—
195C3⋊C8—182no—
196C3⋊C8—181no—
197C3⋊C8—181no—
198C3⋊C8—362no—
199C3⋊C8C2×#15711yescharacteristic
200C2×C32—22no—
201C2×C32—42no—
202C2×C32—61no—
203C2×C32D3×C4⋊C411yescharacteristic
204C3⋊D3—21no—
205C3⋊D3—61no—
206C3⋊D3—122no—
207C3⋊D3—242no—
208C3×D3—42no—
209C3×D3—64no—
210C22×C4—271no—
211M16—31no—
212M16—91no—
213M16—91no—
214M16—271no—
215M16—271no—
216M16—271no—
217M16—271no—
218M16—271no—
219M16—541no—
220M16—541no—
221C2×C8—91no—
222C2×C8—271no—
223C2×C8—541no—
224C2×C8—541no—
225C22×C3—62no—
226C22×C3—91no—
227C2×D3—62no—
228C2×D3—91no—
229C2×D3—182no—
230C2×D3—362no—
231C2×D3—361no—
232C2×D3—364no—
233C2×D3C4⋊#15711yescharacteristic
234C3×C4—22no—
235C3×C4—42no—
236C3×C4—62no—
237C3×C4—91no—
238C3×C4C22×#15711yescharacteristic
239Dic3—62no—
240Dic3—91no—
241Dic3—182no—
242Dic3—362no—
243Dic3C4⋊#15711yescharacteristic
244C32—22no—
245C32—42no—
246C32#489311yescharacteristic, complemented
247C23—271no—
248C2×C4—31no—
249C2×C4—91no—
250C2×C4—271no—
251C2×C4—271no—
252C2×C4—271no—
253C2×C4—271no—
254C8—31no—
255C8—91no—
256C8—181no—
257C8—181no—
258C8—271no—
259C8—541no—
260C8—541no—
261C8D3×#15711yescharacteristic
262C2×C3—22no—
263C2×C3—42no—
264C2×C3—122no—
265C2×C3—181no—
266C2×C3C2×C4⋊#15711yescharacteristic
267D3—21no—
268D3—64no—
269D3—181no—
270D3—362no—
271D3—722no—
272C22—31no—
273C22—91no—
274C22—271no—
275C22—1081no—
276C4—31no—
277C4—91no—
278C4—271no—
279C4C2×D3×#15711yescharacteristic
280C3—22no—
281C3—42no—
282C3#1059377611yescharacteristic, complemented
283C2—61no—
284C2—181no—
285C2—541no—
286C2D3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C4⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
4. #11723095C2semidirect
5. C3×#63771C2semidirect
6. #11722717C2semidirect
7. #11722716C2semidirect
29. #63771D3semidirect
170. C33#430semidirect
246. C32#4893semidirect
282. C3#10593776semidirect

Maximal subgroups

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

subgroupindex
#117242552
D3×#37882
C3⋊D3×ΓL(1,𝔽25)2
#117230952
C3×#637712
#117227172
#117227162
#105937763
#48939