Finite Group Databasebeta

← #13166477

Subgroups — #13166477

Order 1728.

448 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 72 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×#88926C211yesmaximal, characteristic
2C2×#88924C211yesmaximal, characteristic
3#11723469C211yesmaximal, characteristic
4C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6#11722732C211yesmaximal, characteristic, complemented
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593755—31nomaximal
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88926—22no—
15#88924—22no—
16#88859C411yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4C411yescharacteristic
19Dic3⋊(C3⋊Dic3)C411yescharacteristic
20#88804C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C2×#3788—31no—
23C2×#3787—31no—
24#64101—31no—
25#63836—62no—
26#63779—31no—
27#63779D311yescharacteristic, complemented
28C4⋊#622—31no—
29C4⋊#622—31no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3C2×C411yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37#6289D412yes—
38#6289Q812yes—
39C3⋊#622—21no—
40C3⋊#622—21no—
41C3×#622—21no—
42C3×#622—21no—
43#4839—91nomaximal
44C2×C4×C3⋊D3—31no—
45C2×#624—62no—
46C2×#622—31no—
47C2×#622—31no—
48C2×#622C2×D311yescharacteristic
49C2×#622C2×D311yescharacteristic
50#3788—62no—
51#3787—62no—
52C32×C4⋊C4—31no—
53#3752—31no—
54#3752—31no—
55#3752—62no—
56#3752—122no—
57C4⋊(C3⋊Dic3)—31no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C3×Dic3⋊C4—22no—
61C3×C4×Dic3—62no—
62Dic3⋊Dic3—62no—
63#3722—62no—
64Dic32—62no—
65C22×C33C22×C411yescharacteristic
66C2×C3⋊(C3⋊D3)—21no—
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)C4⋊C412yes—
69C33×C4—21no—
70C3⋊(C3⋊Dic3)—21no—
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3C2×D411yescharacteristic
73C3×C3⋊Dic3C2×Q811yescharacteristic
74C32×Dic3—21no—
75C32×Dic3C4⋊C412yes—
76C2×ΓL(1,𝔽25)—91no—
77C2×D3×C8—91no—
78#892—182no—
79#872—91no—
80C3×C4⋊C8—91no—
81Dic3⋊C8—91no—
82Dic3⋊C8—91no—
83C4⋊(C3⋊C8)—91no—
84C22×C3⋊D3—31no—
85C22×C3⋊D3—62no—
86C22×C3⋊D3—122no—
87C2×C32×C4—22no—
88C2×C32×C4—31no—
89C2×C32×C4—31no—
90C2×C32×C4—42no—
91C2×C32×C4C4×D311yescharacteristic
92C2×C3⋊Dic3—31no—
93C2×C3⋊Dic3—62no—
94C2×C3⋊Dic3—122no—
95C2×C3⋊Dic3C4×D311yescharacteristic
96C2×C3⋊Dic3C22×D311yescharacteristic
97C4×C3⋊D3—32no—
98C4×C3⋊D3—32no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103#624—64no—
104#624—64no—
105#622—21no—
106#622—21no—
107#622—61no—
108#622—61no—
109#424—271noSylow
110C2×C33C2×C4⋊C411yescharacteristic
111C2×C33#13111yescharacteristic
112C2×C33#13111yescharacteristic
113C3⋊(C3⋊D3)—22no—
114C2×C4×D3—91no—
115C2×C4×D3—182no—
116C2×C3×C8—91no—
117C2×C3×C8—91no—
118C3×C4⋊C4—62no—
119C3×C42—91no—
120(C2×D3)⋊C4—91no—
121(C2×D3)⋊C4—91no—
122(C2×D3)⋊C4—182no—
123(C2×D3)⋊C4—182no—
124(C2×D3)⋊C4—362no—
125C4⋊Dic3—182no—
126Dic3⋊C4—91no—
127Dic3⋊C4#15711yescharacteristic
128C4×Dic3—62no—
129C4×Dic3—91no—
130C4×Dic3—182no—
131C2×C3⋊C8—91no—
132C2×C3⋊C8—91no—
133ΓL(1,𝔽25)—182no—
134D3×C8—182no—
135C22×C32—22no—
136C22×C32—42no—
137C22×C32C2×C4×D311yescharacteristic
138C2×C3⋊D3—32no—
139C2×C3⋊D3—64no—
140C2×C3⋊D3—61no—
141C2×C3⋊D3—61no—
142C2×C3⋊D3—124no—
143C2×C3⋊D3—122no—
144C2×C3⋊D3—122no—
145C2×C3⋊D3—242no—
146C2×C3⋊D3—242no—
147C32×C4—21no—
148C32×C4—32no—
149C32×C4—42no—
150C32×C4—61no—
151C32×C4—82no—
152C3⋊Dic3—21no—
153C3⋊Dic3—61no—
154C3⋊Dic3—122no—
155C3⋊Dic3—242no—
156C3⋊Dic3D3×D411yescharacteristic
157C3⋊Dic3D3×Q811yescharacteristic
158C3×Dic3—24no—
159C3×Dic3—42no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C3×Dic3—122no—
163C2×M16—271no—
164C22×C8—271no—
165#117—271no—
166C4⋊C8—91no—
167C4⋊C8—271no—
168C4⋊C8—271no—
169C4⋊C8—271no—
170C33#42411yescharacteristic, complemented, Sylow
171C22×D3—91no—
172C22×D3—182no—
173C22×D3—362no—
174C2×C3×C4—22no—
175C2×C3×C4—42no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4C2×#15711yescharacteristic
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—91no—
184C2×Dic3—182no—
185C2×Dic3—362no—
186C2×Dic3C2×#15711yescharacteristic
187C2×Dic3C2×#15711yescharacteristic
188C4×D3—92no—
189C4×D3—92no—
190C4×D3—184no—
191C4×D3—184no—
192C3×C8—181no—
193C3×C8—181no—
194C3⋊C8—181no—
195C3⋊C8—181no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—22no—
199C2×C32—42no—
200C2×C32—42no—
201C2×C32—42no—
202C2×C32D3×C4⋊C411yescharacteristic
203C2×C32#90111yescharacteristic
204C2×C32#90711yescharacteristic
205C3⋊D3—62no—
206C3⋊D3—124no—
207C3⋊D3—244no—
208C22×C4—271no—
209M16—542no—
210C2×C8—91no—
211C2×C8—91no—
212C2×C8—271no—
213C2×C8—271no—
214C2×C8—542no—
215C4⋊C4—31no—
216C4⋊C4—271no—
217C22⋊C4—271no—
218C22⋊C4—271no—
219C42—91no—
220C42—271no—
221C22×C3—22no—
222C22×C3—42no—
223C22×C3C22×#15711yescharacteristic
224C2×D3—92no—
225C2×D3—184no—
226C2×D3—181no—
227C2×D3—181no—
228C2×D3—364no—
229C2×D3—362no—
230C2×D3—362no—
231C2×D3—722no—
232C2×D3—722no—
233C3×C4—21no—
234C3×C4—42no—
235C3×C4—64no—
236C3×C4—82no—
237C3×C4—91no—
238C3×C4—91no—
239C3×C4—122no—
240C3×C4—181no—
241Dic3—21no—
242Dic3—62no—
243Dic3—62no—
244Dic3—122no—
245Dic3—181no—
246Dic3—362no—
247Dic3—722no—
248Dic3C4⋊#15712yes—
249C32—22no—
250C32—42no—
251C32#483911yescharacteristic, complemented
252C23—271no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—272no—
258C2×C4—272no—
259C2×C4—271no—
260C2×C4D3×#15711yescharacteristic
261C8—181no—
262C8—181no—
263C8—541no—
264C8—541no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3C2×C4⋊#15711yescharacteristic
272C2×C3#6428711yescharacteristic
273C2×C3#6428911yescharacteristic
274D3—182no—
275D3—364no—
276D3—724no—
277C22—272no—
278C22—541no—
279C22—541no—
280C22C2×D3×#15711yescharacteristic
281C4—21no—
282C4—32no—
283C4—61no—
284C4—91no—
285C4—91no—
286C4—181no—
287C4—541no—
288C3—22no—
289C3—42no—
290C3#1059375511yescharacteristic, complemented
291C2—542no—
292C2#1172425211yescharacteristic
293C2D3×C4⋊#15711yescharacteristic
294C2#1172426711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C4⋊(C3⋊#622)C2semidirect
5. C3×#63779C2semidirect
6. #11722732C2semidirect
7. Dic3⋊#622C2semidirect
27. #63779D3semidirect
170. C33#424semidirect
251. C32#4839semidirect
290. C3#10593755semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
C2×#889242
#117234692
C4⋊(C3⋊#622)2
C3×#637792
#117227322
Dic3⋊#6222
#105937553
#48399