Finite Group Databasebeta

← #13146560

Subgroups — #13146560

Order 1728.

443 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#11722445C211yesmaximal, characteristic
3#11721065C211yesmaximal, characteristic
4#11720700C214yesmaximal, complemented
5C2×C4×#272C2211yescharacteristic
6C2×#634⋊C3—21no—
7C2×#6199C412yes—
8C2×#6196—21no—
9#88651—21no—
10#88651—21no—
11#88651C412yes—
12#88651C2212yes—
13#88631—21no—
14#88627C2212yes—
15#88626—21no—
16#3748⋊C3—21no—
17#88435C2212yes—
18#88434—21no—
19C2×D3×Dic6—61no—
20D3×#208—61no—
21C22×#272C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×#1084D411yescharacteristic
24C4×#272C2×C411yescharacteristic
25C4×#272D411yescharacteristic
26C4×#272D411yescharacteristic
27C4×#272C2311yescharacteristic
28C2×(C3⋊Dic3)⋊C3—22no—
29#634⋊C3—22no—
30#634⋊C3—22no—
31#6199—22no—
32#6199C2×C414yes—
33#6196—22no—
34#6196—22no—
35#6188—22no—
36(C3⋊(C3⋊C8))⋊C3—22no—
37C2×#824—91nomaximal
38C2×#634—21no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—61no—
41C2×C3×Dic6—61no—
42C2×#627—62no—
43C2×D3×Dic3—62no—
44D3×Dic6—62no—
45D3×Dic6—62no—
46D3×Dic6—62no—
47D3×Dic6—62no—
48#3748—21no—
49C3×#208—61no—
50#3711—62no—
51D3×C3⋊C8—62no—
52C22×H(3,𝔽3)C22⋊C411yescharacteristic
53C2×#272—22no—
54C2×#272C22⋊C411yescharacteristic
55C2×#272C22×C411yescharacteristic
56C4×H(3,𝔽3)C22⋊C411yescharacteristic
57C4×H(3,𝔽3)C2×D411yescharacteristic
58#1084C22⋊C411yescharacteristic
59#1084C2×D411yescharacteristic
60(C3⋊Dic3)⋊C3—24no—
61C2×D3×Q8—181no—
62C22×Dic6—91no—
63C2×C3×M16—91no—
64C2×#208—91no—
65D3×M16—181no—
66#824—94no—
67C22×C3×D3—61no—
68C22×C3×D3—61no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C3⋊Dic3—22no—
72#634—22no—
73#634—22no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—62no—
77C3×C4×D3—61no—
78C3×C4×D3—61no—
79C3×C4×D3—61no—
80C3×C4×D3—61no—
81C3×C4×D3—61no—
82C3×C4×D3—61no—
83C3×C4×D3—61no—
84C3×C4×D3—61no—
85C3×Dic6—62no—
86C3×Dic6—62no—
87#627—64no—
88#627—64no—
89D3×Dic3—64no—
90D3×Dic3—64no—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—62no—
93C2×#101—271noSylow
94C2×H(3,𝔽3)—21no—
95C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
96#272—21no—
97#272#10112yescomplemented
98C2×C3×Q8—181no—
99C22×C3×C4—91no—
100C22×Dic3—92no—
101D3×Q8—182no—
102D3×Q8—182no—
103D3×Q8—182no—
104D3×Q8—182no—
105C2×C4×D3—61no—
106C2×C4×D3—61no—
107C2×C4×D3—182no—
108C2×Dic6—61no—
109C2×Dic6—92no—
110C2×Dic6—92no—
111C2×Dic6—181no—
112C2×Dic6—181no—
113C2×Dic6—181no—
114C2×Dic6—181no—
115C2×Dic6—182no—
116C3×M16—92no—
117C3×M16—181no—
118C3×M16—181no—
119C2×C3×C8—181no—
120#208—61no—
121#208—92no—
122#208—181no—
123C2×C3⋊C8—181no—
124ΓL(1,𝔽25)—182no—
125D3×C8—182no—
126C22×C32—21no—
127C22×C32—21no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C2×C3×D3—61no—
131C2×C3×D3—61no—
132C2×C3×D3—64no—
133C2×C3×D3—122no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C3⋊Dic3—24no—
139C3×Dic3—61no—
140C3×Dic3—61no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143C3×Dic3—64no—
144C22×Q8—271no—
145C2×M16—91no—
146C2×M16—271no—
147#101—274no—
148H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
149C23×C3—91no—
150C22×D3—61no—
151C22×D3—61no—
152C3×Q8—182no—
153C3×Q8—182no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—182no—
162C2×C3×C4GO+(4,𝔽2)11yescharacteristic
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×Dic3—62no—
166C2×Dic3—94no—
167C2×Dic3—182no—
168C2×Dic3—182no—
169C4×D3—61no—
170C4×D3—61no—
171C4×D3—61no—
172C4×D3—61no—
173C4×D3—61no—
174C4×D3—61no—
175C4×D3—61no—
176C4×D3—61no—
177C4×D3—184no—
178C4×D3—184no—
179Dic6—62no—
180Dic6—62no—
181Dic6—182no—
182Dic6—182no—
183Dic6—182no—
184Dic6—182no—
185Dic6—184no—
186Dic6—184no—
187C3×C8—182no—
188C3×C8—182no—
189C3⋊C8—62no—
190C3⋊C8—182no—
191C2×C32—22no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—41no—
195C3×D3—62no—
196C3×D3—62no—
197C3×D3—62no—
198C3×D3—121no—
199C2×Q8—181no—
200C2×Q8—272no—
201C2×Q8—272no—
202C2×Q8—541no—
203C2×Q8—541no—
204C2×Q8—541no—
205C22×C4—91no—
206C22×C4—272no—
207M16—92no—
208M16—181no—
209M16—181no—
210M16—272no—
211C2×C8—181no—
212C2×C8—541no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—91no—
216C22×C3—91no—
217C22×C3—182no—
218C22×C3D32⋊C411yescharacteristic
219C2×D3—61no—
220C2×D3—61no—
221C2×D3—61no—
222C2×D3—61no—
223C2×D3—64no—
224C2×D3—122no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—184no—
232C3×C4D32⋊C411yescharacteristic
233C3×C4C2×GO+(4,𝔽2)11yescharacteristic
234Dic3—61no—
235Dic3—61no—
236Dic3—61no—
237Dic3—61no—
238Dic3—64no—
239Dic3—184no—
240C32—21no—
241C32—21no—
242C23—91no—
243Q8—182no—
244Q8—182no—
245Q8—542no—
246Q8—542no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—91no—
251C2×C4—91no—
252C2×C4—182no—
253C2×C4—274no—
254C2×C4—542no—
255C2×C4#1088⋊C211yescharacteristic
256C8—182no—
257C8—182no—
258C2×C3—21no—
259C2×C3—62no—
260C2×C3—61no—
261C2×C3—61no—
262C2×C3—92no—
263C2×C3—121no—
264C2×C3—181no—
265C2×C3C2×D32⋊C411yescharacteristic
266D3—62no—
267D3—62no—
268D3—62no—
269D3—121no—
270C22—91no—
271C22—91no—
272C22—182no—
273C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
274C4—91no—
275C4—91no—
276C4—184no—
277C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
278C4C2×#1088⋊C211yescharacteristic
279C3—61no—
280C3—61no—
281C3#1059331511yescharacteristic
282C2—21no—
283C2—92no—
284C2—181no—
285C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
centerC2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutatorC2×#272C22×C4
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC2×C3C2×D32⋊C4
prime radicalC22×#272C2×C4
cosocleC4×#272C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)C2×#101
LCSG ▷ C2×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11720700C2semidirect
97. #272#101semidirect
148. H(3,𝔽3)C2×#101semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117224452
#117210652
#117207002
C2×#8249