Finite Group Databasebeta

← #13167365

Subgroups — #13167365

Order 1728.

386 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 49 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#11724333C212yesmaximal
2#11723649C211yesmaximal, characteristic, complemented
3#11723645C211yesmaximal, characteristic
4C3×#63768C211yesmaximal, characteristic, complemented
5#11722784C212yesmaximal, complemented
6#10593289—31nomaximal
7C4×#1113C2211yescharacteristic
8#89040C2212yes—
9C3×C4⋊#157C2212yes—
10#88940—22no—
11#88939—22no—
12C3×#3746—21no—
13C3×#3743—21no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88812—21no—
16#88809—21no—
17#88808C2211yescharacteristic
18D32⋊Q8—32no—
19#63802—31no—
20D3×Dic12—61no—
21#63791—31no—
22D3×ΓL(1,𝔽25)—61no—
23#63768D311yescharacteristic, complemented
24#63730—32no—
25C2×#1113D411yescharacteristic
26C2×C3×#157—22no—
27C3×C4×C3⋊D3—21no—
28C3×C4×C3⋊D3C2311yescharacteristic
29C3×#634—22no—
30#6296—22no—
31#6294—21no—
32#6294—22no—
33#6294D411yescharacteristic
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)—21no—
36C3×C3⋊(C3⋊C8)D411yescharacteristic
37#4867—91nomaximal
38C4×D32—31no—
39C4×D32—61no—
40#3797—32no—
41D3×Dic6—62no—
42C4⋊#157C2×D312yes—
43#3773—62no—
44D32⋊C4—62no—
45#3746—21no—
46#3743—21no—
47C8×C3⋊D3C2×D311yescharacteristic
48C3×Dic12—61no—
49C3×Dic12—61no—
50C3×Dic12—121no—
51C3×ΓL(1,𝔽25)—61no—
52C3×ΓL(1,𝔽25)—61no—
53C3×ΓL(1,𝔽25)—121no—
54C3×D3×C8—61no—
55C3×D3×C8—61no—
56#3719—62no—
57#3718—61no—
58#3712—61no—
59#3711—61no—
60#3711—61no—
61#3710—31no—
62D3×C3⋊C8—61no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3C2×D411yescharacteristic
65#1113—22no—
66C3×#157—42no—
67C33×C4C2×D411yescharacteristic
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—42no—
70C3×C3⋊Dic3#4111yescharacteristic
71D3×Q16—181no—
72D3×M16—181no—
73C2×Dic12—91no—
74C2×ΓL(1,𝔽25)—91no—
75#897—92no—
76C3×#106—91no—
77#810—92no—
78C2×D32—31no—
79C2×D32—61no—
80C2×#157—22no—
81C4×C3⋊D3—21no—
82C4×C3⋊D3C22×D311yescharacteristic
83#634—22no—
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—121no—
89C3×Dic6—62no—
90C3×Dic6—62no—
91C3×Dic6—122no—
92#627—62no—
93#627—122no—
94#624—31no—
95#624—61no—
96#624—62no—
97D3×Dic3—61no—
98D3×Dic3—61no—
99D3×Dic3—61no—
100D3×Dic3—122no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8—41no—
104C32×C8—41no—
105C32×C8D1211yescharacteristic
106C3⋊(C3⋊C8)—21no—
107C3⋊(C3⋊C8)D1211yescharacteristic
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—61no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—121no—
113#470—271noSylow
114C2×C33#12111yescharacteristic
115C3×C3⋊D3—41no—
116C3×C3⋊D3#13712yes—
117D3×Q8—182no—
118C2×C4×D3—91no—
119C2×C4×D3—181no—
120C2×Dic6—92no—
121C3×Q16—181no—
122C3×M16—181no—
123C2×C3×C8—91no—
124C3×C4⋊C4—92no—
125#216—182no—
126(C2×D3)⋊C4—182no—
127C4⋊Dic3—182no—
128#208—181no—
129C2×C3⋊C8—91no—
130Dic12—61no—
131Dic12—61no—
132Dic12—121no—
133Dic12—181no—
134ΓL(1,𝔽25)—61no—
135ΓL(1,𝔽25)—61no—
136ΓL(1,𝔽25)—121no—
137ΓL(1,𝔽25)—181no—
138ΓL(1,𝔽25)—181no—
139D3×C8—61no—
140D3×C8—61no—
141D3×C8—181no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3D3×D411yescharacteristic
144C2×C3×D3—61no—
145C2×C3×D3—61no—
146C2×C3×D3—61no—
147C2×C3×D3—61no—
148C2×C3×D3—121no—
149D32—62no—
150D32—122no—
151#157—42no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4C2×D1211yescharacteristic
157C3⋊Dic3—21no—
158C3⋊Dic3—42no—
159C3⋊Dic3#23911yescharacteristic
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—121no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C3×Dic3—242no—
168C2×Q16—271no—
169C2×M16—271no—
170C22⋊Q8—272no—
171#106—91no—
172#103—272no—
173C33#47011yescharacteristic, complemented, Sylow
174C22×D3—91no—
175C22×D3—181no—
176C3×Q8—182no—
177C2×C3×C4—91no—
178C2×C3×C4—181no—
179C2×C3×C4—182no—
180C2×Dic3—91no—
181C2×Dic3—181no—
182C2×Dic3—182no—
183C4×D3—61no—
184C4×D3—61no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—121no—
188C4×D3—181no—
189C4×D3—181no—
190C4×D3—181no—
191C4×D3—362no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195Dic6—182no—
196Dic6—362no—
197C3×C8—21no—
198C3×C8—21no—
199C3×C8—41no—
200C3×C8—41no—
201C3×C8—91no—
202C3×C8—181no—
203C3×C8GO+(4,𝔽2)11yescharacteristic
204C3⋊C8—61no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C3⋊C8—121no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32#89511yescharacteristic
214C3⋊D3—41no—
215C3⋊D3#91512yes—
216C3×D3—62no—
217C3×D3—62no—
218C3×D3—121no—
219C3×D3—121no—
220C3×D3—241no—
221C2×Q8—272no—
222C22×C4—271no—
223Q16—181no—
224Q16—541no—
225M16—181no—
226M16—541no—
227C2×C8—91no—
228C2×C8—271no—
229C4⋊C4—92no—
230C4⋊C4—542no—
231C22⋊C4—542no—
232C22×C3—91no—
233C22×C3—181no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—121no—
239C2×D3—182no—
240C2×D3—362no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—91no—
246C3×C4—181no—
247C3×C4—362no—
248C3×C4—362no—
249C3×C4C2×GO+(4,𝔽2)11yescharacteristic
250Dic3—61no—
251Dic3—61no—
252Dic3—61no—
253Dic3—61no—
254Dic3—121no—
255Dic3—122no—
256Dic3—122no—
257Dic3—242no—
258C32—21no—
259C32—21no—
260C32—41no—
261C32—41no—
262C32#486711yescharacteristic, complemented
263C23—271no—
264Q8—182no—
265Q8—542no—
266C2×C4—91no—
267C2×C4—181no—
268C2×C4—182no—
269C2×C4—271no—
270C2×C4—541no—
271C2×C4—542no—
272C8—91no—
273C8—181no—
274C8#157⋊D311yescharacteristic
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—92no—
280C2×C3—361no—
281C2×C3#6423011yescharacteristic
282D3—62no—
283D3—62no—
284D3—121no—
285D3—121no—
286D3—241no—
287C22—91no—
288C22—181no—
289C22—542no—
290C4—91no—
291C4—181no—
292C4—362no—
293C4—362no—
294C4C2×#157⋊D311yescharacteristic
295C3—21no—
296C3—21no—
297C3—41no—
298C3—41no—
299C3#1059328911yescharacteristic, complemented
300C2—92no—
301C2—361no—
302C2#1172434311yescharacteristic
subgroupG/H
centerC2#11724343
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#121
prime radicalC2×#1113D4
hypercenterC8#157⋊D3
hypocenterC33#470
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723649C2semidirect
4. C3×#63768C2semidirect
5. #11722784C2semidirect
23. #63768D3semidirect
173. C33#470semidirect
262. C32#4867semidirect
299. C3#10593289semidirect

Maximal subgroups

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

subgroupindex
#117243332
#117236492
#117236452
C3×#637682
#117227842
#105932893
#48679