Finite Group Databasebeta

← #13156653

Subgroups — #13156653

Order 1728.

602 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 112 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#11723536C212yesmaximal
2#11722079C211yesmaximal, characteristic
3#11722018C211yesmaximal, characteristic
4C3×#63574C211yesmaximal, characteristic
5C3×#63573C212yesmaximal
6#10590900—31nomaximal
7#10590416—32nomaximal
8C2×#6293—22no—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88806—22no—
15#64097—32no—
16#63919—32no—
17#63919—32no—
18#63919—62no—
19C8×C3⋊Dic3—31no—
20C3×#817—32no—
21C3×#808—32no—
22C3×#808—32no—
23#63574—31no—
24#63574D311yescharacteristic
25#63573—32no—
26#63573D312yes—
27#63553—32no—
28#63553—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)D412yes—
31C2×C3×C3⋊Dic3—22no—
32C2×C32×Dic3—22no—
33#6293—44no—
34C3×C3⋊(C3⋊C8)Q812yescomplemented
35C32×C3⋊C8Q812yescomplemented
36#4855—91no—
37#4855—92no—
38C2×#634—62no—
39C2×#627—62no—
40C2×#627—62no—
41C2×#627—64no—
42C2×C32×C8—31no—
43C32×C4⋊C4—32no—
44C4⋊(C3⋊Dic3)C2×D312yes—
45#3750—62no—
46C4×C3⋊Dic3—32no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—61no—
49C4×C3⋊Dic3—62no—
50C4×C3⋊Dic3—121no—
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52C3×C4⋊Dic3—22no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3C2×D312yes—
57C3×C4⋊Dic3C2×D312yes—
58C2×C3×C3⋊C8—21no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—61no—
61C2×C3×C3⋊C8C2×D312yes—
62#3724—62no—
63#3724—62no—
64#3724—64no—
65C22×C33C2×D411yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C33×C4C2×Q811yescharacteristic
68C3⋊(C3⋊Dic3)SD1614yes—
69C3×C3⋊Dic3—42no—
70C32×Dic3—42no—
71#888—92no—
72#888—92no—
73#888—92no—
74#888—182no—
75C3×#106—91no—
76C3×#106—92no—
77#817—32no—
78#817—91no—
79C8×Dic3—92no—
80C8×Dic3—91no—
81C8×Dic3—181no—
82#808—32no—
83#808—32no—
84#808—92no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—41no—
88C2×C32×C4—62no—
89C2×C32×C4C22×D312yes—
90C2×C32×C4C22×D311yescharacteristic
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—32no—
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—62no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—122no—
99#634—124no—
100C2×C3×Dic3—24no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106#627—124no—
107#627—124no—
108#627—128no—
109C32×C8—32no—
110C3⋊(C3⋊C8)Dic612yescomplemented
111C3×C3⋊C8—22no—
112C3×C3⋊C8—34no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8Dic614yescomplemented
115#498—271noSylow
116C2×C33C4⋊Q811yescharacteristic
117C2×C33C2×SD1612yes—
118C2×Dic6—182no—
119C2×Dic6—182no—
120C2×Dic6—182no—
121C2×Dic6—184no—
122C2×C3×C8—32no—
123C2×C3×C8—61no—
124C2×C3×C8—91no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—62no—
128C3×C4⋊C4—92no—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131C4⋊Dic3—62no—
132C4⋊Dic3D3212yes—
133Dic3⋊C4—182no—
134Dic3⋊C4—182no—
135Dic3⋊C4—182no—
136Dic3⋊C4—184no—
137C4×Dic3—91no—
138C4×Dic3—92no—
139C4×Dic3—181no—
140C4×Dic3—182no—
141C4×Dic3—361no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—61no—
144C2×C3⋊C8D3211yescharacteristic
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—41no—
148C22×C32D3×D412yes—
149C22×C32D3×D411yescharacteristic
150C32×C4—22no—
151C32×C4—22no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—122no—
157C32×C4C2×Dic611yescharacteristic
158C32×C4C2×Dic612yes—
159C32×C4D3×Q811yescharacteristic
160C32×C4D3×Q812yes—
161C3⋊Dic3—34no—
162C3⋊Dic3—34no—
163C3⋊Dic3—34no—
164C3⋊Dic3—42no—
165C3⋊Dic3—64no—
166C3⋊Dic3—64no—
167C3⋊Dic3—64no—
168C3⋊Dic3—124no—
169C3×Dic3—44no—
170C3×Dic3—42no—
171C3×Dic3—42no—
172C3×Dic3—124no—
173C3×Dic3—122no—
174C3×Dic3—122no—
175C4⋊Q8—272no—
176#106—91no—
177#106—92no—
178#106—271no—
179C4×C8—271no—
180C33#49811yescharacteristic, complemented, Sylow
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—64no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—182no—
188C2×C3×C4C2×D3212yes—
189C2×C3×C4C2×D3211yescharacteristic
190C2×Dic3—22no—
191C2×Dic3—64no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—92no—
195C2×Dic3—92no—
196C2×Dic3—92no—
197C2×Dic3—182no—
198C2×Dic3—182no—
199C2×Dic3—182no—
200C2×Dic3—362no—
201Dic6—364no—
202Dic6—364no—
203Dic6—364no—
204Dic6—368no—
205C3×C8—34no—
206C3×C8—62no—
207C3×C8—92no—
208C3⋊C8—34no—
209C3⋊C8—62no—
210C3⋊C8#62712yescomplemented
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—22no—
215C2×C32—22no—
216C2×C32—41no—
217C2×C32—42no—
218C2×C32#88811yescharacteristic
219C2×C32#88812yes—
220C2×C32D3×SD1612yes—
221C2×C32D3×SD1612yes—
222C2×C32D3×SD1612yes—
223C2×Q8—542no—
224C2×C8—31no—
225C2×C8—91no—
226C4⋊C4—32no—
227C4⋊C4—92no—
228C4⋊C4—542no—
229C42—271no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3#380211yescharacteristic
234C22×C3#381112yes—
235C3×C4—21no—
236C3×C4—22no—
237C3×C4—21no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—124no—
242C3×C4—122no—
243C3×C4—122no—
244C3×C4—362no—
245C3×C4D3×Dic612yes—
246C3×C4D3×Dic612yes—
247C3×C4#379711yescharacteristic
248C3×C4C2×#62711yescharacteristic
249Dic3—42no—
250Dic3—94no—
251Dic3—94no—
252Dic3—94no—
253Dic3—124no—
254Dic3—122no—
255Dic3—122no—
256Dic3—184no—
257Dic3—184no—
258Dic3—184no—
259Dic3—364no—
260C32—21no—
261C32—22no—
262C32—41no—
263C32#485512yescomplemented
264C32#485511yescharacteristic, complemented
265Q8—1084no—
266C2×C4—62no—
267C2×C4—182no—
268C2×C4—272no—
269C2×C4D3311yescharacteristic
270C8—32no—
271C8—92no—
272C2×C3—21no—
273C2×C3—22no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3—41no—
278C2×C3—42no—
279C2×C3#6379612yes—
280C2×C3#6384012yes—
281C2×C3#6391911yescharacteristic
282C2×C3#6393012yes—
283C2×C3#6394012yes—
284C22#8895411yescharacteristic
285C4—122no—
286C4—274no—
287C4—362no—
288C4D3×#62711yescharacteristic
289C4#8895711yescharacteristic
290C3—22no—
291C3—21no—
292C3—41no—
293C3#1059041612yescomplemented
294C3#1059090011yescharacteristic, complemented
295C2#1172284112yes—
296C2#1172303711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#498
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
34. C3×C3⋊(C3⋊C8)Q8semidirect
35. C32×C3⋊C8Q8semidirect
110. C3⋊(C3⋊C8)Dic6semidirect
114. C3×C3⋊C8Dic6semidirect
180. C33#498semidirect
210. C3⋊C8#627semidirect
263. C32#4855semidirect
264. C32#4855semidirect
293. C3#10590416semidirect
294. C3#10590900semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117220792
#117220182
C3×#635742
C3×#635732
#105909003
#105904163