Finite Group Databasebeta

← #13156669

Subgroups — #13156669

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#11722080C211yesmaximal, characteristic
3#11722018C211yesmaximal, characteristic
4C3×#63576C211yesmaximal, characteristic
5C3×#63575C212yesmaximal
6#10590902—31nomaximal
7#10590415—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×#818—32no—
21C3×#807—32no—
22C3×#807—32no—
23#63576—31no—
24#63576D311yescharacteristic
25#63575—32no—
26#63575D312yes—
27#63553—32no—
28#63553—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34#6293—44no—
35C3×C3⋊(C3⋊C8)Q812yescomplemented
36C32×C3⋊C8Q812yescomplemented
37#4874—91no—
38#4874—92no—
39C2×#634—62no—
40C2×#627—62no—
41C2×#627—62no—
42C2×#627—64no—
43C2×C32×C8—31no—
44C32×C4⋊C4—32no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46#3750—62no—
47C4×C3⋊Dic3—32no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3—61no—
50C4×C3⋊Dic3—62no—
51C4×C3⋊Dic3—121no—
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53C3×C4⋊Dic3—22no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C2×D312yes—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—61no—
62C2×C3×C3⋊C8C2×D312yes—
63#3724—62no—
64#3724—62no—
65#3724—64no—
66C22×C33C2×D411yescharacteristic
67C33×C4C2×Q811yescharacteristic
68C33×C4C2×Q811yescharacteristic
69C3⋊(C3⋊Dic3)D812yes—
70C3⋊(C3⋊Dic3)Q1612yes—
71C3×C3⋊Dic3—42no—
72C32×Dic3—42no—
73#888—92no—
74#888—92no—
75#888—92no—
76#888—182no—
77C3×#107—91no—
78C3×#107—92no—
79#818—32no—
80#818—91no—
81C8×Dic3—92no—
82C8×Dic3—91no—
83C8×Dic3—181no—
84#807—32no—
85#807—32no—
86#807—92no—
87C2×C32×C4—21no—
88C2×C32×C4—22no—
89C2×C32×C4—41no—
90C2×C32×C4—62no—
91C2×C32×C4C22×D312yes—
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—31no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3—31no—
97C2×C3⋊Dic3—32no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3—121no—
104#634—124no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111#627—124no—
112#627—124no—
113#627—128no—
114C32×C8—32no—
115C3⋊(C3⋊C8)Dic612yescomplemented
116C3×C3⋊C8—22no—
117C3×C3⋊C8—34no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8Dic614yescomplemented
120#500—271noSylow
121C2×C33C4⋊Q811yescharacteristic
122C2×C33C2×D811yescharacteristic
123C2×C33C2×Q1611yescharacteristic
124C2×Dic6—182no—
125C2×Dic6—182no—
126C2×Dic6—182no—
127C2×Dic6—184no—
128C2×C3×C8—32no—
129C2×C3×C8—61no—
130C2×C3×C8—91no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—92no—
135C4⋊Dic3—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3—62no—
138C4⋊Dic3D3212yes—
139Dic3⋊C4—182no—
140Dic3⋊C4—182no—
141Dic3⋊C4—182no—
142Dic3⋊C4—184no—
143C4×Dic3—91no—
144C4×Dic3—92no—
145C4×Dic3—181no—
146C4×Dic3—182no—
147C4×Dic3—361no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8D3211yescharacteristic
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—41no—
154C22×C32D3×D412yes—
155C22×C32D3×D411yescharacteristic
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—122no—
163C32×C4C2×Dic611yescharacteristic
164C32×C4C2×Dic612yes—
165C32×C4D3×Q811yescharacteristic
166C32×C4D3×Q812yes—
167C3⋊Dic3—32no—
168C3⋊Dic3—34no—
169C3⋊Dic3—34no—
170C3⋊Dic3—32no—
171C3⋊Dic3—42no—
172C3⋊Dic3—64no—
173C3⋊Dic3—62no—
174C3⋊Dic3—62no—
175C3⋊Dic3—64no—
176C3⋊Dic3—122no—
177C3⋊Dic3—122no—
178C3×Dic3—44no—
179C3×Dic3—42no—
180C3×Dic3—42no—
181C3×Dic3—124no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C4⋊Q8—272no—
185#107—91no—
186#107—92no—
187#107—271no—
188C4×C8—271no—
189C33#50011yescharacteristic, complemented, Sylow
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—64no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—182no—
197C2×C3×C4C2×D3212yes—
198C2×C3×C4C2×D3211yescharacteristic
199C2×Dic3—22no—
200C2×Dic3—64no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—91no—
204C2×Dic3—91no—
205C2×Dic3—92no—
206C2×Dic3—92no—
207C2×Dic3—181no—
208C2×Dic3—181no—
209C2×Dic3—182no—
210C2×Dic3—182no—
211C2×Dic3—361no—
212C2×Dic3—361no—
213Dic6—364no—
214Dic6—364no—
215Dic6—364no—
216Dic6—368no—
217C3×C8—34no—
218C3×C8—62no—
219C3×C8—92no—
220C3⋊C8—34no—
221C3⋊C8—62no—
222C3⋊C8#62712yescomplemented
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32#88811yescharacteristic
233C2×C32#88812yes—
234C2×C32D3×D812yes—
235C2×C32D3×D811yescharacteristic
236C2×C32D3×Q1612yes—
237C2×C32D3×Q1611yescharacteristic
238C2×Q8—542no—
239C2×C8—31no—
240C2×C8—91no—
241C4⋊C4—32no—
242C4⋊C4—92no—
243C4⋊C4—542no—
244C42—271no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—41no—
248C22×C3#380211yescharacteristic
249C22×C3#381112yes—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—124no—
257C3×C4—122no—
258C3×C4—122no—
259C3×C4—362no—
260C3×C4D3×Dic612yes—
261C3×C4D3×Dic612yes—
262C3×C4#379711yescharacteristic
263C3×C4C2×#62711yescharacteristic
264Dic3—42no—
265Dic3—92no—
266Dic3—92no—
267Dic3—94no—
268Dic3—94no—
269Dic3—124no—
270Dic3—122no—
271Dic3—122no—
272Dic3—184no—
273Dic3—182no—
274Dic3—182no—
275Dic3—184no—
276Dic3—362no—
277Dic3—362no—
278C32—21no—
279C32—22no—
280C32—41no—
281C32#487412yescomplemented
282C32#487411yescharacteristic, complemented
283Q8—1084no—
284C2×C4—62no—
285C2×C4—182no—
286C2×C4—271no—
287C2×C4—271no—
288C2×C4D3311yescharacteristic
289C8—32no—
290C8—92no—
291C2×C3—21no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—21no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3#6379711yescharacteristic
301C2×C3#6380211yescharacteristic
302C2×C3#6384012yes—
303C2×C3#6391911yescharacteristic
304C2×C3#6392612yes—
305C2×C3#6394412yes—
306C22#8895411yescharacteristic
307C4—122no—
308C4—272no—
309C4—272no—
310C4—362no—
311C4D3×#62711yescharacteristic
312C4#8895711yescharacteristic
313C3—22no—
314C3—21no—
315C3—41no—
316C3#1059041512yescomplemented
317C3#1059090211yescharacteristic, complemented
318C2#1172283211yescharacteristic
319C2#1172284811yescharacteristic
320C2#1172303711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#500
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
35. C3×C3⋊(C3⋊C8)Q8semidirect
36. C32×C3⋊C8Q8semidirect
115. C3⋊(C3⋊C8)Dic6semidirect
119. C3×C3⋊C8Dic6semidirect
189. C33#500semidirect
222. C3⋊C8#627semidirect
281. C32#4874semidirect
282. C32#4874semidirect
316. C3#10590415semidirect
317. C3#10590902semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117220802
#117220182
C3×#635762
C3×#635752
#105909023
#105904153