Finite Group Databasebeta

← #13167423

Subgroups — #13167423

Order 1728.

600 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 79 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×#89040C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3#11723673C211yesmaximal, characteristic
4#11722794C214yesmaximal, complemented
5#10593374—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#634—21no—
8C2×#6296C2212yes—
9C2×#6294—21no—
10#89040—22no—
11#89040C414yescomplemented
12C2×C3×#622C2212yes—
13C3×#3788—22no—
14C3×C4×C3⋊Dic3—21no—
15#88815C2212yes—
16#88814—21no—
17C2×#3797—31no—
18C2×D3×Dic6—61no—
19C2×#3788D311yescharacteristic
20#63874—31no—
21#63855—61no—
22#63740—34no—
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3C2×C412yes—
29C3×C4×C3⋊D3D412yes—
30C3×#634—24no—
31#6296—22no—
32#6296C2×C414yes—
33#6294—24no—
34C3×#622—22no—
35#5120—91nomaximal
36C2×C4×C3⋊D3C2×D311yescharacteristic
37C2×#634—21no—
38C2×C3×C4×D3—61no—
39C2×C3×C4×D3—61no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—61no—
42C2×C3×Dic6—121no—
43C2×#627—32no—
44C2×#627—62no—
45C2×#624—61no—
46C2×D3×Dic3—62no—
47#3797—34no—
48#3797—62no—
49D3×Dic6—64no—
50D3×Dic6—64no—
51C2×#622C2×D312yes—
52#3788—22no—
53C4×C3⋊Dic3—21no—
54C3×C4×Dic3—61no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—121no—
57#3724—32no—
58Dic3⋊Dic3—62no—
59#3722—61no—
60(C2×D3)⋊Dic3—62no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3C22⋊C412yes—
64C33×C4C22⋊C412yes—
65C3×C3⋊Dic3—24no—
66C3×C3⋊Dic3—42no—
67C3×C3⋊Dic3C22×C411yescharacteristic
68C3×C3⋊Dic3C2×D411yescharacteristic
69C2×D3×Q8—181no—
70C22×Dic6—91no—
71C2×C3×M16—91no—
72#926—91no—
73#872—181no—
74#816—94no—
75C22×C3⋊D3D1211yescharacteristic
76C22×C3×D3—61no—
77C22×C3×D3—61no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4D1211yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3C22×D311yescharacteristic
86C4×C3⋊D3C4×D312yes—
87C4×C3⋊D3#6712yes—
88#634—24no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96C2×C3×Dic3—122no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×C4×D3—62no—
100C3×C4×D3—62no—
101C3×Dic6—64no—
102C3×Dic6—64no—
103C3×Dic6—124no—
104#627—34no—
105#627—64no—
106#627—64no—
107#627—62no—
108#624—64no—
109D3×Dic3—68no—
110#622—22no—
111#419—271noSylow
112C2×C33C2×C22⋊C411yescharacteristic
113C2×C33#13712yes—
114C3×C3⋊D3—22no—
115C2×C3×Q8—181no—
116C22×C3×C4—91no—
117C22×Dic3—181no—
118D3×Q8—184no—
119D3×Q8—184no—
120C2×C4×D3—61no—
121C2×C4×D3—61no—
122C2×C4×D3—182no—
123C2×Dic6—61no—
124C2×Dic6—61no—
125C2×Dic6—94no—
126C2×Dic6—92no—
127C2×Dic6—121no—
128C2×Dic6—182no—
129C2×Dic6—182no—
130C3×M16—182no—
131C2×C3×C8—92no—
132C3×C42—181no—
133#217—181no—
134(C2×D3)⋊C4—182no—
135C4⋊Dic3—92no—
136Dic3⋊C4—182no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C4×Dic3—121no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32C2×D1211yescharacteristic
145C2×C3⋊D3—22no—
146C2×C3⋊D3(C2×D3)⋊C412yes—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—64no—
150C2×C3×D3—122no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—42no—
154C32×C4—42no—
155C32×C4(C2×D3)⋊C412yes—
156C3⋊Dic3—24no—
157C3⋊Dic3—42no—
158C3⋊Dic3C2×C4×D311yescharacteristic
159C3⋊Dic3C2×#6711yescharacteristic
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—64no—
165C3×Dic3—64no—
166C3×Dic3—124no—
167C3×Dic3—122no—
168C3×Dic3—124no—
169C3×Dic3—122no—
170C22×Q8—271no—
171C2×M16—91no—
172#117—271no—
173#103—274no—
174C33#41911yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C22×D3—61no—
177C22×D3—61no—
178C3×Q8—184no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—91no—
184C2×C3×C4—92no—
185C2×C3×C4—92no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—122no—
196C2×Dic3—122no—
197C2×Dic3—184no—
198C4×D3—62no—
199C4×D3—62no—
200C4×D3—62no—
201C4×D3—62no—
202C4×D3—188no—
203Dic6—64no—
204Dic6—64no—
205Dic6—94no—
206Dic6—124no—
207Dic6—184no—
208Dic6—182no—
209Dic6—184no—
210C3×C8—182no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—42no—
218C2×C32—42no—
219C2×C32#90912yes—
220C2×C32C2×(C2×D3)⋊C411yescharacteristic
221C3⋊D3—22no—
222C3×D3—64no—
223C3×D3—122no—
224C2×Q8—181no—
225C2×Q8—272no—
226C2×Q8—274no—
227C2×Q8—542no—
228C22×C4—91no—
229C22×C4—541no—
230M16—182no—
231C2×C8—92no—
232C4⋊C4—272no—
233C22⋊C4—541no—
234C42—181no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—92no—
240C22×C3—182no—
241C22×C3C2×GO+(4,𝔽2)11yescharacteristic
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—64no—
245C2×D3—122no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—42no—
249C3×C4—42no—
250C3×C4—91no—
251C3×C4—91no—
252C3×C4—184no—
253C3×C4—362no—
254C3×C4D32⋊C412yes—
255Dic3—61no—
256Dic3—61no—
257Dic3—61no—
258Dic3—61no—
259Dic3—64no—
260Dic3—64no—
261Dic3—124no—
262Dic3—122no—
263Dic3—124no—
264Dic3—122no—
265C32—21no—
266C32—21no—
267C32—41no—
268C32—41no—
269C32#512011yescharacteristic, complemented
270C23—91no—
271Q8—184no—
272Q8—274no—
273Q8—542no—
274C2×C4—91no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—544no—
280C2×C4#157⋊D311yescharacteristic
281C8—182no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—182no—
291C2×C3#6422612yes—
292C2×C3C2×D32⋊C411yescharacteristic
293D3—64no—
294D3—122no—
295C22—92no—
296C22—182no—
297C22C2×#157⋊D311yescharacteristic
298C4—91no—
299C4—91no—
300C4—184no—
301C4—362no—
302C4#8893912yes—
303C3—21no—
304C3—21no—
305C3—41no—
306C3—41no—
307C3#1059337411yescharacteristic, complemented
308C2—182no—
309C2#1172433812yes—
310C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#419
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722794C2semidirect
11. #89040C4semidirect
174. C33#419semidirect
269. C32#5120semidirect
307. C3#10593374semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37882
#117236732
#117227942
#105933743
#51209