Finite Group Databasebeta

← #13167579

Subgroups — #13167579

Order 1728.

492 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 87 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×C3×#3792C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic
3#11723715C211yesmaximal, characteristic
4#11722797C212yesmaximal, complemented
5#11722796C212yesmaximal, complemented
6#10593507—31nomaximal
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C3×#3792C414yescomplemented
10C2×C3×#622C2212yes—
11C3×#3757—21no—
12#88815—21no—
13#88815C2211yescharacteristic
14#88815C2212yes—
15#88815C2211yescharacteristic
16#88813—21no—
17C2×#3792D311yescharacteristic
18C2×#3724—31no—
19C2×Dic3⋊Dic3—61no—
20#63957—31no—
21#63955—61no—
22#63743—32no—
23#63742—32no—
24C23×C33D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3—42no—
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3D411yescharacteristic
31C2×C3×C3⋊Dic3Q812yes—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×#622C2×C414yescomplemented
34#5121—91nomaximal
35C22×C3⋊Dic3—21no—
36C22×C3⋊Dic3C2×D311yescharacteristic
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—121no—
42#3792Dic314yescomplemented
43C2×#622C2×D312yes—
44#3757—21no—
45C3×#217—61no—
46C3×#217—61no—
47C3×#217—121no—
48#3724—31no—
49#3724—31no—
50#3724—32no—
51#3724—61no—
52Dic3⋊Dic3—61no—
53Dic3⋊Dic3—61no—
54Dic3⋊Dic3—62no—
55Dic3⋊Dic3—62no—
56Dic32—61no—
57Dic32—61no—
58Dic32—61no—
59C22×C33—21no—
60C22×C33—21no—
61C22×C33C4⋊C412yes—
62C22×C33C2×D411yescharacteristic
63C3×C3⋊Dic3—42no—
64C3×C3⋊Dic3—41no—
65C3×C3⋊Dic3—41no—
66C3×C3⋊Dic3C4⋊C412yes—
67C3×C3⋊Dic3C22×C411yescharacteristic
68C3×C3⋊Dic3C2×Q811yescharacteristic
69C2×C3×M16—91no—
70#926—91no—
71C2×C4⋊Dic3—91no—
72C2×Dic3⋊C4—181no—
73#877—181no—
74#818—92no—
75#817—92no—
76C23×C32—21no—
77C23×C32—21no—
78C23×C32—41no—
79C23×C32—41no—
80C23×C32D1211yescharacteristic
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3—42no—
85C2×C3⋊Dic3Dic612yes—
86C2×C3⋊Dic3D1211yescharacteristic
87C2×C3⋊Dic3C2×Dic312yes—
88C2×C3⋊Dic3C22×D311yescharacteristic
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—122no—
107C2×C3×Dic3—122no—
108C2×C3×Dic3—124no—
109#622C2×Dic314yescomplemented
110#428—271noSylow
111C2×C33—22no—
112C2×C33C2×C4⋊C411yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C2×C33#13711yescharacteristic
115C22×C3×C4—91no—
116C22×C3×C4—181no—
117C22×Dic3—61no—
118C22×Dic3—61no—
119C22×Dic3—61no—
120C22×Dic3—61no—
121C22×Dic3—121no—
122C3×M16—94no—
123C2×C3×C8—92no—
124C3×C22⋊C4—181no—
125#217—61no—
126#217—61no—
127#217—121no—
128C4⋊Dic3—91no—
129C4⋊Dic3—92no—
130C4⋊Dic3—91no—
131C4⋊Dic3—181no—
132Dic3⋊C4—181no—
133Dic3⋊C4—181no—
134Dic3⋊C4—182no—
135Dic3⋊C4—182no—
136C4×Dic3—181no—
137C4×Dic3—181no—
138C4×Dic3—181no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32—42no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—42no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—81no—
156C22×C32—81no—
157C22×C32C4⋊Dic312yes—
158C22×C32C2×D1211yescharacteristic
159C3⋊Dic3—42no—
160C3⋊Dic3—41no—
161C3⋊Dic3—41no—
162C3⋊Dic3C4⋊Dic312yes—
163C3⋊Dic3C2×Dic611yescharacteristic
164C3⋊Dic3C22×Dic311yescharacteristic
165C3×Dic3—62no—
166C3×Dic3—62no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—121no—
172C3×Dic3—121no—
173C3×Dic3—122no—
174C3×Dic3—124no—
175C3×Dic3—121no—
176C3×Dic3—122no—
177C3×Dic3—121no—
178C3×Dic3—241no—
179C3×Dic3—241no—
180C2×M16—91no—
181#117—271no—
182C2×C4⋊C4—271no—
183#107—272no—
184#106—272no—
185C33#42811yescharacteristic, complemented, Sylow
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3GO+(4,𝔽2)11yescharacteristic
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—92no—
194C2×C3×C4—92no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4—182no—
198C2×C3×C4—362no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—62no—
213C2×Dic3—122no—
214C2×Dic3—121no—
215C2×Dic3—121no—
216C2×Dic3—122no—
217C2×Dic3—122no—
218C2×Dic3—124no—
219C3×C8—94no—
220C2×C32—24no—
221C2×C32—22no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—42no—
235C2×C32—44no—
236C2×C32—82no—
237C2×C32#90811yescharacteristic
238C2×C32#90911yescharacteristic
239C2×C32C2×C4⋊Dic311yescharacteristic
240C22×C4—91no—
241C22×C4—181no—
242M16—94no—
243C2×C8—92no—
244C4⋊C4—271no—
245C4⋊C4—272no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C22⋊C4—181no—
249C42—541no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#377312yes—
269C22×C3C2×GO+(4,𝔽2)11yescharacteristic
270C3×C4—91no—
271C3×C4—92no—
272C3×C4—91no—
273C3×C4—362no—
274C3×C4—361no—
275C3×C4—361no—
276Dic3—62no—
277Dic3—62no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—61no—
282Dic3—122no—
283Dic3—121no—
284Dic3—121no—
285Dic3—124no—
286Dic3—122no—
287Dic3—121no—
288Dic3—121no—
289Dic3—241no—
290Dic3—241no—
291C32—21no—
292C32—21no—
293C32—41no—
294C32—41no—
295C32#512111yescharacteristic, complemented
296C23#157⋊D311yescharacteristic
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—92no—
300C2×C4—92no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4—182no—
304C2×C4—362no—
305C8—94no—
306C2×C3—24no—
307C2×C3—22no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—42no—
315C2×C3—44no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—82no—
323C2×C3C2×#377311yescharacteristic
324C2×C3#6423611yescharacteristic
325C2×C3#6423911yescharacteristic
326C22—21no—
327C22—21no—
328C22#8894012yes—
329C22C2×#157⋊D311yescharacteristic
330C4—91no—
331C4—92no—
332C4—91no—
333C4—362no—
334C4—361no—
335C4—361no—
336C3—21no—
337C3—21no—
338C3—41no—
339C3—41no—
340C3#1059350711yescharacteristic, complemented
341C2—22no—
342C2C2×#8894011yescharacteristic
343C2#1172435011yescharacteristic
344C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23#157⋊D3
hypocenterC33#428
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722797C2semidirect
5. #11722796C2semidirect
9. C3×#3792C4semidirect
33. C3×#622C2×C4semidirect
42. #3792Dic3semidirect
109. #622C2×Dic3semidirect
185. C33#428semidirect
295. C32#5121semidirect
340. C3#10593507semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×#888152
#117237152
#117227972
#117227962
#105935073
#51219