Finite Group Databasebeta

← C3×#10590074 #13154924

Subgroups — C3×#10590074 #13154924

Order 1728.

492 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C3×#63957C211yesmaximal, characteristic
3C2×C3×Dic32C211yesmaximal, characteristic
4#10590074C311yesmaximal, characteristic, direct product factor
5C22×C3×C3⋊Dic3C2211yescharacteristic
6C22×C32×Dic3—21no—
7C3×#3792—21no—
8C3×#3792C412yescomplemented
9C2×C3×#622—21no—
10C32×#217—21no—
11C3×#3724C412yescomplemented
12C3×Dic32—21no—
13C3×Dic32—21no—
14C3×Dic32C412yescomplemented
15C2×#3792C2×C311yescharacteristic
16C2×C3×C4×Dic3—61no—
17C3×#877—61no—
18#63957C2×C311yescharacteristic
19C2×Dic32C2×C311yescharacteristic
20C23×C33D411yescharacteristic
21C2×C3×C3⋊Dic3C2×C411yescharacteristic
22C2×C3×C3⋊Dic3C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3Q811yescharacteristic
27C2×C32×Dic3—24no—
28C2×C32×Dic3—22no—
29C2×C32×Dic3—22no—
30C3×#622—22no—
31C3×#339—91nomaximal
32C22×C32×C4—61no—
33C22×C3⋊Dic3C22×C311yescharacteristic
34C22×C3×Dic3—21no—
35C22×C3×Dic3—21no—
36C22×C3×Dic3—41no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39#3792—21no—
40#3792C3×C412yescomplemented
41C2×#622—21no—
42C32×C22⋊C4—61no—
43C3×#217—21no—
44C3×#217—21no—
45C3×#217—41no—
46C3×Dic3⋊C4—62no—
47C3×C4×Dic3—62no—
48C3×C4×Dic3—62no—
49C3×C4×Dic3—62no—
50#3724C3×C412yescomplemented
51Dic32—21no—
52Dic32—21no—
53Dic32C3×C412yescomplemented
54C22×C33—22no—
55C22×C33C22⋊C411yescharacteristic
56C22×C33C22⋊C411yescharacteristic
57C22×C33C4⋊C411yescharacteristic
58C3×C3⋊Dic3C4211yescharacteristic
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C3×C3⋊Dic3C4⋊C411yescharacteristic
62C32×Dic3—24no—
63C32×Dic3—42no—
64C2×C3×M16—91no—
65C3×#117—91no—
66C2×C3×C42—91no—
67C2×C4×Dic3—61no—
68#877—61no—
69C23×C32—21no—
70C23×C32—21no—
71C23×C32—41no—
72C23×C32—41no—
73C23×C32C3×D411yescharacteristic
74C2×C32×C4—62no—
75C2×C32×C4—64no—
76C2×C32×C4—62no—
77C2×C3⋊Dic3C2×C3×C411yescharacteristic
78C2×C3⋊Dic3C2×C3×C411yescharacteristic
79C2×C3⋊Dic3C2×C3×C411yescharacteristic
80C2×C3⋊Dic3C3×D411yescharacteristic
81C2×C3⋊Dic3C3×D411yescharacteristic
82C2×C3⋊Dic3C3×Q811yescharacteristic
83C2×C3×Dic3—22no—
84C2×C3×Dic3—24no—
85C2×C3×Dic3—24no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—44no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104#622—22no—
105#339—91noSylow
106C2×C33—21no—
107C2×C33—21no—
108C2×C33#9511yescharacteristic
109C2×C33#10412yes—
110C22×C3×C4—61no—
111C22×C3×C4—61no—
112C22×C3×C4—91no—
113C22×C3×C4—121no—
114C22×Dic3—21no—
115C22×Dic3—61no—
116C22×Dic3—61no—
117C3×M16—92no—
118C3×M16—181no—
119C2×C3×C8—181no—
120C3×C4⋊C4—92no—
121C3×C22⋊C4—61no—
122C3×C22⋊C4—61no—
123C3×C22⋊C4—121no—
124C3×C42—92no—
125C3×C42—181no—
126C3×C42—181no—
127#217—21no—
128Dic3⋊C4—62no—
129C4×Dic3—62no—
130C4×Dic3—62no—
131C4×Dic3—62no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32—42no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—82no—
151C22×C32C3×C22⋊C411yescharacteristic
152C22×C32C3×C22⋊C411yescharacteristic
153C22×C32C3×C4⋊C411yescharacteristic
154C32×C4—64no—
155C32×C4—122no—
156C3⋊Dic3C3×C4211yescharacteristic
157C3⋊Dic3C3×C22⋊C411yescharacteristic
158C3⋊Dic3C3×C4⋊C411yescharacteristic
159C3⋊Dic3C3×C4⋊C411yescharacteristic
160C3×Dic3—24no—
161C3×Dic3—24no—
162C3×Dic3—44no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—82no—
174C2×M16—91no—
175#117—91no—
176C2×C42—91no—
177C33#33911yescharacteristic, complemented, Sylow
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3GO+(4,𝔽2)11yescharacteristic
183C2×C3×C4—62no—
184C2×C3×C4—64no—
185C2×C3×C4—64no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—91no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—124no—
196C2×C3×C4—122no—
197C2×C3×C4—122no—
198C2×Dic3—22no—
199C2×Dic3—24no—
200C2×Dic3—22no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C3×C8—182no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—22no—
222C2×C32—22no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—42no—
226C2×C32—42no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32C3×#9511yescharacteristic
235C2×C32C3×#10412yes—
236C22×C4—61no—
237C22×C4—91no—
238M16—92no—
239M16—181no—
240C2×C8—181no—
241C4⋊C4—92no—
242C22⋊C4—61no—
243C42—92no—
244C42—181no—
245C42—181no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—82no—
265C22×C3D32⋊C411yescharacteristic
266C22×C3D32⋊C411yescharacteristic
267C22×C3#377311yescharacteristic
268C3×C4—64no—
269C3×C4—64no—
270C3×C4—91no—
271C3×C4—91no—
272C3×C4—91no—
273C3×C4—91no—
274C3×C4—124no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—242no—
278Dic3—24no—
279Dic3—42no—
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285Dic3—61no—
286Dic3—61no—
287Dic3—61no—
288C32—21no—
289C32—21no—
290C32—41no—
291C32—41no—
292C32C3×#33911yescharacteristic, complemented
293C23C3×GO+(4,𝔽2)11yescharacteristic
294C2×C4—64no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C8—182no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—22no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—22no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—42no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—42no—
321C2×C3—42no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3#6373711yescharacteristic
325C2×C3#6374112yes—
326C22—22no—
327C22C3×D32⋊C411yescharacteristic
328C22C3×D32⋊C411yescharacteristic
329C22C3×#377311yescharacteristic
330C4—64no—
331C4—91no—
332C4—91no—
333C4—91no—
334C4—91no—
335C4—122no—
336C3—21no—
337C3—21no—
338C3—41no—
339C3—41no—
340C3#1059007411yescharacteristic, direct product factor
341C2—21no—
342C2—21no—
343C2C3×#6373711yescharacteristic
344C2C3×#6374112yes—
subgroupG/H
centerC22×C3D32⋊C4
commutatorC3⋊Dic3C3×C42
FrattiniC23C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C22⋊C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#339
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. #10590074C3direct (N × H)
8. C3×#3792C4semidirect
11. C3×#3724C4semidirect
14. C3×Dic32C4semidirect
40. #3792C3×C4semidirect
50. #3724C3×C4semidirect
53. Dic32C3×C4semidirect
177. C33#339semidirect
292. C32C3×#339semidirect
340. C3#10590074direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C3×#639572
C2×C3×Dic322
#105900743
C3×#3399