Finite Group Databasebeta

← C2×C3×#63740 #13166865

Subgroups — C2×C3×#63740 #13166865

Order 1728.

680 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×C3×#627C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic
3C2×C3×#3724C211yesmaximal, characteristic
4C3×#63740C214yesmaximal, direct product factor
5C2×#63740C311yesmaximal, characteristic, direct product factor
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3—21no—
8C22×C32×Dic3—21no—
9C2×C3×#627—21no—
10C2×C3×#627—22no—
11C2×C3×#627C414yescomplemented
12C2×C3×#627C2212yes—
13C2×C3×#622—22no—
14C2×C3×#622C2212yes—
15C3×#3724—21no—
16C3×#3724C2212yes—
17C22×C3×Dic6—61no—
18C22×#627C2×C311yescharacteristic
19C22×#622C2×C311yescharacteristic
20C2×C3×Dic3⋊C4—61no—
21C2×#3724C2×C311yescharacteristic
22#63740C2×C314yesdirect product factor
23C23×C33D411yescharacteristic
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3D412yes—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—24no—
29C2×C32×Dic3—22no—
30C2×C32×Dic3—22no—
31C2×C32×Dic3—42no—
32C3×#627—24no—
33C3×#627—22no—
34C3×#627C2×C414yescomplemented
35C3×#622—22no—
36C2×C3×#103—91nomaximal
37C22×C32×C4—61no—
38C22×C32×C4—61no—
39C22×C3⋊Dic3C22×C311yescharacteristic
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—41no—
45C22×C3×Dic3—41no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C2×C3×Dic6—64no—
49C2×C3×Dic6—62no—
50C2×C3×Dic6—62no—
51C2×C3×Dic6—64no—
52C2×#627—21no—
53C2×#627—22no—
54C2×#627C3×C414yescomplemented
55C2×#627C22×C312yes—
56C2×#622—22no—
57C2×#622C22×C312yes—
58C3×Dic3⋊C4—62no—
59C3×Dic3⋊C4—62no—
60#3724—21no—
61#3724C22×C312yes—
62C22×C33C22⋊C412yes—
63C22×C33SD1612yes—
64C22×C33Q1612yes—
65C22×C33C2×D411yescharacteristic
66C3×C3⋊Dic3C22⋊C412yes—
67C3×C3⋊Dic3C22×C411yescharacteristic
68C3×C3⋊Dic3C2×D411yescharacteristic
69C32×Dic3—24no—
70C32×Dic3—42no—
71C22×C3×Q8—91no—
72C22×Dic6—61no—
73C22×C3×C8—91no—
74C2×C3×C4⋊C4—91no—
75C2×Dic3⋊C4—61no—
76C3×#103—94no—
77C23×C32—21no—
78C23×C32—21no—
79C23×C32—41no—
80C23×C32—41no—
81C23×C32C3×D411yescharacteristic
82C2×C32×C4—62no—
83C2×C32×C4—64no—
84C2×C32×C4—62no—
85C2×C32×C4—122no—
86C2×C3⋊Dic3C2×C3×C412yes—
87C2×C3⋊Dic3C3×D411yescharacteristic
88C2×C3⋊Dic3C3×D412yes—
89C2×C3⋊Dic3C23×C311yescharacteristic
90C2×C3×Dic3—22no—
91C2×C3×Dic3—24no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—44no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—44no—
101C2×C3×Dic3—42no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—61no—
110C3×Dic6—64no—
111C3×Dic6—68no—
112C3×Dic6—64no—
113#627—24no—
114#627—22no—
115#627C2×C3×C414yescomplemented
116#622—22no—
117C2×#103—91noSylow
118C2×C33#10314yescomplemented
119C2×C33C2×C22⋊C411yescharacteristic
120C2×C33C2×SD1611yescharacteristic
121C2×C33C2×Q1611yescharacteristic
122C2×C3×Q8—92no—
123C2×C3×Q8—94no—
124C2×C3×Q8—181no—
125C2×C3×Q8—182no—
126C22×C3×C4—61no—
127C22×C3×C4—61no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4—91no—
131C22×C3×C4—121no—
132C22×C3×C4—121no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—61no—
136C22×Dic3—61no—
137C2×Dic6—64no—
138C2×Dic6—62no—
139C2×Dic6—62no—
140C2×Dic6—64no—
141C2×C3×C8—92no—
142C2×C3×C8—182no—
143C3×C4⋊C4—92no—
144C3×C4⋊C4—181no—
145Dic3⋊C4—62no—
146Dic3⋊C4—62no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—22no—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—22no—
155C22×C32—42no—
156C22×C32—42no—
157C22×C32—42no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32—42no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32C3×C22⋊C412yes—
164C22×C32C3×SD1612yes—
165C22×C32C3×Q1612yes—
166C22×C32C2×C3×D411yescharacteristic
167C32×C4—64no—
168C32×C4—122no—
169C3⋊Dic3C3×C22⋊C412yes—
170C3⋊Dic3C22×C3×C411yescharacteristic
171C3⋊Dic3C2×C3×D411yescharacteristic
172C3×Dic3—24no—
173C3×Dic3—24no—
174C3×Dic3—44no—
175C3×Dic3—42no—
176C3×Dic3—44no—
177C3×Dic3—42no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C22×Q8—91no—
185C22×C8—91no—
186C2×C4⋊C4—91no—
187#103—94no—
188C33C2×#10311yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3GO+(4,𝔽2)11yescharacteristic
194C3×Q8—94no—
195C3×Q8—182no—
196C3×Q8—184no—
197C2×C3×C4—62no—
198C2×C3×C4—64no—
199C2×C3×C4—62no—
200C2×C3×C4—64no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4—122no—
208C2×C3×C4—124no—
209C2×C3×C4—122no—
210C2×C3×C4—122no—
211C2×C3×C4—124no—
212C2×C3×C4—122no—
213C2×Dic3—22no—
214C2×Dic3—24no—
215C2×Dic3—22no—
216C2×Dic3—42no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225Dic6—64no—
226Dic6—68no—
227Dic6—64no—
228C3×C8—182no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—44no—
238C2×C32—44no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32C3×#10314yescomplemented
246C2×C32C2×C3×C22⋊C411yescharacteristic
247C2×C32C2×C3×SD1611yescharacteristic
248C2×C32C2×C3×Q1611yescharacteristic
249C2×Q8—92no—
250C2×Q8—94no—
251C2×Q8—181no—
252C2×Q8—182no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—91no—
256C2×C8—92no—
257C2×C8—182no—
258C4⋊C4—92no—
259C4⋊C4—181no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3D32⋊C412yes—
277C22×C3#377512yes—
278C22×C3#377612yes—
279C22×C3C2×GO+(4,𝔽2)11yescharacteristic
280C3×C4—64no—
281C3×C4—64no—
282C3×C4—91no—
283C3×C4—91no—
284C3×C4—92no—
285C3×C4—124no—
286C3×C4—122no—
287C3×C4—124no—
288C3×C4—122no—
289Dic3—24no—
290Dic3—42no—
291Dic3—62no—
292Dic3—61no—
293Dic3—61no—
294Dic3—62no—
295Dic3—61no—
296Dic3—61no—
297C32—21no—
298C32—21no—
299C32—41no—
300C32—41no—
301C32C2×C3×#10311yescharacteristic, complemented
302C23C3×GO+(4,𝔽2)11yescharacteristic
303Q8—94no—
304Q8—182no—
305Q8—184no—
306C2×C4—62no—
307C2×C4—64no—
308C2×C4—62no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—92no—
312C2×C4—92no—
313C2×C4—122no—
314C8—182no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—44no—
328C2×C3—44no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3#6374014yesdirect product factor
332C2×C3C2×D32⋊C411yescharacteristic
333C2×C3C2×#377511yescharacteristic
334C2×C3C2×#377611yescharacteristic
335C22C3×D32⋊C412yes—
336C22C3×#377512yes—
337C22C3×#377612yes—
338C22C2×C3×GO+(4,𝔽2)11yescharacteristic
339C4—64no—
340C4—91no—
341C4—91no—
342C4—92no—
343C4—122no—
344C3—21no—
345C3—21no—
346C3—41no—
347C3—41no—
348C3C2×#6374011yescharacteristic, direct product factor
349C2C3×#6374014yesdirect product factor
350C2C2×C3×D32⋊C411yescharacteristic
351C2C2×C3×#377511yescharacteristic
352C2C2×C3×#377611yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, socle, prime radicalC23×C33D4
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C2×C3×#103
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#63740C2direct (N × H)
5. C2×#63740C3direct (N × H)
11. C2×C3×#627C4semidirect
22. #63740C2×C3direct (N × H)
34. C3×#627C2×C4semidirect
54. C2×#627C3×C4semidirect
115. #627C2×C3×C4semidirect
118. C2×C33#103semidirect
188. C33C2×#103semidirect
245. C2×C32C3×#103semidirect
301. C32C2×C3×#103semidirect
331. C2×C3#63740direct (N × H)
348. C3C2×#63740direct (N × H)
349. C2C3×#63740direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C22×C3×#6222
C2×C3×#37242
C3×#637402
C2×#637403
C2×C3×#1039