Finite Group Databasebeta

← C3×#10593499 #13166894

Subgroups — C3×#10593499 #13166894

Order 1728.

614 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 78 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
2C2×C3×#3776C212yesmaximal, complemented
3C3×#63982C211yesmaximal, characteristic
4C3×C22⋊#622C211yesmaximal, characteristic
5C3×#63740C212yesmaximal, complemented
6#10593499C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×C3×#627—21no—
10C2×C3×#627—24no—
11C2×C3×#627C2211yescharacteristic
12C2×C3×#627C2212yes—
13C2×C3×#622C2212yes—
14C3×#3776—24no—
15C32×#217—21no—
16C3×#3724C2211yescharacteristic
17C3×Dic3⋊Dic3—21no—
18C22×C3×Dic6—61no—
19C22×#627C2×C311yescharacteristic
20C2×#3776C2×C312yescomplemented
21C3×#878—61no—
22#63982C2×C311yescharacteristic
23C22⋊#622C2×C311yescharacteristic
24#63740C2×C312yescomplemented
25C23×C33D411yescharacteristic
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—24no—
30C2×C32×Dic3—22no—
31C2×C32×Dic3—21no—
32C2×C32×Dic3—21no—
33C3×#627—24no—
34C3×#627—22no—
35C3×#627—22no—
36C3×#627D414yes—
37C3×#622—22no—
38C3×#451—91nomaximal
39C22×C32×C4—61no—
40C22×C3⋊Dic3C22×C311yescharacteristic
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C22×C3×Dic3—41no—
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C2×C3×Dic6—62no—
47C2×C3×Dic6—68no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6—61no—
50C2×#627—21no—
51C2×#627—24no—
52C2×#627C22×C311yescharacteristic
53C2×#627C22×C312yes—
54C2×#622C22×C312yes—
55#3776—24no—
56C32×C22⋊C4—61no—
57C3×#217—21no—
58C3×#217—21no—
59C3×#217—41no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63#3724C22×C311yescharacteristic
64Dic3⋊Dic3—21no—
65C22×C33—21no—
66C22×C33—21no—
67C22×C33Q1612yescomplemented
68C22×C33C2×D411yescharacteristic
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3C2×D412yes—
71C32×Dic3—24no—
72C32×Dic3—41no—
73C32×Dic3—41no—
74C22×C3×Q8—91no—
75C22×Dic6—61no—
76C2×C3×Q16—92no—
77C3×C22⋊Q8—91no—
78#878—61no—
79C3×#103—92no—
80C3×C22⋊C8—91no—
81C23×C32—21no—
82C23×C32—21no—
83C23×C32—41no—
84C23×C32—41no—
85C23×C32C3×D411yescharacteristic
86C2×C32×C4—64no—
87C2×C32×C4—62no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3C3×D411yescharacteristic
92C2×C3⋊Dic3C23×C311yescharacteristic
93C2×C3×Dic3—24no—
94C2×C3×Dic3—24no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—42no—
102C2×C3×Dic3—44no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—122no—
111C3×Dic6—64no—
112C3×Dic6—64no—
113C3×Dic6—68no—
114C3×Dic6—122no—
115#627—24no—
116#627—22no—
117#627—22no—
118#627C3×D414yes—
119#622—22no—
120#451—91noSylow
121C2×C33—22no—
122C2×C33#12011yescharacteristic
123C2×C33C2×Q1611yescharacteristic
124C2×C33#13711yescharacteristic
125C2×C3×Q8—92no—
126C2×C3×Q8—91no—
127C2×C3×Q8—181no—
128C2×C3×Q8—184no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4—91no—
132C22×C3×C4—121no—
133C22×Dic3—21no—
134C22×Dic3—61no—
135C22×Dic3—61no—
136C2×Dic6—62no—
137C2×Dic6—61no—
138C2×Dic6—62no—
139C2×Dic6—68no—
140C3×Q16—184no—
141C2×C3×C8—92no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—181no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—121no—
147#217—21no—
148C4⋊Dic3—61no—
149Dic3⋊C4—61no—
150Dic3⋊C4—61no—
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—22no—
159C22×C32—42no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—42no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32—81no—
168C22×C32—81no—
169C22×C32C3×Q1612yescomplemented
170C22×C32C2×C3×D411yescharacteristic
171C32×C4—64no—
172C32×C4—121no—
173C32×C4—121no—
174C3⋊Dic3—21no—
175C3⋊Dic3C2×C3×D412yes—
176C3×Dic3—24no—
177C3×Dic3—24no—
178C3×Dic3—44no—
179C3×Dic3—41no—
180C3×Dic3—41no—
181C3×Dic3—41no—
182C3×Dic3—41no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—81no—
187C3×Dic3—81no—
188C3×Dic3—121no—
189C22×Q8—91no—
190C2×Q16—92no—
191C22⋊Q8—91no—
192#103—92no—
193C22⋊C8—91no—
194C33#45111yescharacteristic, complemented, Sylow
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3GO+(4,𝔽2)11yescharacteristic
200C3×Q8—94no—
201C3×Q8—182no—
202C3×Q8—184no—
203C3×Q8—182no—
204C2×C3×C4—64no—
205C2×C3×C4—62no—
206C2×C3×C4—64no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—124no—
215C2×C3×C4—122no—
216C2×C3×C4—121no—
217C2×C3×C4—121no—
218C2×C3×C4—182no—
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—64no—
228C2×Dic3—122no—
229Dic6—64no—
230Dic6—64no—
231Dic6—68no—
232Dic6—122no—
233C3×C8—182no—
234C2×C32—24no—
235C2×C32—22no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—42no—
249C2×C32—44no—
250C2×C32—82no—
251C2×C32C3×#12011yescharacteristic
252C2×C32C2×C3×Q1611yescharacteristic
253C2×C32C3×#13711yescharacteristic
254C2×Q8—92no—
255C2×Q8—91no—
256C2×Q8—181no—
257C2×Q8—184no—
258C22×C4—61no—
259C22×C4—91no—
260Q16—184no—
261C2×C8—92no—
262C4⋊C4—91no—
263C4⋊C4—181no—
264C22⋊C4—61no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3#377612yescomplemented
284C22×C3C2×GO+(4,𝔽2)11yescharacteristic
285C3×C4—64no—
286C3×C4—64no—
287C3×C4—92no—
288C3×C4—124no—
289C3×C4—121no—
290C3×C4—121no—
291C3×C4—121no—
292C3×C4—121no—
293C3×C4—181no—
294C3×C4—241no—
295C3×C4—241no—
296Dic3—24no—
297Dic3—41no—
298Dic3—41no—
299Dic3—62no—
300Dic3—62no—
301Dic3—62no—
302Dic3—121no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32—41no—
307C32C3×#45111yescharacteristic, complemented
308C23C3×GO+(4,𝔽2)11yescharacteristic
309Q8—94no—
310Q8—182no—
311Q8—184no—
312Q8—182no—
313C2×C4—64no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—182no—
320C8—182no—
321C2×C3—24no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—44no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—82no—
338C2×C3#6423911yescharacteristic
339C2×C3C2×#377611yescharacteristic
340C2×C3#6424111yescharacteristic
341C22—21no—
342C22—21no—
343C22C3×#377612yescomplemented
344C22C2×C3×GO+(4,𝔽2)11yescharacteristic
345C4—64no—
346C4—92no—
347C4—121no—
348C4—121no—
349C4—181no—
350C3—21no—
351C3—21no—
352C3—41no—
353C3—41no—
354C3#1059349911yescharacteristic, direct product factor
355C2—22no—
356C2C3×#6423911yescharacteristic
357C2C2×C3×#377611yescharacteristic
358C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#451
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3776C2semidirect
5. C3×#63740C2semidirect
6. #10593499C3direct (N × H)
20. C2×#3776C2×C3semidirect
24. #63740C2×C3semidirect
67. C22×C33Q16semidirect
169. C22×C32C3×Q16semidirect
194. C33#451semidirect
283. C22×C3#3776semidirect
307. C32C3×#451semidirect
343. C22C3×#3776semidirect
354. C3#10593499direct (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
C2×C3×#37762
C3×#639822
C3×C22⋊#6222
C3×#637402
#105934993
C3×#4519