Finite Group Databasebeta

← C3×#10593493 #13166860

Subgroups — C3×#10593493 #13166860

Order 1728.

860 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 98 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×#625C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C3×#63957C211yesmaximal, characteristic, complemented
4C3×#63739C214yesmaximal, complemented
5#10593493C311yesmaximal, characteristic, direct product factor
6C23×C32×D3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#625—21no—
9C2×C3×#625—22no—
10C2×C3×#625C414yescomplemented
11C2×C3×#625C2212yes—
12C3×#3792—22no—
13C2×C3×#622C2212yescomplemented
14C32×#217—21no—
15C3×#3724C2212yescomplemented
16C3×Dic32—21no—
17C22×C3×#67—61no—
18C22×#625C2×C311yescharacteristic
19C2×#3792C2×C311yescharacteristic, complemented
20C3×#877—61no—
21#63957C2×C311yescharacteristic, complemented
22#63739C2×C314yescomplemented
23C23×C33D411yescharacteristic
24C22×C32×D3—24no—
25C22×C32×D3—22no—
26C22×C32×D3—44no—
27C2×C3×C3⋊Dic3C2×C412yes—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3D412yes—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C2×C32×Dic3—22no—
32C3×#625—24no—
33C3×#625—22no—
34C3×#625C2×C414yes—
35C3×#622—22no—
36C3×#418—91nomaximal
37C24×C32—61no—
38C23×C3×D3—21no—
39C23×C3×D3—21no—
40C23×C3×D3—41no—
41C22×C3⋊Dic3C22×C311yescharacteristic
42C2×C3×#67—64no—
43C2×C3×#67—62no—
44C2×C3×#67—62no—
45C2×C3×#67—64no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C2×#625—21no—
49C2×#625—22no—
50C2×#625C3×C414yescomplemented
51C2×#625C22×C312yes—
52#3792—22no—
53C2×#622C22×C312yescomplemented
54C32×C22⋊C4—61no—
55C3×#217—21no—
56C3×#217—21no—
57C3×#217—41no—
58C3×Dic3⋊C4—62no—
59C3×C4×Dic3—62no—
60#3724C22×C312yescomplemented
61Dic32—21no—
62C22×C33—22no—
63C22×C33C22⋊C412yes—
64C22×C33C2×D411yescharacteristic
65C2×C32×D3—24no—
66C2×C32×D3—48no—
67C2×C32×D3—44no—
68C3×C3⋊Dic3C22⋊C412yes—
69C3×C3⋊Dic3C22×C411yescharacteristic
70C3×C3⋊Dic3C2×D411yescharacteristic
71C32×Dic3—42no—
72C22×C3×D4—91no—
73C22×#67—61no—
74C2×C3×M16—91no—
75C3×#117—91no—
76#877—61no—
77C3×#102—94no—
78C23×C32—21no—
79C23×C32—21no—
80C23×C32—41no—
81C23×C32—41no—
82C23×C32—62no—
83C23×C32—64no—
84C23×C32—124no—
85C23×C32C3×D411yescharacteristic
86C22×C3×D3—22no—
87C22×C3×D3—24no—
88C22×C3×D3—24no—
89C22×C3×D3—22no—
90C22×C3×D3—44no—
91C22×C3×D3—42no—
92C22×C3×D3—44no—
93C22×C3×D3—44no—
94C22×C3×D3—48no—
95C2×C32×C4—62no—
96C2×C3⋊Dic3C2×C3×C412yes—
97C2×C3⋊Dic3C3×D411yescharacteristic
98C2×C3⋊Dic3C3×D412yes—
99C2×C3⋊Dic3C23×C311yescharacteristic
100C3×#67—64no—
101C3×#67—68no—
102C3×#67—64no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—42no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—61no—
114#625—24no—
115#625—22no—
116#625C2×C3×C414yes—
117#622—22no—
118#418—91noSylow
119C2×C33—22no—
120C2×C33C2×C22⋊C411yescharacteristic
121C2×C33AGL(1,ℤ/8)12yes—
122C32×D3—44no—
123C24×C3—61no—
124C24×C3—61no—
125C24×C3—121no—
126C23×D3—21no—
127C2×C3×D4—92no—
128C2×C3×D4—94no—
129C2×C3×D4—181no—
130C2×C3×D4—182no—
131C22×C3×C4—91no—
132C2×#67—64no—
133C2×#67—62no—
134C2×#67—62no—
135C2×#67—64no—
136C22×Dic3—61no—
137C22×Dic3—61no—
138C3×M16—182no—
139C2×C3×C8—92no—
140C3×C4⋊C4—92no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—61no—
143C3×C22⋊C4—121no—
144C3×C42—181no—
145#217—21no—
146Dic3⋊C4—62no—
147C4×Dic3—62no—
148C22×C32—24no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—42no—
158C22×C32—42no—
159C22×C32—44no—
160C22×C32—64no—
161C22×C32—82no—
162C22×C32—128no—
163C22×C32—124no—
164C22×C32C3×C22⋊C412yes—
165C22×C32C2×C3×D411yescharacteristic
166C2×C3×D3—24no—
167C2×C3×D3—24no—
168C2×C3×D3—44no—
169C2×C3×D3—48no—
170C2×C3×D3—48no—
171C2×C3×D3—44no—
172C2×C3×D3—48no—
173C2×C3×D3—416no—
174C2×C3×D3—44no—
175C32×C4—122no—
176C3⋊Dic3C3×C22⋊C412yes—
177C3⋊Dic3C22×C3×C411yescharacteristic
178C3⋊Dic3C2×C3×D411yescharacteristic
179C3×Dic3—42no—
180C3×Dic3—42no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—82no—
188C22×D4—91no—
189C2×M16—91no—
190#117—91no—
191#102—94no—
192C33#41811yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—62no—
198C23×C3—64no—
199C23×C3—62no—
200C23×C3—64no—
201C23×C3—122no—
202C23×C3—124no—
203C23×C3—124no—
204C23×C3—124no—
205C23×C3—128no—
206C23×C3GO+(4,𝔽2)11yescharacteristic
207C22×D3—22no—
208C22×D3—24no—
209C22×D3—44no—
210C3×D4—94no—
211C3×D4—182no—
212C3×D4—184no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—92no—
218C2×C3×C4—92no—
219C2×C3×C4—122no—
220#67—64no—
221#67—68no—
222#67—64no—
223C2×Dic3—22no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—61no—
227C2×Dic3—62no—
228C2×Dic3—61no—
229C2×Dic3—62no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C3×C8—182no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—42no—
242C2×C32—42no—
243C2×C32—42no—
244C2×C32—44no—
245C2×C32—82no—
246C2×C32—124no—
247C2×C32C2×C3×C22⋊C411yescharacteristic
248C2×C32C3×AGL(1,ℤ/8)12yes—
249C3×D3—44no—
250C3×D3—48no—
251C3×D3—44no—
252C24—61no—
253C2×D4—92no—
254C2×D4—94no—
255C2×D4—181no—
256C2×D4—182no—
257C22×C4—91no—
258M16—182no—
259C2×C8—92no—
260C4⋊C4—92no—
261C22⋊C4—61no—
262C42—181no—
263C22×C3—24no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—42no—
270C22×C3—44no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—64no—
276C22×C3—64no—
277C22×C3—82no—
278C22×C3—124no—
279C22×C3—128no—
280C22×C3—128no—
281C22×C3—124no—
282C22×C3—1216no—
283C22×C3—128no—
284C22×C3—124no—
285C22×C3D32⋊C412yes—
286C22×C3C2×GO+(4,𝔽2)11yescharacteristic
287C2×D3—24no—
288C2×D3—48no—
289C2×D3—44no—
290C3×C4—91no—
291C3×C4—91no—
292C3×C4—92no—
293C3×C4—122no—
294C3×C4—122no—
295C3×C4—242no—
296Dic3—42no—
297Dic3—62no—
298Dic3—61no—
299Dic3—61no—
300Dic3—62no—
301Dic3—61no—
302Dic3—61no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32—41no—
307C32C3×#41811yescharacteristic, complemented
308C23—62no—
309C23—64no—
310C23—124no—
311C23C3×GO+(4,𝔽2)11yescharacteristic
312D4—94no—
313D4—182no—
314D4—184no—
315C2×C4—62no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—92no—
319C2×C4—92no—
320C8—182no—
321C2×C3—24no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—42no—
328C2×C3—44no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—82no—
334C2×C3—124no—
335C2×C3—128no—
336C2×C3—124no—
337C2×C3C2×D32⋊C411yescharacteristic
338C2×C3#6423612yes—
339D3—44no—
340C22—22no—
341C22—64no—
342C22—128no—
343C22—124no—
344C22C3×D32⋊C412yes—
345C22C2×C3×GO+(4,𝔽2)11yescharacteristic
346C4—91no—
347C4—91no—
348C4—92no—
349C4—122no—
350C3—21no—
351C3—21no—
352C3—41no—
353C3—41no—
354C3#1059349311yescharacteristic, direct product factor
355C2—22no—
356C2—124no—
357C2C2×C3×D32⋊C411yescharacteristic
358C2C3×#6423612yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC22×C3×#625C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#418
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
2. C2×C3×#3792C2semidirect
3. C3×#63957C2semidirect
4. C3×#63739C2semidirect
5. #10593493C3direct (N × H)
10. C2×C3×#625C4semidirect
13. C2×C3×#622C22semidirect
15. C3×#3724C22semidirect
19. C2×#3792C2×C3semidirect
21. #63957C2×C3semidirect
22. #63739C2×C3semidirect
50. C2×#625C3×C4semidirect
53. C2×#622C22×C3semidirect
60. #3724C22×C3semidirect
192. C33#418semidirect
307. C32C3×#418semidirect
354. C3#10593493direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×#37922
C3×#639572
C3×#637392
#105934933
C3×#4189