Finite Group Databasebeta

← C3×#10593488 #13166855

Subgroups — C3×#10593488 #13166855

Order 1728.

892 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 114 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×#624C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C3×#63974C211yesmaximal, characteristic
4C3×#63737C214yesmaximal, complemented
5#10593488C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×#157C2212yes—
8C22×C32×Dic3—21no—
9C2×C3×#624—21no—
10C2×C3×#624—22no—
11C2×C3×#624C414yescomplemented
12C2×C3×#624C2212yes—
13C2×C3×#624C2212yes—
14C3×C22⋊#157—22no—
15C32×#217—21no—
16C3×#3722—21no—
17C22×C3×C4×D3—61no—
18C22×#624C2×C311yescharacteristic
19C2×C22⋊#157C2×C311yescharacteristic
20C3×D3×C22⋊C4—61no—
21#63974C2×C311yescharacteristic
22#63737C2×C314yescomplemented
23C23×C33D411yescharacteristic
24C22×C3×C3⋊D3—24no—
25C22×C3×C3⋊D3C2×C412yes—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C3×C3⋊D3D412yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3×#157—22no—
30C2×C3×#157—24no—
31C2×C32×Dic3—24no—
32C2×C32×Dic3—22no—
33C2×C32×Dic3—22no—
34C3×#624—24no—
35C3×#624—22no—
36C3×#624—24no—
37C3×#624C2×C414yes—
38C3×#381—91nomaximal
39C23×C3⋊D3C22×C311yescharacteristic
40C23×C3×D3—61no—
41C23×C3×D3—61no—
42C22×#157C22×C312yes—
43C22×C32×C4—61no—
44C22×C3×Dic3—21no—
45C22×C3×Dic3—21no—
46C22×C3×Dic3—41no—
47C2×C3×C4×D3—64no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—64no—
51C2×C3×C4×D3—62no—
52C2×#624—21no—
53C2×#624—22no—
54C2×#624C3×C414yescomplemented
55C2×#624C22×C312yes—
56C2×#624C22×C312yes—
57C22⋊#157—22no—
58C32×C22⋊C4—61no—
59C3×#217—21no—
60C3×#217—21no—
61C3×#217—41no—
62C3×(C2×D3)⋊C4—62no—
63#3722—21no—
64C22×C33—22no—
65C22×C33C22⋊C412yes—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—22no—
68C2×C3×C3⋊D3—24no—
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3C22⋊C412yes—
71C2×C3×C3⋊D3C22×C411yescharacteristic
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C2×C3×C3⋊D3#4114yes—
74C3×#157—42no—
75C32×Dic3—24no—
76C32×Dic3—42no—
77C23×C3×C4—91no—
78C22×C4×D3—61no—
79C2×C3×C22⋊C4—91no—
80C2×C3×C22⋊C4—91no—
81D3×C22⋊C4—61no—
82C3×#95—94no—
83C23×C32—21no—
84C23×C32—21no—
85C23×C32—41no—
86C23×C32—41no—
87C23×C32C3×D411yescharacteristic
88C22×C3⋊D3—24no—
89C22×C3⋊D3C2×C3×C412yes—
90C22×C3⋊D3C3×D411yescharacteristic
91C22×C3⋊D3C3×D412yes—
92C22×C3⋊D3C23×C311yescharacteristic
93C22×C3×D3—62no—
94C22×C3×D3—61no—
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—61no—
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—68no—
102C22×C3×D3—124no—
103C2×#157—22no—
104C2×#157—24no—
105C2×C32×C4—62no—
106C2×C32×C4—64no—
107C2×C32×C4—62no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—24no—
110C2×C3×Dic3—24no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—44no—
115C2×C3×Dic3—42no—
116C2×C3×Dic3—42no—
117C3×C4×D3—64no—
118C3×C4×D3—68no—
119C3×C4×D3—64no—
120C3×C4×D3—124no—
121#624—24no—
122#624—22no—
123#624—24no—
124#624C2×C3×C414yes—
125#381—91noSylow
126C2×C33—22no—
127C2×C33C2×C22⋊C411yescharacteristic
128C2×C33#12312yes—
129C3×C3⋊D3—22no—
130C3×C3⋊D3#11712yes—
131C3×C3⋊D3#12312yes—
132C24×C3—91no—
133C23×D3—61no—
134C23×D3—61no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C22×C3×C4—92no—
138C22×C3×C4—94no—
139C22×C3×C4—92no—
140C22×C3×C4—92no—
141C22×C3×C4—121no—
142C22×C3×C4—181no—
143C22×C3×C4—182no—
144C22×Dic3—21no—
145C2×C4×D3—64no—
146C2×C4×D3—62no—
147C2×C4×D3—62no—
148C2×C4×D3—62no—
149C2×C4×D3—64no—
150C3×C22⋊C4—61no—
151C3×C22⋊C4—61no—
152C3×C22⋊C4—121no—
153C3×C22⋊C4—181no—
154C3×C22⋊C4—182no—
155#217—21no—
156(C2×D3)⋊C4—62no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—42no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—44no—
169C22×C32—82no—
170C22×C32C3×C22⋊C412yes—
171C22×C32C2×C3×D411yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3—24no—
174C2×C3⋊D3—24no—
175C2×C3⋊D3C3×C22⋊C412yes—
176C2×C3⋊D3C22×C3×C411yescharacteristic
177C2×C3⋊D3C2×C3×D411yescharacteristic
178C2×C3⋊D3C3×#4114yes—
179C2×C3×D3—62no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—64no—
185C2×C3×D3—64no—
186C2×C3×D3—62no—
187C2×C3×D3—68no—
188C2×C3×D3—68no—
189C2×C3×D3—64no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—122no—
193#157—42no—
194C32×C4—64no—
195C32×C4—122no—
196C3×Dic3—24no—
197C3×Dic3—24no—
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—82no—
202C23×C4—91no—
203C2×C22⋊C4—91no—
204C2×C22⋊C4—91no—
205#95—94no—
206C33#38111yescharacteristic, complemented, Sylow
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—91no—
212C23×C3—91no—
213C23×C3—92no—
214C23×C3—92no—
215C23×C3—184no—
216C23×C3GO+(4,𝔽2)11yescharacteristic
217C22×D3—62no—
218C22×D3—61no—
219C22×D3—62no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—62no—
223C22×D3—61no—
224C22×D3—62no—
225C22×D3—68no—
226C22×D3—124no—
227C2×C3×C4—62no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—64no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—94no—
234C2×C3×C4—122no—
235C2×C3×C4—124no—
236C2×C3×C4—122no—
237C2×C3×C4—182no—
238C2×C3×C4—184no—
239C2×C3×C4—184no—
240C2×C3×C4—182no—
241C2×C3×C4—184no—
242C2×Dic3—22no—
243C2×Dic3—24no—
244C2×Dic3—22no—
245C4×D3—64no—
246C4×D3—68no—
247C4×D3—64no—
248C4×D3—124no—
249C2×C32—24no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—42no—
259C2×C32—42no—
260C2×C32—44no—
261C2×C32—82no—
262C2×C32C2×C3×C22⋊C411yescharacteristic
263C2×C32C3×#12312yes—
264C3⋊D3—22no—
265C3⋊D3C3×#11712yes—
266C3⋊D3C3×#12312yes—
267C3×D3—62no—
268C3×D3—62no—
269C3×D3—62no—
270C3×D3—62no—
271C3×D3—64no—
272C3×D3—122no—
273C24—91no—
274C22×C4—61no—
275C22×C4—92no—
276C22×C4—94no—
277C22×C4—92no—
278C22×C4—92no—
279C22×C4—181no—
280C22×C4—182no—
281C22⋊C4—61no—
282C22⋊C4—181no—
283C22⋊C4—182no—
284C22×C3—24no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—42no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—94no—
300C22×C3—92no—
301C22×C3—184no—
302C22×C3—184no—
303C22×C3—182no—
304C22×C3D32⋊C412yes—
305C22×C3C2×GO+(4,𝔽2)11yescharacteristic
306C2×D3—62no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—64no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—64no—
313C2×D3—62no—
314C2×D3—68no—
315C2×D3—68no—
316C2×D3—64no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—122no—
320C3×C4—64no—
321C3×C4—64no—
322C3×C4—124no—
323C3×C4—122no—
324C3×C4—122no—
325C3×C4—242no—
326C3×C4—362no—
327Dic3—24no—
328Dic3—42no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32C3×#38111yescharacteristic, complemented
334C23—91no—
335C23—91no—
336C23—92no—
337C23—92no—
338C23—184no—
339C23C3×GO+(4,𝔽2)11yescharacteristic
340C2×C4—62no—
341C2×C4—64no—
342C2×C4—62no—
343C2×C4—94no—
344C2×C4—182no—
345C2×C4—184no—
346C2×C4—184no—
347C2×C4—182no—
348C2×C4—184no—
349C2×C3—24no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—42no—
356C2×C3—44no—
357C2×C3—42no—
358C2×C3—42no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—82no—
362C2×C3—92no—
363C2×C3—92no—
364C2×C3—182no—
365C2×C3C2×D32⋊C411yescharacteristic
366C2×C3#6423312yes—
367D3—62no—
368D3—62no—
369D3—62no—
370D3—62no—
371D3—64no—
372D3—122no—
373C22—22no—
374C22—94no—
375C22—91no—
376C22—91no—
377C22—92no—
378C22—184no—
379C22—184no—
380C22—182no—
381C22C3×D32⋊C412yes—
382C22C2×C3×GO+(4,𝔽2)11yescharacteristic
383C4—64no—
384C4—122no—
385C4—362no—
386C3—21no—
387C3—21no—
388C3—41no—
389C3—41no—
390C3#1059348811yescharacteristic, direct product factor
391C2—22no—
392C2—92no—
393C2—92no—
394C2—182no—
395C2C2×C3×D32⋊C411yescharacteristic
396C2C3×#6423312yes—
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#381
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#63737C2semidirect
5. #10593488C3direct (N × H)
11. C2×C3×#624C4semidirect
22. #63737C2×C3semidirect
54. C2×#624C3×C4semidirect
206. C33#381semidirect
333. C32C3×#381semidirect
390. C3#10593488direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6242
C2×C3×C22⋊#1572
C3×#639742
C3×#637372
#105934883
C3×#3819