Finite Group Databasebeta

← C3×#10593456 #13166825

Subgroups — C3×#10593456 #13166825

Order 1728.

808 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 90 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×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C212yesmaximal, complemented
4C3×#63921C212yesmaximal, complemented
5C3×#63737C211yesmaximal, characteristic, complemented
6#10593456C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2212yes—
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×C3×GO+(4,𝔽2)—24no—
11C2×C3×GO+(4,𝔽2)—22no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#624C2212yes—
14C3×C4⋊#157—22no—
15C3×D32⋊C4—24no—
16C32×(C2×D3)⋊C4—22no—
17C3×(C2×D3)⋊Dic3—22no—
18C22×GO+(4,𝔽2)C2×C311yescharacteristic
19C2×C4⋊#157C2×C311yescharacteristic, complemented
20C2×D32⋊C4C2×C312yescomplemented
21C3×D3×C22⋊C4—62no—
22#63921C2×C312yescomplemented
23#63737C2×C311yescharacteristic, complemented
24C22×C3×C3⋊D3C2311yescharacteristic
25C22×C32×D3—22no—
26C2×C3×D32—22no—
27C2×C3×D32—22no—
28C2×C3×D32—22no—
29C2×C3×D32—22no—
30C2×C3×D32—22no—
31C2×C3×D32D414yes—
32C2×C3×#157—22no—
33C2×C3×#157—22no—
34C2×C3×#157—21no—
35C2×C3×#157D412yes—
36C3×GO+(4,𝔽2)—28no—
37C2×C33×C4D411yescharacteristic, complemented
38C2×C3×C3⋊Dic3D411yescharacteristic, complemented
39C3×C4×C3⋊D3—21no—
40C3×C4×C3⋊D3—21no—
41C2×C32×Dic3—22no—
42C3×#624—22no—
43C3×#624—22no—
44C3×#394—91nomaximal
45C23×C3×D3—62no—
46C22×D32C22×C312yes—
47C22×#157C22×C311yescharacteristic
48C22×#157C22×C311yescharacteristic
49C2×GO+(4,𝔽2)—24no—
50C2×GO+(4,𝔽2)—22no—
51C2×C4×C3⋊D3C22×C311yescharacteristic
52C2×C3×C4×D3—62no—
53C2×C3×C4×D3—62no—
54C2×#624C22×C312yes—
55C4⋊#157—22no—
56D32⋊C4—24no—
57C32×C22⋊C4—62no—
58C3×#217—62no—
59C3×(C2×D3)⋊C4—22no—
60C3×(C2×D3)⋊C4—22no—
61C3×(C2×D3)⋊C4—42no—
62C3×(C2×D3)⋊C4—62no—
63(C2×D3)⋊Dic3—22no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3C2×D412yes—
66C2×C3×C3⋊D3C2×D411yescharacteristic
67C2×C3×C3⋊D3#4111yescharacteristic
68C2×C3×C3⋊D3#4112yes—
69C2×C32×D3—24no—
70C2×C32×D3—42no—
71C2×C32×D3—42no—
72C3×D32—24no—
73C3×D32—24no—
74C3×D32—42no—
75C3×D32—42no—
76C3×#157—22no—
77C3×#157—41no—
78C33×C4—21no—
79C3×C3⋊Dic3—21no—
80C32×Dic3—42no—
81C22×C3×D4—91no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C22⋊C4—92no—
84C2×C3×C22⋊C4—92no—
85D3×C22⋊C4—62no—
86C3×#95—91no—
87C23×C32—62no—
88C22×C3⋊D3C23×C311yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3—42no—
92C22×C3×D3—64no—
93C22×C3×D3—62no—
94C22×C3×D3—64no—
95C22×C3×D3—122no—
96C22×C3×D3—122no—
97C22×C3×D3—122no—
98C22×C3×D3—122no—
99C2×D32—22no—
100C2×D32—22no—
101C2×D32—22no—
102C2×D32—22no—
103C2×D32—22no—
104C2×D32C3×D414yes—
105C2×#157—22no—
106C2×#157—22no—
107C2×#157—21no—
108C2×#157C3×D412yes—
109GO+(4,𝔽2)—28no—
110C2×C32×C4—22no—
111C2×C32×C4—42no—
112C2×C32×C4—62no—
113C2×C32×C4C3×D411yescharacteristic, complemented
114C2×C3⋊Dic3C3×D411yescharacteristic, complemented
115C4×C3⋊D3—21no—
116C4×C3⋊D3—21no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—62no—
121C3×C4×D3—122no—
122C3×C4×D3—122no—
123C3×C4×D3—122no—
124C3×C4×D3—122no—
125#624—22no—
126#624—22no—
127#394—91noSylow
128C2×C33#12112yes—
129C2×C33#12111yescharacteristic
130C3×C3⋊D3#12011yescharacteristic
131C3×C3⋊D3#12312yes—
132C3×C3⋊D3#12411yescharacteristic
133C32×D3—44no—
134C24×C3—92no—
135C23×D3—62no—
136C2×C3×D4—182no—
137C2×C3×D4—184no—
138C22×C3×C4—92no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C2×C4×D3—62no—
143C2×C4×D3—62no—
144C3×C4⋊C4—182no—
145C3×C22⋊C4—62no—
146C3×C22⋊C4—62no—
147C3×C22⋊C4—122no—
148C3×C22⋊C4—182no—
149C3×C22⋊C4—184no—
150#217—62no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—62no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32—64no—
156C22×C32—122no—
157C22×C32—122no—
158C22×C32C2×C3×D411yescharacteristic
159C2×C3⋊D3C2×C3×D412yes—
160C2×C3⋊D3C2×C3×D411yescharacteristic
161C2×C3⋊D3C3×#4111yescharacteristic
162C2×C3⋊D3C3×#4112yes—
163C2×C3×D3—24no—
164C2×C3×D3—24no—
165C2×C3×D3—44no—
166C2×C3×D3—42no—
167C2×C3×D3—42no—
168C2×C3×D3—42no—
169C2×C3×D3—42no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—82no—
177C2×C3×D3—82no—
178C2×C3×D3—124no—
179C2×C3×D3—124no—
180C2×C3×D3—124no—
181C2×C3×D3—124no—
182D32—24no—
183D32—24no—
184D32—42no—
185D32—42no—
186#157—22no—
187#157—41no—
188C32×C4—21no—
189C32×C4—42no—
190C32×C4—82no—
191C32×C4—122no—
192C3⋊Dic3—21no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C3×Dic3—82no—
196C3×Dic3—122no—
197C22×D4—91no—
198C2×C4⋊C4—91no—
199C2×C22⋊C4—92no—
200C2×C22⋊C4—92no—
201#95—91no—
202C33#39411yescharacteristic, complemented, Sylow
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—94no—
206C23×C3—91no—
207C23×C3—122no—
208C23×C3—182no—
209C23×C3—182no—
210C23×C3—182no—
211C23×C3—182no—
212C23×C3—182no—
213C22×D3—22no—
214C22×D3—64no—
215C22×D3—62no—
216C22×D3—64no—
217C22×D3—122no—
218C22×D3—122no—
219C22×D3—122no—
220C22×D3—122no—
221C3×D4—188no—
222C2×C3×C4—22no—
223C2×C3×C4—42no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—92no—
227C2×C3×C4—91no—
228C2×C3×C4—122no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—182no—
232C2×C3×C4—182no—
233C2×C3×C4—182no—
234C2×C3×C4—182no—
235C2×C3×C4—181no—
236C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
237C2×Dic3—22no—
238C2×Dic3—62no—
239C4×D3—122no—
240C4×D3—122no—
241C4×D3—122no—
242C4×D3—122no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—42no—
247C2×C32—42no—
248C2×C32—42no—
249C2×C32—124no—
250C2×C32C3×#12112yes—
251C2×C32C3×#12111yescharacteristic
252C3⋊D3C3×#12011yescharacteristic
253C3⋊D3C3×#12312yes—
254C3⋊D3C3×#12411yescharacteristic
255C3×D3—44no—
256C3×D3—44no—
257C3×D3—62no—
258C3×D3—62no—
259C3×D3—62no—
260C3×D3—62no—
261C3×D3—84no—
262C24—92no—
263C2×D4—182no—
264C2×D4—184no—
265C22×C4—92no—
266C22×C4—91no—
267C22×C4—91no—
268C22×C4—91no—
269C4⋊C4—182no—
270C22⋊C4—62no—
271C22⋊C4—182no—
272C22⋊C4—184no—
273C22×C3—22no—
274C22×C3—42no—
275C22×C3—64no—
276C22×C3—64no—
277C22×C3—92no—
278C22×C3—92no—
279C22×C3—91no—
280C22×C3—91no—
281C22×C3—124no—
282C22×C3—122no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—184no—
287C22×C3—184no—
288C22×C3—242no—
289C22×C3—242no—
290C22×C3—362no—
291C22×C3—362no—
292C22×C3C2×GO+(4,𝔽2)11yescharacteristic
293C2×D3—24no—
294C2×D3—42no—
295C2×D3—42no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—124no—
303C2×D3—124no—
304C2×D3—124no—
305C2×D3—124no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—82no—
309C3×C4—122no—
310C3×C4—122no—
311C3×C4—182no—
312C3×C4—181no—
313C3×C4—242no—
314C3×C4—361no—
315Dic3—42no—
316Dic3—122no—
317C32—22no—
318C32—42no—
319C32C3×#39411yescharacteristic, complemented
320C23—62no—
321C23—94no—
322C23—91no—
323C23—182no—
324C23—182no—
325C23—182no—
326C23—182no—
327C23—182no—
328D4—188no—
329C2×C4—62no—
330C2×C4—92no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—181no—
339C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
340C2×C3—22no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—42no—
344C2×C3—42no—
345C2×C3—42no—
346C2×C3—91no—
347C2×C3—92no—
348C2×C3—91no—
349C2×C3—124no—
350C2×C3—124no—
351C2×C3—244no—
352C2×C3#6423012yes—
353C2×C3#6424211yescharacteristic
354D3—44no—
355D3—62no—
356D3—62no—
357D3—62no—
358D3—62no—
359C22—64no—
360C22—91no—
361C22—91no—
362C22—92no—
363C22—92no—
364C22—122no—
365C22—122no—
366C22—184no—
367C22—184no—
368C22—362no—
369C22—362no—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—21no—
372C4—122no—
373C4—182no—
374C4—181no—
375C4—361no—
376C3—22no—
377C3—42no—
378C3#1059345611yescharacteristic, direct product factor
379C2—91no—
380C2—92no—
381C2—91no—
382C2—124no—
383C2C3×#6423012yes—
384C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×GO+(4,𝔽2)C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#394
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊#157C2semidirect
3. C2×C3×D32⋊C4C2semidirect
4. C3×#63921C2semidirect
5. C3×#63737C2semidirect
6. #10593456C3direct (N × H)
19. C2×C4⋊#157C2×C3semidirect
20. C2×D32⋊C4C2×C3semidirect
22. #63921C2×C3semidirect
23. #63737C2×C3semidirect
37. C2×C33×C4D4semidirect
38. C2×C3×C3⋊Dic3D4semidirect
113. C2×C32×C4C3×D4semidirect
114. C2×C3⋊Dic3C3×D4semidirect
202. C33#394semidirect
236. C2×C3×C4GO+(4,𝔽2)semidirect
319. C32C3×#394semidirect
339. C2×C4C3×GO+(4,𝔽2)semidirect
378. C3#10593456direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×C4⋊#1572
C2×C3×D32⋊C42
C3×#639212
C3×#637372
#105934563
C3×#3949