Finite Group Databasebeta

← C3×#10593368 #13166707

Subgroups — C3×#10593368 #13166707

Order 1728.

474 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×#63912C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3C3×#63778C211yesmaximal, characteristic, complemented
4C3×#63743C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593368C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#626—21no—
9C2×C3×#625C2211yescharacteristic
10C2×C3×#624—21no—
11C2×C3×#622C2212yes—
12C32×(C2×D3)⋊C4—21no—
13C32×C4⋊Dic3—21no—
14C3×#3724C2212yes—
15C3×#3724C2211yescharacteristic
16C3×D3×C4⋊C4—61no—
17C3×#884—61no—
18#63912C2×C311yescharacteristic
19#63900C2×C311yescharacteristic, complemented
20#63778C2×C311yescharacteristic, complemented
21#63743C2×C312yescomplemented
22#63739C2×C312yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C22×C32×D3—21no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—22no—
28C2×C32×Dic3—22no—
29C2×C32×Dic3—21no—
30C3×#626—42no—
31C3×#625—22no—
32C3×#624—42no—
33C3×#622—22no—
34C3×#480—91nomaximal
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×#67—61no—
37C2×C3×#67—61no—
38C2×C3×D12—61no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—61no—
42C2×#626—21no—
43C2×#625C22×C311yescharacteristic
44C2×#624—21no—
45C2×#622C22×C312yes—
46C32×C4⋊C4—61no—
47C32×C22⋊C4—61no—
48C3×(C2×D3)⋊C4—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×(C2×D3)⋊C4—41no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—41no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—62no—
56#3724C22×C312yes—
57#3724C22×C311yescharacteristic
58C22×C33C2×D411yescharacteristic
59C2×C3×C3⋊D3—21no—
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3D812yes—
62C2×C32×D3—41no—
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C33×C4—21no—
66C3×C3⋊Dic3#4112yes—
67C32×Dic3—42no—
68C32×Dic3—41no—
69C3×#121—91no—
70C2×C3×C4⋊C4—91no—
71D3×C4⋊C4—61no—
72#884—61no—
73C3×#107—92no—
74C3×#102—92no—
75C3×C22⋊C8—91no—
76C23×C32—61no—
77C22×C3⋊D3C3×D411yescharacteristic
78C22×C3×D3—21no—
79C22×C3×D3—21no—
80C22×C3×D3—41no—
81C22×C3×D3—61no—
82C22×C3×D3—61no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—61no—
88C2×C32×C4—62no—
89C2×C32×C4C3×D411yescharacteristic
90C2×C3⋊Dic3C23×C311yescharacteristic
91C4×C3⋊D3—22no—
92C3×#67—122no—
93C3×#67—122no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—21no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C3×D12—122no—
103C3×C4×D3—64no—
104C3×C4×D3—122no—
105C3×C4×D3—124no—
106#626—42no—
107#625—22no—
108#624—42no—
109#622—22no—
110#480—91noSylow
111C2×C33#12311yescharacteristic
112C2×C33C2×D811yescharacteristic
113C2×C33#13711yescharacteristic
114C3×C3⋊D3—22no—
115C32×D3—81no—
116C2×C3×D4—91no—
117C2×C3×D4—181no—
118C22×C3×C4—91no—
119C22×C3×C4—181no—
120C2×#67—61no—
121C2×#67—61no—
122C2×D12—61no—
123C2×C4×D3—61no—
124C2×C4×D3—61no—
125C2×C4×D3—62no—
126C2×C3×C8—92no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—92no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—121no—
132C3×C22⋊C4—61no—
133C3×C22⋊C4—61no—
134C3×C22⋊C4—121no—
135(C2×D3)⋊C4—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—61no—
138Dic3⋊C4—62no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—121no—
144C22×C32—121no—
145C22×C32—121no—
146C22×C32C2×C3×D411yescharacteristic
147C2×C3⋊D3—21no—
148C2×C3⋊D3—21no—
149C2×C3⋊D3C3×D812yes—
150C2×C3×D3—41no—
151C2×C3×D3—41no—
152C2×C3×D3—41no—
153C2×C3×D3—42no—
154C2×C3×D3—41no—
155C2×C3×D3—41no—
156C2×C3×D3—41no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—81no—
162C2×C3×D3—81no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C32×C4—22no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—42no—
169C32×C4—81no—
170C32×C4—122no—
171C32×C4—121no—
172C3⋊Dic3C3×#4112yes—
173C3×Dic3—42no—
174C3×Dic3—41no—
175C3×Dic3—42no—
176C3×Dic3—41no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—82no—
180C3×Dic3—81no—
181#121—91no—
182C2×C4⋊C4—91no—
183#107—92no—
184#102—92no—
185C22⋊C8—91no—
186C33#48011yescharacteristic, complemented, Sylow
187C23×C3—61no—
188C23×C3—61no—
189C23×C3—91no—
190C23×C3—121no—
191C22×D3—21no—
192C22×D3—61no—
193C22×D3—61no—
194C3×D4—182no—
195C3×D4—362no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4—61no—
203C2×C3×C4—62no—
204C2×C3×C4—91no—
205C2×C3×C4—121no—
206C2×C3×C4—122no—
207C2×C3×C4—182no—
208C2×C3×C4—362no—
209C2×C3×C4GO+(4,𝔽2)11yescharacteristic
210#67—122no—
211#67—122no—
212C2×Dic3—22no—
213C2×Dic3—21no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216D12—122no—
217C4×D3—64no—
218C4×D3—122no—
219C4×D3—124no—
220C3×C8—182no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—241no—
234C2×C32C3×#12311yescharacteristic
235C2×C32C2×C3×D811yescharacteristic
236C2×C32C3×#13711yescharacteristic
237C3⋊D3—22no—
238C3×D3—64no—
239C3×D3—81no—
240C3×D3—82no—
241C3×D3—81no—
242C3×D3—122no—
243C2×D4—91no—
244C2×D4—181no—
245C22×C4—91no—
246C22×C4—181no—
247C2×C8—92no—
248C4⋊C4—61no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C22⋊C4—61no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—92no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3—122no—
261C22×C3—121no—
262C22×C3—121no—
263C22×C3—121no—
264C22×C3—181no—
265C22×C3—181no—
266C22×C3—241no—
267C22×C3—241no—
268C22×C3C2×GO+(4,𝔽2)11yescharacteristic
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—121no—
277C2×D3—121no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—42no—
282C3×C4—81no—
283C3×C4—92no—
284C3×C4—122no—
285C3×C4—121no—
286C3×C4—122no—
287C3×C4—121no—
288C3×C4—242no—
289C3×C4—241no—
290Dic3—42no—
291Dic3—41no—
292Dic3—62no—
293Dic3—62no—
294C32—21no—
295C32—21no—
296C32—41no—
297C32—41no—
298C32C3×#48011yescharacteristic, complemented
299C23—61no—
300C23—91no—
301D4—182no—
302D4—362no—
303C2×C4—61no—
304C2×C4—62no—
305C2×C4—91no—
306C2×C4—182no—
307C2×C4—362no—
308C2×C4C3×GO+(4,𝔽2)11yescharacteristic
309C8—182no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—182no—
323C2×C3—241no—
324C2×C3—242no—
325C2×C3—241no—
326C2×C3#6422511yescharacteristic
327C2×C3#6422611yescharacteristic
328C2×C3#6423311yescharacteristic
329D3—64no—
330D3—81no—
331D3—122no—
332C22—92no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22—181no—
337C22—181no—
338C22C2×C3×GO+(4,𝔽2)11yescharacteristic
339C4—21no—
340C4—92no—
341C4—122no—
342C4—121no—
343C3—21no—
344C3—21no—
345C3—41no—
346C3—41no—
347C3#1059336811yescharacteristic, direct product factor
348C2—182no—
349C2—241no—
350C2C3×#6422511yescharacteristic
351C2C3×#6422611yescharacteristic
352C2C3×#6423311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63912C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#480
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63900C2semidirect
3. C3×#63778C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. #10593368C3direct (N × H)
19. #63900C2×C3semidirect
20. #63778C2×C3semidirect
21. #63743C2×C3semidirect
22. #63739C2×C3semidirect
186. C33#480semidirect
298. C32C3×#480semidirect
347. C3#10593368direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639122
C3×#639002
C3×#637782
C3×#637432
C3×#637392
#105933683
C3×#4809