Finite Group Databasebeta

← C3×#10593437 #13166801

Subgroups — C3×#10593437 #13166801

Order 1728.

482 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×C3×#3775C212yesmaximal
2C3×#63919C211yesmaximal, characteristic, complemented
3C3×C4×#625C211yesmaximal, characteristic
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593437C311yesmaximal, characteristic, direct product factor
7C2×C32×C4×D3—21no—
8C2×C3×#627C2212yes—
9C2×C3×#625C2211yescharacteristic
10C2×C3×#622C2212yes—
11C3×#3775—24no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3—21no—
14C3×#3724—21no—
15C3×#3724C2211yescharacteristic
16C3×(C2×D3)⋊Dic3—21no—
17C2×#3775C2×C312yes—
18C3×C4×#67—61no—
19C3×#888—61no—
20#63919C2×C311yescharacteristic, complemented
21C4×#625C2×C311yescharacteristic
22C4⋊#622C2×C311yescharacteristic, complemented
23#63740C2×C312yescomplemented
24C22×C32×D3—21no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29C2×C32×Dic3—21no—
30C32×C4×D3—42no—
31C3×#627—22no—
32C3×#627—22no—
33C3×#625—21no—
34C3×#625D412yes—
35C3×#622—22no—
36C3×#461—91nomaximal
37C22×C32×C4—61no—
38C2×C3×#67—61no—
39C2×C3×C4×D3—21no—
40C2×C3×C4×D3—21no—
41C2×C3×C4×D3—41no—
42C2×C3×Dic6—62no—
43C2×#627C22×C312yes—
44C2×#625C22×C311yescharacteristic
45C2×#622C22×C312yes—
46#3775—24no—
47C32×C4⋊C4—61no—
48C4×C3⋊Dic3C22×C311yescharacteristic
49C3×#217—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—41no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—61no—
57C3×C4×Dic3—61no—
58#3724—21no—
59#3724C22×C311yescharacteristic
60(C2×D3)⋊Dic3—21no—
61C22×C33C2×D411yescharacteristic
62C2×C32×D3—22no—
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C33×C4SD1612yescomplemented
66C3×C3⋊Dic3—21no—
67C3×C3⋊Dic3C2×D411yescharacteristic
68C3×C3⋊Dic3#4111yescharacteristic
69C32×Dic3—42no—
70C32×Dic3—41no—
71C2×C3×SD16—92no—
72C3×C4⋊Q8—91no—
73C3×C4×D4—91no—
74C4×#67—61no—
75#888—61no—
76C3×C4⋊C8—91no—
77C3×#103—92no—
78C23×C32—61no—
79C22×C3×D3—21no—
80C22×C3×D3—21no—
81C22×C3×D3—41no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—61no—
87C2×C32×C4—62no—
88C2×C32×C4—122no—
89C2×C32×C4C3×D411yescharacteristic
90C2×C3⋊Dic3C3×D411yescharacteristic
91C2×C3⋊Dic3C23×C311yescharacteristic
92C3×#67—62no—
93C3×#67—62no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—21no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C3×C4×D3—42no—
105C3×C4×D3—44no—
106C3×C4×D3—42no—
107C3×Dic6—122no—
108C3×Dic6—122no—
109#627—22no—
110#627—22no—
111#625—21no—
112#625C3×D412yes—
113#622—22no—
114#461—91noSylow
115C2×C33#12111yescharacteristic
116C2×C33C2×SD1611yescharacteristic
117C2×C33#13711yescharacteristic
118C32×D3—42no—
119C2×C3×Q8—92no—
120C2×C3×D4—91no—
121C22×C3×C4—61no—
122C22×C3×C4—61no—
123C22×C3×C4—121no—
124C2×#67—61no—
125C2×C4×D3—21no—
126C2×Dic6—62no—
127C3×SD16—184no—
128C2×C3×C8—92no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—121no—
133C3×C4⋊C4—181no—
134C3×C22⋊C4—181no—
135C3×C42—91no—
136#217—61no—
137(C2×D3)⋊C4—61no—
138C4⋊Dic3—21no—
139Dic3⋊C4—61no—
140Dic3⋊C4—62no—
141C4×Dic3—61no—
142C4×Dic3—61no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32—62no—
148C22×C32—121no—
149C22×C32—121no—
150C22×C32C2×C3×D411yescharacteristic
151C2×C3×D3—22no—
152C2×C3×D3—22no—
153C2×C3×D3—42no—
154C2×C3×D3—41no—
155C2×C3×D3—41no—
156C2×C3×D3—42no—
157C2×C3×D3—42no—
158C2×C3×D3—41no—
159C2×C3×D3—41no—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—42no—
163C32×C4—42no—
164C32×C4—121no—
165C32×C4—122no—
166C32×C4C3×SD1612yescomplemented
167C3⋊Dic3—21no—
168C3⋊Dic3C2×C3×D411yescharacteristic
169C3⋊Dic3C3×#4111yescharacteristic
170C3×Dic3—42no—
171C3×Dic3—41no—
172C3×Dic3—42no—
173C3×Dic3—42no—
174C3×Dic3—41no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—62no—
180C3×Dic3—82no—
181C3×Dic3—121no—
182C2×SD16—92no—
183C4⋊Q8—91no—
184C4×D4—91no—
185C4⋊C8—91no—
186#103—92no—
187C33#46111yescharacteristic, complemented, Sylow
188C23×C3—61no—
189C23×C3—61no—
190C23×C3—121no—
191C22×D3—21no—
192C3×Q8—182no—
193C3×Q8—182no—
194C3×D4—92no—
195C3×D4—181no—
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—91no—
206C2×C3×C4—121no—
207C2×C3×C4—122no—
208C2×C3×C4—122no—
209C2×C3×C4—124no—
210C2×C3×C4—122no—
211C2×C3×C4GO+(4,𝔽2)11yescharacteristic
212#67—62no—
213#67—62no—
214C2×Dic3—22no—
215C2×Dic3—21no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C4×D3—42no—
221Dic6—122no—
222Dic6—122no—
223C3×C8—182no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—122no—
237C2×C32C3×#12111yescharacteristic
238C2×C32C2×C3×SD1611yescharacteristic
239C2×C32C3×#13711yescharacteristic
240C3×D3—42no—
241C3×D3—44no—
242C3×D3—42no—
243C2×Q8—92no—
244C2×D4—91no—
245C22×C4—61no—
246SD16—184no—
247C2×C8—92no—
248C4⋊C4—61no—
249C4⋊C4—91no—
250C4⋊C4—181no—
251C22⋊C4—181no—
252C42—91no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—122no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3—122no—
263C22×C3—122no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3C2×GO+(4,𝔽2)11yescharacteristic
267C2×D3—22no—
268C2×D3—41no—
269C2×D3—41no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—42no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—122no—
277C3×C4—121no—
278C3×C4—122no—
279C3×C4—122no—
280C3×C4—121no—
281C3×C4—181no—
282C3×C4—242no—
283C3×C4#377512yescomplemented
284Dic3—42no—
285Dic3—41no—
286Dic3—61no—
287Dic3—61no—
288Dic3—61no—
289Dic3—61no—
290Dic3—62no—
291Dic3—121no—
292C32—21no—
293C32—21no—
294C32—41no—
295C32—41no—
296C32C3×#46111yescharacteristic, complemented
297C23—61no—
298Q8—182no—
299Q8—182no—
300D4—92no—
301D4—181no—
302C2×C4—61no—
303C2×C4—62no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—122no—
307C2×C4C3×GO+(4,𝔽2)11yescharacteristic
308C8—182no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—122no—
322C2×C3—124no—
323C2×C3—122no—
324C2×C3#6422611yescharacteristic
325C2×C3#6423011yescharacteristic
326C2×C3C2×#377511yescharacteristic
327D3—42no—
328C22—62no—
329C22—121no—
330C22—121no—
331C22C2×C3×GO+(4,𝔽2)11yescharacteristic
332C4—91no—
333C4—91no—
334C4—121no—
335C4—122no—
336C4—181no—
337C4C3×#377512yescomplemented
338C3—21no—
339C3—21no—
340C3—41no—
341C3—41no—
342C3#1059343711yescharacteristic, direct product factor
343C2—122no—
344C2C3×#6422611yescharacteristic
345C2C3×#6423011yescharacteristic
346C2C2×C3×#377511yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#461
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×#63919C2semidirect
4. C3×C4⋊#622C2semidirect
5. C3×#63740C2semidirect
6. #10593437C3direct (N × H)
20. #63919C2×C3semidirect
22. C4⋊#622C2×C3semidirect
23. #63740C2×C3semidirect
65. C33×C4SD16semidirect
166. C32×C4C3×SD16semidirect
187. C33#461semidirect
283. C3×C4#3775semidirect
296. C32C3×#461semidirect
337. C4C3×#3775semidirect
342. C3#10593437direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C3×#639192
C3×C4×#6252
C3×C4⋊#6222
C3×#637402
#105934373
C3×#4619