Finite Group Databasebeta

← C3×#10593367 #13166709

Subgroups — C3×#10593367 #13166709

Order 1728.

438 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 70 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×#63919C211yesmaximal, characteristic, complemented
2C3×#63902C211yesmaximal, characteristic
3C3×#63779C211yesmaximal, characteristic, complemented
4C3×#63743C212yesmaximal, complemented
5C3×#63739C212yesmaximal
6#10593367C311yesmaximal, characteristic, direct product factor
7C2×C3×#627—21no—
8C2×C3×#625C2211yescharacteristic
9C2×C3×D3×Dic3—21no—
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×(C2×D3)⋊C4—21no—
13C32×C4⋊Dic3—21no—
14C3×#3724C2211yescharacteristic
15C3×#3724C2212yes—
16C3×#888—61no—
17C3×#67⋊C4—61no—
18#63919C2×C311yescharacteristic, complemented
19#63902C2×C311yescharacteristic
20#63779C2×C311yescharacteristic, complemented
21#63743C2×C312yescomplemented
22#63739C2×C312yes—
23C22×C32×D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—22no—
29C3×#627—42no—
30C3×#625—21no—
31C3×#625Q812yescomplemented
32C3×D3×Dic3—42no—
33C3×#622—22no—
34C3×#474—91nomaximal
35C2×C3×#67—61no—
36C22×C3×Dic3—61no—
37C2×C3×C4×D3—61no—
38C2×C3×Dic6—62no—
39C2×#627—21no—
40C2×#625C22×C311yescharacteristic
41C2×D3×Dic3—21no—
42C2×#622C22×C312yes—
43C32×C4⋊C4—61no—
44C32×C22⋊C4—61no—
45C4×C3⋊Dic3C22×C311yescharacteristic
46C3×(C2×D3)⋊C4—21no—
47C3×(C2×D3)⋊C4—21no—
48C3×(C2×D3)⋊C4—41no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—41no—
52C3×Dic3⋊C4—61no—
53C3×Dic3⋊C4—62no—
54C3×C4×Dic3—61no—
55C3×C4×Dic3—61no—
56#3724C22×C311yescharacteristic
57#3724C22×C312yes—
58C22×C33C2×D411yescharacteristic
59C2×C32×D3—22no—
60C2×C32×D3—41no—
61C2×C32×D3—41no—
62C33×C4—21no—
63C3×C3⋊Dic3D812yes—
64C3×C3⋊Dic3C2×Q811yescharacteristic
65C3×C3⋊Dic3#4111yescharacteristic
66C32×Dic3—41no—
67C32×Dic3—42no—
68C3×C4⋊Q8—91no—
69C3×C4×D4—91no—
70#888—61no—
71#67⋊C4—61no—
72C3×#107—92no—
73C3×C4⋊C8—91no—
74C3×#102—92no—
75C23×C32—61no—
76C22×C3×D3—21no—
77C22×C3×D3—21no—
78C22×C3×D3—41no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—61no—
84C2×C32×C4—62no—
85C2×C32×C4C3×D411yescharacteristic
86C2×C3⋊Dic3C3×D411yescharacteristic
87C2×C3⋊Dic3C23×C311yescharacteristic
88C3×#67—62no—
89C3×#67—62no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—122no—
101C3×C4×D3—122no—
102C3×Dic6—124no—
103#627—42no—
104#625—21no—
105#625C3×Q812yescomplemented
106D3×Dic3—42no—
107#622—22no—
108#474—91noSylow
109C2×C33C22⋊Q811yescharacteristic
110C2×C33C2×D811yescharacteristic
111C2×C33#13711yescharacteristic
112C32×D3—42no—
113C2×C3×Q8—181no—
114C2×C3×D4—91no—
115C22×C3×C4—181no—
116C2×#67—61no—
117C22×Dic3—61no—
118C2×C4×D3—61no—
119C2×Dic6—62no—
120C2×C3×C8—92no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—61no—
123C3×C4⋊C4—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—121no—
126C3×C22⋊C4—61no—
127C3×C22⋊C4—61no—
128C3×C22⋊C4—121no—
129C3×C42—91no—
130(C2×D3)⋊C4—21no—
131C4⋊Dic3—21no—
132Dic3⋊C4—61no—
133Dic3⋊C4—62no—
134C4×Dic3—61no—
135C4×Dic3—61no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—62no—
141C22×C32—121no—
142C22×C32—121no—
143C22×C32C2×C3×D411yescharacteristic
144C2×C3×D3—22no—
145C2×C3×D3—22no—
146C2×C3×D3—42no—
147C2×C3×D3—41no—
148C2×C3×D3—41no—
149C2×C3×D3—41no—
150C2×C3×D3—41no—
151C2×C3×D3—81no—
152C2×C3×D3—81no—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—41no—
156C32×C4—42no—
157C32×C4—81no—
158C32×C4—122no—
159C32×C4—121no—
160C3⋊Dic3C3×D812yes—
161C3⋊Dic3C2×C3×Q811yescharacteristic
162C3⋊Dic3C3×#4111yescharacteristic
163C3×Dic3—42no—
164C3×Dic3—41no—
165C3×Dic3—41no—
166C3×Dic3—42no—
167C3×Dic3—62no—
168C3×Dic3—62no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—82no—
174C3×Dic3—81no—
175C4⋊Q8—91no—
176C4×D4—91no—
177#107—92no—
178C4⋊C8—91no—
179#102—92no—
180C33#47411yescharacteristic, complemented, Sylow
181C23×C3—61no—
182C23×C3—61no—
183C23×C3—121no—
184C22×D3—21no—
185C3×Q8—362no—
186C3×D4—92no—
187C3×D4—181no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—61no—
193C2×C3×C4—62no—
194C2×C3×C4—61no—
195C2×C3×C4—62no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—121no—
199C2×C3×C4—122no—
200C2×C3×C4—362no—
201C2×C3×C4GO+(4,𝔽2)11yescharacteristic
202#67—62no—
203#67—62no—
204C2×Dic3—22no—
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—122no—
211C4×D3—122no—
212Dic6—124no—
213C3×C8—182no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—122no—
227C2×C32C3×C22⋊Q811yescharacteristic
228C2×C32C2×C3×D811yescharacteristic
229C2×C32C3×#13711yescharacteristic
230C3×D3—42no—
231C3×D3—42no—
232C3×D3—82no—
233C2×Q8—181no—
234C2×D4—91no—
235C22×C4—181no—
236C2×C8—92no—
237C4⋊C4—61no—
238C4⋊C4—92no—
239C4⋊C4—91no—
240C22⋊C4—61no—
241C42—91no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—62no—
247C22×C3—62no—
248C22×C3—122no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—241no—
254C22×C3—241no—
255C22×C3C2×GO+(4,𝔽2)11yescharacteristic
256C2×D3—22no—
257C2×D3—41no—
258C2×D3—41no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—41no—
262C3×C4—42no—
263C3×C4—81no—
264C3×C4—91no—
265C3×C4—91no—
266C3×C4—92no—
267C3×C4—122no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—122no—
271C3×C4—241no—
272C3×C4—242no—
273Dic3—42no—
274Dic3—41no—
275Dic3—62no—
276Dic3—62no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281C32—21no—
282C32—21no—
283C32—41no—
284C32—41no—
285C32C3×#47411yescharacteristic, complemented
286C23—61no—
287Q8—362no—
288D4—92no—
289D4—181no—
290C2×C4—61no—
291C2×C4—62no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—362no—
295C2×C4C3×GO+(4,𝔽2)11yescharacteristic
296C8—182no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—122no—
310C2×C3—122no—
311C2×C3—242no—
312C2×C3D32⋊Q811yescharacteristic
313C2×C3#6422511yescharacteristic
314C2×C3#6423911yescharacteristic
315D3—42no—
316C22—62no—
317C22—121no—
318C22—121no—
319C22C2×C3×GO+(4,𝔽2)11yescharacteristic
320C4—21no—
321C4—91no—
322C4—91no—
323C4—92no—
324C4—121no—
325C4—122no—
326C3—21no—
327C3—21no—
328C3—41no—
329C3—41no—
330C3#1059336711yescharacteristic, direct product factor
331C2—122no—
332C2C3×D32⋊Q811yescharacteristic
333C2C3×#6422511yescharacteristic
334C2C3×#6423911yescharacteristic
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×#474
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
1. C3×#63919C2semidirect
3. C3×#63779C2semidirect
4. C3×#63743C2semidirect
6. #10593367C3direct (N × H)
18. #63919C2×C3semidirect
20. #63779C2×C3semidirect
21. #63743C2×C3semidirect
31. C3×#625Q8semidirect
105. #625C3×Q8semidirect
180. C33#474semidirect
285. C32C3×#474semidirect
330. C3#10593367direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639192
C3×#639022
C3×#637792
C3×#637432
C3×#637392
#105933673
C3×#4749