Finite Group Databasebeta

← C3×#10591062 #13158090

Subgroups — C3×#10591062 #13158090

Order 1728.

614 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64135C211yesmaximal, characteristic
2C32×#830C212yesmaximal
3C3×#63572C212yesmaximal, complemented
4C3×#63555C211yesmaximal, characteristic, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#4720—32nomaximal
7#10591062C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C3×#3757C412yes—
10C3×C4⋊(C3⋊Dic3)C412yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14C2×C32×C3⋊C8C2212yes—
15#64135C2×C311yescharacteristic
16C3×#926—32no—
17C3×#926—61no—
18C32×C22⋊C8—32no—
19C3×#830—22no—
20C3×#830D312yes—
21C3×#830C2×C312yes—
22C3×#815—32no—
23C3×Dic3⋊C8—32no—
24C3×Dic3⋊C8—32no—
25C3×C8×Dic3—32no—
26#63572C2×C312yescomplemented
27#63555C2×C311yescharacteristic, complemented
28#63553C2×C311yescharacteristic, complemented
29C23×C33C2×C411yescharacteristic
30C2×C33×C4—22no—
31C2×C33×C4C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C412yes—
34C2×C3×C3⋊Dic3C2×C412yes—
35C32×C3⋊C8—22no—
36C32×C3⋊C8—22no—
37C3×#433—91no—
38#4720—32no—
39C22×C32×C4—21no—
40C22×C32×C4—22no—
41C22×C32×C4—41no—
42C22×C32×C4C2×D312yes—
43C22×C32×C4C22×C311yescharacteristic
44C2×C32×C8—32no—
45C2×C32×C8—32no—
46#3757C3×C412yes—
47C4⋊(C3⋊Dic3)C3×C412yes—
48C4×C3⋊Dic3C22×C311yescharacteristic
49C4×C3⋊Dic3C22×C311yescharacteristic
50C3×#217—32no—
51C3×#217—32no—
52C3×#217—62no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—62no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8C2×D312yes—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C22×C312yes—
65C2×C3×C3⋊C8C22×C312yes—
66C22×C33—21no—
67C22×C33—22no—
68C22×C33C22×C411yescharacteristic
69C33×C4—21no—
70C33×C4#4112yes—
71C3×C3⋊Dic3—22no—
72C3×C3⋊Dic3—22no—
73C3×#117—91no—
74#926—32no—
75#926—61no—
76C3×C4⋊C8—92no—
77C3×C22⋊C8—32no—
78C3×C22⋊C8—32no—
79C3×C22⋊C8—62no—
80C3×#97—91no—
81C3×C4×C8—91no—
82#830C3×D312yes—
83#815—32no—
84Dic3⋊C8—32no—
85Dic3⋊C8—32no—
86C8×Dic3—32no—
87C23×C32—21no—
88C23×C32—22no—
89C23×C32—41no—
90C23×C32C4×D312yes—
91C23×C32C2×C3×C411yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—24no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—42no—
103C2×C32×C4—42no—
104C2×C32×C4—44no—
105C2×C32×C4C4×D312yes—
106C2×C32×C4C2×C3×C411yescharacteristic
107C2×C32×C4C22×D312yes—
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3C2×C3×C412yes—
110C2×C3⋊Dic3C2×C3×C412yes—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C32×C8—62no—
118C32×C8—62no—
119C3×C3⋊C8—22no—
120C3×C3⋊C8—22no—
121C3×C3⋊C8—22no—
122C3×C3⋊C8—22no—
123C3×C3⋊C8—42no—
124C3×C3⋊C8—42no—
125#433—91noSylow
126C2×C33—41no—
127C2×C33#11711yescharacteristic
128C2×C33#13112yes—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—41no—
132C22×C3×C4D3211yescharacteristic
133C22×C3×C4C2×C3×D312yes—
134C2×C3×C8—32no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—32no—
138C2×C3×C8—62no—
139C2×C3×C8—62no—
140C3×C4⋊C4—92no—
141C3×C22⋊C4—92no—
142C3×C42—91no—
143C3×C42—91no—
144#217—32no—
145#217—32no—
146#217—62no—
147C4⋊Dic3—32no—
148C4⋊Dic3—32no—
149C4⋊Dic3—62no—
150C4×Dic3—32no—
151C4×Dic3—32no—
152C4×Dic3—61no—
153C4×Dic3—61no—
154C2×C3⋊C8C2×C3×D312yes—
155C2×C3⋊C8C2×C3×D312yes—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—22no—
159C22×C32—24no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—42no—
169C22×C32—42no—
170C22×C32—44no—
171C22×C32C2×C4×D312yes—
172C22×C32C22×C3×C411yescharacteristic
173C32×C4—21no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4—42no—
182C32×C4#23712yes—
183C32×C4#23712yes—
184C32×C4C3×#4112yes—
185C3⋊Dic3—22no—
186C3⋊Dic3—22no—
187C3×Dic3—62no—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—122no—
192C3×Dic3—122no—
193#117—91no—
194C4⋊C8—92no—
195C22⋊C8—32no—
196#97—91no—
197C4×C8—91no—
198C33#43311yescharacteristic, complemented, Sylow
199C23×C3—22no—
200C23×C3—21no—
201C23×C3—41no—
202C23×C3#62411yescharacteristic
203C23×C3C3×C4×D312yes—
204C2×C3×C4—24no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—92no—
218C2×C3×C4—92no—
219C2×C3×C4#62411yescharacteristic
220C2×C3×C4C3×C4×D312yes—
221C2×C3×C4C2×D3211yescharacteristic
222C2×C3×C4C22×C3×D312yes—
223C2×Dic3—32no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C3×C8—62no—
230C3×C8—62no—
231C3×C8—62no—
232C3×C8—62no—
233C3×C8—122no—
234C3×C8—122no—
235C3⋊C8—22no—
236C3⋊C8—22no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—42no—
244C2×C32—44no—
245C2×C32—42no—
246C2×C32—44no—
247C2×C32—42no—
248C2×C32—41no—
249C2×C32#87712yes—
250C2×C32#90112yes—
251C2×C32#90712yes—
252C2×C32C3×#11711yescharacteristic
253C2×C32C3×#13112yes—
254C22×C4C3×D3211yescharacteristic
255C2×C8—32no—
256C2×C8—32no—
257C4⋊C4—92no—
258C22⋊C4—92no—
259C42—91no—
260C42—91no—
261C22×C3—22no—
262C22×C3—24no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—44no—
275C22×C3—41no—
276C22×C3C2×#62411yescharacteristic
277C22×C3C2×C3×C4×D312yes—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4—42no—
285C3×C4—42no—
286C3×C4—42no—
287C3×C4—182no—
288C3×C4—182no—
289C3×C4#380512yes—
290C3×C4C3×#23712yes—
291C3×C4C3×#23712yes—
292Dic3—62no—
293Dic3—62no—
294Dic3—62no—
295Dic3—62no—
296Dic3—122no—
297Dic3—122no—
298C32—22no—
299C32—21no—
300C32—41no—
301C32#472012yescomplemented
302C32C3×#43311yescharacteristic, complemented
303C23C3×#62411yescharacteristic
304C2×C4—22no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4C3×#62411yescharacteristic
308C2×C4C2×C3×D3211yescharacteristic
309C8—62no—
310C8—62no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3—42no—
318C2×C3—44no—
319C2×C3—42no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—42no—
323C2×C3#6381712yes—
324C2×C3#6395711yescharacteristic
325C2×C3C3×#87712yes—
326C2×C3C3×#90112yes—
327C2×C3C3×#90712yes—
328C22—21no—
329C22—22no—
330C22C2×C3×#62411yescharacteristic
331C4—21no—
332C4—182no—
333C4—182no—
334C4C3×#380512yes—
335C3—22no—
336C3—21no—
337C3—41no—
338C3#1059106211yescharacteristic, direct product factor
339C3C3×#472012yescomplemented
340C2—41no—
341C2C3×#6381712yes—
342C2C3×#6395711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63572C2semidirect
4. C3×#63555C2semidirect
5. C3×#63553C2semidirect
7. #10591062C3direct (N × H)
26. #63572C2×C3semidirect
27. #63555C2×C3semidirect
28. #63553C2×C3semidirect
198. C33#433semidirect
301. C32#4720semidirect
302. C32C3×#433semidirect
338. C3#10591062direct (N × H)
339. C3C3×#4720semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C32×#8302
C3×#635722
C3×#635552
C3×#635532
C3×#47203
#105910623