Finite Group Databasebeta

← C3×#10590901 #13157869

Subgroups — C3×#10590901 #13157869

Order 1728.

542 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C3×#63873C211yesmaximal, characteristic
3C3×C8×C3⋊Dic3C211yesmaximal, characteristic
4C32×#816C212yesmaximal
5C3×#63565C212yesmaximal
6C3×#4792—32nomaximal
7#10590901C311yesmaximal, characteristic, direct product factor
8C2×C32×Dic6C2212yes—
9C2×C3×#627—21no—
10C2×C33×C8C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14C3×#3724—21no—
15C3×#3724—21no—
16C3×#3724—21no—
17C3×#944—32no—
18C3×#888—32no—
19#63919C2×C311yescharacteristic
20#63873C2×C311yescharacteristic
21C32×#103—32no—
22C8×C3⋊Dic3C2×C311yescharacteristic
23C3×#835—32no—
24C3×#816—22no—
25C3×#816D312yes—
26C3×#816C2×C312yes—
27C3×C8×Dic3—32no—
28C3×C8×Dic3—61no—
29C3×#810—32no—
30#63565C2×C312yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C32×Dic3—22no—
35C2×C32×Dic3—22no—
36C32×Dic6—22no—
37C32×Dic6—22no—
38C3×#627—42no—
39C33×C8—21no—
40C3×C3⋊(C3⋊C8)—21no—
41C3×#487—91no—
42#4792—32no—
43C2×C32×Q8—32no—
44C2×C3×Dic6—22no—
45C2×C3×Dic6—62no—
46C2×C3×Dic6C2×D312yes—
47C2×C3×Dic6C22×C312yes—
48C2×#627—21no—
49C2×C32×C8—21no—
50C2×C32×C8—22no—
51C2×C32×C8—41no—
52C2×C32×C8C2×D312yes—
53C2×C32×C8C22×C311yescharacteristic
54C32×C4⋊C4—32no—
55C4×C3⋊Dic3C22×C311yescharacteristic
56C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3C2×D312yes—
59C3×C4⋊Dic3C22×C312yes—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—61no—
67#3724—21no—
68#3724—21no—
69#3724—21no—
70C22×C33C2×D411yescharacteristic
71C33×C4#4111yescharacteristic
72C33×C4#4111yescharacteristic
73C3×C3⋊Dic3SD1612yes—
74C3×C3⋊Dic3Q1612yes—
75C32×Dic3—42no—
76C32×Dic3—42no—
77C3×C4⋊Q8—91no—
78C3×C4⋊Q8—91no—
79#944—32no—
80#888—32no—
81C3×#103—32no—
82C3×#103—32no—
83C3×#103—62no—
84C3×#103—92no—
85C3×C4×C8—91no—
86#835—32no—
87#816C3×D312yes—
88C8×Dic3—32no—
89C8×Dic3—61no—
90#810—32no—
91C32×Q8—62no—
92C32×Q8—62no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—41no—
96C2×C32×C4—62no—
97C2×C32×C4—62no—
98C2×C32×C4C22×D312yes—
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3C3×D411yescharacteristic
101C2×C3⋊Dic3C3×D411yescharacteristic
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—42no—
109C2×C3×Dic3—42no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C3×Dic6—24no—
113C3×Dic6—22no—
114C3×Dic6—22no—
115C3×Dic6—22no—
116C3×Dic6—22no—
117C3×Dic6—42no—
118C3×Dic6—124no—
119#627—42no—
120C32×C8—21no—
121C32×C8—22no—
122C32×C8—22no—
123C32×C8—42no—
124C32×C8—42no—
125C3⋊(C3⋊C8)—21no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—62no—
128#487—91noSylow
129C2×C33#12411yescharacteristic
130C2×C33C2×SD1611yescharacteristic
131C2×C33C2×Q1611yescharacteristic
132C2×C3×Q8—32no—
133C2×C3×Q8—32no—
134C2×C3×Q8—62no—
135C2×C3×Q8—181no—
136C2×Dic6—62no—
137C2×Dic6C2×C3×D312yes—
138C2×C3×C8—21no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C2×C3×C8—91no—
142C2×C3×C8D3211yescharacteristic
143C2×C3×C8C2×C3×D312yes—
144C3×C4⋊C4—32no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C42—91no—
151C4⋊Dic3C2×C3×D312yes—
152Dic3⋊C4—62no—
153Dic3⋊C4—62no—
154Dic3⋊C4—62no—
155C4×Dic3—32no—
156C4×Dic3—61no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—61no—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—41no—
162C22×C32D3×D412yes—
163C22×C32C2×C3×D411yescharacteristic
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4—122no—
171C32×C4—122no—
172C32×C4#23512yes—
173C32×C4#23712yes—
174C32×C4C3×#4111yescharacteristic
175C32×C4C3×#4111yescharacteristic
176C3⋊Dic3C3×SD1612yes—
177C3⋊Dic3C3×Q1612yes—
178C3×Dic3—34no—
179C3×Dic3—34no—
180C3×Dic3—44no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—42no—
184C3×Dic3—42no—
185C3×Dic3—62no—
186C3×Dic3—62no—
187C3×Dic3—82no—
188C4⋊Q8—91no—
189C4⋊Q8—91no—
190#103—32no—
191#103—92no—
192C4×C8—91no—
193C33#48711yescharacteristic, complemented, Sylow
194C3×Q8—64no—
195C3×Q8—62no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—62no—
199C3×Q8—122no—
200C3×Q8—362no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—122no—
211C2×C3×C4—122no—
212C2×C3×C4C2×D3211yescharacteristic
213C2×C3×C4C22×C3×D312yes—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220Dic6—22no—
221Dic6—22no—
222Dic6—124no—
223C3×C8—22no—
224C3×C8—21no—
225C3×C8—22no—
226C3×C8—42no—
227C3×C8—42no—
228C3×C8—181no—
229C3⋊C8—62no—
230C3⋊C8—62no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32#88512yes—
241C2×C32D3×SD1612yes—
242C2×C32D3×Q1612yes—
243C2×C32C3×#12411yescharacteristic
244C2×C32C2×C3×SD1611yescharacteristic
245C2×C32C2×C3×Q1611yescharacteristic
246C2×Q8—32no—
247C2×Q8—181no—
248C2×C8—91no—
249C2×C8C3×D3211yescharacteristic
250C4⋊C4—32no—
251C4⋊C4—181no—
252C4⋊C4—181no—
253C4⋊C4—181no—
254C42—91no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3#380211yescharacteristic
259C22×C3C3×D3×D412yes—
260C3×C4—21no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—92no—
267C3×C4—92no—
268C3×C4—124no—
269C3×C4—122no—
270C3×C4—122no—
271C3×C4—122no—
272C3×C4—122no—
273C3×C4—242no—
274C3×C4#379811yescharacteristic
275C3×C4#380511yescharacteristic
276C3×C4C3×#23512yes—
277C3×C4C3×#23712yes—
278Dic3—34no—
279Dic3—34no—
280Dic3—42no—
281Dic3—42no—
282Dic3—62no—
283Dic3—62no—
284C32—22no—
285C32—21no—
286C32—41no—
287C32#479212yescomplemented
288C32C3×#48711yescharacteristic, complemented
289Q8—62no—
290Q8—62no—
291Q8—362no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4C2×C3×D3211yescharacteristic
297C8—21no—
298C8—181no—
299C2×C3—21no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3#6379611yescharacteristic
309C2×C3#6380211yescharacteristic
310C2×C3#6391511yescharacteristic
311C2×C3C3×#88512yes—
312C2×C3C3×D3×SD1612yes—
313C2×C3C3×D3×Q1612yes—
314C22C3×#380211yescharacteristic
315C4—92no—
316C4—92no—
317C4—122no—
318C4—122no—
319C4C3×#379811yescharacteristic
320C4C3×#380511yescharacteristic
321C3—22no—
322C3—21no—
323C3—41no—
324C3#1059090111yescharacteristic, direct product factor
325C3C3×#479212yescomplemented
326C2C3×#6379611yescharacteristic
327C2C3×#6380211yescharacteristic
328C2C3×#6391511yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#487
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
7. #10590901C3direct (N × H)
193. C33#487semidirect
287. C32#4792semidirect
288. C32C3×#487semidirect
324. C3#10590901direct (N × H)
325. C3C3×#4792semidirect

Maximal subgroups

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

subgroupindex
C3×#639192
C3×#638732
C3×C8×C3⋊Dic32
C32×#8162
C3×#635652
C3×#47923
#105909013