Finite Group Databasebeta

← C3×#10590568 #13157561

Subgroups — C3×#10590568 #13157561

Order 1728.

526 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64099C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C32×#810C212yesmaximal
4C3×#63566C212yesmaximal
5C3×#63555C211yesmaximal, characteristic
6C3×#4797—32nomaximal
7#10590568C311yesmaximal, characteristic, direct product factor
8C2×C32×Dic6C2212yes—
9C33×C4⋊C4C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×#3750—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—21no—
16C3×#3724—21no—
17#64099C2×C311yescharacteristic
18C3×#944—32no—
19C3×#890—32no—
20C3×#890—61no—
21#63873C2×C311yescharacteristic
22C32×#103—32no—
23C3×#835—32no—
24C3×#816—32no—
25C3×#815—32no—
26C3×#810—22no—
27C3×#810D312yes—
28C3×#810C2×C312yes—
29#63566C2×C312yes—
30#63555C2×C311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C32×C3⋊C8—22no—
40C3×#491—91no—
41#4797—32no—
42C2×C32×Q8—32no—
43C2×C3×Dic6—22no—
44C2×C3×Dic6C2×D312yes—
45C2×C3×Dic6C22×C312yes—
46C2×C32×C8—32no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—22no—
49C32×C4⋊C4—41no—
50C32×C4⋊C4C2×D312yes—
51C32×C4⋊C4C22×C311yescharacteristic
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)C22×C311yescharacteristic
54#3750—21no—
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—62no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C22×C312yes—
69#3724—21no—
70#3724—21no—
71C22×C33C2×D411yescharacteristic
72C33×C4—41no—
73C33×C4#4112yes—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—21no—
76C3×C3⋊Dic3—41no—
77C32×Dic3—42no—
78C3×C4⋊Q8—91no—
79C3×#125—91no—
80#944—32no—
81#890—32no—
82#890—61no—
83C3×#103—32no—
84C3×#103—32no—
85C3×#103—62no—
86C3×#103—92no—
87C3×#97—91no—
88#835—32no—
89#816—32no—
90#815—32no—
91#810C3×D312yes—
92C32×Q8—62no—
93C32×Q8—62no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—41no—
100C2×C32×C4—42no—
101C2×C32×C4—42no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3C3×D411yescharacteristic
107C2×C3⋊Dic3C3×D411yescharacteristic
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C3×Dic6—24no—
118C3×Dic6—22no—
119C3×Dic6—22no—
120C3×Dic6—22no—
121C3×Dic6—22no—
122C3×Dic6—42no—
123C32×C8—62no—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—42no—
127#491—91noSylow
128C2×C33#12411yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—32no—
132C2×C3×Q8—32no—
133C2×C3×Q8—62no—
134C2×Dic6C2×C3×D312yes—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—62no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4D3211yescharacteristic
147C3×C4⋊C4C2×C3×D312yes—
148C3×C42—91no—
149C4⋊Dic3—32no—
150C4⋊Dic3—61no—
151C4⋊Dic3—62no—
152C4⋊Dic3—62no—
153Dic3⋊C4—62no—
154Dic3⋊C4—62no—
155Dic3⋊C4—62no—
156Dic3⋊C4—62no—
157C4×Dic3—32no—
158C4×Dic3—61no—
159C2×C3⋊C8C2×C3×D312yes—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—41no—
163C22×C32D3×D412yes—
164C22×C32C2×C3×D411yescharacteristic
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4—41no—
170C32×C4—42no—
171C32×C4—42no—
172C32×C4—82no—
173C32×C4—82no—
174C32×C4—122no—
175C32×C4#23512yes—
176C32×C4#23712yes—
177C32×C4C3×#4112yes—
178C3⋊Dic3—21no—
179C3⋊Dic3—21no—
180C3⋊Dic3—41no—
181C3×Dic3—44no—
182C3×Dic3—42no—
183C3×Dic3—42no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—121no—
187C3×Dic3—121no—
188C3×Dic3—122no—
189C3×Dic3—122no—
190C4⋊Q8—91no—
191#125—91no—
192#103—32no—
193#103—92no—
194#97—91no—
195C33#49111yescharacteristic, complemented, Sylow
196C3×Q8—64no—
197C3×Q8—62no—
198C3×Q8—62no—
199C3×Q8—62no—
200C3×Q8—62no—
201C3×Q8—122no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—122no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×C3×C4C22×C3×D312yes—
218C2×Dic3—22no—
219C2×Dic3—32no—
220C2×Dic3—32no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225Dic6—22no—
226Dic6—22no—
227C3×C8—62no—
228C3×C8—62no—
229C3×C8—122no—
230C3⋊C8—22no—
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×C32#91512yes—
242C2×C32#91812yes—
243C2×C32C3×#12411yescharacteristic
244C2×C32C3×#13711yescharacteristic
245C2×C32C3×#13711yescharacteristic
246C2×Q8—32no—
247C2×C8—32no—
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4—181no—
251C4⋊C4—181no—
252C4⋊C4—181no—
253C4⋊C4C3×D3211yescharacteristic
254C42—91no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3#381111yescharacteristic
259C22×C3C3×D3×D412yes—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—42no—
265C3×C4—41no—
266C3×C4—42no—
267C3×C4—82no—
268C3×C4—82no—
269C3×C4—124no—
270C3×C4—122no—
271C3×C4—122no—
272C3×C4—181no—
273C3×C4—181no—
274C3×C4—361no—
275C3×C4#379512yes—
276C3×C4C3×#23512yes—
277C3×C4C3×#23712yes—
278Dic3—42no—
279Dic3—62no—
280Dic3—62no—
281Dic3—121no—
282Dic3—121no—
283Dic3—122no—
284Dic3—122no—
285C32—21no—
286C32—22no—
287C32—41no—
288C32#479712yescomplemented
289C32C3×#49111yescharacteristic, complemented
290Q8—62no—
291Q8—62no—
292C2×C4—21no—
293C2×C4—62no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—181no—
297C2×C4C2×C3×D3211yescharacteristic
298C8—62no—
299C2×C3—21no—
300C2×C3—22no—
301C2×C3—22no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—22no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3#6391311yescharacteristic
309C2×C3#6393111yescharacteristic
310C2×C3#6394511yescharacteristic
311C2×C3C3×#88512yes—
312C2×C3C3×#91512yes—
313C2×C3C3×#91812yes—
314C22C3×#381111yescharacteristic
315C4—41no—
316C4—122no—
317C4—181no—
318C4—181no—
319C4—361no—
320C4C3×#379512yes—
321C3—22no—
322C3—21no—
323C3—41no—
324C3#1059056811yescharacteristic, direct product factor
325C3C3×#479712yescomplemented
326C2C3×#6391311yescharacteristic
327C2C3×#6393111yescharacteristic
328C2C3×#6394511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#491
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10590568C3direct (N × H)
195. C33#491semidirect
288. C32#4797semidirect
289. C32C3×#491semidirect
324. C3#10590568direct (N × H)
325. C3C3×#4797semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
C3×#638732
C32×#8102
C3×#635662
C3×#635552
C3×#47973
#105905683