Finite Group Databasebeta

← C3×#10591538 #13158632

Subgroups — C3×#10591538 #13158632

Order 1728.

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

normal subgroups only

This group has 582 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×#64153C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3C32×#835C212yesmaximal
4C3×#63567C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6C3×#4792—32nomaximal
7#10591538C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C3×#627—21no—
11C3×#3750—21no—
12C3×#3750—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64153C2×C311yescharacteristic
18C3×#944—32no—
19C3×#944—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic
22C32×#103—32no—
23C3×#835—22no—
24C3×#835D312yes—
25C3×#835C2×C312yes—
26C3×#816—32no—
27C3×C8×Dic3—32no—
28C3×#810—32no—
29#63567C2×C312yes—
30#63553C2×C311yescharacteristic
31C33×Q8—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×#634—22no—
38C2×C32×Dic3—22no—
39C3×#627—42no—
40C32×C3⋊C8—22no—
41C3×#487—91no—
42#4792—32no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C22×C311yescharacteristic
48C2×#634C22×C311yescharacteristic
49C2×C3×Dic6—32no—
50C2×C3×Dic6—62no—
51C2×C3×Dic6—61no—
52C2×#627—21no—
53C2×C32×C8—32no—
54C32×C4⋊C4—32no—
55#3750—21no—
56#3750—21no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C22×C312yes—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C33×C4—41no—
74C33×C4#4112yes—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3SD1612yes—
77C3×C3⋊Dic3Q1612yes—
78C32×Dic3—42no—
79C3×C4⋊Q8—91no—
80C3×C4⋊Q8—91no—
81#944—32no—
82#944—61no—
83#888—32no—
84C3×#103—32no—
85C3×#103—32no—
86C3×#103—62no—
87C3×#103—92no—
88C3×C4×C8—91no—
89#835C3×D312yes—
90#816—32no—
91C8×Dic3—32no—
92#810—32no—
93C32×Q8—22no—
94C32×Q8—24no—
95C32×Q8—22no—
96C32×Q8—22no—
97C32×Q8—42no—
98C32×Q8—42no—
99C32×Q8—44no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—41no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4—62no—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C23×C311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3C3×D411yescharacteristic
113C2×C3⋊Dic3C3×D411yescharacteristic
114#634—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126C3×Dic6—122no—
127C3×Dic6—124no—
128#627—42no—
129C32×C8—62no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—42no—
133#487—91noSylow
134C2×C33#12411yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33C2×Q1611yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—41no—
140C2×C3×Q8—91no—
141C2×C3×Q8—181no—
142C2×C3×Q8D3211yescharacteristic
143C2×C3×Q8C2×C3×D312yes—
144C2×Dic6—32no—
145C2×Dic6—61no—
146C2×Dic6—62no—
147C2×C3×C8—32no—
148C2×C3×C8—32no—
149C2×C3×C8—62no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—62no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—181no—
155C3×C4⋊C4—181no—
156C3×C42—91no—
157C4⋊Dic3C2×C3×D312yes—
158Dic3⋊C4—62no—
159Dic3⋊C4—62no—
160Dic3⋊C4—62no—
161Dic3⋊C4—62no—
162Dic3⋊C4—62no—
163C4×Dic3—32no—
164C4×Dic3—61no—
165C2×C3⋊C8C2×C3×D312yes—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×C3×D411yescharacteristic
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—44no—
177C32×C4—44no—
178C32×C4—42no—
179C32×C4—41no—
180C32×C4—122no—
181C32×C4#23512yes—
182C32×C4#23712yes—
183C32×C4C3×#4112yes—
184C3⋊Dic3—41no—
185C3⋊Dic3C3×SD1612yes—
186C3⋊Dic3C3×Q1612yes—
187C3×Dic3—34no—
188C3×Dic3—34no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—82no—
194C3×Dic3—122no—
195C3×Dic3—122no—
196C4⋊Q8—91no—
197C4⋊Q8—91no—
198#103—32no—
199#103—92no—
200C4×C8—91no—
201C33#48711yescharacteristic, complemented, Sylow
202C3×Q8—24no—
203C3×Q8—22no—
204C3×Q8—22no—
205C3×Q8—22no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—44no—
209C3×Q8—182no—
210C3×Q8—362no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—122no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3211yescharacteristic
226C2×C3×C4C22×C3×D312yes—
227C2×Dic3—22no—
228C2×Dic3—32no—
229C2×Dic3—32no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234Dic6—62no—
235Dic6—62no—
236Dic6—122no—
237Dic6—124no—
238C3×C8—62no—
239C3×C8—62no—
240C3×C8—122no—
241C3⋊C8—22no—
242C2×C32—22no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#88512yes—
252C2×C32D3×SD1612yes—
253C2×C32D3×Q1612yes—
254C2×C32C3×#12411yescharacteristic
255C2×C32C2×C3×SD1611yescharacteristic
256C2×C32C2×C3×Q1611yescharacteristic
257C2×Q8—91no—
258C2×Q8—181no—
259C2×Q8C3×D3211yescharacteristic
260C2×C8—32no—
261C4⋊C4—32no—
262C4⋊C4—181no—
263C4⋊C4—181no—
264C4⋊C4—181no—
265C42—91no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3#381111yescharacteristic
270C22×C3C3×D3×D412yes—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—42no—
280C3×C4—92no—
281C3×C4—92no—
282C3×C4—122no—
283C3×C4—122no—
284C3×C4—242no—
285C3×C4—361no—
286C3×C4#380412yes—
287C3×C4C3×#23512yes—
288C3×C4C3×#23712yes—
289Dic3—34no—
290Dic3—34no—
291Dic3—42no—
292Dic3—62no—
293Dic3—62no—
294Dic3—122no—
295Dic3—122no—
296C32—21no—
297C32—22no—
298C32—41no—
299C32#479212yescomplemented
300C32C3×#48711yescharacteristic, complemented
301Q8—22no—
302Q8—182no—
303Q8—362no—
304C2×C4—21no—
305C2×C4—62no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4C2×C3×D3211yescharacteristic
310C8—62no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3#6394011yescharacteristic
321C2×C3#6394411yescharacteristic
322C2×C3#6395311yescharacteristic
323C2×C3C3×#88512yes—
324C2×C3C3×D3×SD1612yes—
325C2×C3C3×D3×Q1612yes—
326C22C3×#381111yescharacteristic
327C4—41no—
328C4—92no—
329C4—92no—
330C4—122no—
331C4—361no—
332C4C3×#380412yes—
333C3—22no—
334C3—21no—
335C3—41no—
336C3#1059153811yescharacteristic, direct product factor
337C3C3×#479212yescomplemented
338C2C3×#6394011yescharacteristic
339C2C3×#6394411yescharacteristic
340C2C3×#6395311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#487
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
7. #10591538C3direct (N × H)
201. C33#487semidirect
299. C32#4792semidirect
300. C32C3×#487semidirect
336. C3#10591538direct (N × H)
337. C3C3×#4792semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C3×#639192
C32×#8352
C3×#635672
C3×#635532
C3×#47923
#105915383