Finite Group Databasebeta

← C3×#10591496 #13158508

Subgroups — C3×#10591496 #13158508

Order 1728.

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

normal subgroups only

This group has 630 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×#64145C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3C32×#832C212yesmaximal
4C3×#63567C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#4765—32nomaximal
7#10591496C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C3×#627—21no—
11C3×#3757—21no—
12C3×#3757—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64145C2×C311yescharacteristic
18C3×#936—32no—
19C3×#936—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic, complemented
22C32×#102—32no—
23C3×#832—22no—
24C3×#832D312yes—
25C3×#832C2×C312yes—
26C3×#816—32no—
27C3×#815—32no—
28C3×#810—32no—
29#63567C2×C312yescomplemented
30#63555C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C33×D4—22no—
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×#489—91no—
42#4765—32no—
43C2×C32×D4—22no—
44C2×C32×D4—21no—
45C2×C32×D4—41no—
46C2×C32×D4C2×D312yes—
47C2×C32×D4C22×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#3757—21no—
56#3757—21no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×#217—62no—
59C3×#217—62no—
60C3×#217—62no—
61C3×#217—62no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4⋊Dic3C22×C312yes—
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×C33—41no—
73C22×C33—41no—
74C22×C33—41no—
75C22×C33C2×D411yescharacteristic
76C33×C4#4112yes—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—42no—
81C3×C4⋊Q8—91no—
82C3×#124—91no—
83#936—32no—
84#936—61no—
85#888—32no—
86C3×#103—92no—
87C3×#102—32no—
88C3×#102—32no—
89C3×#102—62no—
90C3×#97—91no—
91#832C3×D312yes—
92#816—32no—
93#815—32no—
94#810—32no—
95C23×C32—21no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—42no—
99C23×C32—42no—
100C32×D4—22no—
101C32×D4—24no—
102C32×D4—22no—
103C32×D4—22no—
104C32×D4—42no—
105C32×D4—42no—
106C32×D4—44no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3C3×D411yescharacteristic
115C2×C3⋊Dic3C3×D411yescharacteristic
116#634—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—122no—
129C3×Dic6—124no—
130#627—42no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#489—91noSylow
136C2×C33—81no—
137C2×C33#12411yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C2×C33#13711yescharacteristic
140C2×C3×Q8—91no—
141C2×C3×Q8—181no—
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4—41no—
145C2×C3×D4D3211yescharacteristic
146C2×C3×D4C2×C3×D312yes—
147C2×Dic6—32no—
148C2×Dic6—61no—
149C2×Dic6—62no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—62no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—181no—
157C3×C22⋊C4—181no—
158C3×C22⋊C4—181no—
159C3×C42—91no—
160#217—62no—
161#217—62no—
162#217—62no—
163#217—62no—
164C4⋊Dic3C2×C3×D312yes—
165Dic3⋊C4—62no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8C2×C3×D312yes—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32—44no—
173C22×C32—42no—
174C22×C32—44no—
175C22×C32—42no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—82no—
184C22×C32—82no—
185C22×C32—82no—
186C22×C32—82no—
187C22×C32D3×D412yes—
188C22×C32C2×C3×D411yescharacteristic
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—122no—
194C32×C4#23512yes—
195C32×C4#23712yes—
196C32×C4C3×#4112yes—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—41no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—82no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—122no—
208C3×Dic3—122no—
209C4⋊Q8—91no—
210#124—91no—
211#103—92no—
212#102—32no—
213#97—91no—
214C33#48911yescharacteristic, complemented, Sylow
215C23×C3—22no—
216C23×C3—21no—
217C23×C3—22no—
218C23×C3—42no—
219C23×C3—42no—
220C3×Q8—182no—
221C3×Q8—362no—
222C3×D4—24no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—44no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—122no—
237C2×C3×C4—181no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×C3×C4C22×C3×D312yes—
240C2×Dic3—22no—
241C2×Dic3—32no—
242C2×Dic3—32no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247Dic6—62no—
248Dic6—62no—
249Dic6—122no—
250Dic6—124no—
251C3×C8—62no—
252C3×C8—62no—
253C3×C8—122no—
254C3⋊C8—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—82no—
265C2×C32—84no—
266C2×C32—84no—
267C2×C32—82no—
268C2×C32—81no—
269C2×C32#88512yes—
270C2×C32#91112yes—
271C2×C32#91512yes—
272C2×C32C3×#12411yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
274C2×C32C3×#13711yescharacteristic
275C2×Q8—91no—
276C2×Q8—181no—
277C2×D4C3×D3211yescharacteristic
278C2×C8—32no—
279C4⋊C4—32no—
280C4⋊C4—181no—
281C22⋊C4—181no—
282C22⋊C4—181no—
283C42—91no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3#381111yescharacteristic
303C22×C3C3×D3×D412yes—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—122no—
309C3×C4—122no—
310C3×C4—181no—
311C3×C4—181no—
312C3×C4—242no—
313C3×C4—361no—
314C3×C4#380412yes—
315C3×C4C3×#23512yes—
316C3×C4C3×#23712yes—
317Dic3—42no—
318Dic3—62no—
319Dic3—62no—
320Dic3—121no—
321Dic3—121no—
322Dic3—122no—
323Dic3—122no—
324C32—21no—
325C32—22no—
326C32—41no—
327C32#476512yescomplemented
328C32C3×#48911yescharacteristic, complemented
329C23—21no—
330Q8—182no—
331Q8—362no—
332D4—22no—
333C2×C4—62no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4C2×C3×D3211yescharacteristic
338C8—62no—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—82no—
349C2×C3—84no—
350C2×C3—84no—
351C2×C3—81no—
352C2×C3—82no—
353C2×C3#6392711yescharacteristic
354C2×C3#6393111yescharacteristic
355C2×C3#6395311yescharacteristic
356C2×C3C3×#88512yes—
357C2×C3C3×#91112yes—
358C2×C3C3×#91512yes—
359C22—41no—
360C22—41no—
361C22—41no—
362C22C3×#381111yescharacteristic
363C4—122no—
364C4—181no—
365C4—181no—
366C4—361no—
367C4C3×#380412yes—
368C3—22no—
369C3—21no—
370C3—41no—
371C3#1059149611yescharacteristic, direct product factor
372C3C3×#476512yescomplemented
373C2—81no—
374C2C3×#6392711yescharacteristic
375C2C3×#6393111yescharacteristic
376C2C3×#6395311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
4. C3×#63567C2semidirect
5. C3×#63555C2semidirect
7. #10591496C3direct (N × H)
21. #63919C2×C3semidirect
29. #63567C2×C3semidirect
30. #63555C2×C3semidirect
214. C33#489semidirect
327. C32#4765semidirect
328. C32C3×#489semidirect
371. C3#10591496direct (N × H)
372. C3C3×#4765semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C3×#639192
C32×#8322
C3×#635672
C3×#635552
C3×#47653
#105914963