Finite Group Databasebeta

← C3×#68210 #11780285

Subgroups — C3×#68210 #11780285

Order 1152.

509 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Dic6C211yesmaximal, characteristic
2C3×#5019C211yesmaximal, characteristic
3C3×#4851C211yesmaximal, characteristic, complemented
4C3×#4680C211yesmaximal, characteristic, complemented
5C32×D4⋊C8C211yesmaximal, characteristic
6C3×#4482C211yesmaximal, characteristic, complemented
7C3×#4456C211yesmaximal, characteristic, complemented
8C3×#1651—31nomaximal
9#68210C311yesmaximal, characteristic, direct product factor
10C22×C3×Dic6—21no—
11C32×C4×D4C2211yescharacteristic
12C3×D4×Dic3—21no—
13C2×C3×#214—21no—
14C3×#924—21no—
15C3×#888—21no—
16C3×#878—22no—
17C3×#869C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20C32×C4⋊C8C2211yescharacteristic
21C32×C4×C8C2211yescharacteristic
22C3×#817—22no—
23C3×#817—21no—
24C3×#816—21no—
25C3×#810—21no—
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×D4×Q8—31no—
28D4×Dic6C2×C311yescharacteristic
29C3×#498—31no—
30C3×#475—31no—
31C3×#461—31no—
32#5019C2×C311yescharacteristic
33#4851C2×C311yescharacteristic, complemented
34#4680C2×C311yescharacteristic, complemented
35C3×#332—31no—
36C3×Q8⋊C8—31no—
37C3×D4⋊C8—21no—
38C3×D4⋊C8D311yescharacteristic
39C3×D4⋊C8C2×C311yescharacteristic
40#4482C2×C311yescharacteristic, complemented
41#4456C2×C311yescharacteristic, complemented
42C2×C32×D4D411yescharacteristic
43C22×C32×C4—21no—
44C22×C3×Dic3—22no—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—21no—
47C2×C3×Dic6—21no—
48C2×C3×Dic6—42no—
49C2×C3×Dic6—42no—
50C2×C3×Dic6D411yescharacteristic
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C32×C4⋊C4D411yescharacteristic
54C32×C22⋊C4—21no—
55C32×C42C2311yescharacteristic
56C3×#217—22no—
57C3×#214—42no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3D412yes—
62C3×C4⋊Dic3D411yescharacteristic
63C3×Dic3⋊C4—22no—
64C3×Dic3⋊C4—21no—
65C3×C4×Dic3—21no—
66C2×C3×C3⋊C8—21no—
67#1651—31noSylow
68C22×C3×Q8—61no—
69C22×Dic6—21no—
70C2×C3×SD16—61no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8—31no—
73C3×C4⋊Q8—61no—
74C3×C22⋊Q8—62no—
75C3×C22⋊Q8—61no—
76C3×C4×Q8—31no—
77C3×C4×D4—21no—
78C3×C4×D4—61no—
79C3×C4×D4C2×D311yescharacteristic
80C3×C4×D4C22×C311yescharacteristic
81D4×Dic3—21no—
82C2×#214—21no—
83#924—21no—
84#888—21no—
85#878—22no—
86#869C22×C311yescharacteristic
87#869C22×C311yescharacteristic
88C4×Dic6C22×C311yescharacteristic
89C3×#106—61no—
90C3×#106—62no—
91C3×C4⋊C8—21no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8C2×D311yescharacteristic
94C3×C4⋊C8C22×C311yescharacteristic
95C3×#103—61no—
96C3×#103—61no—
97C3×C4×C8—21no—
98C3×C4×C8C2×D311yescharacteristic
99C3×C4×C8C22×C311yescharacteristic
100#817—22no—
101#817—21no—
102#816—21no—
103#810—21no—
104C4⋊(C3⋊C8)C22×C311yescharacteristic
105C23×C32—21no—
106C32×D4—21no—
107C32×D4SD1612yes—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—44no—
118C2×C3×Dic3—41no—
119C3×Dic6—21no—
120C3×Dic6—44no—
121C3×Dic6—41no—
122C3×Dic6—41no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6SD1612yescomplemented
126C32×C8—22no—
127C32×C8—41no—
128C3×C3⋊C8—41no—
129D4×Q8—31no—
130#498—31no—
131#475—31no—
132#461—31no—
133#332—31no—
134Q8⋊C8—31no—
135D4⋊C8C3×D311yescharacteristic
136C2×C3×Q8—31no—
137C2×C3×Q8—62no—
138C2×C3×Q8—61no—
139C2×C3×Q8—61no—
140C2×C3×Q8—122no—
141C2×C3×Q8—122no—
142C2×C3×D4—21no—
143C2×C3×D4D1211yescharacteristic
144C2×C3×D4C3×D411yescharacteristic
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—41no—
148C22×C3×C4—62no—
149C22×Dic3—22no—
150C2×Dic6—22no—
151C2×Dic6—21no—
152C2×Dic6—21no—
153C2×Dic6—42no—
154C2×Dic6—42no—
155C2×Dic6C3×D411yescharacteristic
156C3×SD16—122no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—61no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—62no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4—61no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4D1211yescharacteristic
173C3×C4⋊C4C3×D411yescharacteristic
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—62no—
178C3×C42—21no—
179C3×C42—61no—
180C3×C42C22×D311yescharacteristic
181C3×C42C23×C311yescharacteristic
182#217—22no—
183#214—42no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3C3×D411yescharacteristic
188C4⋊Dic3C3×D412yes—
189Dic3⋊C4—22no—
190Dic3⋊C4—21no—
191C4×Dic3—21no—
192C2×C3⋊C8—21no—
193C22×C32—22no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32#12011yescharacteristic
197C32×C4—21no—
198C32×C4—41no—
199C32×C4C2×SD1611yescharacteristic
200C32×C4C2×SD1611yescharacteristic
201C32×C4AGL(1,ℤ/8)11yescharacteristic
202C32×C4#13711yescharacteristic
203C3×Dic3—22no—
204C3×Dic3—42no—
205C3×Dic3—41no—
206C3×Dic3—81no—
207C22×Q8—61no—
208C2×SD16—61no—
209C4⋊Q8—31no—
210C4⋊Q8—31no—
211C4⋊Q8—61no—
212C22⋊Q8—62no—
213C22⋊Q8—61no—
214C4×Q8—31no—
215C4×D4—61no—
216C4×D4C2×C3×D311yescharacteristic
217#106—61no—
218#106—62no—
219C4⋊C8—31no—
220C4⋊C8C2×C3×D311yescharacteristic
221#103—61no—
222#103—61no—
223C4×C8C2×C3×D311yescharacteristic
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C3×Q8—32no—
228C3×Q8—61no—
229C3×Q8—124no—
230C3×Q8—121no—
231C3×Q8—121no—
232C3×Q8—121no—
233C3×Q8—121no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—22no—
237C3×D4—41no—
238C3×D4#20412yes—
239C3×D4C3×SD1612yes—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4—124no—
256C2×C3×C4—121no—
257C2×C3×C4C2×D1211yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4C2×C3×D411yescharacteristic
261C2×C3×C4C2×C3×D411yescharacteristic
262C2×C3×C4C2×C3×D411yescharacteristic
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—22no—
266C2×Dic3—44no—
267C2×Dic3—41no—
268Dic6—21no—
269Dic6—44no—
270Dic6—41no—
271Dic6—41no—
272Dic6—41no—
273Dic6—41no—
274Dic6C3×SD1612yescomplemented
275C3×C8—22no—
276C3×C8—22no—
277C3×C8—42no—
278C3×C8—41no—
279C3×C8—41no—
280C3×C8—81no—
281C3×C8—121no—
282C3⋊C8—41no—
283C2×C32—42no—
284C2×C32#44811yescharacteristic
285C2×C32#45011yescharacteristic
286C2×C32#45611yescharacteristic
287C2×Q8—31no—
288C2×Q8—62no—
289C2×Q8—61no—
290C2×Q8—61no—
291C2×Q8—122no—
292C2×Q8—122no—
293C2×D4C3×D1211yescharacteristic
294C22×C4—21no—
295C22×C4—62no—
296SD16—122no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—61no—
300C4⋊C4—31no—
301C4⋊C4—32no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4C3×D1211yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—62no—
310C42—61no—
311C42C22×C3×D311yescharacteristic
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—21no—
315C22×C3—42no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3#88211yescharacteristic
323C22×C3C3×#12011yescharacteristic
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—62no—
334C3×C4—81no—
335C3×C4—122no—
336C3×C4—121no—
337C3×C4—241no—
338C3×C4C2×#20411yescharacteristic
339C3×C4#90911yescharacteristic
340C3×C4#91111yescharacteristic
341C3×C4D3×SD1611yescharacteristic
342C3×C4C2×C3×SD1611yescharacteristic
343C3×C4C2×C3×SD1611yescharacteristic
344C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
345C3×C4C3×#13711yescharacteristic
346Dic3—22no—
347Dic3—42no—
348Dic3—41no—
349Dic3—81no—
350C32#165111yescharacteristic, complemented, Sylow
351C23—21no—
352Q8—32no—
353Q8—61no—
354Q8—124no—
355Q8—121no—
356Q8—121no—
357Q8—121no—
358Q8—121no—
359D4—21no—
360D4C3×#20412yes—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—124no—
368C2×C4—121no—
369C2×C4C2×C3×D1211yescharacteristic
370C2×C4C3×D3×D411yescharacteristic
371C2×C4C3×D3×D411yescharacteristic
372C8—22no—
373C8—41no—
374C8—121no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—42no—
379C2×C3—42no—
380C2×C3—82no—
381C2×C3#473811yescharacteristic
382C2×C3#477611yescharacteristic
383C2×C3#482611yescharacteristic
384C2×C3C3×#44811yescharacteristic
385C2×C3C3×#45011yescharacteristic
386C2×C3C3×#45611yescharacteristic
387C22—22no—
388C22—41no—
389C22—41no—
390C22C3×#88211yescharacteristic
391C4—21no—
392C4—41no—
393C4—62no—
394C4—122no—
395C4—121no—
396C4—241no—
397C4C2×C3×#20411yescharacteristic
398C4C3×#90911yescharacteristic
399C4C3×#91111yescharacteristic
400C4C3×D3×SD1611yescharacteristic
401C3—21no—
402C3#6821011yescharacteristic, direct product factor
403C3C3×#165111yescharacteristic, complemented
404C2—42no—
405C2C3×#473811yescharacteristic
406C2C3×#477611yescharacteristic
407C2C3×#482611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×D4⋊C8C2
socleC22×C32#120
prime radicalC2×C32×D4D4
hypercenterC3×D4⋊C8D3
hypocenterC3C3×#1651
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×D4⋊C8

Complements

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

NH ≅ G/Ntype
3. C3×#4851C2semidirect
4. C3×#4680C2semidirect
6. C3×#4482C2semidirect
7. C3×#4456C2semidirect
9. #68210C3direct (N × H)
33. #4851C2×C3semidirect
34. #4680C2×C3semidirect
40. #4482C2×C3semidirect
41. #4456C2×C3semidirect
125. C3×Dic6SD16semidirect
274. Dic6C3×SD16semidirect
350. C32#1651semidirect
402. C3#68210direct (N × H)
403. C3C3×#1651semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Dic62
C3×#50192
C3×#48512
C3×#46802
C32×D4⋊C82
C3×#44822
C3×#44562
C3×#16513
#682103