Finite Group Databasebeta

← C3×#68397 #11781424

Subgroups — C3×#68397 #11781424

Order 1152.

406 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5072C211yesmaximal, characteristic
2C3×#5070C211yesmaximal, characteristic, complemented
3C3×#4841C211yesmaximal, characteristic, complemented
4C3×#4834C211yesmaximal, characteristic, complemented
5C32×#331C211yesmaximal, characteristic, complemented
6C3×#4486C211yesmaximal, characteristic, complemented
7C3×#4479C211yesmaximal, characteristic, complemented
8C3×#1681—31nomaximal
9#68397C311yesmaximal, characteristic, direct product factor
10C32×#127C2211yescharacteristic
11C2×C3×#213—21no—
12C2×C3×#204—21no—
13C2×C3×D3×C8—21no—
14C3×C4×D12C2211yescharacteristic
15C3×#869C2211yescharacteristic
16C32×C4⋊C8C2211yescharacteristic
17C32×C4⋊C8C2211yescharacteristic
18C3×#832—21no—
19C3×#832—21no—
20C3×#832—21no—
21C3×(C2×D3)⋊C8—21no—
22C3×#816—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#486—31no—
26C3×#461—31no—
27C3×#459—31no—
28C3×C8×D4—31no—
29#5072C2×C311yescharacteristic
30#5070C2×C311yescharacteristic, complemented
31#4841C2×C311yescharacteristic, complemented
32#4834C2×C311yescharacteristic, complemented
33C3×#335—31no—
34C3×#332—31no—
35C3×#331—21no—
36C3×#331D311yescharacteristic, complemented
37C3×#331C2×C311yescharacteristic, complemented
38#4486C2×C311yescharacteristic, complemented
39#4479C2×C311yescharacteristic, complemented
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C3×D12D411yescharacteristic
44C2×C3×C4×D3—21no—
45C2×C3×Dic6—21no—
46C2×C32×C8—21no—
47C2×C32×C8—21no—
48C32×C42C2311yescharacteristic
49C3×#213—42no—
50C3×(C2×D3)⋊C4—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57C3×#204—42no—
58C3×D3×C8—42no—
59#1681—31noSylow
60C2×C3×SD16—61no—
61C2×C3×D8—61no—
62C22×C3×C8—61no—
63C3×C4⋊Q8—31no—
64C3×#127—21no—
65C3×#127C2×D311yescharacteristic
66C3×#127C22×C311yescharacteristic
67C3×C4×D4—31no—
68C2×#213—21no—
69C2×#204—21no—
70C2×D3×C8—21no—
71C4×D12C22×C311yescharacteristic
72#869C22×C311yescharacteristic
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—31no—
76C3×C4⋊C8C2×D311yescharacteristic
77C3×C4⋊C8C2×D311yescharacteristic
78C3×C4⋊C8C22×C311yescharacteristic
79C3×C4⋊C8C22×C311yescharacteristic
80C3×#103—61no—
81C3×#102—61no—
82C3×#102—61no—
83C3×#102—61no—
84C3×C22⋊C8—61no—
85C3×C4×C8—31no—
86#832—21no—
87#832—21no—
88#832—21no—
89(C2×D3)⋊C8—21no—
90#816—21no—
91C4⋊(C3⋊C8)C22×C311yescharacteristic
92C4×C3⋊C8C22×C311yescharacteristic
93C23×C32—41no—
94C22×C3×D3—21no—
95C32×D4—41no—
96C32×D4—41no—
97C32×D4—41no—
98C32×D4—41no—
99C32×D4—81no—
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4#4111yescharacteristic
103C2×C3×Dic3—21no—
104C2×C3×Dic3—41no—
105C3×D12—21no—
106C3×D12D812yes—
107C3×C4×D3—41no—
108C3×C4×D3—41no—
109C3×Dic6—41no—
110C3×Dic6—41no—
111C32×C8—41no—
112C32×C8—41no—
113C3×C3⋊C8—21no—
114C3×C3⋊C8—21no—
115C3×C3⋊C8—41no—
116#486—31no—
117#461—31no—
118#459—31no—
119C8×D4—31no—
120#335—31no—
121#332—31no—
122#331C3×D311yescharacteristic, complemented
123C2×C3×Q8—61no—
124C2×C3×D4—22no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—31no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C22×C3×C4—61no—
135C2×D12C3×D411yescharacteristic
136C2×C4×D3—21no—
137C2×Dic6—21no—
138C3×SD16—122no—
139C3×D8—122no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—31no—
145C2×C3×C8—31no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—61no—
149C2×C3×C8—122no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—61no—
153C3×C22⋊C4—61no—
154C3×C42—21no—
155C3×C42C22×D311yescharacteristic
156C3×C42C23×C311yescharacteristic
157#213—42no—
158(C2×D3)⋊C4—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3C3×D411yescharacteristic
162C2×C3⋊C8—21no—
163C2×C3⋊C8C3×D411yescharacteristic
164C2×C3⋊C8C3×D411yescharacteristic
165#204—42no—
166D3×C8—42no—
167C22×C32—81no—
168C22×C32—81no—
169C22×C32—81no—
170C22×C32#12111yescharacteristic
171C2×C3×D3—22no—
172C2×C3×D3—41no—
173C2×C3×D3—41no—
174C32×C4—21no—
175C32×C4C2×D811yescharacteristic
176C32×C4#13511yescharacteristic
177C32×C4#13511yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C3×Dic3—41no—
180C3×Dic3—81no—
181C2×SD16—61no—
182C2×D8—61no—
183C22×C8—61no—
184C4⋊Q8—31no—
185#127C2×C3×D311yescharacteristic
186C4×D4—31no—
187C4⋊C8—31no—
188C4⋊C8C2×C3×D311yescharacteristic
189C4⋊C8C2×C3×D311yescharacteristic
190#103—61no—
191#102—61no—
192#102—61no—
193#102—61no—
194C22⋊C8—61no—
195C4×C8—31no—
196C23×C3—42no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—61no—
200C22×D3—21no—
201C3×Q8—121no—
202C3×Q8—121no—
203C3×D4—32no—
204C3×D4—41no—
205C3×D4—41no—
206C3×D4—41no—
207C3×D4—41no—
208C3×D4—41no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—61no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—81no—
219C3×D4—82no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—61no—
224C2×C3×C4—121no—
225C2×C3×C4—121no—
226C2×C3×C4—121no—
227C2×C3×C4#23511yescharacteristic
228C2×C3×C4D3×D411yescharacteristic
229C2×C3×C4D3×D411yescharacteristic
230C2×C3×C4C2×C3×D411yescharacteristic
231C2×C3×C4C2×C3×D411yescharacteristic
232C2×C3×C4C3×#4111yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—41no—
235D12—21no—
236D12C3×D812yes—
237C4×D3—41no—
238C4×D3—41no—
239Dic6—41no—
240Dic6—41no—
241C3×C8—41no—
242C3×C8—41no—
243C3×C8—41no—
244C3×C8—41no—
245C3×C8—61no—
246C3×C8—61no—
247C3×C8—81no—
248C3×C8—81no—
249C3×C8—121no—
250C3⋊C8—21no—
251C3⋊C8—21no—
252C3⋊C8—41no—
253C2×C32—161no—
254C2×C32#46311yescharacteristic
255C2×C32#46611yescharacteristic
256C2×C32#47111yescharacteristic
257C3×D3—42no—
258C2×Q8—61no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—31no—
263C22×C4—61no—
264SD16—122no—
265D8—122no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—31no—
269C2×C8—31no—
270C2×C8—61no—
271C2×C8—122no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C22⋊C4—61no—
276C42C22×C3×D311yescharacteristic
277C22×C3—21no—
278C22×C3—62no—
279C22×C3—82no—
280C22×C3—82no—
281C22×C3—82no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—121no—
289C22×C3—121no—
290C22×C3#88411yescharacteristic
291C22×C3C3×#12111yescharacteristic
292C2×D3—22no—
293C2×D3—41no—
294C2×D3—41no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—121no—
303C3×C4—241no—
304C3×C4D3×D811yescharacteristic
305C3×C4#91111yescharacteristic
306C3×C4#91611yescharacteristic
307C3×C4#91611yescharacteristic
308C3×C4C2×C3×D811yescharacteristic
309C3×C4C3×#13511yescharacteristic
310C3×C4C3×#13511yescharacteristic
311C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
312Dic3—41no—
313Dic3—81no—
314C32#168111yescharacteristic, complemented, Sylow
315C23—41no—
316C23—61no—
317Q8—121no—
318Q8—121no—
319D4—32no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—61no—
325D4—81no—
326C2×C4—61no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4C3×#23511yescharacteristic
331C2×C4C3×D3×D411yescharacteristic
332C2×C4C3×D3×D411yescharacteristic
333C8—41no—
334C8—41no—
335C8—61no—
336C8—61no—
337C8—121no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—122no—
342C2×C3—162no—
343C2×C3—161no—
344C2×C3—161no—
345C2×C3#474811yescharacteristic
346C2×C3#477511yescharacteristic
347C2×C3#478711yescharacteristic
348C2×C3C3×#46311yescharacteristic
349C2×C3C3×#46611yescharacteristic
350C2×C3C3×#47111yescharacteristic
351D3—42no—
352C22—62no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—121no—
357C22—121no—
358C22C3×#88411yescharacteristic
359C4—21no—
360C4—121no—
361C4—241no—
362C4C3×D3×D811yescharacteristic
363C4C3×#91111yescharacteristic
364C4C3×#91611yescharacteristic
365C4C3×#91611yescharacteristic
366C3—21no—
367C3#6839711yescharacteristic, direct product factor
368C3C3×#168111yescharacteristic, complemented
369C2—122no—
370C2—161no—
371C2C3×#474811yescharacteristic
372C2C3×#477511yescharacteristic
373C2C3×#478711yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#331C2
socleC22×C32#121
prime radicalC3×#5072C2
hypercenterC3×#331D3
hypocenterC3C3×#1681
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#331

Complements

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

NH ≅ G/Ntype
2. C3×#5070C2semidirect
3. C3×#4841C2semidirect
4. C3×#4834C2semidirect
5. C32×#331C2semidirect
6. C3×#4486C2semidirect
7. C3×#4479C2semidirect
9. #68397C3direct (N × H)
30. #5070C2×C3semidirect
31. #4841C2×C3semidirect
32. #4834C2×C3semidirect
36. C3×#331D3semidirect
37. C3×#331C2×C3semidirect
38. #4486C2×C3semidirect
39. #4479C2×C3semidirect
122. #331C3×D3semidirect
314. C32#1681semidirect
367. C3#68397direct (N × H)
368. C3C3×#1681semidirect

Maximal subgroups

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

subgroupindex
C3×#50722
C3×#50702
C3×#48412
C3×#48342
C32×#3312
C3×#44862
C3×#44792
C3×#16813
#683973