Finite Group Databasebeta

← C3×#83588 #11898295

Subgroups — C3×#83588 #11898295

Order 1152.

964 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#550C211yesmaximal, characteristic
2C3×#5570C211yesmaximal, characteristic, complemented
3C3×#5076C213yesmaximal, complemented
4C3×#5075C213yesmaximal, complemented
5C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
6C3×#5019C213yesmaximal, complemented
7C3×C4×#214C213yesmaximal, complemented
8C3×#3392—31nomaximal
9#83588C311yesmaximal, characteristic, direct product factor
10C2×C32×#41—21no—
11C32×#127C2213yes—
12C32×#121—23no—
13C32×C4×Q8C2211yescharacteristic
14C32×C4×D4C2213yes—
15C3×Q8×Dic3—21no—
16C2×C3×#214C2213yes—
17C2×C3×#214C2216yes—
18C3×#890—23no—
19C3×#869C2213yes—
20C3×C4×Dic6C2213yes—
21C3×#832C2213yes—
22C3×#810C2213yes—
23C3×#808C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2213yes—
25C3×C4×C3⋊C8C2213yes—
26C3×#557—31no—
27C3×#550—21no—
28C3×#550D311yescharacteristic
29C3×#550C2×C311yescharacteristic
30#5570C2×C311yescharacteristic, complemented
31C3×#492—33no—
32C3×#461—33no—
33C3×#460—33no—
34C3×C8×Q8—31no—
35C3×C4×SD16—33no—
36#5076C2×C313yescomplemented
37#5075C2×C313yescomplemented
38Q8×C3⋊C8C2×C311yescharacteristic, complemented
39#5019C2×C313yescomplemented
40C4×#214C2×C313yescomplemented
41C32×#41—44no—
42C2×C32×Q8D411yescharacteristic
43C2×C32×D4—23no—
44C2×C32×D4—23no—
45C2×C32×D4C2313yes—
46C22×C32×C4—23no—
47C2×C3×Dic6C2313yes—
48C32×C4⋊C4D413yes—
49C32×C4⋊C4C2311yescharacteristic
50C32×C22⋊C4—23no—
51C32×C42C2313yes—
52C3×#214—26no—
53C3×#214—46no—
54C3×C4⋊Dic3—23no—
55C3×C4⋊Dic3—23no—
56C3×C4⋊Dic3C2311yescharacteristic
57C3×Dic3⋊C4—23no—
58C3×C4×Dic3—23no—
59C2×C3×C3⋊C8C2313yes—
60C2×C3×C3⋊C8C2311yescharacteristic
61#3392—31noSylow
62C2×C3×#41—22no—
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C2×C3×SD16—33no—
66C2×C3×SD16—36no—
67C3×C4⋊Q8—33no—
68C3×#127—23no—
69C3×#127C2×D313yes—
70C3×#127C22×C313yes—
71C3×#125—63no—
72C3×#121—26no—
73C3×#121—23no—
74C3×#121—23no—
75C3×C4×Q8—21no—
76C3×C4×Q8—33no—
77C3×C4×Q8—61no—
78C3×C4×Q8C2×D311yescharacteristic
79C3×C4×Q8C22×C311yescharacteristic
80C3×C4×D4—23no—
81C3×C4×D4C2×D313yes—
82C3×C4×D4C22×C313yes—
83Q8×Dic3—21no—
84C2×#214C22×C313yes—
85C2×#214C22×C316yes—
86#890—23no—
87#869C22×C313yes—
88C4×Dic6C22×C313yes—
89C3×#106—31no—
90C3×C4⋊C8—33no—
91C3×#103—33no—
92C3×#102—33no—
93C3×C4×C8—33no—
94#832C22×C313yes—
95#810C22×C313yes—
96#808C22×C311yescharacteristic
97C4⋊(C3⋊C8)C22×C313yes—
98C4×C3⋊C8C22×C313yes—
99C23×C32—23no—
100C32×Q8SD1614yescomplemented
101C32×D4—23no—
102C32×D4—23no—
103C32×D4—46no—
104C32×D4—83no—
105C2×C32×C4—21no—
106C2×C32×C4—46no—
107C2×C32×C4C2×D413yes—
108C2×C32×C4C2×D413yes—
109C2×C32×C4C2411yescharacteristic
110C2×C3×Dic3—21no—
111C2×C3×Dic3—23no—
112C3×Dic6—23no—
113C3×Dic6—23no—
114C3×C3⋊C8—23no—
115C3×C3⋊C8#4112yescomplemented
116#557—31no—
117#550C3×D311yescharacteristic
118#492—33no—
119#461—33no—
120#460—33no—
121C8×Q8—31no—
122C4×SD16—33no—
123C3×#41—48no—
124C3×#41—44no—
125C3×#41—44no—
126C2×C3×Q8—21no—
127C2×C3×Q8—33no—
128C2×C3×Q8#6711yescharacteristic
129C2×C3×Q8C3×D411yescharacteristic
130C2×C3×D4—26no—
131C2×C3×D4—26no—
132C2×C3×D4—23no—
133C2×C3×D4—23no—
134C2×C3×D4—23no—
135C2×C3×D4—23no—
136C2×C3×D4—23no—
137C2×C3×D4C22×D313yes—
138C2×C3×D4C23×C313yes—
139C22×C3×C4—26no—
140C22×C3×C4—23no—
141C22×C3×C4—23no—
142C2×Dic6C23×C313yes—
143C3×SD16—66no—
144C3×SD16—126no—
145C2×C3×C8—31no—
146C2×C3×C8—33no—
147C3×C4⋊C4—23no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—63no—
151C3×C4⋊C4—63no—
152C3×C4⋊C4—63no—
153C3×C4⋊C4#6713yes—
154C3×C4⋊C4C3×D413yes—
155C3×C4⋊C4C22×D311yescharacteristic
156C3×C4⋊C4C23×C311yescharacteristic
157C3×C22⋊C4—26no—
158C3×C22⋊C4—23no—
159C3×C22⋊C4—23no—
160C3×C42—23no—
161C3×C42—63no—
162C3×C42C22×D313yes—
163C3×C42C23×C313yes—
164#214—26no—
165#214—46no—
166C4⋊Dic3—23no—
167C4⋊Dic3—23no—
168C4⋊Dic3C23×C311yescharacteristic
169Dic3⋊C4—23no—
170C4×Dic3—23no—
171C2×C3⋊C8C23×C313yes—
172C2×C3⋊C8C23×C311yescharacteristic
173C22×C32—43no—
174C22×C32—43no—
175C22×C32—43no—
176C22×C32C22×D411yescharacteristic
177C32×C4—23no—
178C32×C4—41no—
179C32×C4C2×SD1616yes—
180C32×C4C2×#4111yescharacteristic
181C32×C4H(5,𝔽2)11yescharacteristic
182C3×Dic3—43no—
183C3×Dic3—41no—
184C2×#41—21no—
185C2×SD16—33no—
186C2×SD16—36no—
187C4⋊Q8—33no—
188#127C2×C3×D313yes—
189#125—63no—
190#121—23no—
191C4×Q8—33no—
192C4×Q8—61no—
193C4×Q8C2×C3×D311yescharacteristic
194C4×D4C2×C3×D313yes—
195#106—31no—
196C4⋊C8—33no—
197#103—33no—
198#102—33no—
199C4×C8—33no—
200C23×C3—26no—
201C23×C3—23no—
202C23×C3—23no—
203C3×Q8—24no—
204C3×Q8—63no—
205C3×Q8—63no—
206C3×Q8#21414yescomplemented
207C3×Q8C3×SD1614yescomplemented
208C3×D4—23no—
209C3×D4—23no—
210C3×D4—23no—
211C3×D4—23no—
212C3×D4—43no—
213C3×D4—43no—
214C3×D4—412no—
215C3×D4—46no—
216C3×D4—46no—
217C3×D4—83no—
218C3×D4—83no—
219C3×D4—86no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—23no—
225C2×C3×C4—23no—
226C2×C3×C4—412no—
227C2×C3×C4—46no—
228C2×C3×C4—46no—
229C2×C3×C4—61no—
230C2×C3×C4—63no—
231C2×C3×C4C2×#6713yes—
232C2×C3×C4C2×#6713yes—
233C2×C3×C4C2×C3×D413yes—
234C2×C3×C4C2×C3×D413yes—
235C2×C3×C4C23×D311yescharacteristic
236C2×C3×C4C24×C311yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—23no—
239Dic6—23no—
240Dic6—23no—
241C3×C8—32no—
242C3×C8—63no—
243C3⋊C8—23no—
244C3⋊C8C3×#4112yescomplemented
245C2×C32—83no—
246C2×C32#55011yescharacteristic
247C2×C32C22×SD1611yescharacteristic
248C2×C32#57711yescharacteristic
249#41—44no—
250C2×Q8—33no—
251C2×Q8C3×#6711yescharacteristic
252C2×D4—23no—
253C2×D4—23no—
254C2×D4C22×C3×D313yes—
255C22×C4—23no—
256SD16—66no—
257SD16—126no—
258C2×C8—31no—
259C2×C8—33no—
260C4⋊C4—31no—
261C4⋊C4—63no—
262C4⋊C4—63no—
263C4⋊C4—63no—
264C4⋊C4C3×#6713yes—
265C4⋊C4C22×C3×D311yescharacteristic
266C22⋊C4—23no—
267C42—63no—
268C42C22×C3×D313yes—
269C22×C3—21no—
270C22×C3—46no—
271C22×C3—46no—
272C22×C3—46no—
273C22×C3—43no—
274C22×C3—43no—
275C22×C3—43no—
276C22×C3—43no—
277C22×C3—43no—
278C22×C3—43no—
279C22×C3C22×#6711yescharacteristic
280C22×C3C22×C3×D411yescharacteristic
281C3×C4—23no—
282C3×C4—23no—
283C3×C4—26no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—43no—
287C3×C4—42no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—123no—
291C3×C4—121no—
292C3×C4C2×#21416yes—
293C3×C4C2×C3×SD1616yes—
294C3×C4D3×#4111yescharacteristic
295C3×C4#100911yescharacteristic
296C3×C4C2×C3×#4111yescharacteristic
297C3×C4C3×H(5,𝔽2)11yescharacteristic
298Dic3—41no—
299Dic3—43no—
300C32#339211yescharacteristic, complemented, Sylow
301C23—23no—
302Q8—63no—
303Q8—63no—
304Q8C3×#21414yescomplemented
305D4—23no—
306D4—23no—
307D4—46no—
308D4—83no—
309C2×C4—21no—
310C2×C4—46no—
311C2×C4—61no—
312C2×C4—63no—
313C2×C4C2×C3×#6713yes—
314C2×C4C2×C3×#6713yes—
315C2×C4C23×C3×D311yescharacteristic
316C8—32no—
317C8—63no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—86no—
322C2×C3—83no—
323C2×C3—83no—
324C2×C3C22×#21411yescharacteristic
325C2×C3#583411yescharacteristic
326C2×C3#583611yescharacteristic
327C2×C3C3×#55011yescharacteristic
328C2×C3C22×C3×SD1611yescharacteristic
329C2×C3C3×#57711yescharacteristic
330C22—43no—
331C22—43no—
332C22—43no—
333C22C22×C3×#6711yescharacteristic
334C4—23no—
335C4—41no—
336C4—123no—
337C4—121no—
338C4C2×C3×#21416yes—
339C4C3×D3×#4111yescharacteristic
340C4C3×#100911yescharacteristic
341C3—21no—
342C3#8358811yescharacteristic, direct product factor
343C3C3×#339211yescharacteristic, complemented
344C2—83no—
345C2C22×C3×#21411yescharacteristic
346C2C3×#583411yescharacteristic
347C2C3×#583611yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#550C2
socleC22×C32C22×D4
hypercenterC3×#550D3
hypocenterC3C3×#3392
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#550

Complements

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

NH ≅ G/Ntype
2. C3×#5570C2semidirect
3. C3×#5076C2semidirect
4. C3×#5075C2semidirect
5. C3×Q8×C3⋊C8C2semidirect
6. C3×#5019C2semidirect
7. C3×C4×#214C2semidirect
9. #83588C3direct (N × H)
30. #5570C2×C3semidirect
36. #5076C2×C3semidirect
37. #5075C2×C3semidirect
38. Q8×C3⋊C8C2×C3semidirect
39. #5019C2×C3semidirect
40. C4×#214C2×C3semidirect
100. C32×Q8SD16semidirect
115. C3×C3⋊C8#41semidirect
206. C3×Q8#214semidirect
207. C3×Q8C3×SD16semidirect
244. C3⋊C8C3×#41semidirect
300. C32#3392semidirect
304. Q8C3×#214semidirect
342. C3#83588direct (N × H)
343. C3C3×#3392semidirect

Maximal subgroups

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

subgroupindex
C32×#5502
C3×#55702
C3×#50762
C3×#50752
C3×Q8×C3⋊C82
C3×#50192
C3×C4×#2142
C3×#33923
#835883