Finite Group Databasebeta

← C3×#3225 #85089

Subgroups — C3×#3225 #85089

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic, complemented
2C3×#536C211yesmaximal, characteristic, complemented
3C3×#497C211yesmaximal, characteristic, complemented
4C3×#495C211yesmaximal, characteristic, complemented
5C3×#489C211yesmaximal, characteristic, complemented
6C3×#487C211yesmaximal, characteristic, complemented
7C3×#464C211yesmaximal, characteristic, complemented
8C3×#463C211yesmaximal, characteristic, complemented
9C3×#462C211yesmaximal, characteristic, complemented
10C3×#461C211yesmaximal, characteristic, complemented
11C3×#452C211yesmaximal, characteristic, complemented
12C3×#451C211yesmaximal, characteristic, complemented
13C3×#449C211yesmaximal, characteristic
14C3×#448C211yesmaximal, characteristic, complemented
15C3×#433C211yesmaximal, characteristic, complemented
16#3225C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic
24C2×C3×SD16C2211yescharacteristic
25C2×C3×SD16C2211yescharacteristic
26C2×C3×D8C2211yescharacteristic
27C3×C4⋊Q8C2211yescharacteristic
28C3×C4⋊Q8C2211yescharacteristic
29C3×C4⋊Q8C2211yescharacteristic
30C3×#126—21no—
31C3×#124—21no—
32C3×#124C2211yescharacteristic
33C3×#124C2211yescharacteristic
34C3×#124C2211yescharacteristic
35C3×#123—21no—
36C3×#123—21no—
37C3×#123—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8C2211yescharacteristic
42C3×#121C2211yescharacteristic
43C3×C4×Q8C2211yescharacteristic
44C3×C4×D4C2211yescharacteristic
45C3×#117C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×C22⋊C8C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×#97C2211yescharacteristic
59C3×C4×C8C2211yescharacteristic
60#542C2×C311yescharacteristic, complemented
61#536C2×C311yescharacteristic, complemented
62#497C2×C311yescharacteristic, complemented
63#495C2×C311yescharacteristic, complemented
64#489C2×C311yescharacteristic, complemented
65#487C2×C311yescharacteristic, complemented
66#464C2×C311yescharacteristic, complemented
67#463C2×C311yescharacteristic, complemented
68#462C2×C311yescharacteristic, complemented
69#461C2×C311yescharacteristic, complemented
70#452C2×C311yescharacteristic, complemented
71#451C2×C311yescharacteristic, complemented
72#449C2×C311yescharacteristic
73#448C2×C311yescharacteristic, complemented
74#433C2×C311yescharacteristic, complemented
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8—21no—
78C2×C3×Q8—21no—
79C2×C3×Q8—41no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2311yescharacteristic
93C3×Q16—41no—
94C3×Q16—41no—
95C3×Q16—41no—
96C3×SD16—41no—
97C3×SD16—41no—
98C3×SD16—41no—
99C3×SD16—41no—
100C3×D8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4D411yescharacteristic
113C3×C4⋊C4D411yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C4⋊C4C2311yescharacteristic
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4D411yescharacteristic
126C3×C22⋊C4D411yescharacteristic
127C3×C42—21no—
128C3×C42C2311yescharacteristic
129C3×C42C2311yescharacteristic
130C2×#41C22×C311yescharacteristic
131C22×Q8C22×C311yescharacteristic
132C2×Q16C22×C311yescharacteristic
133C2×Q16C22×C311yescharacteristic
134C2×Q16C22×C311yescharacteristic
135C2×SD16C22×C311yescharacteristic
136C2×SD16C22×C311yescharacteristic
137C2×SD16C22×C311yescharacteristic
138C2×SD16C22×C311yescharacteristic
139C2×D8C22×C311yescharacteristic
140C4⋊Q8C22×C311yescharacteristic
141C4⋊Q8C22×C311yescharacteristic
142C4⋊Q8C22×C311yescharacteristic
143#126—21no—
144#124—21no—
145#124C22×C311yescharacteristic
146#124C22×C311yescharacteristic
147#124C22×C311yescharacteristic
148#123—21no—
149#123—21no—
150#123—21no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154C22⋊Q8C22×C311yescharacteristic
155#121C22×C311yescharacteristic
156C4×Q8C22×C311yescharacteristic
157C4×D4C22×C311yescharacteristic
158#117C22×C311yescharacteristic
159C4⋊C8C22×C311yescharacteristic
160C4⋊C8C22×C311yescharacteristic
161#103C22×C311yescharacteristic
162#103C22×C311yescharacteristic
163#103C22×C311yescharacteristic
164#103C22×C311yescharacteristic
165#103C22×C311yescharacteristic
166#103C22×C311yescharacteristic
167#102C22×C311yescharacteristic
168#102C22×C311yescharacteristic
169C22⋊C8C22×C311yescharacteristic
170C22⋊C8C22×C311yescharacteristic
171#97C22×C311yescharacteristic
172C4×C8C22×C311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3C2×D411yescharacteristic
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—42no—
185C3×Q8—81no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—42no—
191C3×D4—81no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2411yescharacteristic
209C3×C8—21no—
210C3×C8—21no—
211C3×C8—21no—
212C3×C8—21no—
213#41—41no—
214#41—41no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8C23×C311yescharacteristic
220C2×Q8C23×C311yescharacteristic
221C2×Q8C23×C311yescharacteristic
222C2×Q8C23×C311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4C23×C311yescharacteristic
226C2×D4C23×C311yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4C23×C311yescharacteristic
231Q16—41no—
232Q16—41no—
233Q16—41no—
234SD16—41no—
235SD16—41no—
236SD16—41no—
237SD16—41no—
238D8—41no—
239C2×C8C23×C311yescharacteristic
240C2×C8C23×C311yescharacteristic
241C2×C8C23×C311yescharacteristic
242C2×C8C23×C311yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4C3×D411yescharacteristic
251C4⋊C4C3×D411yescharacteristic
252C4⋊C4C23×C311yescharacteristic
253C4⋊C4C23×C311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4C3×D411yescharacteristic
264C22⋊C4C3×D411yescharacteristic
265C42—21no—
266C42C23×C311yescharacteristic
267C42C23×C311yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4H(5,𝔽2)11yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291C23—21no—
292C23—21no—
293C23C2×C3×D411yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—81no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4C2×C3×D411yescharacteristic
322C2×C4C2×C3×D411yescharacteristic
323C2×C4C2×C3×D411yescharacteristic
324C2×C4C2×C3×D411yescharacteristic
325C2×C4C2×C3×D411yescharacteristic
326C2×C4C24×C311yescharacteristic
327C8—21no—
328C8—21no—
329C8—21no—
330C8—21no—
331C2×C3—41no—
332C2×C3—81no—
333C2×C3—81no—
334C2×C3#53511yescharacteristic
335C2×C3#57711yescharacteristic
336C2×C3#57811yescharacteristic
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C22×C3×D411yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4C3×H(5,𝔽2)11yescharacteristic
359C4C3×H(5,𝔽2)11yescharacteristic
360C3#322511yescharacteristic, direct product factor, Sylow
361C2—41no—
362C2—81no—
363C2—81no—
364C2C3×#53511yescharacteristic
365C2C3×#57711yescharacteristic
366C2C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#542C2semidirect
2. C3×#536C2semidirect
3. C3×#497C2semidirect
4. C3×#495C2semidirect
5. C3×#489C2semidirect
6. C3×#487C2semidirect
7. C3×#464C2semidirect
8. C3×#463C2semidirect
9. C3×#462C2semidirect
10. C3×#461C2semidirect
11. C3×#452C2semidirect
12. C3×#451C2semidirect
14. C3×#448C2semidirect
15. C3×#433C2semidirect
16. #3225C3direct (N × H)
60. #542C2×C3semidirect
61. #536C2×C3semidirect
62. #497C2×C3semidirect
63. #495C2×C3semidirect
64. #489C2×C3semidirect
65. #487C2×C3semidirect
66. #464C2×C3semidirect
67. #463C2×C3semidirect
68. #462C2×C3semidirect
69. #461C2×C3semidirect
70. #452C2×C3semidirect
71. #451C2×C3semidirect
73. #448C2×C3semidirect
74. #433C2×C3semidirect
360. C3#3225direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5422
C3×#5362
C3×#4972
C3×#4952
C3×#4892
C3×#4872
C3×#4642
C3×#4632
C3×#4622
C3×#4612
C3×#4522
C3×#4512
C3×#4492
C3×#4482
C3×#4332
#32253