Finite Group Databasebeta

← C3×#3201 #85065

Subgroups — C3×#3201 #85065

Order 384.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#537C211yesmaximal, characteristic
2C3×#536C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4C3×#484C211yesmaximal, characteristic, complemented
5C3×#483C211yesmaximal, characteristic, complemented
6C3×#482C211yesmaximal, characteristic, complemented
7C3×#470C211yesmaximal, characteristic, complemented
8C3×#468C211yesmaximal, characteristic, complemented
9C3×#467C211yesmaximal, characteristic, complemented
10C3×#465C211yesmaximal, characteristic, complemented
11C3×#452C211yesmaximal, characteristic, complemented
12C3×#452C211yesmaximal, characteristic, complemented
13C3×#451C211yesmaximal, characteristic, complemented
14C3×#448C211yesmaximal, characteristic, complemented
15C3×#408C211yesmaximal, characteristic, complemented
16#3201C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C22×C3×Q8C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic
25C22×C3×C8C2211yescharacteristic
26C3×C4⋊Q8—21no—
27C3×#124—21no—
28C3×#123—21no—
29C3×#123—21no—
30C3×C22⋊Q8—21no—
31C3×C22⋊Q8C2211yescharacteristic
32C3×C22⋊Q8C2211yescharacteristic
33C3×C22⋊Q8C2211yescharacteristic
34C3×C22⋊Q8C2211yescharacteristic
35C3×C22⋊Q8C2211yescharacteristic
36C3×C22⋊Q8C2211yescharacteristic
37C3×#121—21no—
38C3×#121—21no—
39C3×#121—21no—
40C3×#121C2211yescharacteristic
41C3×#121C2211yescharacteristic
42C3×#117C2211yescharacteristic
43C2×C3×C4⋊C4C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#107C2211yescharacteristic
46C3×#106C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×C22⋊C8C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×C22⋊C8C2211yescharacteristic
60#537C2×C311yescharacteristic
61#536C2×C311yescharacteristic, complemented
62#485C2×C311yescharacteristic, complemented
63#484C2×C311yescharacteristic, complemented
64#483C2×C311yescharacteristic, complemented
65#482C2×C311yescharacteristic, complemented
66#470C2×C311yescharacteristic, complemented
67#468C2×C311yescharacteristic, complemented
68#467C2×C311yescharacteristic, complemented
69#465C2×C311yescharacteristic, complemented
70#452C2×C311yescharacteristic, complemented
71#452C2×C311yescharacteristic, complemented
72#451C2×C311yescharacteristic, complemented
73#448C2×C311yescharacteristic, complemented
74#408C2×C311yescharacteristic, complemented
75C3×#41—41no—
76C3×#41—41no—
77C3×#41—41no—
78C3×#41—41no—
79C2×C3×Q8—21no—
80C2×C3×Q8—41no—
81C2×C3×Q8—41no—
82C2×C3×Q8D411yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4—21no—
87C2×C3×D4—21no—
88C2×C3×D4—21no—
89C2×C3×D4D411yescharacteristic
90C2×C3×D4D411yescharacteristic
91C2×C3×D4D411yescharacteristic
92C2×C3×D4C2311yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C3×Q16—41no—
100C3×Q16—41no—
101C3×SD16—41no—
102C3×SD16—41no—
103C3×M16—41no—
104C2×C3×C8—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4C2311yescharacteristic
117C3×C4⋊C4C2311yescharacteristic
118C3×C4⋊C4C2311yescharacteristic
119C3×C4⋊C4C2311yescharacteristic
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—21no—
126C3×C22⋊C4—21no—
127C3×C22⋊C4—21no—
128C3×C22⋊C4—21no—
129C3×C42—21no—
130C2×#41C22×C311yescharacteristic
131C2×#41C22×C311yescharacteristic
132C22×Q8C22×C311yescharacteristic
133C2×Q16C22×C311yescharacteristic
134C2×Q16C22×C311yescharacteristic
135C2×SD16C22×C311yescharacteristic
136C2×SD16C22×C311yescharacteristic
137C2×M16C22×C311yescharacteristic
138C22×C8C22×C311yescharacteristic
139C4⋊Q8—21no—
140#124—21no—
141#123—21no—
142#123—21no—
143C22⋊Q8—21no—
144C22⋊Q8C22×C311yescharacteristic
145C22⋊Q8C22×C311yescharacteristic
146C22⋊Q8C22×C311yescharacteristic
147C22⋊Q8C22×C311yescharacteristic
148C22⋊Q8C22×C311yescharacteristic
149C22⋊Q8C22×C311yescharacteristic
150#121—21no—
151#121—21no—
152#121—21no—
153#121C22×C311yescharacteristic
154#121C22×C311yescharacteristic
155#117C22×C311yescharacteristic
156C2×C4⋊C4C22×C311yescharacteristic
157#107C22×C311yescharacteristic
158#107C22×C311yescharacteristic
159#106C22×C311yescharacteristic
160#106C22×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
171C22⋊C8C22×C311yescharacteristic
172C22⋊C8C22×C311yescharacteristic
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C23×C3C2×D411yescharacteristic
176C23×C3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—41no—
184C3×Q8—42no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—41no—
190C3×D4—42no—
191C3×D4—81no—
192C3×D4—81no—
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—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2411yescharacteristic
213C3×C8—21no—
214C3×C8—21no—
215C3×C8—21no—
216C3×C8—21no—
217#41—41no—
218#41—41no—
219#41—41no—
220#41—41no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8C3×D411yescharacteristic
225C2×Q8C23×C311yescharacteristic
226C2×Q8C23×C311yescharacteristic
227C2×Q8C23×C311yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4C3×D411yescharacteristic
232C2×D4C3×D411yescharacteristic
233C2×D4C3×D411yescharacteristic
234C2×D4C23×C311yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4C23×C311yescharacteristic
239C22×C4C23×C311yescharacteristic
240C22×C4C23×C311yescharacteristic
241Q16—41no—
242Q16—41no—
243SD16—41no—
244SD16—41no—
245M16—41no—
246C2×C8—41no—
247C2×C8C23×C311yescharacteristic
248C2×C8C23×C311yescharacteristic
249C2×C8C23×C311yescharacteristic
250C2×C8C23×C311yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4C23×C311yescharacteristic
259C4⋊C4C23×C311yescharacteristic
260C4⋊C4C23×C311yescharacteristic
261C4⋊C4C23×C311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C42—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297C23—21no—
298C23C2×C3×D411yescharacteristic
299C23C2×C3×D411yescharacteristic
300C23C2×C3×D411yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—41no—
308Q8—42no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—42no—
315D4—81no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4C2×C3×D411yescharacteristic
334C2×C4C2×C3×D411yescharacteristic
335C2×C4C2×C3×D411yescharacteristic
336C2×C4C24×C311yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—81no—
345C2×C3#53411yescharacteristic
346C2×C3#57711yescharacteristic
347C2×C3#57811yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22C22×C3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4C3×H(5,𝔽2)11yescharacteristic
372C4C3×H(5,𝔽2)11yescharacteristic
373C3#320111yescharacteristic, direct product factor, Sylow
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2C3×#53411yescharacteristic
379C2C3×#57711yescharacteristic
380C2C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#536C2semidirect
3. C3×#485C2semidirect
4. C3×#484C2semidirect
5. C3×#483C2semidirect
6. C3×#482C2semidirect
7. C3×#470C2semidirect
8. C3×#468C2semidirect
9. C3×#467C2semidirect
10. C3×#465C2semidirect
11. C3×#452C2semidirect
12. C3×#452C2semidirect
13. C3×#451C2semidirect
14. C3×#448C2semidirect
15. C3×#408C2semidirect
16. #3201C3direct (N × H)
61. #536C2×C3semidirect
62. #485C2×C3semidirect
63. #484C2×C3semidirect
64. #483C2×C3semidirect
65. #482C2×C3semidirect
66. #470C2×C3semidirect
67. #468C2×C3semidirect
68. #467C2×C3semidirect
69. #465C2×C3semidirect
70. #452C2×C3semidirect
71. #452C2×C3semidirect
72. #451C2×C3semidirect
73. #448C2×C3semidirect
74. #408C2×C3semidirect
373. C3#3201direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5372
C3×#5362
C3×#4852
C3×#4842
C3×#4832
C3×#4822
C3×#4702
C3×#4682
C3×#4672
C3×#4652
C3×#4522
C3×#4512
C3×#4482
C3×#4082
#32013