Finite Group Databasebeta

← C3×#3337 #85201

Subgroups — C3×#3337 #85201

Order 384.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#553C211yesmaximal, characteristic
2C3×#552C211yesmaximal, characteristic, complemented
3C3×#499C211yesmaximal, characteristic, complemented
4C3×#485C211yesmaximal, characteristic, complemented
5C3×#484C211yesmaximal, characteristic, complemented
6C3×#482C211yesmaximal, characteristic, complemented
7C3×#480C211yesmaximal, characteristic, complemented
8C3×#479C211yesmaximal, characteristic, complemented
9C3×#478C211yesmaximal, characteristic, complemented
10C3×#465C211yesmaximal, characteristic, complemented
11C3×#465C211yesmaximal, characteristic, complemented
12C3×C4×SD16C211yesmaximal, characteristic, complemented
13C3×C8×D4C211yesmaximal, characteristic, complemented
14C3×#427C211yesmaximal, characteristic, complemented
15C2×C3×#106C211yesmaximal, characteristic, complemented
16#3337C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×SD16C2211yescharacteristic
18C22×C3×C8C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×#126—21no—
21C3×#126—21no—
22C3×#125—21no—
23C3×#125C2211yescharacteristic
24C3×#125C2211yescharacteristic
25C3×#123—21no—
26C3×#123—21no—
27C3×C22⋊Q8C2211yescharacteristic
28C3×C22⋊Q8C2211yescharacteristic
29C3×#121—21no—
30C3×#121C2211yescharacteristic
31C3×#121C2211yescharacteristic
32C3×C4×Q8C2211yescharacteristic
33C3×C4×D4—21no—
34C3×C4×D4C2211yescharacteristic
35C3×C4×D4C2211yescharacteristic
36C3×#117—21no—
37C3×#117C2211yescharacteristic
38C3×#117C2211yescharacteristic
39C2×C3×C4⋊C4C2211yescharacteristic
40C2×C3×C4⋊C4C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×#106C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×#106C2211yescharacteristic
49C3×#106C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×C4×C8C2211yescharacteristic
60#553C2×C311yescharacteristic
61#552C2×C311yescharacteristic, complemented
62#499C2×C311yescharacteristic, complemented
63#485C2×C311yescharacteristic, complemented
64#484C2×C311yescharacteristic, complemented
65#482C2×C311yescharacteristic, complemented
66#480C2×C311yescharacteristic, complemented
67#479C2×C311yescharacteristic, complemented
68#478C2×C311yescharacteristic, complemented
69#465C2×C311yescharacteristic, complemented
70#465C2×C311yescharacteristic, complemented
71C4×SD16C2×C311yescharacteristic, complemented
72C8×D4C2×C311yescharacteristic, complemented
73#427C2×C311yescharacteristic, complemented
74C2×#106C2×C311yescharacteristic, complemented
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4D411yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C3×SD16—22no—
86C2×C3×C8—22no—
87C2×C3×C8—22no—
88C2×C3×C8C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4D411yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115C3×C22⋊C4D411yescharacteristic
116C3×C22⋊C4D411yescharacteristic
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42C2311yescharacteristic
121C2×SD16C22×C311yescharacteristic
122C22×C8C22×C311yescharacteristic
123C22×C8C22×C311yescharacteristic
124#126—21no—
125#126—21no—
126#125—21no—
127#125C22×C311yescharacteristic
128#125C22×C311yescharacteristic
129#123—21no—
130#123—21no—
131C22⋊Q8C22×C311yescharacteristic
132C22⋊Q8C22×C311yescharacteristic
133#121—21no—
134#121C22×C311yescharacteristic
135#121C22×C311yescharacteristic
136C4×Q8C22×C311yescharacteristic
137C4×D4—21no—
138C4×D4C22×C311yescharacteristic
139C4×D4C22×C311yescharacteristic
140#117—21no—
141#117C22×C311yescharacteristic
142#117C22×C311yescharacteristic
143C2×C4⋊C4C22×C311yescharacteristic
144C2×C4⋊C4C22×C311yescharacteristic
145#107C22×C311yescharacteristic
146#107C22×C311yescharacteristic
147#107C22×C311yescharacteristic
148#107C22×C311yescharacteristic
149#106C22×C311yescharacteristic
150#106C22×C311yescharacteristic
151#106C22×C311yescharacteristic
152#106C22×C311yescharacteristic
153#106C22×C311yescharacteristic
154C4⋊C8C22×C311yescharacteristic
155#103C22×C311yescharacteristic
156#103C22×C311yescharacteristic
157#103C22×C311yescharacteristic
158#102C22×C311yescharacteristic
159#102C22×C311yescharacteristic
160#102C22×C311yescharacteristic
161C22⋊C8C22×C311yescharacteristic
162C22⋊C8C22×C311yescharacteristic
163C4×C8C22×C311yescharacteristic
164C23×C3—21no—
165C23×C3C2×D411yescharacteristic
166C23×C3C2×D411yescharacteristic
167C3×Q8—21no—
168C3×Q8—21no—
169C3×D4—22no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—42no—
173C3×D4—81no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193C3×C8—21no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8#4112yes—
197C2×Q8C23×C311yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4C3×D411yescharacteristic
201C2×D4C23×C311yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4C23×C311yescharacteristic
206C22×C4C23×C311yescharacteristic
207SD16—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8C23×C311yescharacteristic
211C2×C8C23×C311yescharacteristic
212C2×C8C23×C311yescharacteristic
213C2×C8C23×C311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4C3×D411yescharacteristic
224C4⋊C4C23×C311yescharacteristic
225C4⋊C4C23×C311yescharacteristic
226C4⋊C4C23×C311yescharacteristic
227C4⋊C4C23×C311yescharacteristic
228C4⋊C4C23×C311yescharacteristic
229C4⋊C4C23×C311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4C3×D411yescharacteristic
238C22⋊C4C3×D411yescharacteristic
239C42—21no—
240C42—21no—
241C42—21no—
242C42C23×C311yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3#13512yes—
252C22×C3C22×D411yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4C2×#4111yescharacteristic
265C3×C42−1+411yescharacteristic
266C23—21no—
267C23C2×C3×D411yescharacteristic
268C23C2×C3×D411yescharacteristic
269Q8—21no—
270Q8—21no—
271D4—22no—
272D4—21no—
273D4—21no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4C2×C3×D411yescharacteristic
291C2×C4C2×C3×D411yescharacteristic
292C2×C4C2×C3×D411yescharacteristic
293C2×C4C2×C3×D411yescharacteristic
294C2×C4C24×C311yescharacteristic
295C8—21no—
296C8—21no—
297C8—21no—
298C8C3×#4112yes—
299C2×C3—22no—
300C2×C3—41no—
301C2×C3—81no—
302C2×C3#54711yescharacteristic
303C2×C3C2×#13511yescharacteristic
304C2×C3#57711yescharacteristic
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C3×#13512yes—
314C22C22×C3×D411yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4C2×C3×#4111yescharacteristic
327C4C3×2−1+411yescharacteristic
328C3#333711yescharacteristic, direct product factor, Sylow
329C2—22no—
330C2—41no—
331C2—81no—
332C2C3×#54711yescharacteristic
333C2C2×C3×#13511yescharacteristic
334C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#553C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#552C2semidirect
3. C3×#499C2semidirect
4. C3×#485C2semidirect
5. C3×#484C2semidirect
6. C3×#482C2semidirect
7. C3×#480C2semidirect
8. C3×#479C2semidirect
9. C3×#478C2semidirect
10. C3×#465C2semidirect
11. C3×#465C2semidirect
12. C3×C4×SD16C2semidirect
13. C3×C8×D4C2semidirect
14. C3×#427C2semidirect
15. C2×C3×#106C2semidirect
16. #3337C3direct (N × H)
61. #552C2×C3semidirect
62. #499C2×C3semidirect
63. #485C2×C3semidirect
64. #484C2×C3semidirect
65. #482C2×C3semidirect
66. #480C2×C3semidirect
67. #479C2×C3semidirect
68. #478C2×C3semidirect
69. #465C2×C3semidirect
70. #465C2×C3semidirect
71. C4×SD16C2×C3semidirect
72. C8×D4C2×C3semidirect
73. #427C2×C3semidirect
74. C2×#106C2×C3semidirect
328. C3#3337direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5532
C3×#5522
C3×#4992
C3×#4852
C3×#4842
C3×#4822
C3×#4802
C3×#4792
C3×#4782
C3×#4652
C3×C4×SD162
C3×C8×D42
C3×#4272
C2×C3×#1062
#33373