Finite Group Databasebeta

← C3×#3350 #85214

Subgroups — C3×#3350 #85214

Order 384.

348 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×#556C211yesmaximal, characteristic, complemented
2C3×#553C211yesmaximal, characteristic
3C3×#501C211yesmaximal, characteristic, complemented
4C3×#485C211yesmaximal, characteristic, complemented
5C3×#483C211yesmaximal, characteristic, complemented
6C3×#482C211yesmaximal, characteristic, complemented
7C3×#482C211yesmaximal, characteristic, complemented
8C3×#479C211yesmaximal, characteristic, complemented
9C3×#476C211yesmaximal, characteristic, complemented
10C3×#468C211yesmaximal, characteristic, complemented
11C3×#465C211yesmaximal, characteristic, complemented
12C3×#440C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#428C211yesmaximal, characteristic, complemented
15C3×#427C211yesmaximal, characteristic, complemented
16#3350C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic
21C3×#126—21no—
22C3×#126—21no—
23C3×#126—21no—
24C3×#125C2211yescharacteristic
25C3×#124—21no—
26C3×#123—21no—
27C3×C22⋊Q8C2211yescharacteristic
28C3×C22⋊Q8C2211yescharacteristic
29C3×#121—21no—
30C3×#121C2211yescharacteristic
31C3×#121C2211yescharacteristic
32C3×C4×Q8—21no—
33C3×C4×Q8C2211yescharacteristic
34C3×C4×D4—21no—
35C3×C4×D4C2211yescharacteristic
36C3×C4×D4C2211yescharacteristic
37C3×#117C2211yescharacteristic
38C3×#117C2211yescharacteristic
39C3×#117C2211yescharacteristic
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×#97C2211yescharacteristic
60#556C2×C311yescharacteristic, complemented
61#553C2×C311yescharacteristic
62#501C2×C311yescharacteristic, complemented
63#485C2×C311yescharacteristic, complemented
64#483C2×C311yescharacteristic, complemented
65#482C2×C311yescharacteristic, complemented
66#482C2×C311yescharacteristic, complemented
67#479C2×C311yescharacteristic, complemented
68#476C2×C311yescharacteristic, complemented
69#468C2×C311yescharacteristic, complemented
70#465C2×C311yescharacteristic, complemented
71#440C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#428C2×C311yescharacteristic, complemented
74#427C2×C311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4D411yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C3×SD16—22no—
86C3×M16—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⋊C4D411yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
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⋊C4D411yescharacteristic
115C3×C22⋊C4D411yescharacteristic
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42C2311yescharacteristic
121C2×SD16C22×C311yescharacteristic
122C2×M16C22×C311yescharacteristic
123C22×C8C22×C311yescharacteristic
124C4⋊Q8C22×C311yescharacteristic
125#126—21no—
126#126—21no—
127#126—21no—
128#125C22×C311yescharacteristic
129#124—21no—
130#123—21no—
131C22⋊Q8C22×C311yescharacteristic
132C22⋊Q8C22×C311yescharacteristic
133#121—21no—
134#121C22×C311yescharacteristic
135#121C22×C311yescharacteristic
136C4×Q8—21no—
137C4×Q8C22×C311yescharacteristic
138C4×D4—21no—
139C4×D4C22×C311yescharacteristic
140C4×D4C22×C311yescharacteristic
141#117C22×C311yescharacteristic
142#117C22×C311yescharacteristic
143#117C22×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
163#97C22×C311yescharacteristic
164C23×C3—21no—
165C23×C3C2×D411yescharacteristic
166C23×C3C2×D411yescharacteristic
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—42no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—41no—
173C3×D4—42no—
174C3×D4—81no—
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—21no—
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×Q8—21no—
198C2×Q8C23×C311yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4C3×D411yescharacteristic
202C2×D4C23×C311yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4C23×C311yescharacteristic
206C22×C4C23×C311yescharacteristic
207SD16—22no—
208M16—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⋊C4C3×D411yescharacteristic
222C4⋊C4C23×C311yescharacteristic
223C4⋊C4C23×C311yescharacteristic
224C4⋊C4C23×C311yescharacteristic
225C4⋊C4C23×C311yescharacteristic
226C4⋊C4C23×C311yescharacteristic
227C4⋊C4C23×C311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4C3×D411yescharacteristic
237C22⋊C4C3×D411yescharacteristic
238C42—21no—
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—21no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
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—
271Q8—42no—
272D4—21no—
273D4—21no—
274D4—41no—
275D4—42no—
276D4—81no—
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—21no—
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—41no—
300C2×C3—41no—
301C2×C3—81no—
302C2×C3#54711yescharacteristic
303C2×C3#57511yescharacteristic
304C2×C3#57711yescharacteristic
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
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#335011yescharacteristic, direct product factor, Sylow
329C2—41no—
330C2—41no—
331C2—81no—
332C2C3×#54711yescharacteristic
333C2C3×#57511yescharacteristic
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
1. C3×#556C2semidirect
3. C3×#501C2semidirect
4. C3×#485C2semidirect
5. C3×#483C2semidirect
6. C3×#482C2semidirect
7. C3×#482C2semidirect
8. C3×#479C2semidirect
9. C3×#476C2semidirect
10. C3×#468C2semidirect
11. C3×#465C2semidirect
12. C3×#440C2semidirect
13. C3×#435C2semidirect
14. C3×#428C2semidirect
15. C3×#427C2semidirect
16. #3350C3direct (N × H)
60. #556C2×C3semidirect
62. #501C2×C3semidirect
63. #485C2×C3semidirect
64. #483C2×C3semidirect
65. #482C2×C3semidirect
66. #482C2×C3semidirect
67. #479C2×C3semidirect
68. #476C2×C3semidirect
69. #468C2×C3semidirect
70. #465C2×C3semidirect
71. #440C2×C3semidirect
72. #435C2×C3semidirect
73. #428C2×C3semidirect
74. #427C2×C3semidirect
328. C3#3350direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5562
C3×#5532
C3×#5012
C3×#4852
C3×#4832
C3×#4822
C3×#4792
C3×#4762
C3×#4682
C3×#4652
C3×#4402
C3×#4352
C3×#4282
C3×#4272
#33503