Finite Group Databasebeta

← C3×#3317 #85181

Subgroups — C3×#3317 #85181

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#552C211yesmaximal, characteristic, complemented
2C3×#550C211yesmaximal, characteristic
3C3×#490C211yesmaximal, characteristic, complemented
4C3×#482C211yesmaximal, characteristic, complemented
5C3×#481C211yesmaximal, characteristic, complemented
6C3×#479C211yesmaximal, characteristic, complemented
7C3×#468C211yesmaximal, characteristic, complemented
8C3×#465C211yesmaximal, characteristic, complemented
9C3×#460C211yesmaximal, characteristic, complemented
10C3×#452C211yesmaximal, characteristic, complemented
11C3×#452C211yesmaximal, characteristic, complemented
12C3×#441C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#417C211yesmaximal, characteristic, complemented
15C3×#416C211yesmaximal, characteristic, complemented
16#3317C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic
23C22×C3×C8C2211yescharacteristic
24C3×#127—21no—
25C3×#127C2211yescharacteristic
26C3×#126—21no—
27C3×#125—21no—
28C3×#125C2211yescharacteristic
29C3×#123—21no—
30C3×C22⋊Q8C2211yescharacteristic
31C3×C22⋊Q8C2211yescharacteristic
32C3×#121—21no—
33C3×#121—21no—
34C3×#121C2211yescharacteristic
35C3×#121C2211yescharacteristic
36C3×C4×Q8C2211yescharacteristic
37C3×C4×Q8C2211yescharacteristic
38C3×C4×D4—21no—
39C3×C4×D4C2211yescharacteristic
40C3×#117—21no—
41C3×#117C2211yescharacteristic
42C2×C3×C4⋊C4C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×#97C2211yescharacteristic
60#552C2×C311yescharacteristic, complemented
61#550C2×C311yescharacteristic
62#490C2×C311yescharacteristic, complemented
63#482C2×C311yescharacteristic, complemented
64#481C2×C311yescharacteristic, complemented
65#479C2×C311yescharacteristic, complemented
66#468C2×C311yescharacteristic, complemented
67#465C2×C311yescharacteristic, complemented
68#460C2×C311yescharacteristic, complemented
69#452C2×C311yescharacteristic, complemented
70#452C2×C311yescharacteristic, complemented
71#441C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#417C2×C311yescharacteristic, complemented
74#416C2×C311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—22no—
77C3×#41—41no—
78C3×#41—41no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4D411yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C3×Q16—22no—
96C3×SD16—41no—
97C3×SD16—41no—
98C3×M16—41no—
99C2×C3×C8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
122C3×C22⋊C4D411yescharacteristic
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42C2311yescharacteristic
127C2×#41C22×C311yescharacteristic
128C2×#41C22×C311yescharacteristic
129C2×Q16C22×C311yescharacteristic
130C2×SD16C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic
132C2×M16C22×C311yescharacteristic
133C22×C8C22×C311yescharacteristic
134#127—21no—
135#127C22×C311yescharacteristic
136#126—21no—
137#125—21no—
138#125C22×C311yescharacteristic
139#123—21no—
140C22⋊Q8C22×C311yescharacteristic
141C22⋊Q8C22×C311yescharacteristic
142#121—21no—
143#121—21no—
144#121C22×C311yescharacteristic
145#121C22×C311yescharacteristic
146C4×Q8C22×C311yescharacteristic
147C4×Q8C22×C311yescharacteristic
148C4×D4—21no—
149C4×D4C22×C311yescharacteristic
150#117—21no—
151#117C22×C311yescharacteristic
152C2×C4⋊C4C22×C311yescharacteristic
153#107C22×C311yescharacteristic
154#106C22×C311yescharacteristic
155#106C22×C311yescharacteristic
156C4⋊C8C22×C311yescharacteristic
157#103C22×C311yescharacteristic
158#103C22×C311yescharacteristic
159#103C22×C311yescharacteristic
160#103C22×C311yescharacteristic
161#102C22×C311yescharacteristic
162#102C22×C311yescharacteristic
163#102C22×C311yescharacteristic
164#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167C22⋊C8C22×C311yescharacteristic
168C22⋊C8C22×C311yescharacteristic
169#97C22×C311yescharacteristic
170C23×C3—21no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C23×C3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8#4112yes—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—41no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—42no—
186C3×D4—81no—
187C3×D4—81no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
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—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207C3×C8—21no—
208C3×C8—21no—
209C3×C8—21no—
210C3×C8—21no—
211#41—22no—
212#41—22no—
213#41—41no—
214#41—41no—
215C2×Q8C23×C311yescharacteristic
216C2×Q8C23×C311yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4C3×D411yescharacteristic
224C2×D4C23×C311yescharacteristic
225C2×D4C23×C311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C23×C311yescharacteristic
230C22×C4C23×C311yescharacteristic
231Q16—22no—
232SD16—41no—
233SD16—41no—
234M16—41no—
235C2×C8—41no—
236C2×C8C23×C311yescharacteristic
237C2×C8C23×C311yescharacteristic
238C2×C8C23×C311yescharacteristic
239C2×C8C23×C311yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4C3×D411yescharacteristic
248C4⋊C4C23×C311yescharacteristic
249C4⋊C4C23×C311yescharacteristic
250C4⋊C4C23×C311yescharacteristic
251C4⋊C4C23×C311yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4C3×D411yescharacteristic
258C22⋊C4C3×D411yescharacteristic
259C42—21no—
260C42—21no—
261C42—21no—
262C42C23×C311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4C2×#4111yescharacteristic
287C3×C4H(5,𝔽2)11yescharacteristic
288C23—21no—
289C23—21no—
290C23C2×C3×D411yescharacteristic
291C23C2×C3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8C3×#4112yes—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—81no—
305D4—81no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4C2×C3×D411yescharacteristic
321C2×C4C2×C3×D411yescharacteristic
322C2×C4C2×C3×D411yescharacteristic
323C2×C4C2×C3×D411yescharacteristic
324C2×C4C24×C311yescharacteristic
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3#54611yescharacteristic
334C2×C3#57511yescharacteristic
335C2×C3#57711yescharacteristic
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22C22×C3×D411yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4C2×C3×#4111yescharacteristic
360C4C3×H(5,𝔽2)11yescharacteristic
361C3#331711yescharacteristic, direct product factor, Sylow
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2C3×#54611yescharacteristic
367C2C3×#57511yescharacteristic
368C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#550C2
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×#552C2semidirect
3. C3×#490C2semidirect
4. C3×#482C2semidirect
5. C3×#481C2semidirect
6. C3×#479C2semidirect
7. C3×#468C2semidirect
8. C3×#465C2semidirect
9. C3×#460C2semidirect
10. C3×#452C2semidirect
11. C3×#452C2semidirect
12. C3×#441C2semidirect
13. C3×#435C2semidirect
14. C3×#417C2semidirect
15. C3×#416C2semidirect
16. #3317C3direct (N × H)
60. #552C2×C3semidirect
62. #490C2×C3semidirect
63. #482C2×C3semidirect
64. #481C2×C3semidirect
65. #479C2×C3semidirect
66. #468C2×C3semidirect
67. #465C2×C3semidirect
68. #460C2×C3semidirect
69. #452C2×C3semidirect
70. #452C2×C3semidirect
71. #441C2×C3semidirect
72. #435C2×C3semidirect
73. #417C2×C3semidirect
74. #416C2×C3semidirect
361. C3#3317direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5522
C3×#5502
C3×#4902
C3×#4822
C3×#4812
C3×#4792
C3×#4682
C3×#4652
C3×#4602
C3×#4522
C3×#4412
C3×#4352
C3×#4172
C3×#4162
#33173