Finite Group Databasebeta

← C3×#3401 #85265

Subgroups — C3×#3401 #85265

Order 384.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#559C211yesmaximal, characteristic, complemented
2C3×#556C211yesmaximal, characteristic, complemented
3C3×#497C211yesmaximal, characteristic, complemented
4C3×#496C211yesmaximal, characteristic
5C3×#495C211yesmaximal, characteristic, complemented
6C3×#464C211yesmaximal, characteristic, complemented
7C3×#464C211yesmaximal, characteristic, complemented
8C3×#463C211yesmaximal, characteristic, complemented
9C3×#463C211yesmaximal, characteristic, complemented
10C3×#461C211yesmaximal, characteristic, complemented
11C3×#460C211yesmaximal, characteristic, complemented
12C3×#446C211yesmaximal, characteristic, complemented
13C3×#442C211yesmaximal, characteristic, complemented
14C3×#441C211yesmaximal, characteristic, complemented
15C3×C4×SD16C211yesmaximal, characteristic, complemented
16#3401C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic
24C2×C3×D8C2211yescharacteristic
25C2×C3×D8C2211yescharacteristic
26C3×C4⋊Q8C2211yescharacteristic
27C3×#127C2211yescharacteristic
28C3×#126—21no—
29C3×#126—21no—
30C3×#124—21no—
31C3×#124C2211yescharacteristic
32C3×#124C2211yescharacteristic
33C3×#124C2211yescharacteristic
34C3×#124C2211yescharacteristic
35C3×#123—21no—
36C3×#123—21no—
37C3×C22⋊Q8—21no—
38C3×C4×Q8C2211yescharacteristic
39C3×C4×Q8C2211yescharacteristic
40C3×C4×Q8C2211yescharacteristic
41C3×C4×Q8C2211yescharacteristic
42C3×C4×D4—21no—
43C3×C4×D4C2211yescharacteristic
44C3×C4×D4C2211yescharacteristic
45C3×C4×D4C2211yescharacteristic
46C3×#117—21no—
47C3×#106C2211yescharacteristic
48C3×C4⋊C8C2211yescharacteristic
49C3×C4⋊C8C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×#97C2211yescharacteristic
58C3×#97C2211yescharacteristic
59C3×C4×C8C2211yescharacteristic
60#559C2×C311yescharacteristic, complemented
61#556C2×C311yescharacteristic, complemented
62#497C2×C311yescharacteristic, complemented
63#496C2×C311yescharacteristic
64#495C2×C311yescharacteristic, complemented
65#464C2×C311yescharacteristic, complemented
66#464C2×C311yescharacteristic, complemented
67#463C2×C311yescharacteristic, complemented
68#463C2×C311yescharacteristic, complemented
69#461C2×C311yescharacteristic, complemented
70#460C2×C311yescharacteristic, complemented
71#446C2×C311yescharacteristic, complemented
72#442C2×C311yescharacteristic, complemented
73#441C2×C311yescharacteristic, complemented
74C4×SD16C2×C311yescharacteristic, complemented
75C2×C3×Q8D411yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C3×Q16—22no—
88C3×Q16—41no—
89C3×SD16—22no—
90C3×SD16—41no—
91C3×SD16—41no—
92C3×SD16—41no—
93C3×SD16—41no—
94C3×D8—22no—
95C3×D8—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4D411yescharacteristic
106C3×C4⋊C4D411yescharacteristic
107C3×C4⋊C4D411yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120C3×C22⋊C4—21no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42C2311yescharacteristic
125C3×C42C2311yescharacteristic
126C3×C42C2311yescharacteristic
127C2×Q16C22×C311yescharacteristic
128C2×Q16C22×C311yescharacteristic
129C2×SD16C22×C311yescharacteristic
130C2×SD16C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic
132C2×SD16C22×C311yescharacteristic
133C2×SD16C22×C311yescharacteristic
134C2×D8C22×C311yescharacteristic
135C2×D8C22×C311yescharacteristic
136C4⋊Q8C22×C311yescharacteristic
137#127C22×C311yescharacteristic
138#126—21no—
139#126—21no—
140#124—21no—
141#124C22×C311yescharacteristic
142#124C22×C311yescharacteristic
143#124C22×C311yescharacteristic
144#124C22×C311yescharacteristic
145#123—21no—
146#123—21no—
147C22⋊Q8—21no—
148C4×Q8C22×C311yescharacteristic
149C4×Q8C22×C311yescharacteristic
150C4×Q8C22×C311yescharacteristic
151C4×Q8C22×C311yescharacteristic
152C4×D4—21no—
153C4×D4C22×C311yescharacteristic
154C4×D4C22×C311yescharacteristic
155C4×D4C22×C311yescharacteristic
156#117—21no—
157#106C22×C311yescharacteristic
158C4⋊C8C22×C311yescharacteristic
159C4⋊C8C22×C311yescharacteristic
160C4⋊C8C22×C311yescharacteristic
161#103C22×C311yescharacteristic
162#103C22×C311yescharacteristic
163#103C22×C311yescharacteristic
164#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167#97C22×C311yescharacteristic
168#97C22×C311yescharacteristic
169C4×C8C22×C311yescharacteristic
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—41no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C3×D4—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2411yescharacteristic
203C3×C8—21no—
204C3×C8—21no—
205C3×C8—21no—
206C3×C8#4112yes—
207C2×Q8C3×D411yescharacteristic
208C2×Q8C23×C311yescharacteristic
209C2×Q8C23×C311yescharacteristic
210C2×Q8C23×C311yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4C23×C311yescharacteristic
214C2×D4C23×C311yescharacteristic
215C2×D4C23×C311yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219Q16—22no—
220Q16—41no—
221SD16—22no—
222SD16—41no—
223SD16—41no—
224SD16—41no—
225SD16—41no—
226D8—22no—
227D8—41no—
228C2×C8C23×C311yescharacteristic
229C2×C8C23×C311yescharacteristic
230C2×C8C23×C311yescharacteristic
231C2×C8C23×C311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4C3×D411yescharacteristic
238C4⋊C4C3×D411yescharacteristic
239C4⋊C4C3×D411yescharacteristic
240C4⋊C4C23×C311yescharacteristic
241C4⋊C4C23×C311yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42C23×C311yescharacteristic
257C42C23×C311yescharacteristic
258C42C23×C311yescharacteristic
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C22×D411yescharacteristic
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282C23—21no—
283C23—21no—
284C23—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—41no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—81no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4C2×C3×D411yescharacteristic
309C2×C4C2×C3×D411yescharacteristic
310C2×C4C2×C3×D411yescharacteristic
311C2×C4C2×C3×D411yescharacteristic
312C2×C4C2×C3×D411yescharacteristic
313C2×C4C2×C3×D411yescharacteristic
314C2×C4C24×C311yescharacteristic
315C8—21no—
316C8—21no—
317C8—21no—
318C8C3×#4112yes—
319C2×C3—81no—
320C2×C3—81no—
321C2×C3—81no—
322C2×C3#55011yescharacteristic
323C2×C3#57511yescharacteristic
324C2×C3#57711yescharacteristic
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C22×C3×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4C2×C3×#4111yescharacteristic
347C4C3×H(5,𝔽2)11yescharacteristic
348C3#340111yescharacteristic, direct product factor, Sylow
349C2—81no—
350C2—81no—
351C2—81no—
352C2C3×#55011yescharacteristic
353C2C3×#57511yescharacteristic
354C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#496C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#559C2semidirect
2. C3×#556C2semidirect
3. C3×#497C2semidirect
5. C3×#495C2semidirect
6. C3×#464C2semidirect
7. C3×#464C2semidirect
8. C3×#463C2semidirect
9. C3×#463C2semidirect
10. C3×#461C2semidirect
11. C3×#460C2semidirect
12. C3×#446C2semidirect
13. C3×#442C2semidirect
14. C3×#441C2semidirect
15. C3×C4×SD16C2semidirect
16. #3401C3direct (N × H)
60. #559C2×C3semidirect
61. #556C2×C3semidirect
62. #497C2×C3semidirect
64. #495C2×C3semidirect
65. #464C2×C3semidirect
66. #464C2×C3semidirect
67. #463C2×C3semidirect
68. #463C2×C3semidirect
69. #461C2×C3semidirect
70. #460C2×C3semidirect
71. #446C2×C3semidirect
72. #442C2×C3semidirect
73. #441C2×C3semidirect
74. C4×SD16C2×C3semidirect
348. C3#3401direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5592
C3×#5562
C3×#4972
C3×#4962
C3×#4952
C3×#4642
C3×#4632
C3×#4612
C3×#4602
C3×#4462
C3×#4422
C3×#4412
C3×C4×SD162
#34013