Finite Group Databasebeta

← C3×#3370 #85234

Subgroups — C3×#3370 #85234

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#559C211yesmaximal, characteristic, complemented
2C3×#554C211yesmaximal, characteristic, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4C3×#489C211yesmaximal, characteristic, complemented
5C3×#488C211yesmaximal, characteristic, complemented
6C3×#479C211yesmaximal, characteristic, complemented
7C3×#478C211yesmaximal, characteristic, complemented
8C3×#474C211yesmaximal, characteristic, complemented
9C3×#463C211yesmaximal, characteristic, complemented
10C3×#463C211yesmaximal, characteristic, complemented
11C3×#459C211yesmaximal, characteristic
12C3×#446C211yesmaximal, characteristic, complemented
13C3×#442C211yesmaximal, characteristic, complemented
14C3×#440C211yesmaximal, characteristic, complemented
15C3×C4×D8C211yesmaximal, characteristic, complemented
16#3370C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic
21C3×#127C2211yescharacteristic
22C3×#125C2211yescharacteristic
23C3×#125C2211yescharacteristic
24C3×#124—21no—
25C3×#124C2211yescharacteristic
26C3×#124C2211yescharacteristic
27C3×#123—21no—
28C3×#123—21no—
29C3×C22⋊Q8—21no—
30C3×C22⋊Q8—21no—
31C3×C22⋊Q8—21no—
32C3×C4×Q8C2211yescharacteristic
33C3×C4×Q8C2211yescharacteristic
34C3×C4×D4—21no—
35C3×C4×D4C2211yescharacteristic
36C3×C4×D4C2211yescharacteristic
37C3×C4×D4C2211yescharacteristic
38C3×C4×D4C2211yescharacteristic
39C3×C4×D4C2211yescharacteristic
40C2×C3×C4⋊C4—21no—
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×C4⋊C8C2211yescharacteristic
45C3×C4⋊C8C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#102C2211yescharacteristic
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×#97C2211yescharacteristic
58C3×#97C2211yescharacteristic
59C3×C4×C8C2211yescharacteristic
60#559C2×C311yescharacteristic, complemented
61#554C2×C311yescharacteristic, complemented
62#490C2×C311yescharacteristic, complemented
63#489C2×C311yescharacteristic, complemented
64#488C2×C311yescharacteristic, complemented
65#479C2×C311yescharacteristic, complemented
66#478C2×C311yescharacteristic, complemented
67#474C2×C311yescharacteristic, complemented
68#463C2×C311yescharacteristic, complemented
69#463C2×C311yescharacteristic, complemented
70#459C2×C311yescharacteristic
71#446C2×C311yescharacteristic, complemented
72#442C2×C311yescharacteristic, complemented
73#440C2×C311yescharacteristic, complemented
74C4×D8C2×C311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8D411yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C3×SD16—22no—
89C3×D8—22no—
90C3×D8—22no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
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⋊C4D411yescharacteristic
103C3×C4⋊C4D411yescharacteristic
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⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C42—21no—
118C3×C42C2311yescharacteristic
119C3×C42C2311yescharacteristic
120C3×C42C2311yescharacteristic
121C2×SD16C22×C311yescharacteristic
122C2×D8C22×C311yescharacteristic
123C2×D8C22×C311yescharacteristic
124C4⋊Q8C22×C311yescharacteristic
125#127C22×C311yescharacteristic
126#125C22×C311yescharacteristic
127#125C22×C311yescharacteristic
128#124—21no—
129#124C22×C311yescharacteristic
130#124C22×C311yescharacteristic
131#123—21no—
132#123—21no—
133C22⋊Q8—21no—
134C22⋊Q8—21no—
135C22⋊Q8—21no—
136C4×Q8C22×C311yescharacteristic
137C4×Q8C22×C311yescharacteristic
138C4×D4—21no—
139C4×D4C22×C311yescharacteristic
140C4×D4C22×C311yescharacteristic
141C4×D4C22×C311yescharacteristic
142C4×D4C22×C311yescharacteristic
143C4×D4C22×C311yescharacteristic
144C2×C4⋊C4—21no—
145#107C22×C311yescharacteristic
146#107C22×C311yescharacteristic
147#106C22×C311yescharacteristic
148C4⋊C8C22×C311yescharacteristic
149C4⋊C8C22×C311yescharacteristic
150C4⋊C8C22×C311yescharacteristic
151#103C22×C311yescharacteristic
152#103C22×C311yescharacteristic
153#103C22×C311yescharacteristic
154#102C22×C311yescharacteristic
155#102C22×C311yescharacteristic
156#102C22×C311yescharacteristic
157#102C22×C311yescharacteristic
158#102C22×C311yescharacteristic
159#102C22×C311yescharacteristic
160#102C22×C311yescharacteristic
161#97C22×C311yescharacteristic
162#97C22×C311yescharacteristic
163C4×C8C22×C311yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—41no—
170C3×Q8—81no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—42no—
177C3×D4—81no—
178C3×D4#4112yes—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—42no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196C3×C8—21no—
197C3×C8—21no—
198C3×C8—21no—
199C3×C8—21no—
200C2×Q8—21no—
201C2×Q8C3×D411yescharacteristic
202C2×Q8C23×C311yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4C23×C311yescharacteristic
206C2×D4C23×C311yescharacteristic
207C2×D4C23×C311yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213SD16—22no—
214D8—22no—
215D8—22no—
216C2×C8C23×C311yescharacteristic
217C2×C8C23×C311yescharacteristic
218C2×C8C23×C311yescharacteristic
219C2×C8C23×C311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4C3×D411yescharacteristic
227C4⋊C4C3×D411yescharacteristic
228C4⋊C4C3×D411yescharacteristic
229C4⋊C4C23×C311yescharacteristic
230C4⋊C4C23×C311yescharacteristic
231C4⋊C4C23×C311yescharacteristic
232C4⋊C4C23×C311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—21no—
242C42—21no—
243C42C23×C311yescharacteristic
244C42C23×C311yescharacteristic
245C42C23×C311yescharacteristic
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3C22×D411yescharacteristic
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4C2×#4111yescharacteristic
268C3×C42−1+411yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—41no—
275Q8—81no—
276D4—21no—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283D4C3×#4112yes—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4C2×C3×D411yescharacteristic
295C2×C4C2×C3×D411yescharacteristic
296C2×C4C2×C3×D411yescharacteristic
297C2×C4C2×C3×D411yescharacteristic
298C2×C4C2×C3×D411yescharacteristic
299C2×C4C2×C3×D411yescharacteristic
300C2×C4C24×C311yescharacteristic
301C8—21no—
302C8—21no—
303C8—21no—
304C8—21no—
305C2×C3—42no—
306C2×C3—81no—
307C2×C3—81no—
308C2×C3#54811yescharacteristic
309C2×C3#57511yescharacteristic
310C2×C3#57611yescharacteristic
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C22×C3×D411yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4C2×C3×#4111yescharacteristic
333C4C3×2−1+411yescharacteristic
334C3#337011yescharacteristic, direct product factor, Sylow
335C2—42no—
336C2—81no—
337C2—81no—
338C2C3×#54811yescharacteristic
339C2C3×#57511yescharacteristic
340C2C3×#57611yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#459C2
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×#554C2semidirect
3. C3×#490C2semidirect
4. C3×#489C2semidirect
5. C3×#488C2semidirect
6. C3×#479C2semidirect
7. C3×#478C2semidirect
8. C3×#474C2semidirect
9. C3×#463C2semidirect
10. C3×#463C2semidirect
12. C3×#446C2semidirect
13. C3×#442C2semidirect
14. C3×#440C2semidirect
15. C3×C4×D8C2semidirect
16. #3370C3direct (N × H)
60. #559C2×C3semidirect
61. #554C2×C3semidirect
62. #490C2×C3semidirect
63. #489C2×C3semidirect
64. #488C2×C3semidirect
65. #479C2×C3semidirect
66. #478C2×C3semidirect
67. #474C2×C3semidirect
68. #463C2×C3semidirect
69. #463C2×C3semidirect
71. #446C2×C3semidirect
72. #442C2×C3semidirect
73. #440C2×C3semidirect
74. C4×D8C2×C3semidirect
334. C3#3370direct (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×#5542
C3×#4902
C3×#4892
C3×#4882
C3×#4792
C3×#4782
C3×#4742
C3×#4632
C3×#4592
C3×#4462
C3×#4422
C3×#4402
C3×C4×D82
#33703