Finite Group Databasebeta

← C3×#3345 #85209

Subgroups — C3×#3345 #85209

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#553C211yesmaximal, characteristic, complemented
2C3×#547C211yesmaximal, characteristic, complemented
3C3×#501C211yesmaximal, characteristic, complemented
4C3×#482C211yesmaximal, characteristic, complemented
5C3×#481C211yesmaximal, characteristic, complemented
6C3×#481C211yesmaximal, characteristic, complemented
7C3×#480C211yesmaximal, characteristic, complemented
8C3×#478C211yesmaximal, characteristic, complemented
9C3×#476C211yesmaximal, characteristic, complemented
10C3×#469C211yesmaximal, characteristic, complemented
11C3×#466C211yesmaximal, characteristic, complemented
12C3×#442C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#428C211yesmaximal, characteristic, complemented
15C2×C3×#106C211yesmaximal, characteristic, complemented
16#3345C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41—21no—
18C2×C3×D8C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×#126—21no—
23C3×#125C2211yescharacteristic
24C3×#123—21no—
25C3×#123—21no—
26C3×#123—21no—
27C3×C22⋊Q8—21no—
28C3×#121—21no—
29C3×#121C2211yescharacteristic
30C3×#121C2211yescharacteristic
31C3×#121C2211yescharacteristic
32C3×#121C2211yescharacteristic
33C3×C4×D4—21no—
34C3×C4×D4C2211yescharacteristic
35C3×C4×D4C2211yescharacteristic
36C3×C4×D4C2211yescharacteristic
37C3×#117C2211yescharacteristic
38C2×C3×C4⋊C4C2211yescharacteristic
39C2×C3×C4⋊C4C2211yescharacteristic
40C2×C3×C4⋊C4C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×#106C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×#106C2211yescharacteristic
49C3×#106C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×#97C2211yescharacteristic
60#553C2×C311yescharacteristic, complemented
61#547C2×C311yescharacteristic, complemented
62#501C2×C311yescharacteristic, complemented
63#482C2×C311yescharacteristic, complemented
64#481C2×C311yescharacteristic, complemented
65#481C2×C311yescharacteristic, complemented
66#480C2×C311yescharacteristic, complemented
67#478C2×C311yescharacteristic, complemented
68#476C2×C311yescharacteristic, complemented
69#469C2×C311yescharacteristic, complemented
70#466C2×C311yescharacteristic, complemented
71#442C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#428C2×C311yescharacteristic, complemented
74C2×#106C2×C311yescharacteristic, complemented
75C3×#41—44no—
76C2×C3×Q8—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4D411yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C3×D8—22no—
92C3×M16—22no—
93C2×C3×C8—22no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4D411yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
119C3×C22⋊C4D411yescharacteristic
120C3×C42—21no—
121C3×C42C2311yescharacteristic
122C2×#41—21no—
123C2×D8C22×C311yescharacteristic
124C2×M16C22×C311yescharacteristic
125C22×C8C22×C311yescharacteristic
126C4⋊Q8C22×C311yescharacteristic
127#126—21no—
128#125C22×C311yescharacteristic
129#123—21no—
130#123—21no—
131#123—21no—
132C22⋊Q8—21no—
133#121—21no—
134#121C22×C311yescharacteristic
135#121C22×C311yescharacteristic
136#121C22×C311yescharacteristic
137#121C22×C311yescharacteristic
138C4×D4—21no—
139C4×D4C22×C311yescharacteristic
140C4×D4C22×C311yescharacteristic
141C4×D4C22×C311yescharacteristic
142#117C22×C311yescharacteristic
143C2×C4⋊C4C22×C311yescharacteristic
144C2×C4⋊C4C22×C311yescharacteristic
145C2×C4⋊C4C22×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
154#106C22×C311yescharacteristic
155C4⋊C8C22×C311yescharacteristic
156#102C22×C311yescharacteristic
157#102C22×C311yescharacteristic
158#102C22×C311yescharacteristic
159#102C22×C311yescharacteristic
160#102C22×C311yescharacteristic
161#102C22×C311yescharacteristic
162C22⋊C8C22×C311yescharacteristic
163C22⋊C8C22×C311yescharacteristic
164#97C22×C311yescharacteristic
165C23×C3—21no—
166C23×C3—21no—
167C23×C3C2×D411yescharacteristic
168C23×C3C2×D411yescharacteristic
169C3×Q8—42no—
170C3×D4—22no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—42no—
176C3×D4—42no—
177C3×D4—42no—
178C3×D4—81no—
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—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2411yescharacteristic
199C3×C8—21no—
200C3×C8—21no—
201C3×C8—21no—
202C3×C8#4112yescomplemented
203#41—44no—
204C2×Q8—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4C3×D411yescharacteristic
210C2×D4C23×C311yescharacteristic
211C2×D4C23×C311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C23×C311yescharacteristic
218C22×C4C23×C311yescharacteristic
219D8—22no—
220M16—22no—
221C2×C8—22no—
222C2×C8C23×C311yescharacteristic
223C2×C8C23×C311yescharacteristic
224C2×C8C23×C311yescharacteristic
225C2×C8C23×C311yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4C3×D411yescharacteristic
233C4⋊C4C23×C311yescharacteristic
234C4⋊C4C23×C311yescharacteristic
235C4⋊C4C23×C311yescharacteristic
236C4⋊C4C23×C311yescharacteristic
237C4⋊C4C23×C311yescharacteristic
238C4⋊C4C23×C311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—21no—
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4C3×D411yescharacteristic
247C22⋊C4C3×D411yescharacteristic
248C42—21no—
249C42C23×C311yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3AGL(1,ℤ/8)12yes—
262C22×C3C22×D411yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4C2×#4111yescharacteristic
274C3×C42−1+411yescharacteristic
275C23—21no—
276C23—21no—
277C23C2×C3×D411yescharacteristic
278C23C2×C3×D411yescharacteristic
279Q8—42no—
280D4—22no—
281D4—21no—
282D4—21no—
283D4—21no—
284D4—21no—
285D4—42no—
286D4—42no—
287D4—42no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4C2×C3×D411yescharacteristic
305C2×C4C2×C3×D411yescharacteristic
306C2×C4C2×C3×D411yescharacteristic
307C2×C4C2×C3×D411yescharacteristic
308C2×C4C24×C311yescharacteristic
309C8—21no—
310C8—21no—
311C8—21no—
312C8C3×#4112yescomplemented
313C2×C3—22no—
314C2×C3—41no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3#54711yescharacteristic
318C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
319C2×C3#57711yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C3×AGL(1,ℤ/8)12yes—
332C22C22×C3×D411yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4C2×C3×#4111yescharacteristic
344C4C3×2−1+411yescharacteristic
345C3#334511yescharacteristic, direct product factor, Sylow
346C2—22no—
347C2—41no—
348C2—81no—
349C2—81no—
350C2C3×#54711yescharacteristic
351C2C2×C3×AGL(1,ℤ/8)11yescharacteristic
352C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
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×#553C2semidirect
2. C3×#547C2semidirect
3. C3×#501C2semidirect
4. C3×#482C2semidirect
5. C3×#481C2semidirect
6. C3×#481C2semidirect
7. C3×#480C2semidirect
8. C3×#478C2semidirect
9. C3×#476C2semidirect
10. C3×#469C2semidirect
11. C3×#466C2semidirect
12. C3×#442C2semidirect
13. C3×#435C2semidirect
14. C3×#428C2semidirect
15. C2×C3×#106C2semidirect
16. #3345C3direct (N × H)
60. #553C2×C3semidirect
61. #547C2×C3semidirect
62. #501C2×C3semidirect
63. #482C2×C3semidirect
64. #481C2×C3semidirect
65. #481C2×C3semidirect
66. #480C2×C3semidirect
67. #478C2×C3semidirect
68. #476C2×C3semidirect
69. #469C2×C3semidirect
70. #466C2×C3semidirect
71. #442C2×C3semidirect
72. #435C2×C3semidirect
73. #428C2×C3semidirect
74. C2×#106C2×C3semidirect
202. C3×C8#41semidirect
312. C8C3×#41semidirect
345. C3#3345direct (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×#5472
C3×#5012
C3×#4822
C3×#4812
C3×#4802
C3×#4782
C3×#4762
C3×#4692
C3×#4662
C3×#4422
C3×#4352
C3×#4282
C2×C3×#1062
#33453