Finite Group Databasebeta

← C3×#85680 #11900878

Subgroups — C3×#85680 #11900878

Order 1152.

2673 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C42C211yesmaximal, characteristic, complemented
2C3×#5791C216yesmaximal, complemented
3C3×#5479C2112yesmaximal, complemented
4C3×#5477C213yesmaximal, complemented
5C3×C4×#235C216yesmaximal, complemented
6C3×#5470C213yesmaximal, complemented
7C3×#3444—31nomaximal
8#85680C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2213yes—
10C2×C3×#235C2216yes—
11C2×C32×C42C2213yes—
12C2×C32×C42C2213yes—
13C2×C32×C42C2216yes—
14C3×#953C2214yes—
15C3×#930C22112yes—
16C3×#929C22112yes—
17C3×C4×#67C22124yes—
18C3×#926C2216yes—
19C3×#924C22112yes—
20C3×#876C2218yescomplemented
21C3×#875C2216yescomplemented
22C3×#874C2211yescharacteristic, complemented
23C3×C4×D12C22112yescomplemented
24C3×#872C22112yescomplemented
25C3×C42×D3C2214yescomplemented
26C3×#870C2216yescomplemented
27C3×#869C2213yescomplemented
28C3×C4×Dic6C22112yescomplemented
29C3×#533—33no—
30C3×#532—33no—
31C3×#529—312no—
32C3×#525—36no—
33C3×C4×#41—36no—
34C22×C3×C42—21no—
35C22×C3×C42D311yescharacteristic, complemented
36C22×C3×C42C2×C311yescharacteristic, complemented
37#5791C2×C316yescomplemented
38#5479C2×C3112yescomplemented
39#5477C2×C313yescomplemented
40C4×#235C2×C316yescomplemented
41#5470C2×C313yescomplemented
42C24×C32C2311yescharacteristic
43C22×C32×C4—212no—
44C22×C32×C4C2316yes—
45C22×C32×C4C2316yes—
46C22×C32×C4C2313yes—
47C22×C32×C4C2313yes—
48C2×C3×#67C23112yes—
49C3×#235—224no—
50C2×C3×D12C2316yes—
51C2×C3×C4×D3C23112yes—
52C2×C3×Dic6C2316yes—
53C32×C42C2316yes—
54C32×C42C2311yescharacteristic
55C32×C42C2313yes—
56C32×C42C2313yes—
57C32×C42C2312yes—
58C32×C42C2311yescharacteristic
59C3×#217C23112yes—
60C3×(C2×D3)⋊C4C23124yes—
61C3×C4⋊Dic3C23112yes—
62C3×Dic3⋊C4C23124yes—
63C3×C4×Dic3C23112yes—
64#3444—31noSylow
65C2×C3×#41—36no—
66C23×C3×C4—23no—
67C23×C3×C4C2×D313yes—
68C23×C3×C4C22×C313yes—
69C2×#235C22×C316yes—
70C3×C4⋊Q8—33no—
71C3×#127—31no—
72C3×#126—38no—
73C3×#125—36no—
74C3×#124—36no—
75C3×#123—312no—
76C3×C22⋊Q8—312no—
77C3×#121—312no—
78C3×#120—34no—
79C3×C4×Q8—312no—
80C3×C4×D4—324no—
81C3×C4×D4—312no—
82C3×#117—36no—
83C3×#117—312no—
84C2×C3×C42—23no—
85C2×C3×C42—26no—
86C2×C3×C42—23no—
87C2×C3×C42—34no—
88C2×C3×C42C2×D316yes—
89C2×C3×C42C2×D313yes—
90C2×C3×C42C2×D313yes—
91C2×C3×C42C22×C313yes—
92C2×C3×C42C22×C313yes—
93C2×C3×C42C22×C316yes—
94#953C22×C314yes—
95#930C22×C3112yes—
96#929C22×C3112yes—
97C4×#67C22×C3124yes—
98#926C22×C316yes—
99#924C22×C3112yes—
100#876C22×C318yescomplemented
101#875C22×C316yescomplemented
102#874C22×C311yescharacteristic, complemented
103C4×D12C22×C3112yescomplemented
104#872C22×C3112yescomplemented
105C42×D3C22×C314yescomplemented
106#870C22×C316yescomplemented
107#869C22×C313yescomplemented
108C4×Dic6C22×C3112yescomplemented
109C23×C32—26no—
110C23×C32C2413yes—
111C22×C3×D3C2414yes—
112C2×C32×C4—26no—
113C2×C32×C4—212no—
114C2×C32×C4—212no—
115C2×C32×C4#41112yescomplemented
116C2×C32×C4C2416yes—
117C2×C32×C4C2413yes—
118C2×C32×C4C2413yes—
119C3×#67—224no—
120C2×C3×Dic3C24112yes—
121C3×D12—212no—
122C3×C4×D3—224no—
123C3×Dic6—212no—
124#533—33no—
125#532—33no—
126#529—312no—
127#525—36no—
128C4×#41—36no—
129C22×C42C3×D311yescharacteristic, complemented
130C24×C3—21no—
131C24×C3C22×D311yescharacteristic
132C24×C3C23×C311yescharacteristic
133C3×#41—624no—
134C2×C3×Q8—36no—
135C2×C3×D4—312no—
136C2×C3×D4—36no—
137C22×C3×C4—224no—
138C22×C3×C4—212no—
139C22×C3×C4—212no—
140C22×C3×C4—26no—
141C22×C3×C4—26no—
142C22×C3×C4—23no—
143C22×C3×C4—23no—
144C22×C3×C4—312no—
145C22×C3×C4C22×D316yes—
146C22×C3×C4C22×D316yes—
147C22×C3×C4C22×D313yes—
148C22×C3×C4C22×D313yes—
149C22×C3×C4C23×C316yes—
150C22×C3×C4C23×C316yes—
151C22×C3×C4C23×C313yes—
152C22×C3×C4C23×C313yes—
153C2×#67C23×C3112yes—
154#235—224no—
155C2×D12C23×C316yes—
156C2×C4×D3C23×C3112yes—
157C2×Dic6C23×C316yes—
158C3×C4⋊C4—324no—
159C3×C4⋊C4—312no—
160C3×C22⋊C4—312no—
161C3×C22⋊C4—324no—
162C3×C42—21no—
163C3×C42—23no—
164C3×C42—26no—
165C3×C42—23no—
166C3×C42—22no—
167C3×C42—21no—
168C3×C42—312no—
169C3×C42C22×D313yes—
170C3×C42C22×D316yes—
171C3×C42C22×D313yes—
172C3×C42C22×D312yes—
173C3×C42C22×D311yescharacteristic
174C3×C42C22×D311yescharacteristic
175C3×C42C23×C316yes—
176C3×C42C23×C311yescharacteristic
177C3×C42C23×C313yes—
178C3×C42C23×C311yescharacteristic
179C3×C42C23×C313yes—
180C3×C42C23×C312yes—
181#217C23×C3112yes—
182(C2×D3)⋊C4C23×C3124yes—
183C4⋊Dic3C23×C3112yes—
184Dic3⋊C4C23×C3124yes—
185C4×Dic3C23×C3112yes—
186C22×C32—28no—
187C22×C32—26no—
188C22×C32C2×#4116yes—
189C22×C32C2511yescharacteristic
190C2×C3×D3—212no—
191C32×C4—26no—
192C32×C4C2×#4116yes—
193C32×C4C2×#4116yes—
194C3×Dic3—212no—
195C2×#41—36no—
196C23×C4C2×C3×D313yes—
197C4⋊Q8—33no—
198#127—31no—
199#126—38no—
200#125—36no—
201#124—36no—
202#123—312no—
203C22⋊Q8—312no—
204#121—312no—
205#120—34no—
206C4×Q8—312no—
207C4×D4—324no—
208C4×D4—312no—
209#117—36no—
210#117—312no—
211C2×C42—34no—
212C2×C42C2×C3×D313yes—
213C2×C42C2×C3×D316yes—
214C2×C42C2×C3×D313yes—
215C23×C3—212no—
216C23×C3—26no—
217C23×C3—26no—
218C23×C3—23no—
219C23×C3—34no—
220C23×C3C23×D313yes—
221C23×C3C24×C313yes—
222C22×D3C24×C314yes—
223C3×Q8—612no—
224C3×D4—624no—
225C3×D4—612no—
226C2×C3×C4—212no—
227C2×C3×C4—224no—
228C2×C3×C4—224no—
229C2×C3×C4—26no—
230C2×C3×C4—26no—
231C2×C3×C4—212no—
232C2×C3×C4—212no—
233C2×C3×C4—212no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—23no—
237C2×C3×C4—23no—
238C2×C3×C4—212no—
239C2×C3×C4—312no—
240C2×C3×C4—624no—
241C2×C3×C4#235112yescomplemented
242C2×C3×C4C3×#41112yescomplemented
243C2×C3×C4C23×D316yes—
244C2×C3×C4C23×D313yes—
245C2×C3×C4C23×D313yes—
246C2×C3×C4C24×C316yes—
247C2×C3×C4C24×C313yes—
248C2×C3×C4C24×C313yes—
249#67—224no—
250C2×Dic3C24×C3112yes—
251D12—212no—
252C4×D3—224no—
253Dic6—212no—
254C2×C32—26no—
255C2×C32C22×#4113yes—
256C3×D3—44no—
257C24C22×C3×D311yescharacteristic
258#41—624no—
259C2×Q8—36no—
260C2×D4—312no—
261C2×D4—36no—
262C22×C4—212no—
263C22×C4—312no—
264C22×C4C22×C3×D316yes—
265C22×C4C22×C3×D316yes—
266C22×C4C22×C3×D313yes—
267C22×C4C22×C3×D313yes—
268C4⋊C4—312no—
269C4⋊C4—324no—
270C22⋊C4—312no—
271C22⋊C4—324no—
272C42—312no—
273C42C22×C3×D311yescharacteristic
274C42C22×C3×D313yes—
275C42C22×C3×D316yes—
276C42C22×C3×D313yes—
277C42C22×C3×D312yes—
278C42C22×C3×D311yescharacteristic
279C22×C3—212no—
280C22×C3—216no—
281C22×C3—28no—
282C22×C3—28no—
283C22×C3—26no—
284C22×C3—26no—
285C22×C3—26no—
286C22×C3—21no—
287C22×C3—612no—
288C22×C3C2×#23516yes—
289C22×C3C2×C3×#4116yes—
290C22×C3C24×D311yescharacteristic
291C22×C3C25×C311yescharacteristic
292C2×D3—212no—
293C3×C4—212no—
294C3×C4—26no—
295C3×C4—26no—
296C3×C4—26no—
297C3×C4—26no—
298C3×C4—612no—
299C3×C4C2×#23516yes—
300C3×C4C2×#23516yes—
301C3×C4C2×C3×#4116yes—
302C3×C4C2×C3×#4116yes—
303Dic3—212no—
304C32#344411yescharacteristic, complemented, Sylow
305C23—26no—
306C23—34no—
307C23C23×C3×D313yes—
308Q8—612no—
309D4—624no—
310D4—612no—
311C2×C4—26no—
312C2×C4—212no—
313C2×C4—212no—
314C2×C4—312no—
315C2×C4—624no—
316C2×C4C3×#235112yescomplemented
317C2×C4C23×C3×D316yes—
318C2×C4C23×C3×D313yes—
319C2×C4C23×C3×D313yes—
320C2×C3—212no—
321C2×C3—26no—
322C2×C3—26no—
323C2×C3—23no—
324C2×C3—124no—
325C2×C3C22×#23513yes—
326C2×C3C22×C3×#4113yes—
327D3—44no—
328C22—28no—
329C22—26no—
330C22—612no—
331C22C2×C3×#23516yes—
332C22C24×C3×D311yescharacteristic
333C4—26no—
334C4—612no—
335C4C2×C3×#23516yes—
336C4C2×C3×#23516yes—
337C3—21no—
338C3#8568011yescharacteristic, direct product factor
339C3C3×#344411yescharacteristic, complemented
340C2—26no—
341C2—124no—
342C2C22×C3×#23513yes—
subgroupG/H
centerC3×C42C22×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC22×C32×C42C2
socleC22×C32C25
hypercenterC22×C3×C42D3
hypocenterC3C3×#3444
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C3×C42 ◁ C22×C3×C42

Complements

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

NH ≅ G/Ntype
1. C22×C32×C42C2semidirect
2. C3×#5791C2semidirect
3. C3×#5479C2semidirect
4. C3×#5477C2semidirect
5. C3×C4×#235C2semidirect
6. C3×#5470C2semidirect
8. #85680C3direct (N × H)
20. C3×#876C22semidirect
21. C3×#875C22semidirect
22. C3×#874C22semidirect
23. C3×C4×D12C22semidirect
24. C3×#872C22semidirect
25. C3×C42×D3C22semidirect
26. C3×#870C22semidirect
27. C3×#869C22semidirect
28. C3×C4×Dic6C22semidirect
35. C22×C3×C42D3semidirect
36. C22×C3×C42C2×C3semidirect
37. #5791C2×C3semidirect
38. #5479C2×C3semidirect
39. #5477C2×C3semidirect
40. C4×#235C2×C3semidirect
41. #5470C2×C3semidirect
100. #876C22×C3semidirect
101. #875C22×C3semidirect
102. #874C22×C3semidirect
103. C4×D12C22×C3semidirect
104. #872C22×C3semidirect
105. C42×D3C22×C3semidirect
106. #870C22×C3semidirect
107. #869C22×C3semidirect
108. C4×Dic6C22×C3semidirect
115. C2×C32×C4#41semidirect
129. C22×C42C3×D3semidirect
241. C2×C3×C4#235semidirect
242. C2×C3×C4C3×#41semidirect
304. C32#3444semidirect
316. C2×C4C3×#235semidirect
338. C3#85680direct (N × H)
339. C3C3×#3444semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C422
C3×#57912
C3×#54792
C3×#54772
C3×C4×#2352
C3×#54702
C3×#34443
#856803