Finite Group Databasebeta

← C3×#84125 #11865303

Subgroups — C3×#84125 #11865303

Order 1152.

1937 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1937 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#585C211yesmaximal, characteristic, complemented
2C3×#5828C213yesmaximal, complemented
3C3×#5244C214yesmaximal, complemented
4C3×#5243C214yesmaximal, complemented
5C3×#5240C213yesmaximal, complemented
6C3×#3025—31nomaximal
7#84125C311yesmaximal, characteristic, direct product factor
8C32×2−1+4—23no—
9C32×H(5,𝔽2)—23no—
10C32×H(5,𝔽2)C2214yes—
11C2×C32×#41—26no—
12C2×C32×#41C2213yes—
13C22×C3×C4×D3C2211yescharacteristic, complemented
14C3×#946—23no—
15C3×#938—23no—
16C3×#937C2216yescomplemented
17C3×#935C2216yescomplemented
18C3×C4×#67—26no—
19C3×#926C2213yescomplemented
20C3×#834C22112yescomplemented
21C3×#585—21no—
22C3×#585D311yescharacteristic, complemented
23C3×#585C2×C311yescharacteristic, complemented
24C3×#525—33no—
25#5828C2×C313yescomplemented
26C3×#458—34no—
27C3×B(4,𝔽2)—34no—
28C3×#410—33no—
29#5244C2×C314yescomplemented
30#5243C2×C314yescomplemented
31#5240C2×C313yescomplemented
32C23×C3×D3C2311yescharacteristic
33C32×#41—24no—
34C32×#41—212no—
35C32×#41—26no—
36C32×#41—212no—
37C32×#41—26no—
38C2×C32×Q8—26no—
39C2×C32×Q8D413yescomplemented
40C2×C32×D4—26no—
41C2×C32×D4—212no—
42C2×C32×D4D413yescomplemented
43C2×C32×D4C2316yes—
44C22×C32×C4—24no—
45C22×C32×C4D416yescomplemented
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—26no—
48C22×C3×Dic3C2311yescharacteristic
49C2×C3×C4×D3—23no—
50C2×C3×C4×D3—23no—
51C3×#217—26no—
52C3×#217C2316yes—
53C3×(C2×D3)⋊C4—26no—
54C3×C4⋊Dic3—23no—
55C3×Dic3⋊C4—26no—
56C3×C4×Dic3—23no—
57#3025—31noSylow
58C3×2−1+4—26no—
59C3×2−1+4—23no—
60C3×2−1+4—23no—
61C3×H(5,𝔽2)—26no—
62C3×H(5,𝔽2)—23no—
63C3×H(5,𝔽2)—23no—
64C3×H(5,𝔽2)—24no—
65C3×H(5,𝔽2)C2×D314yes—
66C3×H(5,𝔽2)C22×C314yes—
67C2×C3×#41—212no—
68C2×C3×#41—26no—
69C2×C3×#41—26no—
70C2×C3×#41—23no—
71C2×C3×#41C2×D313yes—
72C2×C3×#41C22×C313yes—
73C23×C3×C4—31no—
74C22×C4×D3C22×C311yescharacteristic, complemented
75C3×#123—36no—
76C3×C22⋊Q8—63no—
77C3×#121—63no—
78C3×#120—36no—
79C3×C4×D4—66no—
80C3×#117—33no—
81#946—23no—
82#938—23no—
83#937C22×C316yescomplemented
84#935C22×C316yescomplemented
85C4×#67—26no—
86#926C22×C313yescomplemented
87C3×#99—312no—
88#834C22×C3112yescomplemented
89C23×C32—24no—
90C23×C32C2×D416yes—
91C23×C32C2411yescharacteristic
92C22×C3×D3—23no—
93C22×C3×D3—42no—
94C32×Q8—24no—
95C32×Q8—26no—
96C32×D4—212no—
97C32×D4—212no—
98C32×D4—26no—
99C2×C32×C4—212no—
100C2×C32×C4—26no—
101C2×C32×C4—26no—
102C2×C32×C4C2×D416yes—
103C2×C32×C4C2×D413yes—
104C2×C32×C4C2×D413yes—
105C3×#67—412no—
106C2×C3×Dic3—26no—
107C2×C3×Dic3—23no—
108C3×C4×D3—43no—
109C3×C4×D3—41no—
110#585C3×D311yescharacteristic, complemented
111#525—33no—
112#458—34no—
113B(4,𝔽2)—34no—
114#410—33no—
115C24×C3—31no—
116C23×D3C23×C311yescharacteristic
117C3×#41—28no—
118C3×#41—24no—
119C3×#41—24no—
120C3×#41—212no—
121C3×#41—224no—
122C3×#41—212no—
123C3×#41—212no—
124C3×#41—224no—
125C3×#41—212no—
126C3×#41—26no—
127C3×#41—212no—
128C3×#41—26no—
129C3×#41—26no—
130C3×#41—212no—
131C3×#41—26no—
132C2×C3×Q8—212no—
133C2×C3×Q8—26no—
134C2×C3×Q8—26no—
135C2×C3×Q8—23no—
136C2×C3×Q8#6713yescomplemented
137C2×C3×Q8C3×D413yescomplemented
138C2×C3×D4—212no—
139C2×C3×D4—26no—
140C2×C3×D4—26no—
141C2×C3×D4—224no—
142C2×C3×D4—212no—
143C2×C3×D4—212no—
144C2×C3×D4—23no—
145C2×C3×D4—26no—
146C2×C3×D4—66no—
147C2×C3×D4#6713yescomplemented
148C2×C3×D4C3×D413yescomplemented
149C2×C3×D4C22×D316yes—
150C2×C3×D4C23×C316yes—
151C22×C3×C4—28no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—26no—
155C22×C3×C4—21no—
156C22×C3×C4—31no—
157C22×C3×C4—63no—
158C22×C3×C4—63no—
159C22×C3×C4#6716yescomplemented
160C22×C3×C4C3×D416yescomplemented
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C23×C311yescharacteristic
163C2×#67—26no—
164C22×Dic3C23×C311yescharacteristic
165C2×C4×D3—23no—
166C2×C4×D3—23no—
167C3×C4⋊C4—66no—
168C3×C4⋊C4—63no—
169C3×C22⋊C4—36no—
170C3×C22⋊C4—66no—
171C3×C22⋊C4—66no—
172C3×C42—63no—
173#217—26no—
174#217C23×C316yes—
175(C2×D3)⋊C4—26no—
176C4⋊Dic3—23no—
177Dic3⋊C4—26no—
178C4×Dic3—23no—
179C22×C32—26no—
180C22×C32—41no—
181C22×C32—46no—
182C22×C32—84no—
183C22×C32C22×D413yes—
184C2×C3×D3—26no—
185C2×C3×D3—43no—
186C2×C3×D3—41no—
187C32×C4—26no—
188C32×C4#12013yes—
189C32×C4#12011yescharacteristic
190C3×Dic3—46no—
191C3×Dic3—41no—
1922−1+4—23no—
193H(5,𝔽2)—23no—
194H(5,𝔽2)C2×C3×D314yes—
195C2×#41—26no—
196C2×#41C2×C3×D313yes—
197C23×C4—31no—
198#123—36no—
199C22⋊Q8—63no—
200#121—63no—
201#120—36no—
202C4×D4—66no—
203#117—33no—
204#99—312no—
205C23×C3—28no—
206C23×C3—24no—
207C23×C3—24no—
208C23×C3—21no—
209C23×C3—26no—
210C23×C3—63no—
211C23×C3—122no—
212C23×C3C2×#6716yes—
213C23×C3C2×C3×D416yes—
214C23×C3C23×D311yescharacteristic
215C23×C3C24×C311yescharacteristic
216C22×D3—23no—
217C22×D3—42no—
218C3×Q8—28no—
219C3×Q8—24no—
220C3×Q8—24no—
221C3×Q8—212no—
222C3×Q8—26no—
223C3×Q8—26no—
224C3×D4—224no—
225C3×D4—212no—
226C3×D4—212no—
227C3×D4—224no—
228C3×D4—212no—
229C3×D4—212no—
230C3×D4—212no—
231C3×D4—26no—
232C3×D4—26no—
233C3×D4—1212no—
234C2×C3×C4—212no—
235C2×C3×C4—224no—
236C2×C3×C4—212no—
237C2×C3×C4—212no—
238C2×C3×C4—212no—
239C2×C3×C4—26no—
240C2×C3×C4—26no—
241C2×C3×C4—26no—
242C2×C3×C4—26no—
243C2×C3×C4—23no—
244C2×C3×C4—26no—
245C2×C3×C4—23no—
246C2×C3×C4—66no—
247C2×C3×C4—63no—
248C2×C3×C4—123no—
249C2×C3×C4—121no—
250C2×C3×C4C2×#6716yes—
251C2×C3×C4C2×#6713yes—
252C2×C3×C4C2×#6713yes—
253C2×C3×C4C2×C3×D416yes—
254C2×C3×C4C2×C3×D413yes—
255C2×C3×C4C2×C3×D413yes—
256#67—412no—
257C2×Dic3—26no—
258C2×Dic3—23no—
259C4×D3—41no—
260C4×D3—43no—
261C2×C32—23no—
262C2×C32—46no—
263C2×C32C2×#12011yescharacteristic
264C3×D3—42no—
265C24—31no—
266#41—24no—
267#41—212no—
268#41—26no—
269#41—212no—
270#41—26no—
271C2×Q8—26no—
272C2×Q8C3×#6713yescomplemented
273C2×D4—26no—
274C2×D4—212no—
275C2×D4—66no—
276C2×D4C3×#6713yescomplemented
277C2×D4C22×C3×D316yes—
278C22×C4—24no—
279C22×C4—31no—
280C22×C4—63no—
281C22×C4—63no—
282C22×C4C3×#6716yescomplemented
283C22×C4C22×C3×D311yescharacteristic
284C4⋊C4—63no—
285C4⋊C4—66no—
286C22⋊C4—36no—
287C22⋊C4—66no—
288C22⋊C4—66no—
289C42—63no—
290C22×C3—212no—
291C22×C3—26no—
292C22×C3—26no—
293C22×C3—23no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—46no—
297C22×C3—46no—
298C22×C3—66no—
299C22×C3—81no—
300C22×C3—86no—
301C22×C3—88no—
302C22×C3—84no—
303C22×C3—84no—
304C22×C3—123no—
305C22×C3—121no—
306C22×C3C22×#6713yes—
307C22×C3C22×C3×D413yes—
308C2×D3—26no—
309C2×D3—43no—
310C2×D3—41no—
311C3×C4—212no—
312C3×C4—26no—
313C3×C4—26no—
314C3×C4—21no—
315C3×C4—23no—
316C3×C4—126no—
317C3×C4—121no—
318C3×C4#95313yes—
319C3×C4#95311yescharacteristic
320C3×C4C3×#12011yescharacteristic
321C3×C4C3×#12013yes—
322Dic3—46no—
323Dic3—41no—
324C32#302511yescharacteristic, complemented, Sylow
325C23—24no—
326C23—63no—
327C23—122no—
328C23C2×C3×#6716yes—
329C23C23×C3×D311yescharacteristic
330Q8—24no—
331Q8—26no—
332D4—212no—
333D4—212no—
334D4—26no—
335D4—1212no—
336C2×C4—212no—
337C2×C4—26no—
338C2×C4—26no—
339C2×C4—66no—
340C2×C4—63no—
341C2×C4—121no—
342C2×C4—123no—
343C2×C4C2×C3×#6716yes—
344C2×C4C2×C3×#6713yes—
345C2×C4C2×C3×#6713yes—
346C2×C3—23no—
347C2×C3—23no—
348C2×C3—21no—
349C2×C3—43no—
350C2×C3—412no—
351C2×C3—46no—
352C2×C3—46no—
353C2×C3—122no—
354C2×C3C2×#95311yescharacteristic
355C2×C3C2×C3×#12011yescharacteristic
356D3—42no—
357C22—26no—
358C22—41no—
359C22—46no—
360C22—66no—
361C22—84no—
362C22—123no—
363C22—121no—
364C22C22×C3×#6713yes—
365C4—26no—
366C4—126no—
367C4—121no—
368C4C3×#95311yescharacteristic
369C4C3×#95313yes—
370C3—21no—
371C3#8412511yescharacteristic, direct product factor
372C3C3×#302511yescharacteristic, complemented
373C2—23no—
374C2—46no—
375C2—122no—
376C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC3×C4#953
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×#585C2
socleC2×C32C2×#120
hypercenterC3×#585D3
hypocenterC3C3×#3025
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#585

Complements

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

NH ≅ G/Ntype
1. C32×#585C2semidirect
2. C3×#5828C2semidirect
3. C3×#5244C2semidirect
4. C3×#5243C2semidirect
5. C3×#5240C2semidirect
7. #84125C3direct (N × H)
13. C22×C3×C4×D3C22semidirect
16. C3×#937C22semidirect
17. C3×#935C22semidirect
19. C3×#926C22semidirect
20. C3×#834C22semidirect
22. C3×#585D3semidirect
23. C3×#585C2×C3semidirect
25. #5828C2×C3semidirect
29. #5244C2×C3semidirect
30. #5243C2×C3semidirect
31. #5240C2×C3semidirect
39. C2×C32×Q8D4semidirect
42. C2×C32×D4D4semidirect
45. C22×C32×C4D4semidirect
74. C22×C4×D3C22×C3semidirect
83. #937C22×C3semidirect
84. #935C22×C3semidirect
86. #926C22×C3semidirect
88. #834C22×C3semidirect
110. #585C3×D3semidirect
136. C2×C3×Q8#67semidirect
137. C2×C3×Q8C3×D4semidirect
147. C2×C3×D4#67semidirect
148. C2×C3×D4C3×D4semidirect
159. C22×C3×C4#67semidirect
160. C22×C3×C4C3×D4semidirect
272. C2×Q8C3×#67semidirect
276. C2×D4C3×#67semidirect
282. C22×C4C3×#67semidirect
324. C32#3025semidirect
371. C3#84125direct (N × H)
372. C3C3×#3025semidirect

Maximal subgroups

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

subgroupindex
C32×#5852
C3×#58282
C3×#52442
C3×#52432
C3×#52402
C3×#30253
#841253