Finite Group Databasebeta

← C3×#72112 #11798268

Subgroups — C3×#72112 #11798268

Order 1152.

583 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 106 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×#575C211yesmaximal, characteristic, complemented
2C3×#5193C212yesmaximal, complemented
3C3×#5135C211yesmaximal, characteristic, complemented
4C3×#4566C212yesmaximal, complemented
5C3×#4553C211yesmaximal, characteristic, complemented
6C3×#2190—31nomaximal
7#72112C311yesmaximal, characteristic, direct product factor
8C2×C32×#41—21no—
9C32×#137—22no—
10C32×AGL(1,ℤ/8)—22no—
11C32×#135—21no—
12C32×#135C2212yes—
13C2×C32×M16C2211yescharacteristic
14C3×#930—21no—
15C2×C3×D24C2211yescharacteristic, complemented
16C3×#828—22no—
17C3×#826—22no—
18C3×#821—21no—
19C3×#817C2211yescharacteristic, complemented
20C3×#812C2212yescomplemented
21C3×#575—21no—
22C3×#575D311yescharacteristic, complemented
23C3×#575C2×C311yescharacteristic, complemented
24C3×#509—32no—
25C3×#469—31no—
26#5193C2×C312yescomplemented
27#5135C2×C311yescharacteristic, complemented
28C3×#360—32no—
29C3×#349—31no—
30#4566C2×C312yescomplemented
31#4553C2×C311yescharacteristic, complemented
32C32×#41—22no—
33C32×#41—22no—
34C32×#41—41no—
35C32×#41—41no—
36C2×C32×Q8—21no—
37C2×C32×D4—21no—
38C2×C32×D4—41no—
39C22×C32×C4—41no—
40C22×C32×C4D411yescharacteristic
41C2×C3×#67—41no—
42C2×C3×D12—21no—
43C32×Q16—21no—
44C32×Q16D412yescomplemented
45C32×SD16—22no—
46C32×SD16—22no—
47C32×D8—21no—
48C32×D8D412yescomplemented
49C32×M16—22no—
50C2×C32×C8D411yescharacteristic
51C2×C32×C8C2311yescharacteristic
52C3×(C2×D3)⋊C4—41no—
53C3×C4⋊Dic3—21no—
54C3×D24—22no—
55C3×C3⋊C16—22no—
56#2190—31noSylow
57C2×C3×#41—22no—
58C2×C3×#41—21no—
59C2×C3×#41—21no—
60C3×#137—24no—
61C3×#137—22no—
62C3×#137—22no—
63C3×AGL(1,ℤ/8)—24no—
64C3×AGL(1,ℤ/8)—22no—
65C3×AGL(1,ℤ/8)—22no—
66C3×#135—22no—
67C3×#135—21no—
68C3×#135—21no—
69C3×#135—22no—
70C3×#135C2×D312yes—
71C3×#135C22×C312yes—
72C2×C3×D8—31no—
73C2×C3×M16—21no—
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×M16C22×C311yescharacteristic
76C3×#121—61no—
77#930—21no—
78C2×D24C22×C311yescharacteristic, complemented
79C3×SD32—62no—
80C3×D16—62no—
81C3×M32—32no—
82C3×#106—31no—
83C3×#102—61no—
84#828—22no—
85#826—22no—
86#821—21no—
87#817C22×C311yescharacteristic, complemented
88#812C22×C312yescomplemented
89C23×C32—41no—
90C23×C32D811yescharacteristic
91C22×C3×D3—41no—
92C32×Q8—22no—
93C32×Q8—41no—
94C32×D4—22no—
95C32×D4—41no—
96C32×D4—42no—
97C32×D4—42no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—42no—
101C2×C32×C4—41no—
102C2×C32×C4—42no—
103C2×C32×C4D811yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C3×#67—82no—
106C2×C3×Dic3—41no—
107C3×D12—41no—
108C3×D12—41no—
109C32×C8—21no—
110C32×C8C2×D412yes—
111#575C3×D311yescharacteristic, complemented
112#509—32no—
113#469—31no—
114#360—32no—
115#349—31no—
116C3×#41—24no—
117C3×#41—24no—
118C3×#41—22no—
119C3×#41—22no—
120C3×#41—22no—
121C3×#41—22no—
122C3×#41—42no—
123C3×#41—41no—
124C3×#41—41no—
125C3×#41—42no—
126C3×#41—41no—
127C3×#41—41no—
128C2×C3×Q8—22no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—42no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—61no—
138C2×C3×D4—121no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4#6711yescharacteristic
144C22×C3×C4C3×D411yescharacteristic
145C2×#67—41no—
146C2×D12—21no—
147C3×Q16—22no—
148C3×Q16—21no—
149C3×Q16—21no—
150C3×Q16—22no—
151C3×Q16#6712yescomplemented
152C3×Q16C3×D412yescomplemented
153C3×SD16—24no—
154C3×SD16—22no—
155C3×SD16—22no—
156C3×SD16—24no—
157C3×SD16—22no—
158C3×SD16—22no—
159C3×D8—22no—
160C3×D8—21no—
161C3×D8—21no—
162C3×D8—22no—
163C3×D8—62no—
164C3×D8#6712yescomplemented
165C3×D8C3×D412yescomplemented
166C3×M16—24no—
167C3×M16—22no—
168C3×M16—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8#6711yescharacteristic
172C2×C3×C8C3×D411yescharacteristic
173C2×C3×C8C22×D311yescharacteristic
174C2×C3×C8C23×C311yescharacteristic
175C3×C4⋊C4—61no—
176C3×C22⋊C4—121no—
177(C2×D3)⋊C4—41no—
178C4⋊Dic3—21no—
179D24—22no—
180C3×C16—62no—
181C3⋊C16—22no—
182C22×C32—21no—
183C22×C32—42no—
184C22×C32—41no—
185C22×C32—82no—
186C22×C32C2×D811yescharacteristic
187C2×C3×D3—81no—
188C2×C3×D3—81no—
189C2×C3×D3—81no—
190C32×C4—21no—
191C32×C4—42no—
192C32×C4#12011yescharacteristic
193C32×C4AGL(1,ℤ/8)11yescharacteristic
194C3×Dic3—81no—
195C2×#41—21no—
196#137—22no—
197AGL(1,ℤ/8)—22no—
198#135—21no—
199#135C2×C3×D312yes—
200C2×D8—31no—
201C2×M16C2×C3×D311yescharacteristic
202#121—61no—
203SD32—62no—
204D16—62no—
205M32—32no—
206#106—31no—
207#102—61no—
208C23×C3—21no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—121no—
213C23×C3#21311yescharacteristic
214C23×C3C3×D811yescharacteristic
215C22×D3—41no—
216C3×Q8—24no—
217C3×Q8—22no—
218C3×Q8—22no—
219C3×Q8—42no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×D4—24no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—44no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—44no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—121no—
235C3×D4—121no—
236C3×D4—242no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—44no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—121no—
255C2×C3×C4#21311yescharacteristic
256C2×C3×C4C3×D811yescharacteristic
257C2×C3×C4C2×#6711yescharacteristic
258C2×C3×C4C2×C3×D411yescharacteristic
259#67—82no—
260C2×Dic3—41no—
261D12—41no—
262D12—41no—
263C3×C8—22no—
264C3×C8—21no—
265C3×C8—21no—
266C3×C8—22no—
267C3×C8C2×#6712yes—
268C3×C8C2×C3×D412yes—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—82no—
272C2×C32#44711yescharacteristic
273C3×D3—161no—
274#41—22no—
275#41—22no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—61no—
282C2×D4—121no—
283C22×C4—41no—
284C22×C4C3×#6711yescharacteristic
285Q16—21no—
286Q16C3×#6712yescomplemented
287SD16—22no—
288SD16—22no—
289D8—21no—
290D8—62no—
291D8C3×#6712yescomplemented
292M16—22no—
293C2×C8C3×#6711yescharacteristic
294C2×C8C22×C3×D311yescharacteristic
295C4⋊C4—61no—
296C22⋊C4—121no—
297C16—62no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—44no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—81no—
308C22×C3—84no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3—241no—
312C22×C3—241no—
313C22×C3—241no—
314C22×C3C2×#21311yescharacteristic
315C22×C3C2×C3×D811yescharacteristic
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—81no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—44no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4—241no—
328C3×C4#93211yescharacteristic
329C3×C4#95311yescharacteristic
330C3×C4C3×#12011yescharacteristic
331C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
332Dic3—81no—
333C32#219011yescharacteristic, complemented, Sylow
334C23—41no—
335C23—121no—
336C23C3×#21311yescharacteristic
337Q8—22no—
338Q8—41no—
339D4—22no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—121no—
344D4—121no—
345D4—242no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—121no—
352C2×C4C3×#21311yescharacteristic
353C2×C4C2×C3×#6711yescharacteristic
354C8—21no—
355C8C2×C3×#6712yes—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—41no—
360C2×C3—42no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—84no—
364C2×C3—82no—
365C2×C3—82no—
366C2×C3—481no—
367C2×C3#521711yescharacteristic
368C2×C3C3×#44711yescharacteristic
369D3—161no—
370C22—21no—
371C22—41no—
372C22—42no—
373C22—82no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22C2×C3×#21311yescharacteristic
378C4—21no—
379C4—42no—
380C4—241no—
381C4C3×#93211yescharacteristic
382C4C3×#95311yescharacteristic
383C3—21no—
384C3#7211211yescharacteristic, direct product factor
385C3C3×#219011yescharacteristic, complemented
386C2—21no—
387C2—41no—
388C2—82no—
389C2—481no—
390C2C3×#521711yescharacteristic
subgroupG/H
centerC2×C3#5217
commutator, cosocleC2×C3×C8C23×C3
FrattiniC2×C8C22×C3×D3
FittingC32×#575C2
socleC2×C32#447
hypercenterC3×#575D3
hypocenterC3C3×#2190
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ C3×#575

Complements

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

NH ≅ G/Ntype
1. C32×#575C2semidirect
2. C3×#5193C2semidirect
3. C3×#5135C2semidirect
4. C3×#4566C2semidirect
5. C3×#4553C2semidirect
7. #72112C3direct (N × H)
15. C2×C3×D24C22semidirect
19. C3×#817C22semidirect
20. C3×#812C22semidirect
22. C3×#575D3semidirect
23. C3×#575C2×C3semidirect
26. #5193C2×C3semidirect
27. #5135C2×C3semidirect
30. #4566C2×C3semidirect
31. #4553C2×C3semidirect
44. C32×Q16D4semidirect
48. C32×D8D4semidirect
78. C2×D24C22×C3semidirect
87. #817C22×C3semidirect
88. #812C22×C3semidirect
111. #575C3×D3semidirect
151. C3×Q16#67semidirect
152. C3×Q16C3×D4semidirect
164. C3×D8#67semidirect
165. C3×D8C3×D4semidirect
286. Q16C3×#67semidirect
291. D8C3×#67semidirect
333. C32#2190semidirect
384. C3#72112direct (N × H)
385. C3C3×#2190semidirect

Maximal subgroups

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

subgroupindex
C32×#5752
C3×#51932
C3×#51352
C3×#45662
C3×#45532
C3×#21903
#721123