Finite Group Databasebeta

← C3×#71467 #11798046

Subgroups — C3×#71467 #11798046

Order 1152.

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

normal subgroups only

This group has 581 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
1C2×C3×D3×D8C211yesmaximal, characteristic, complemented
2C2×C32×D16C211yesmaximal, characteristic, complemented
3C3×#5157C211yesmaximal, characteristic, complemented
4C2×C3×#826C211yesmaximal, characteristic, complemented
5C3×#4562C211yesmaximal, characteristic, complemented
6C3×#4509C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊C16C211yesmaximal, characteristic, complemented
8C3×#2184—31nomaximal
9#71467C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D4—21no—
11C2×C32×D8C2211yescharacteristic
12C2×C32×D8C2211yescharacteristic, complemented
13C3×#938—21no—
14C2×C3×#213—21no—
15C3×D3×D8—22no—
16C3×D3×D8—22no—
17C2×C3×D24C2211yescharacteristic
18C2×C3×D3×C8C2211yescharacteristic
19C32×D16—22no—
20C2×C32×C16C2211yescharacteristic, complemented
21C3×#832—21no—
22C3×#826—22no—
23C3×#818C2211yescharacteristic, complemented
24C2×C3×C3⋊C16C2211yescharacteristic, complemented
25C22×C3×D8—31no—
26C2×D3×D8C2×C311yescharacteristic, complemented
27C2×C3×D16—21no—
28C2×C3×D16—31no—
29C2×C3×D16D311yescharacteristic, complemented
30C2×C3×D16C2×C311yescharacteristic, complemented
31C3×#466—31no—
32#5157C2×C311yescharacteristic, complemented
33C2×#826C2×C311yescharacteristic, complemented
34C3×#357—31no—
35C3×#357—31no—
36C3×C22⋊C16—31no—
37#4562C2×C311yescharacteristic, complemented
38#4509C2×C311yescharacteristic, complemented
39(C2×D3)⋊C16C2×C311yescharacteristic, complemented
40C23×C3×D3—41no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C3×#67—41no—
44C2×C3×#67—41no—
45C3×D3×D4—24no—
46C3×D3×D4—42no—
47C2×C3×D12—21no—
48C2×C3×C4×D3D411yescharacteristic
49C32×D8—21no—
50C32×D8—21no—
51C32×D8—21no—
52C32×D8D412yescomplemented
53C2×C32×C8C2311yescharacteristic
54C3×#217—41no—
55C3×#213—24no—
56C3×C4⋊Dic3—21no—
57C2×C3×C3⋊C8D411yescharacteristic
58C3×D24—21no—
59C3×D24D412yescomplemented
60C3×D3×C8—21no—
61C3×D3×C8—21no—
62C32×C16—21no—
63C3×C3⋊C16—21no—
64#2184—31noSylow
65C22×C3×D4—61no—
66C2×D3×D4—21no—
67C2×C3×D8—21no—
68C2×C3×D8—21no—
69C2×C3×D8—31no—
70C2×C3×D8—61no—
71C2×C3×D8—62no—
72C2×C3×D8—62no—
73C2×C3×D8C2×D311yescharacteristic
74C2×C3×D8C2×D311yescharacteristic, complemented
75C2×C3×D8C22×C311yescharacteristic
76C2×C3×D8C22×C311yescharacteristic, complemented
77C22×C3×C8—31no—
78C3×#121—61no—
79#938—21no—
80C2×#213—21no—
81D3×D8—22no—
82D3×D8—22no—
83C2×D24C22×C311yescharacteristic
84C2×D3×C8C22×C311yescharacteristic
85C3×D16—24no—
86C3×D16—22no—
87C3×D16—22no—
88C3×D16—62no—
89C2×C3×C16—21no—
90C2×C3×C16—31no—
91C2×C3×C16C2×D311yescharacteristic, complemented
92C2×C3×C16C22×C311yescharacteristic, complemented
93C3×#107—31no—
94C3×#102—61no—
95#832—21no—
96#826—22no—
97#818C22×C311yescharacteristic, complemented
98C2×C3⋊C16C22×C311yescharacteristic, complemented
99C23×C32—41no—
100C23×C32—41no—
101C22×C3×D3—44no—
102C22×C3×D3—41no—
103C22×C3×D3—82no—
104C22×C3×D3—82no—
105C22×C3×D3D811yescharacteristic
106C32×D4—22no—
107C32×D4—41no—
108C32×D4—41no—
109C32×D4—41no—
110C2×C32×C4C2×D411yescharacteristic
111C3×#67—44no—
112C3×#67—82no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3D811yescharacteristic
115C3×D12—22no—
116C3×D12—41no—
117C3×C4×D3—21no—
118C3×C4×D3—21no—
119C32×C8C2×D411yescharacteristic
120C32×C8C2×D411yescharacteristic
121C3×C3⋊C8—21no—
122C22×D8—31no—
123C2×D16—31no—
124C2×D16C3×D311yescharacteristic, complemented
125#466—31no—
126#357—31no—
127#357—31no—
128C22⋊C16—31no—
129C24×C3—121no—
130C23×D3—41no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—64no—
138C2×C3×D4—61no—
139C2×C3×D4—121no—
140C2×C3×D4—122no—
141C2×C3×D4—121no—
142C22×C3×C4—31no—
143C2×#67—41no—
144C2×#67—41no—
145D3×D4—24no—
146D3×D4—42no—
147C2×D12—21no—
148C2×C4×D3C3×D411yescharacteristic
149C3×D8—22no—
150C3×D8—21no—
151C3×D8—21no—
152C3×D8—21no—
153C3×D8—21no—
154C3×D8—21no—
155C3×D8—21no—
156C3×D8—22no—
157C3×D8—32no—
158C3×D8—41no—
159C3×D8—41no—
160C3×D8—64no—
161C3×D8—61no—
162C3×D8#6712yescomplemented
163C3×D8C3×D412yescomplemented
164C2×C3×C8—21no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—61no—
168C2×C3×C8C22×D311yescharacteristic
169C2×C3×C8C23×C311yescharacteristic
170C3×C4⋊C4—61no—
171C3×C22⋊C4—121no—
172#217—41no—
173#213—24no—
174C4⋊Dic3—21no—
175C2×C3⋊C8C3×D411yescharacteristic
176D24—21no—
177D24C3×D412yescomplemented
178D3×C8—21no—
179D3×C8—21no—
180C3×C16—22no—
181C3×C16—21no—
182C3×C16—21no—
183C3×C16—61no—
184C3⋊C16—21no—
185C22×C32—42no—
186C22×C32—81no—
187C22×C32—81no—
188C22×C32—81no—
189C22×C32—81no—
190C22×C32—81no—
191C22×C32C2×D811yescharacteristic
192C2×C3×D3—21no—
193C2×C3×D3—21no—
194C2×C3×D3—42no—
195C2×C3×D3—88no—
196C2×C3×D3—81no—
197C2×C3×D3—81no—
198C2×C3×D3D1612yescomplemented
199C32×C4#12011yescharacteristic
200C32×C4AGL(1,ℤ/8)11yescharacteristic
201C3×Dic3—21no—
202C3×Dic3—81no—
203C22×D4—61no—
204C2×D8—31no—
205C2×D8—61no—
206C2×D8—62no—
207C2×D8—62no—
208C2×D8C2×C3×D311yescharacteristic
209C2×D8C2×C3×D311yescharacteristic, complemented
210C22×C8—31no—
211#121—61no—
212D16—22no—
213D16—62no—
214C2×C16—31no—
215C2×C16C2×C3×D311yescharacteristic, complemented
216#107—31no—
217#102—61no—
218C23×C3—31no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3—124no—
226C23×C3—121no—
227C23×C3—242no—
228C23×C3—242no—
229C22×D3—44no—
230C22×D3—41no—
231C22×D3—82no—
232C22×D3—82no—
233C22×D3C3×D811yescharacteristic
234C3×D4—22no—
235C3×D4—22no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—62no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—124no—
248C3×D4—121no—
249C3×D4—242no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—121no—
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4C2×C3×D411yescharacteristic
257#67—44no—
258#67—82no—
259C2×Dic3—41no—
260C2×Dic3C3×D811yescharacteristic
261D12—22no—
262D12—41no—
263C4×D3—21no—
264C4×D3—21no—
265C3×C8—21no—
266C3×C8—21no—
267C3×C8—61no—
268C3×C8D3×D411yescharacteristic
269C3×C8C2×#6711yescharacteristic
270C3×C8C2×C3×D411yescharacteristic
271C3×C8C2×C3×D411yescharacteristic
272C3⋊C8—21no—
273C2×C32—82no—
274C2×C32—161no—
275C2×C32#44711yescharacteristic
276C2×C32C2×D1611yescharacteristic
277C2×C32#50911yescharacteristic
278C3×D3—22no—
279C3×D3—82no—
280C24—121no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—64no—
284C2×D4—61no—
285C2×D4—121no—
286C2×D4—122no—
287C2×D4—121no—
288C22×C4—31no—
289D8—21no—
290D8—21no—
291D8—21no—
292D8—32no—
293D8—64no—
294D8—61no—
295D8C3×#6712yescomplemented
296C2×C8—31no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8C22×C3×D311yescharacteristic
300C4⋊C4—61no—
301C22⋊C4—121no—
302C16—21no—
303C16—61no—
304C22×C3—21no—
305C22×C3—32no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—122no—
323C22×C3—161no—
324C22×C3—161no—
325C22×C3—161no—
326C22×C3—161no—
327C22×C3—248no—
328C22×C3—241no—
329C22×C3—241no—
330C22×C3D3×D811yescharacteristic
331C22×C3C2×C3×D811yescharacteristic
332C2×D3—21no—
333C2×D3—21no—
334C2×D3—42no—
335C2×D3—88no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3C3×D1612yescomplemented
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—61no—
342C3×C4—241no—
343C3×C4#91111yescharacteristic
344C3×C4#93711yescharacteristic
345C3×C4C3×#12011yescharacteristic
346C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
347Dic3—21no—
348Dic3—81no—
349C32#218411yescharacteristic, complemented, Sylow
350C23—31no—
351C23—41no—
352C23—41no—
353C23—124no—
354C23—121no—
355C23—242no—
356C23—242no—
357D4—22no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—62no—
362D4—124no—
363D4—121no—
364D4—242no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—121no—
369C2×C4C3×D3×D411yescharacteristic
370C8—61no—
371C8C3×D3×D411yescharacteristic
372C8C2×C3×#6711yescharacteristic
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—62no—
377C2×C3—82no—
378C2×C3—82no—
379C2×C3—162no—
380C2×C3—162no—
381C2×C3—161no—
382C2×C3—161no—
383C2×C3—242no—
384C2×C3D3×D1611yescharacteristic
385C2×C3#491111yescharacteristic
386C2×C3#515611yescharacteristic
387C2×C3C3×#44711yescharacteristic
388C2×C3C2×C3×D1611yescharacteristic
389C2×C3C3×#50911yescharacteristic
390D3—22no—
391D3—82no—
392C22—32no—
393C22—42no—
394C22—61no—
395C22—61no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—122no—
402C22—248no—
403C22—241no—
404C22—241no—
405C22C3×D3×D811yescharacteristic
406C4—61no—
407C4—241no—
408C4C3×#91111yescharacteristic
409C4C3×#93711yescharacteristic
410C3—21no—
411C3#7146711yescharacteristic, direct product factor
412C3C3×#218411yescharacteristic, complemented
413C2—62no—
414C2—82no—
415C2—161no—
416C2—242no—
417C2C3×D3×D1611yescharacteristic
418C2C3×#491111yescharacteristic
419C2C3×#515611yescharacteristic
subgroupG/H
centerC22×C3D3×D8
commutator, cosocleC2×C3×C8C23×C3
FrattiniC2×C8C22×C3×D3
FittingC2×C32×D16C2
socleC22×C32C2×D8
hypercenterC2×C3×D16D3
hypocenterC3C3×#2184
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×D16

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×D8C2semidirect
2. C2×C32×D16C2semidirect
3. C3×#5157C2semidirect
4. C2×C3×#826C2semidirect
5. C3×#4562C2semidirect
6. C3×#4509C2semidirect
7. C3×(C2×D3)⋊C16C2semidirect
9. #71467C3direct (N × H)
12. C2×C32×D8C22semidirect
20. C2×C32×C16C22semidirect
23. C3×#818C22semidirect
24. C2×C3×C3⋊C16C22semidirect
26. C2×D3×D8C2×C3semidirect
29. C2×C3×D16D3semidirect
30. C2×C3×D16C2×C3semidirect
32. #5157C2×C3semidirect
33. C2×#826C2×C3semidirect
37. #4562C2×C3semidirect
38. #4509C2×C3semidirect
39. (C2×D3)⋊C16C2×C3semidirect
52. C32×D8D4semidirect
59. C3×D24D4semidirect
74. C2×C3×D8C2×D3semidirect
76. C2×C3×D8C22×C3semidirect
91. C2×C3×C16C2×D3semidirect
92. C2×C3×C16C22×C3semidirect
97. #818C22×C3semidirect
98. C2×C3⋊C16C22×C3semidirect
124. C2×D16C3×D3semidirect
162. C3×D8#67semidirect
163. C3×D8C3×D4semidirect
177. D24C3×D4semidirect
198. C2×C3×D3D16semidirect
209. C2×D8C2×C3×D3semidirect
215. C2×C16C2×C3×D3semidirect
295. D8C3×#67semidirect
338. C2×D3C3×D16semidirect
349. C32#2184semidirect
411. C3#71467direct (N × H)
412. C3C3×#2184semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×D82
C2×C32×D162
C3×#51572
C2×C3×#8262
C3×#45622
C3×#45092
C3×(C2×D3)⋊C162
C3×#21843
#714673