Finite Group Databasebeta

← C3×#71091 #11798297

Subgroups — C3×#71091 #11798297

Order 1152.

487 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C3×#5748C211yesmaximal, characteristic
2C32×#470C211yesmaximal, characteristic, complemented
3C3×#5179C212yesmaximal, complemented
4C3×#4553C211yesmaximal, characteristic, complemented
5C3×#4488C212yesmaximal, complemented
6C3×#2191—31nomaximal
7#71091C311yesmaximal, characteristic, direct product factor
8C2×C3×#235—21no—
9C2×C32×Q16C2211yescharacteristic, complemented
10C2×C32×M16C2211yescharacteristic
11C32×C22⋊Q8—21no—
12C3×#909—22no—
13C3×#908—22no—
14C3×#904—21no—
15C3×#904C2212yes—
16C32×#106C2211yescharacteristic, complemented
17C32×#103—21no—
18C3×#829—22no—
19C3×#828—22no—
20C3×#812C2212yescomplemented
21C3×#575—31no—
22#5748C2×C311yescharacteristic
23C3×#510—32no—
24C3×#470—21no—
25C3×#470D311yescharacteristic, complemented
26C3×#470C2×C311yescharacteristic, complemented
27#5179C2×C312yescomplemented
28C3×#360—32no—
29C3×#349—31no—
30#4553C2×C311yescharacteristic, complemented
31#4488C2×C312yescomplemented
32C2×C32×Q8—21no—
33C22×C32×C4D411yescharacteristic
34C2×C3×#67—41no—
35C3×#235—22no—
36C3×#235—22no—
37C3×#235—41no—
38C3×#235—41no—
39C2×C3×D12—21no—
40C2×C3×C4×D3—41no—
41C2×C3×Dic6—21no—
42C32×Q16—22no—
43C32×M16—22no—
44C2×C32×C8D411yescharacteristic
45C2×C32×C8C2311yescharacteristic
46C32×C4⋊C4—21no—
47C32×C4⋊C4—41no—
48C32×C22⋊C4—41no—
49C3×Dic12—21no—
50C3×Dic12D412yes—
51C3×D24—21no—
52C3×D24D412yes—
53C3×#204—22no—
54C3×#204—22no—
55C3×C3⋊C16—22no—
56#2191—31noSylow
57C2×C3×#41—61no—
58C2×#235—21no—
59C3×#137—62no—
60C3×AGL(1,ℤ/8)—62no—
61C3×#135—32no—
62C3×#135—61no—
63C2×C3×Q16—21no—
64C2×C3×Q16C2×D311yescharacteristic, complemented
65C2×C3×Q16C22×C311yescharacteristic, complemented
66C2×C3×M16—21no—
67C2×C3×M16C2×D311yescharacteristic
68C2×C3×M16C22×C311yescharacteristic
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8—21no—
72#909—22no—
73#908—22no—
74#904—21no—
75#904C22×C312yes—
76C3×Q32—62no—
77C3×SD32—62no—
78C3×M32—32no—
79C3×#106—21no—
80C3×#106C2×D311yescharacteristic, complemented
81C3×#106C22×C311yescharacteristic, complemented
82C3×#103—22no—
83C3×#103—21no—
84C3×#103—21no—
85#829—22no—
86#828—22no—
87#812C22×C312yescomplemented
88C23×C32D811yescharacteristic
89C22×C3×D3—41no—
90C32×Q8—41no—
91C32×Q8—41no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4D811yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C3×#67—42no—
99C3×#67—42no—
100C2×C3×Dic3—41no—
101C3×D12—22no—
102C3×D12—41no—
103C3×C4×D3—42no—
104C3×C4×D3—42no—
105C3×Dic6—22no—
106C3×Dic6—41no—
107C32×C8—21no—
108C32×C8C2×D412yes—
109#575—31no—
110#510—32no—
111#470C3×D311yescharacteristic, complemented
112#360—32no—
113#349—31no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—121no—
117C3×#41—121no—
118C2×C3×Q8—22no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—61no—
122C2×C3×D4—61no—
123C2×C3×D4—121no—
124C22×C3×C4—21no—
125C22×C3×C4—121no—
126C22×C3×C4#6711yescharacteristic
127C22×C3×C4C3×D411yescharacteristic
128C2×#67—41no—
129#235—22no—
130#235—22no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C3×Q16—22no—
137C3×Q16—22no—
138C3×Q16—32no—
139C3×Q16—42no—
140C3×Q16—61no—
141C3×SD16—62no—
142C3×SD16—62no—
143C3×D8—32no—
144C3×D8—61no—
145C3×M16—22no—
146C3×M16—22no—
147C3×M16—42no—
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8#6711yescharacteristic
151C2×C3×C8C3×D411yescharacteristic
152C2×C3×C8C22×D311yescharacteristic
153C2×C3×C8C23×C311yescharacteristic
154C3×C4⋊C4—22no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163Dic12—21no—
164Dic12C3×D412yes—
165D24—21no—
166D24C3×D412yes—
167#204—22no—
168#204—22no—
169C3×C16—62no—
170C3⋊C16—22no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32C2×D811yescharacteristic
174C2×C3×D3—42no—
175C2×C3×D3—82no—
176C32×C4—21no—
177C32×C4—81no—
178C32×C4—81no—
179C32×C4#12011yescharacteristic
180C32×C4AGL(1,ℤ/8)11yescharacteristic
181C3×Dic3—42no—
182C2×#41—61no—
183#137—62no—
184AGL(1,ℤ/8)—62no—
185#135—32no—
186#135—61no—
187C2×Q16C2×C3×D311yescharacteristic, complemented
188C2×M16C2×C3×D311yescharacteristic
189C22⋊Q8—21no—
190Q32—62no—
191SD32—62no—
192M32—32no—
193#106C2×C3×D311yescharacteristic, complemented
194#103—21no—
195C23×C3—21no—
196C23×C3—121no—
197C23×C3#21311yescharacteristic
198C23×C3C3×D811yescharacteristic
199C22×D3—41no—
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—41no—
206C3×Q8—62no—
207C3×Q8—121no—
208C3×D4—62no—
209C3×D4—121no—
210C3×D4—122no—
211C3×D4—122no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—122no—
227C2×C3×C4—121no—
228C2×C3×C4—122no—
229C2×C3×C4#21311yescharacteristic
230C2×C3×C4C3×D811yescharacteristic
231C2×C3×C4C2×#6711yescharacteristic
232C2×C3×C4C2×C3×D411yescharacteristic
233#67—42no—
234#67—42no—
235C2×Dic3—41no—
236D12—22no—
237D12—41no—
238C4×D3—42no—
239C4×D3—42no—
240Dic6—22no—
241Dic6—41no—
242C3×C8—21no—
243C3×C8—21no—
244C3×C8—22no—
245C3×C8—41no—
246C3×C8D3×D412yes—
247C3×C8C2×C3×D412yes—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32#44711yescharacteristic
251C3×D3—82no—
252#41—62no—
253#41—62no—
254#41—121no—
255#41—121no—
256C2×Q8—21no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—121no—
261C22×C4C3×#6711yescharacteristic
262Q16—22no—
263Q16—32no—
264Q16—61no—
265SD16—62no—
266SD16—62no—
267D8—32no—
268D8—61no—
269M16—22no—
270C2×C8C3×#6711yescharacteristic
271C2×C8C22×C3×D311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—41no—
274C22⋊C4—41no—
275C16—62no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—42no—
283C22×C3—122no—
284C22×C3—242no—
285C22×C3C2×#21311yescharacteristic
286C22×C3C2×C3×D811yescharacteristic
287C2×D3—42no—
288C2×D3—82no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—82no—
295C3×C4—82no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—122no—
301C3×C4#93711yescharacteristic
302C3×C4#94911yescharacteristic
303C3×C4C3×#12011yescharacteristic
304C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
305Dic3—42no—
306C32#219111yescharacteristic, complemented, Sylow
307C23—121no—
308C23C3×#21311yescharacteristic
309Q8—41no—
310Q8—41no—
311Q8—62no—
312Q8—121no—
313D4—62no—
314D4—121no—
315D4—122no—
316D4—122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—122no—
324C2×C4C3×#21311yescharacteristic
325C2×C4C2×C3×#6711yescharacteristic
326C8—21no—
327C8C3×D3×D412yes—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3—242no—
336C2×C3#503611yescharacteristic
337C2×C3C3×#44711yescharacteristic
338D3—82no—
339C22—21no—
340C22—41no—
341C22—122no—
342C22—242no—
343C22C2×C3×#21311yescharacteristic
344C4—21no—
345C4—81no—
346C4—81no—
347C4—122no—
348C4C3×#93711yescharacteristic
349C4C3×#94911yescharacteristic
350C3—21no—
351C3#7109111yescharacteristic, direct product factor
352C3C3×#219111yescharacteristic, complemented
353C2—21no—
354C2—41no—
355C2—242no—
356C2C3×#503611yescharacteristic
subgroupG/H
centerC2×C3#5036
commutator, cosocleC2×C3×C8C23×C3
FrattiniC2×C8C22×C3×D3
FittingC32×#470C2
socleC2×C32#447
prime radicalC3×#5748C2
hypercenterC3×#470D3
hypocenterC3C3×#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ C3×#470

Complements

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

NH ≅ G/Ntype
2. C32×#470C2semidirect
3. C3×#5179C2semidirect
4. C3×#4553C2semidirect
5. C3×#4488C2semidirect
7. #71091C3direct (N × H)
9. C2×C32×Q16C22semidirect
16. C32×#106C22semidirect
20. C3×#812C22semidirect
25. C3×#470D3semidirect
26. C3×#470C2×C3semidirect
27. #5179C2×C3semidirect
30. #4553C2×C3semidirect
31. #4488C2×C3semidirect
64. C2×C3×Q16C2×D3semidirect
65. C2×C3×Q16C22×C3semidirect
80. C3×#106C2×D3semidirect
81. C3×#106C22×C3semidirect
87. #812C22×C3semidirect
111. #470C3×D3semidirect
187. C2×Q16C2×C3×D3semidirect
193. #106C2×C3×D3semidirect
306. C32#2191semidirect
351. C3#71091direct (N × H)
352. C3C3×#2191semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C32×#4702
C3×#51792
C3×#45532
C3×#44882
C3×#21913
#710913