Finite Group Databasebeta

← C3×#68133 #11778329

Subgroups — C3×#68133 #11778329

Order 1152.

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

normal subgroups only

This group has 603 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
1C3×#5510C211yesmaximal, characteristic
2C3×#4979C211yesmaximal, characteristic, complemented
3C3×#4738C212yesmaximal, complemented
4C32×#324C211yesmaximal, characteristic, complemented
5C3×#4470C212yesmaximal, complemented
6C3×#1604—31nomaximal
7#68133C311yesmaximal, characteristic, direct product factor
8C2×C3×#235—21no—
9C22×C3×Dic6C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C3×#930—21no—
12C3×#930C2212yes—
13C2×C3×Dic3⋊C4—21no—
14C2×C3×#204—22no—
15C3×#897—22no—
16C3×#895—22no—
17C32×C22⋊C8C2212yes—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2211yescharacteristic
20C3×#816—22no—
21C3×#537—31no—
22#5510C2×C311yescharacteristic
23C3×#448—32no—
24C3×#395—31no—
25#4979C2×C311yescharacteristic, complemented
26#4738C2×C312yescomplemented
27C3×#327—32no—
28C3×#324—21no—
29C3×#324D311yescharacteristic, complemented
30C3×#324C2×C311yescharacteristic, complemented
31#4470C2×C312yescomplemented
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C2×C3×#67—22no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C3×#235—42no—
39C3×#235—42no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12D412yes—
43C2×C3×C4×D3—22no—
44C2×C3×Dic6—22no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6—42no—
47C2×C32×C8—22no—
48C32×C4⋊C4—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×(C2×D3)⋊C4—22no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D412yes—
53C3×Dic3⋊C4—42no—
54C3×#204—44no—
55#1604—31noSylow
56C2×C3×#41—61no—
57C22×C3×Q8—31no—
58C2×#235—21no—
59C22×Dic6C22×C311yescharacteristic
60C2×C3×SD16—62no—
61C3×C22⋊Q8—62no—
62C3×#121—32no—
63C3×#121—62no—
64C3×#121—61no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—61no—
67C2×C3×C4⋊C4C2×D311yescharacteristic
68C2×C3×C4⋊C4C22×C311yescharacteristic
69#930—21no—
70#930C22×C312yes—
71C2×Dic3⋊C4—21no—
72C2×#204—22no—
73#897—22no—
74#895—22no—
75C3×#103—62no—
76C3×C22⋊C8—22no—
77C3×C22⋊C8C2×D312yes—
78C3×C22⋊C8C22×C312yes—
79C3×#95—31no—
80C3×#95—61no—
81C3⋊#95—21no—
82C3⋊#95C22×C311yescharacteristic
83#816—22no—
84C23×C32C2×D411yescharacteristic
85C22×C3×D3—22no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4SD1612yescomplemented
92C2×C32×C4C2×D412yes—
93C3×#67—42no—
94C3×#67—42no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C3×D12—42no—
103C3×D12—42no—
104C3×D12—81no—
105C3×C4×D3—44no—
106C3×Dic6—24no—
107C3×Dic6—42no—
108C3×Dic6—42no—
109C32×C8—42no—
110#537—31no—
111#448—32no—
112#395—31no—
113#327—32no—
114#324C3×D311yescharacteristic, complemented
115C3×#41—122no—
116C3×#41—122no—
117C2×C3×Q8—61no—
118C2×C3×Q8—62no—
119C2×C3×Q8—122no—
120C2×C3×D4—32no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—62no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—62no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4D1211yescharacteristic
131C22×C3×C4D1211yescharacteristic
132C22×C3×C4C3×D411yescharacteristic
133C22×C3×C4C3×D411yescharacteristic
134C22×C3×C4C22×D311yescharacteristic
135C22×C3×C4C23×C311yescharacteristic
136C2×#67—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139#235—42no—
140#235—42no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12C3×D412yes—
144C2×C4×D3—22no—
145C2×Dic6—22no—
146C2×Dic6—21no—
147C2×Dic6—42no—
148C3×SD16—124no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—42no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—61no—
157C3×C4⋊C4—122no—
158C3×C22⋊C4—62no—
159C3×C22⋊C4—62no—
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3C3×D412yes—
164Dic3⋊C4—42no—
165#204—44no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32#12011yescharacteristic
169C22×C32C2×SD1611yescharacteristic
170C22×C32AGL(1,ℤ/8)11yescharacteristic
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—42no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—41no—
177C3×Dic3—42no—
178C3×Dic3—42no—
179C3×Dic3—42no—
180C2×#41—61no—
181C22×Q8—31no—
182C2×SD16—62no—
183C22⋊Q8—62no—
184#121—32no—
185#121—62no—
186#121—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4C2×C3×D311yescharacteristic
189#103—62no—
190C22⋊C8C2×C3×D312yes—
191#95—31no—
192#95—61no—
193C23×C3—21no—
194C23×C3—62no—
195C23×C3C2×D1211yescharacteristic
196C23×C3C2×C3×D411yescharacteristic
197C22×D3—22no—
198C3×Q8—64no—
199C3×Q8—122no—
200C3×Q8—122no—
201C3×D4—122no—
202C3×D4—122no—
203C3×D4—122no—
204C3×D4—122no—
205C3×D4—241no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—124no—
227C2×C3×C4—121no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—121no—
231C2×C3×C4#20412yescomplemented
232C2×C3×C4C3×SD1612yescomplemented
233C2×C3×C4D3×D412yes—
234C2×C3×C4C2×C3×D412yes—
235#67—42no—
236#67—42no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244D12—42no—
245D12—42no—
246D12—81no—
247C4×D3—44no—
248Dic6—24no—
249Dic6—42no—
250Dic6—42no—
251C3×C8—42no—
252C3×C8—42no—
253C3×C8—82no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32#45011yescharacteristic
257C2×C32#45611yescharacteristic
258C2×C32B(4,𝔽2)11yescharacteristic
259C3×D3—82no—
260#41—122no—
261#41—122no—
262C2×Q8—61no—
263C2×Q8—62no—
264C2×Q8—122no—
265C2×D4—32no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—62no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4C3×D1211yescharacteristic
273C22×C4C3×D1211yescharacteristic
274C22×C4C22×C3×D311yescharacteristic
275SD16—124no—
276C2×C8—22no—
277C4⋊C4—22no—
278C4⋊C4—32no—
279C4⋊C4—61no—
280C4⋊C4—122no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3#88211yescharacteristic
296C22×C3C2×#20411yescharacteristic
297C22×C3#90811yescharacteristic
298C22×C3C3×#12011yescharacteristic
299C22×C3C2×C3×SD1611yescharacteristic
300C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—42no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—81no—
313C3×C4—122no—
314C3×C4—122no—
315C3×C4—122no—
316Dic3—42no—
317Dic3—42no—
318Dic3—42no—
319C32#160411yescharacteristic, complemented, Sylow
320C23—62no—
321C23C2×C3×D1211yescharacteristic
322Q8—64no—
323Q8—122no—
324Q8—122no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—124no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4C3×#20412yescomplemented
344C2×C4C3×D3×D412yes—
345C8—42no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—242no—
356C2×C3#473111yescharacteristic
357C2×C3#474111yescharacteristic
358C2×C3#475211yescharacteristic
359C2×C3C3×#45011yescharacteristic
360C2×C3C3×#45611yescharacteristic
361C2×C3C3×B(4,𝔽2)11yescharacteristic
362D3—82no—
363C22—21no—
364C22—21no—
365C22—122no—
366C22—122no—
367C22—122no—
368C22C3×#88211yescharacteristic
369C22C2×C3×#20411yescharacteristic
370C22C3×#90811yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—122no—
375C4—122no—
376C4—122no—
377C3—21no—
378C3#6813311yescharacteristic, direct product factor
379C3C3×#160411yescharacteristic, complemented
380C2—21no—
381C2—21no—
382C2—242no—
383C2C3×#473111yescharacteristic
384C2C3×#474111yescharacteristic
385C2C3×#475211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#324C2
socleC22×C32#120
prime radicalC3×#5510C2
hypercenterC3×#324D3
hypocenterC3C3×#1604
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#324

Complements

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

NH ≅ G/Ntype
2. C3×#4979C2semidirect
3. C3×#4738C2semidirect
4. C32×#324C2semidirect
5. C3×#4470C2semidirect
7. #68133C3direct (N × H)
25. #4979C2×C3semidirect
26. #4738C2×C3semidirect
29. C3×#324D3semidirect
30. C3×#324C2×C3semidirect
31. #4470C2×C3semidirect
91. C2×C32×C4SD16semidirect
114. #324C3×D3semidirect
231. C2×C3×C4#204semidirect
232. C2×C3×C4C3×SD16semidirect
319. C32#1604semidirect
343. C2×C4C3×#204semidirect
378. C3#68133direct (N × H)
379. C3C3×#1604semidirect

Maximal subgroups

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

subgroupindex
C3×#55102
C3×#49792
C3×#47382
C32×#3242
C3×#44702
C3×#16043
#681333