Finite Group Databasebeta

← C3×#68137 #11778781

Subgroups — C3×#68137 #11778781

Order 1152.

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

normal subgroups only

This group has 531 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×#4986C211yesmaximal, characteristic, complemented
3C3×#4735C212yesmaximal, complemented
4C32×#324C211yesmaximal, characteristic, complemented
5C3×#4470C212yesmaximal, complemented
6C3×#1616—31nomaximal
7#68137C311yesmaximal, characteristic, direct product factor
8C2×C3×#235—21no—
9C2×C32×C4⋊C4C2211yescharacteristic
10C3×#930—21no—
11C3×#930C2212yes—
12C2×C3×C4⋊Dic3C2211yescharacteristic
13C3×#897—22no—
14C3×#895—22no—
15C32×C22⋊C8C2212yes—
16C3×C3⋊#95—21no—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#821—22no—
20C3×#817—22no—
21C3×#537—31no—
22#5510C2×C311yescharacteristic
23C3×#481—32no—
24C3×#400—31no—
25#4986C2×C311yescharacteristic, complemented
26#4735C2×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—21no—
45C2×C32×C8—22no—
46C32×C4⋊C4—22no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—22no—
51C3×C4⋊Dic3D412yes—
52#1616—31noSylow
53C2×C3×#41—61no—
54C2×#235—21no—
55C3×C22⋊Q8—62no—
56C3×#121—32no—
57C3×#121—62no—
58C3×#121—61no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—31no—
61C2×C3×C4⋊C4C2×D311yescharacteristic
62C2×C3×C4⋊C4C22×C311yescharacteristic
63#930—21no—
64#930C22×C312yes—
65C2×C4⋊Dic3C22×C311yescharacteristic
66#897—22no—
67#895—22no—
68C3×#106—62no—
69C3×#102—62no—
70C3×C22⋊C8—22no—
71C3×C22⋊C8C2×D312yes—
72C3×C22⋊C8C22×C312yes—
73C3×#95—31no—
74C3×#95—61no—
75C3×#95—61no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95C22×C311yescharacteristic
79#821—22no—
80#817—22no—
81C23×C32C2×D411yescharacteristic
82C22×C3×D3—22no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4SD1612yes—
89C2×C32×C4C2×D412yes—
90C3×#67—42no—
91C3×#67—42no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C3×D12—42no—
99C3×D12—42no—
100C3×D12—81no—
101C3×C4×D3—44no—
102C3×Dic6—42no—
103C32×C8—42no—
104#537—31no—
105#481—32no—
106#400—31no—
107#327—32no—
108#324C3×D311yescharacteristic, complemented
109C3×#41—122no—
110C3×#41—122no—
111C2×C3×Q8—61no—
112C2×C3×D4—32no—
113C2×C3×D4—61no—
114C2×C3×D4—61no—
115C2×C3×D4—62no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—62no—
120C22×C3×C4—61no—
121C22×C3×C4—61no—
122C22×C3×C4D1211yescharacteristic
123C22×C3×C4D1211yescharacteristic
124C22×C3×C4C3×D411yescharacteristic
125C22×C3×C4C3×D411yescharacteristic
126C22×C3×C4C22×D311yescharacteristic
127C22×C3×C4C23×C311yescharacteristic
128C2×#67—22no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131#235—42no—
132#235—42no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12C3×D412yes—
136C2×C4×D3—22no—
137C2×Dic6—21no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—42no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—32no—
144C3×C4⋊C4—42no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—62no—
147C3×C22⋊C4—62no—
148C3×C22⋊C4—62no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3C3×D412yes—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32#12011yescharacteristic
157C22×C32C2×SD1611yescharacteristic
158C22×C32AGL(1,ℤ/8)11yescharacteristic
159C2×C3×D3—42no—
160C2×C3×D3—42no—
161C2×C3×D3—42no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—41no—
165C3×Dic3—42no—
166C3×Dic3—81no—
167C3×Dic3—81no—
168C2×#41—61no—
169C22⋊Q8—62no—
170#121—32no—
171#121—62no—
172#121—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4C2×C3×D311yescharacteristic
175#106—62no—
176#102—62no—
177C22⋊C8C2×C3×D312yes—
178#95—31no—
179#95—61no—
180#95—61no—
181C23×C3—21no—
182C23×C3—62no—
183C23×C3C2×D1211yescharacteristic
184C23×C3C2×C3×D411yescharacteristic
185C22×D3—22no—
186C3×Q8—122no—
187C3×D4—122no—
188C3×D4—122no—
189C3×D4—122no—
190C3×D4—122no—
191C3×D4—241no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—62no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—124no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4—122no—
214C2×C3×C4—121no—
215C2×C3×C4—121no—
216C2×C3×C4#20412yes—
217C2×C3×C4C3×SD1612yes—
218C2×C3×C4D3×D412yes—
219C2×C3×C4C2×C3×D412yes—
220#67—42no—
221#67—42no—
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—42no—
229D12—42no—
230D12—81no—
231C4×D3—44no—
232Dic6—42no—
233C3×C8—42no—
234C3×C8—42no—
235C3×C8—82no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32#45011yescharacteristic
239C2×C32#45411yescharacteristic
240C2×C32#45811yescharacteristic
241C3×D3—82no—
242#41—122no—
243#41—122no—
244C2×Q8—61no—
245C2×D4—32no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—62no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4C3×D1211yescharacteristic
253C22×C4C3×D1211yescharacteristic
254C22×C4C22×C3×D311yescharacteristic
255C2×C8—22no—
256C4⋊C4—22no—
257C4⋊C4—32no—
258C4⋊C4—61no—
259C4⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—122no—
272C22×C3—122no—
273C22×C3—122no—
274C22×C3#88211yescharacteristic
275C22×C3C2×#20411yescharacteristic
276C22×C3#90811yescharacteristic
277C22×C3C3×#12011yescharacteristic
278C22×C3C2×C3×SD1611yescharacteristic
279C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
280C2×D3—42no—
281C2×D3—42no—
282C2×D3—42no—
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—42no—
288C3×C4—42no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4—122no—
293C3×C4—241no—
294C3×C4—241no—
295Dic3—42no—
296Dic3—81no—
297Dic3—81no—
298C32#161611yescharacteristic, complemented, Sylow
299C23—62no—
300C23C2×C3×D1211yescharacteristic
301Q8—122no—
302D4—122no—
303D4—122no—
304D4—122no—
305D4—122no—
306D4—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—62no—
313C2×C4—124no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4C3×#20412yes—
320C2×C4C3×D3×D412yes—
321C8—42no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—242no—
332C2×C3#473111yescharacteristic
333C2×C3#474211yescharacteristic
334C2×C3#475311yescharacteristic
335C2×C3C3×#45011yescharacteristic
336C2×C3C3×#45411yescharacteristic
337C2×C3C3×#45811yescharacteristic
338D3—82no—
339C22—21no—
340C22—21no—
341C22—122no—
342C22—122no—
343C22—122no—
344C22C3×#88211yescharacteristic
345C22C2×C3×#20411yescharacteristic
346C22C3×#90811yescharacteristic
347C4—22no—
348C4—22no—
349C4—41no—
350C4—122no—
351C4—241no—
352C4—241no—
353C3—21no—
354C3#6813711yescharacteristic, direct product factor
355C3C3×#161611yescharacteristic, complemented
356C2—21no—
357C2—21no—
358C2—242no—
359C2C3×#473111yescharacteristic
360C2C3×#474211yescharacteristic
361C2C3×#475311yescharacteristic
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×#1616
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×#4986C2semidirect
3. C3×#4735C2semidirect
4. C32×#324C2semidirect
5. C3×#4470C2semidirect
7. #68137C3direct (N × H)
25. #4986C2×C3semidirect
26. #4735C2×C3semidirect
29. C3×#324D3semidirect
30. C3×#324C2×C3semidirect
31. #4470C2×C3semidirect
108. #324C3×D3semidirect
298. C32#1616semidirect
354. C3#68137direct (N × H)
355. C3C3×#1616semidirect

Maximal subgroups

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

subgroupindex
C3×#55102
C3×#49862
C3×#47352
C32×#3242
C3×#44702
C3×#16163
#681373