Finite Group Databasebeta

← C3×#68112 #11778668

Subgroups — C3×#68112 #11778668

Order 1152.

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

normal subgroups only

This group has 625 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×#5493C211yesmaximal, characteristic
2C3×#4949C211yesmaximal, characteristic, complemented
3C3×#4735C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic, complemented
5C3×#4470C212yesmaximal, complemented
6C3×#1613—31nomaximal
7#68112C311yesmaximal, characteristic, direct product factor
8C22×C3×#67—21no—
9C2×C32×C22⋊C4C2211yescharacteristic
10C2×C3×#217—21no—
11C3×#930—21no—
12C3×#930C2212yes—
13C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
14C2×C3×Dic3⋊C4—21no—
15C3×#886—22no—
16C3×#882—22no—
17C32×C22⋊C8C2212yescomplemented
18C3×C3⋊#95C2211yescharacteristic, complemented
19C3×#821—22no—
20C3×#817—22no—
21C3×#534—31no—
22#5493C2×C311yescharacteristic
23C3×#481—32no—
24C3×#399—31no—
25#4949C2×C311yescharacteristic, complemented
26#4735C2×C312yescomplemented
27C3×#327—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic, complemented
30C3×#323C2×C311yescharacteristic, complemented
31#4470C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C23×C3×D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—22no—
37C2×C3×#67—22no—
38C2×C3×#67—42no—
39C2×C3×#67—42no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C2×C3×D12—21no—
44C2×C3×D12D412yes—
45C2×C32×C8—22no—
46C32×C22⋊C4—22no—
47C3×#217—42no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3D412yes—
53C3×Dic3⋊C4—42no—
54#1613—31noSylow
55C22×C3×D4—61no—
56C22×#67—21no—
57C3×#123—62no—
58C3×#121—32no—
59C3×#121—61no—
60C3×#120—62no—
61C2×C3×C4⋊C4—31no—
62C2×C3×C4⋊C4—61no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—61no—
65C2×C3×C22⋊C4C2×D311yescharacteristic
66C2×C3×C22⋊C4C22×C311yescharacteristic
67C2×#217—21no—
68#930—21no—
69#930C22×C312yes—
70C2×C4⋊Dic3C22×C311yescharacteristic, complemented
71C2×Dic3⋊C4—21no—
72#886—22no—
73#882—22no—
74C3×#106—62no—
75C3×#102—62no—
76C3×C22⋊C8—22no—
77C3×C22⋊C8C2×D312yescomplemented
78C3×C22⋊C8C22×C312yescomplemented
79C3×#95—31no—
80C3⋊#95C22×C311yescharacteristic, complemented
81#821—22no—
82#817—22no—
83C23×C32—21no—
84C23×C32—21no—
85C23×C32—41no—
86C23×C32—41no—
87C23×C32SD1612yes—
88C23×C32C2×D411yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—42no—
91C22×C3×D3—81no—
92C22×C3×D3—81no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D412yes—
97C3×#67—44no—
98C3×#67—42no—
99C3×#67—42no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C3×D12—42no—
109C3×D12—81no—
110C32×C8—42no—
111#534—31no—
112#481—32no—
113#399—31no—
114#327—32no—
115#323C3×D311yescharacteristic, complemented
116C24×C3—21no—
117C24×C3—61no—
118C24×C3D1211yescharacteristic
119C24×C3C3×D411yescharacteristic
120C23×D3—21no—
121C2×C3×D4—32no—
122C2×C3×D4—61no—
123C2×C3×D4—62no—
124C2×C3×D4—62no—
125C2×C3×D4—122no—
126C2×C3×D4—122no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4—61no—
132C22×C3×C4D1211yescharacteristic
133C22×C3×C4C3×D411yescharacteristic
134C22×C3×C4C22×D311yescharacteristic
135C22×C3×C4C23×C311yescharacteristic
136C2×#67—22no—
137C2×#67—22no—
138C2×#67—42no—
139C2×#67—42no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C2×D12—21no—
144C2×D12C3×D412yes—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—42no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—62no—
151C3×C4⋊C4—122no—
152C3×C22⋊C4—22no—
153C3×C22⋊C4—22no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—62no—
157C3×C22⋊C4—122no—
158#217—42no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—22no—
161C4⋊Dic3—21no—
162C4⋊Dic3—22no—
163C4⋊Dic3C3×D412yes—
164Dic3⋊C4—42no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—42no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—81no—
172C22×C32#12011yescharacteristic
173C22×C32C2×SD1611yescharacteristic
174C22×C32AGL(1,ℤ/8)11yescharacteristic
175C2×C3×D3—42no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—84no—
179C32×C4—22no—
180C32×C4—41no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—81no—
184C22×D4—61no—
185#123—62no—
186#121—32no—
187#121—61no—
188#120—62no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4C2×C3×D311yescharacteristic
193#106—62no—
194#102—62no—
195C22⋊C8C2×C3×D312yescomplemented
196#95—31no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—22no—
202C23×C3—21no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—62no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3—122no—
213C23×C3—241no—
214C23×C3—241no—
215C23×C3#20412yes—
216C23×C3C3×SD1612yes—
217C23×C3C2×D1211yescharacteristic
218C23×C3C2×C3×D411yescharacteristic
219C22×D3—22no—
220C22×D3—42no—
221C22×D3—81no—
222C22×D3—81no—
223C3×D4—122no—
224C3×D4—124no—
225C3×D4—122no—
226C3×D4—122no—
227C3×D4—241no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—121no—
241C2×C3×C4—122no—
242C2×C3×C4—121no—
243C2×C3×C4—121no—
244C2×C3×C4—121no—
245C2×C3×C4—121no—
246C2×C3×C4D3×D412yes—
247C2×C3×C4C2×C3×D412yes—
248#67—44no—
249#67—42no—
250#67—42no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259D12—42no—
260D12—81no—
261C3×C8—42no—
262C3×C8—42no—
263C3×C8—82no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—42no—
267C2×C32#45011yescharacteristic
268C2×C32#45311yescharacteristic
269C2×C32#45811yescharacteristic
270C3×D3—82no—
271C24—61no—
272C24C3×D1211yescharacteristic
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—122no—
278C2×D4—122no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4C3×D1211yescharacteristic
283C22×C4C22×C3×D311yescharacteristic
284C2×C8—22no—
285C4⋊C4—32no—
286C4⋊C4—61no—
287C4⋊C4—62no—
288C4⋊C4—122no—
289C22⋊C4—22no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—122no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—82no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—82no—
317C22×C3—122no—
318C22×C3—122no—
319C22×C3—122no—
320C22×C3—244no—
321C22×C3#88211yescharacteristic
322C22×C3C2×#20411yescharacteristic
323C22×C3#90811yescharacteristic
324C22×C3C3×#12011yescharacteristic
325C22×C3C2×C3×SD1611yescharacteristic
326C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—84no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—81no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4—241no—
340Dic3—42no—
341Dic3—42no—
342Dic3—81no—
343C32#161311yescharacteristic, complemented, Sylow
344C23—21no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—62no—
349C23—122no—
350C23—241no—
351C23—241no—
352C23C3×#20412yes—
353C23C2×C3×D1211yescharacteristic
354D4—122no—
355D4—124no—
356D4—122no—
357D4—122no—
358D4—241no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4C3×D3×D412yes—
371C8—42no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—42no—
382C2×C3—42no—
383C2×C3—82no—
384C2×C3—242no—
385C2×C3#473111yescharacteristic
386C2×C3#474211yescharacteristic
387C2×C3#474711yescharacteristic
388C2×C3C3×#45011yescharacteristic
389C2×C3C3×#45311yescharacteristic
390C2×C3C3×#45811yescharacteristic
391D3—82no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—122no—
400C22—122no—
401C22—122no—
402C22—244no—
403C22C3×#88211yescharacteristic
404C22C2×C3×#20411yescharacteristic
405C22C3×#90811yescharacteristic
406C4—22no—
407C4—41no—
408C4—122no—
409C4—122no—
410C4—241no—
411C3—21no—
412C3#6811211yescharacteristic, direct product factor
413C3C3×#161311yescharacteristic, complemented
414C2—21no—
415C2—21no—
416C2—42no—
417C2—242no—
418C2C3×#473111yescharacteristic
419C2C3×#474211yescharacteristic
420C2C3×#474711yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC3×#5493C2
hypercenterC3×#323D3
hypocenterC3C3×#1613
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
2. C3×#4949C2semidirect
3. C3×#4735C2semidirect
4. C32×#323C2semidirect
5. C3×#4470C2semidirect
7. #68112C3direct (N × H)
13. C2×C3×C4⋊Dic3C22semidirect
17. C32×C22⋊C8C22semidirect
18. C3×C3⋊#95C22semidirect
25. #4949C2×C3semidirect
26. #4735C2×C3semidirect
29. C3×#323D3semidirect
30. C3×#323C2×C3semidirect
31. #4470C2×C3semidirect
70. C2×C4⋊Dic3C22×C3semidirect
77. C3×C22⋊C8C2×D3semidirect
78. C3×C22⋊C8C22×C3semidirect
80. C3⋊#95C22×C3semidirect
115. #323C3×D3semidirect
195. C22⋊C8C2×C3×D3semidirect
343. C32#1613semidirect
412. C3#68112direct (N × H)
413. C3C3×#1613semidirect

Maximal subgroups

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

subgroupindex
C3×#54932
C3×#49492
C3×#47352
C32×#3232
C3×#44702
C3×#16133
#681123