Finite Group Databasebeta

← C3×#68116 #11778474

Subgroups — C3×#68116 #11778474

Order 1152.

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

normal subgroups only

This group has 505 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×#5483C211yesmaximal, characteristic
2C3×#4953C211yesmaximal, characteristic
3C3×#4721C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic
5C3×#4467C212yesmaximal, complemented
6C3×#1608—31nomaximal
7#68116C311yesmaximal, characteristic, direct product factor
8C2×C32×C22⋊C4C2211yescharacteristic
9C2×C3×#217—21no—
10C2×C3×#217—21no—
11C2×C3×C4⋊Dic3C2211yescharacteristic
12C3×#924—21no—
13C3×#924C2212yes—
14C3×#878—22no—
15C3×#878—22no—
16C32×C22⋊C8C2212yes—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#817—22no—
20C3×#816—22no—
21C3×#543—31no—
22#5483C2×C311yescharacteristic
23C3×#483—32no—
24C3×#397—31no—
25#4953C2×C311yescharacteristic
26#4721C2×C312yescomplemented
27C3×#328—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic
30C3×#323C2×C311yescharacteristic
31#4467C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C22×C3×Dic3—21no—
39C2×C3×Dic6—21no—
40C2×C3×Dic6D412yes—
41C2×C32×C8—22no—
42C32×C22⋊C4—22no—
43C3×#217—22no—
44C3×#217—22no—
45C3×#217—42no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3D412yes—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—22no—
51#1608—31noSylow
52C3×C22⋊Q8—32no—
53C3×C22⋊Q8—62no—
54C3×C22⋊Q8—62no—
55C3×C22⋊Q8—61no—
56C2×C3×C4⋊C4—31no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—61no—
59C2×C3×C22⋊C4—61no—
60C2×C3×C22⋊C4C2×D311yescharacteristic
61C2×C3×C22⋊C4C22×C311yescharacteristic
62C2×#217—21no—
63C2×#217—21no—
64C2×C4⋊Dic3C22×C311yescharacteristic
65#924—21no—
66#924C22×C312yes—
67#878—22no—
68#878—22no—
69C3×#106—62no—
70C3×#103—62no—
71C3×C22⋊C8—22no—
72C3×C22⋊C8C2×D312yes—
73C3×C22⋊C8C22×C312yes—
74C3×#95—31no—
75C3×#95—61no—
76C3⋊#95—21no—
77C3⋊#95C22×C311yescharacteristic
78#817—22no—
79#816—22no—
80C23×C32—21no—
81C23×C32—21no—
82C23×C32—41no—
83C23×C32—41no—
84C23×C32SD1612yes—
85C23×C32C2×D411yescharacteristic
86C2×C32×C4—21no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D412yes—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—81no—
99C2×C3×Dic3—81no—
100C3×Dic6—42no—
101C3×Dic6—81no—
102C32×C8—42no—
103#543—31no—
104#483—32no—
105#397—31no—
106#328—32no—
107#323C3×D311yescharacteristic
108C24×C3—21no—
109C24×C3D1211yescharacteristic
110C24×C3C3×D411yescharacteristic
111C2×C3×Q8—32no—
112C2×C3×Q8—61no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—61no—
116C22×C3×C4—61no—
117C22×C3×C4—61no—
118C22×C3×C4—61no—
119C22×C3×C4D1211yescharacteristic
120C22×C3×C4C3×D411yescharacteristic
121C22×C3×C4C22×D311yescharacteristic
122C22×C3×C4C23×C311yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×Dic6—21no—
128C2×Dic6C3×D412yes—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—42no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—62no—
136C3×C4⋊C4—62no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—62no—
142C3×C22⋊C4—122no—
143#217—22no—
144#217—22no—
145#217—42no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3C3×D412yes—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—81no—
158C22×C32#12011yescharacteristic
159C22×C32C2×SD1611yescharacteristic
160C22×C32#13711yescharacteristic
161C32×C4—22no—
162C32×C4—41no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—81no—
166C3×Dic3—81no—
167C22⋊Q8—32no—
168C22⋊Q8—62no—
169C22⋊Q8—62no—
170C22⋊Q8—61no—
171C2×C4⋊C4—31no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4C2×C3×D311yescharacteristic
175#106—62no—
176#103—62no—
177C22⋊C8C2×C3×D312yes—
178#95—31no—
179#95—61no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—22no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—81no—
193C23×C3—81no—
194C23×C3#20412yes—
195C23×C3C3×SD1612yes—
196C23×C3C2×D1211yescharacteristic
197C23×C3C2×C3×D411yescharacteristic
198C3×Q8—122no—
199C3×Q8—241no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—121no—
213C2×C3×C4—121no—
214C2×C3×C4—121no—
215C2×C3×C4—121no—
216C2×C3×C4—121no—
217C2×C3×C4—121no—
218C2×C3×C4—241no—
219C2×C3×C4—241no—
220C2×C3×C4D3×D412yes—
221C2×C3×C4C2×C3×D412yes—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232Dic6—42no—
233Dic6—81no—
234C3×C8—42no—
235C3×C8—42no—
236C3×C8—82no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—42no—
240C2×C32#44811yescharacteristic
241C2×C32#45511yescharacteristic
242C2×C32#45811yescharacteristic
243C24C3×D1211yescharacteristic
244C2×Q8—32no—
245C2×Q8—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C3×D1211yescharacteristic
251C22×C4C22×C3×D311yescharacteristic
252C2×C8—22no—
253C4⋊C4—32no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C22⋊C4—22no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—122no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—82no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—82no—
286C22×C3#88211yescharacteristic
287C22×C3C2×#20411yescharacteristic
288C22×C3#90911yescharacteristic
289C22×C3C3×#12011yescharacteristic
290C22×C3C2×C3×SD1611yescharacteristic
291C22×C3C3×#13711yescharacteristic
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—42no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—81no—
298C3×C4—122no—
299C3×C4—122no—
300C3×C4—241no—
301C3×C4—241no—
302Dic3—42no—
303Dic3—42no—
304Dic3—81no—
305Dic3—81no—
306C32#160811yescharacteristic, complemented, Sylow
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C3×#20412yes—
312C23C2×C3×D1211yescharacteristic
313Q8—122no—
314Q8—241no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4C3×D3×D412yes—
329C8—42no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—42no—
340C2×C3—42no—
341C2×C3—82no—
342C2×C3#473811yescharacteristic
343C2×C3#474211yescharacteristic
344C2×C3#474611yescharacteristic
345C2×C3C3×#44811yescharacteristic
346C2×C3C3×#45511yescharacteristic
347C2×C3C3×#45811yescharacteristic
348C22—21no—
349C22—21no—
350C22—22no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22C3×#88211yescharacteristic
356C22C2×C3×#20411yescharacteristic
357C22C3×#90911yescharacteristic
358C4—22no—
359C4—41no—
360C4—122no—
361C4—122no—
362C4—241no—
363C4—241no—
364C3—21no—
365C3#6811611yescharacteristic, direct product factor
366C3C3×#160811yescharacteristic, complemented
367C2—21no—
368C2—21no—
369C2—42no—
370C2C3×#473811yescharacteristic
371C2C3×#474211yescharacteristic
372C2C3×#474611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC24×C32D4
hypercenterC3×#323D3
hypocenterC3C3×#1608
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
3. C3×#4721C2semidirect
5. C3×#4467C2semidirect
7. #68116C3direct (N × H)
26. #4721C2×C3semidirect
31. #4467C2×C3semidirect
306. C32#1608semidirect
365. C3#68116direct (N × H)
366. C3C3×#1608semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
C3×#49532
C3×#47212
C32×#3232
C3×#44672
C3×#16083
#681163