Finite Group Databasebeta

← C3×#69713 #11778578

Subgroups — C3×#69713 #11778578

Order 1152.

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

normal subgroups only

This group has 495 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×#5500C211yesmaximal, characteristic
2C32×#398C211yesmaximal, characteristic
3C3×#4971C212yesmaximal
4C3×#4528C211yesmaximal, characteristic
5C3×#4452C212yesmaximal
6C3×#1611—31nomaximal
7#69713C311yesmaximal, characteristic, direct product factor
8C2×C32×C4⋊C4—21no—
9C2×C32×C4⋊C4C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C3×#926—21no—
12C3×#924—21no—
13C3×#924C2212yes—
14C3×#890—22no—
15C3×#888—22no—
16C32×#95—21no—
17C32×#95C2211yescharacteristic
18C3×#830C2212yes—
19C3×#810—22no—
20C3×#808—22no—
21C3×#544—31no—
22#5500C2×C311yescharacteristic
23C3×#483—32no—
24C3×#398—21no—
25C3×#398D311yescharacteristic
26C3×#398C2×C311yescharacteristic
27#4971C2×C312yes—
28C3×#328—32no—
29C3×#324—31no—
30#4528C2×C311yescharacteristic
31#4452C2×C312yes—
32C22×C32×C4—21no—
33C22×C32×C4—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C2×C3×Dic6—21no—
38C2×C3×Dic6D412yes—
39C32×C4⋊C4—22no—
40C32×C4⋊C4—22no—
41C32×C4⋊C4—42no—
42C3×#217—22no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—21no—
46C3×C4⋊Dic3D412yes—
47C3×Dic3⋊C4—22no—
48C3×Dic3⋊C4—22no—
49C3×C4×Dic3—22no—
50C2×C3×C3⋊C8—22no—
51#1611—31noSylow
52C3×C4⋊Q8—62no—
53C3×#125—62no—
54C3×C22⋊Q8—32no—
55C3×C22⋊Q8—61no—
56C3×#117—61no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2×D311yescharacteristic
63C2×C3×C4⋊C4C2×D311yescharacteristic
64C2×C3×C4⋊C4C22×C311yescharacteristic
65C2×C3×C4⋊C4C22×C311yescharacteristic
66#926—21no—
67#924—21no—
68#924C22×C312yes—
69#890—22no—
70#888—22no—
71C3×#106—62no—
72C3×#103—62no—
73C3×C22⋊C8—32no—
74C3×#95—22no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—21no—
78C3×#95C2×D311yescharacteristic
79C3×#95C22×C311yescharacteristic
80#830C22×C312yes—
81#810—22no—
82#808—22no—
83C23×C32C2×D411yescharacteristic
84C2×C32×C4—22no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4SD1612yes—
96C2×C32×C4C2×D412yes—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C3×Dic6—42no—
100C3×Dic6—81no—
101C3×C3⋊C8—42no—
102#544—31no—
103#483—32no—
104#398C3×D311yescharacteristic
105#328—32no—
106#324—31no—
107C2×C3×Q8—32no—
108C2×C3×Q8—61no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4#6711yescharacteristic
119C22×C3×C4#6711yescharacteristic
120C22×C3×C4C3×D411yescharacteristic
121C22×C3×C4C3×D411yescharacteristic
122C22×C3×C4C22×D311yescharacteristic
123C22×C3×C4C23×C311yescharacteristic
124C2×Dic6—21no—
125C2×Dic6C3×D412yes—
126C2×C3×C8—62no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4—44no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—62no—
142C3×C22⋊C4—62no—
143C3×C42—62no—
144#217—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3C3×D412yes—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151C4×Dic3—22no—
152C2×C3⋊C8—22no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32#12011yescharacteristic
156C22×C32C2×SD1611yescharacteristic
157C22×C32#13711yescharacteristic
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—42no—
161C32×C4—41no—
162C32×C4—81no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C4⋊Q8—62no—
166#125—62no—
167C22⋊Q8—32no—
168C22⋊Q8—61no—
169#117—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C2×C3×D311yescharacteristic
172C2×C4⋊C4C2×C3×D311yescharacteristic
173#106—62no—
174#103—62no—
175C22⋊C8—32no—
176#95—21no—
177#95C2×C3×D311yescharacteristic
178C23×C3—21no—
179C23×C3C2×#6711yescharacteristic
180C23×C3C2×C3×D411yescharacteristic
181C3×Q8—122no—
182C3×Q8—241no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—24no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4#21412yes—
221C2×C3×C4C3×SD1612yes—
222C2×C3×C4D3×D412yes—
223C2×C3×C4C2×C3×D412yes—
224C2×Dic3—22no—
225C2×Dic3—22no—
226Dic6—42no—
227Dic6—81no—
228C3×C8—122no—
229C3⋊C8—42no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32#44811yescharacteristic
233C2×C32#45611yescharacteristic
234C2×C32#45811yescharacteristic
235C2×Q8—32no—
236C2×Q8—61no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C3×#6711yescharacteristic
240C22×C4C3×#6711yescharacteristic
241C22×C4C22×C3×D311yescharacteristic
242C2×C8—62no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—32no—
246C4⋊C4—42no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C22⋊C4—62no—
253C42—62no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3C2×#21411yescharacteristic
264C22×C3#93711yescharacteristic
265C22×C3#95111yescharacteristic
266C22×C3C3×#12011yescharacteristic
267C22×C3C2×C3×SD1611yescharacteristic
268C22×C3C3×#13711yescharacteristic
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—42no—
275C3×C4—44no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—82no—
282C3×C4—81no—
283C3×C4—81no—
284C3×C4—122no—
285C3×C4—122no—
286Dic3—42no—
287Dic3—42no—
288C32#161111yescharacteristic, complemented, Sylow
289C23C2×C3×#6711yescharacteristic
290Q8—122no—
291Q8—241no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4C3×#21412yes—
306C2×C4C3×D3×D412yes—
307C8—122no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3#504011yescharacteristic
318C2×C3#505311yescharacteristic
319C2×C3#509511yescharacteristic
320C2×C3C3×#44811yescharacteristic
321C2×C3C3×#45611yescharacteristic
322C2×C3C3×#45811yescharacteristic
323C22—21no—
324C22—21no—
325C22C2×C3×#21411yescharacteristic
326C22C3×#93711yescharacteristic
327C22C3×#95111yescharacteristic
328C4—22no—
329C4—22no—
330C4—42no—
331C4—41no—
332C4—81no—
333C4—122no—
334C4—122no—
335C3—21no—
336C3#6971311yescharacteristic, direct product factor
337C3C3×#161111yescharacteristic, complemented
338C2—21no—
339C2—21no—
340C2C3×#504011yescharacteristic
341C2C3×#505311yescharacteristic
342C2C3×#509511yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#398C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC3×#398D3
hypocenterC3C3×#1611
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#398

Complements

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

NH ≅ G/Ntype
7. #69713C3direct (N × H)
288. C32#1611semidirect
336. C3#69713direct (N × H)
337. C3C3×#1611semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C32×#3982
C3×#49712
C3×#45282
C3×#44522
C3×#16113
#697133