Finite Group Databasebeta

← #13049537

Subgroups — #13049537

Order 1296.

555 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 74 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⋊#6296C211yesmaximal, characteristic, complemented
2#10620154C212yesmaximal
3C3×#6296C211yesmaximal, characteristic, complemented
4C3×#6293C212yesmaximal, complemented
5C3×#6289C211yesmaximal, characteristic
6#89040—31nomaximal
7#88958—31nomaximal
8#88957—32nomaximal
9C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
10C3×C3⋊(C3⋊Dic3)C2212yes—
11C32×C3⋊Dic3C2211yescharacteristic
12C32×C3⋊Dic3C2212yes—
13C33×Dic3C2211yescharacteristic
14C3×C4×C3⋊D3—31no—
15C3×#634—32no—
16#6296—31no—
17#6296—31no—
18#6296—61no—
19#6296D311yescharacteristic, complemented
20#6294—32no—
21#6294—32no—
22#6294—62no—
23#6293—32no—
24#6293D312yescomplemented
25#6289—32no—
26#6289D311yescharacteristic
27Dic3×C3⋊D3—32no—
28C3×#627—32no—
29C3×#627—31no—
30C3×#627—32no—
31C3×#627—32no—
32C3×#627—62no—
33C3×#624—32no—
34C3×#624—61no—
35C2×C34C2311yescharacteristic
36C3×C3⋊(C3⋊D3)Q812yescomplemented
37#3797—92no—
38#3797—91no—
39D3×Dic6—91no—
40D3×Dic6—92no—
41C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
42C2×C3×C3⋊D3—32no—
43C2×C3×C3⋊D3—31no—
44C2×C3×C3⋊D3—61no—
45C2×C3×C3⋊D3—62no—
46C2×C3×C3⋊D3—121no—
47C33×C4—31no—
48C3⋊(C3⋊Dic3)C2×D312yes—
49C3×C3⋊Dic3—22no—
50C3×C3⋊Dic3—21no—
51C3×C3⋊Dic3—32no—
52C3×C3⋊Dic3—32no—
53C3×C3⋊Dic3—32no—
54C3×C3⋊Dic3—62no—
55C3×C3⋊Dic3—62no—
56C3×C3⋊Dic3—62no—
57C3×C3⋊Dic3—122no—
58C3×C3⋊Dic3C2×D311yescharacteristic
59C3×C3⋊Dic3C2×D312yes—
60C3×C3⋊Dic3C2×D312yes—
61C3×C3⋊Dic3C2×D311yescharacteristic
62C32×Dic3—21no—
63C32×Dic3—22no—
64C32×Dic3—32no—
65C32×Dic3—32no—
66C32×Dic3—32no—
67C32×Dic3—41no—
68C32×Dic3—61no—
69C32×Dic3—62no—
70C32×Dic3C2×D312yes—
71C32×Dic3C2×D311yescharacteristic
72C34C2×Q811yescharacteristic, complemented, Sylow
73C4×C3⋊D3—31no—
74C4×C3⋊D3—92no—
75#634—32no—
76#634—91no—
77C3×C4×D3—92no—
78C3×C4×D3—91no—
79C3×C4×D3—181no—
80C3×Dic6—91no—
81C3×Dic6—92no—
82C3×Dic6—92no—
83C3×Dic6—92no—
84C3×Dic6—92no—
85C3×Dic6—182no—
86#627—32no—
87#627—31no—
88#627—32no—
89#627—32no—
90#627—62no—
91#627—92no—
92#627—92no—
93#627—91no—
94#627—182no—
95#624—32no—
96#624—61no—
97#624—92no—
98#624—92no—
99#624—182no—
100D3×Dic3—92no—
101D3×Dic3—92no—
102D3×Dic3—92no—
103D3×Dic3—182no—
104C2×C33—22no—
105C2×C33—21no—
106C2×C33—21no—
107C2×C33—22no—
108C2×C33—41no—
109C2×C33—42no—
110C2×C33—41no—
111C2×C33—81no—
112C2×C33C22×D311yescharacteristic
113C2×C33C22×D312yes—
114C2×C33C22×D311yescharacteristic
115C3⋊(C3⋊D3)Dic612yescomplemented
116C3×C3⋊D3—34no—
117C3×C3⋊D3—32no—
118C3×C3⋊D3—62no—
119C3×C3⋊D3—64no—
120C3×C3⋊D3—122no—
121D3×Q8—272no—
122D3×Q8—271no—
123C2×Dic6—271no—
124C2×C3⋊D3—32no—
125C2×C3⋊D3—31no—
126C2×C3⋊D3—61no—
127C2×C3⋊D3—62no—
128C2×C3⋊D3—121no—
129C2×C3×D3—92no—
130C2×C3×D3—91no—
131C2×C3×D3—182no—
132C2×C3×D3—181no—
133C2×C3×D3—361no—
134C32×C4—31no—
135C32×C4—32no—
136C32×C4—62no—
137C32×C4—61no—
138C32×C4—92no—
139C32×C4—91no—
140C32×C4—121no—
141C3⋊Dic3—32no—
142C3⋊Dic3—32no—
143C3⋊Dic3—32no—
144C3⋊Dic3—62no—
145C3⋊Dic3—62no—
146C3⋊Dic3—62no—
147C3⋊Dic3—122no—
148C3⋊Dic3D3212yes—
149C3⋊Dic3D3211yescharacteristic
150C3×Dic3—21no—
151C3×Dic3—22no—
152C3×Dic3—32no—
153C3×Dic3—32no—
154C3×Dic3—32no—
155C3×Dic3—32no—
156C3×Dic3—32no—
157C3×Dic3—32no—
158C3×Dic3—41no—
159C3×Dic3—62no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C3×Dic3—61no—
163C3×Dic3—62no—
164C3×Dic3—62no—
165C3×Dic3—61no—
166C3×Dic3—92no—
167C3×Dic3—92no—
168C3×Dic3—92no—
169C3×Dic3—122no—
170C3×Dic3—121no—
171C3×Dic3—182no—
172C3×Dic3—182no—
173C3×Dic3—182no—
174C3×Dic3—362no—
175C3×Dic3D3212yes—
176C3×Dic3D3211yescharacteristic
177C33—21no—
178C33—22no—
179C33—22no—
180C33—21no—
181C33—41no—
182C33—42no—
183C33—41no—
184C33—81no—
185C33C2×Dic611yescharacteristic, complemented
186C33D3×Q811yescharacteristic, complemented
187C33D3×Q812yescomplemented
188C3×Q8—272no—
189C3×Q8—271no—
190C2×C3×C4—271no—
191C2×Dic3—272no—
192C4×D3—92no—
193C4×D3—91no—
194C4×D3—181no—
195C4×D3—272no—
196C4×D3—272no—
197C4×D3—272no—
198C4×D3—542no—
199Dic6—91no—
200Dic6—92no—
201Dic6—92no—
202Dic6—92no—
203Dic6—92no—
204Dic6—182no—
205Dic6—272no—
206Dic6—271no—
207Dic6—271no—
208Dic6—541no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—42no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—42no—
220C2×C32—41no—
221C2×C32—42no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—82no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32—82no—
228C2×C32—81no—
229C2×C32C2×D3211yescharacteristic
230C2×C32C2×D3212yes—
231C2×C32C2×D3212yes—
232C2×C32C2×D3211yescharacteristic
233C3⋊D3—32no—
234C3⋊D3—34no—
235C3⋊D3—64no—
236C3⋊D3—62no—
237C3⋊D3—122no—
238C3×D3—94no—
239C3×D3—92no—
240C3×D3—184no—
241C3×D3—182no—
242C3×D3—362no—
243C2×Q8—811noSylow
244C22×C3—271no—
245C2×D3—92no—
246C2×D3—91no—
247C2×D3—181no—
248C2×D3—182no—
249C2×D3—361no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—62no—
253C3×C4—61no—
254C3×C4—91no—
255C3×C4—92no—
256C3×C4—92no—
257C3×C4—91no—
258C3×C4—121no—
259C3×C4—181no—
260C3×C4—182no—
261C3×C4—272no—
262Dic3—32no—
263Dic3—32no—
264Dic3—32no—
265Dic3—62no—
266Dic3—61no—
267Dic3—92no—
268Dic3—92no—
269Dic3—92no—
270Dic3—182no—
271Dic3—182no—
272Dic3—182no—
273Dic3—362no—
274Dic3#111311yescharacteristic
275C32—22no—
276C32—22no—
277C32—22no—
278C32—22no—
279C32—21no—
280C32—21no—
281C32—22no—
282C32—41no—
283C32—42no—
284C32—41no—
285C32—41no—
286C32—42no—
287C32—42no—
288C32—41no—
289C32—41no—
290C32—81no—
291C32—82no—
292C32—82no—
293C32—81no—
294C32—81no—
295C32D3×Dic611yescharacteristic, complemented
296C32D3×Dic612yescomplemented
297C32#379712yescomplemented
298C32#379711yescharacteristic, complemented
299Q8—272no—
300Q8—271no—
301Q8—811no—
302C2×C4—271no—
303C2×C4—812no—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—22no—
308C2×C3—42no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—81no—
312C2×C3—272no—
313C2×C3D3312yes—
314C2×C3D3311yescharacteristic
315C2×C3C2×#111311yescharacteristic
316D3—94no—
317D3—92no—
318D3—182no—
319D3—184no—
320D3—362no—
321C22—271no—
322C4—31no—
323C4—91no—
324C4—92no—
325C4—272no—
326C3—21no—
327C3—22no—
328C3—22no—
329C3—21no—
330C3—41no—
331C3—42no—
332C3—41no—
333C3—81no—
334C3#8895712yescomplemented
335C3#8895811yescharacteristic, complemented
336C3#8904011yescharacteristic, complemented
337C2—272no—
338C2D3×#111311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×#1113
commutator, Fitting, socle, cosocleC2×C34C23
prime radicalC2×C3×C3⋊(C3⋊D3)C22
hypocenterC34C2×Q8
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3⋊#6296C2semidirect
3. C3×#6296C2semidirect
4. C3×#6293C2semidirect
19. #6296D3semidirect
24. #6293D3semidirect
36. C3×C3⋊(C3⋊D3)Q8semidirect
72. C34C2×Q8semidirect
115. C3⋊(C3⋊D3)Dic6semidirect
185. C33C2×Dic6semidirect
186. C33D3×Q8semidirect
187. C33D3×Q8semidirect
295. C32D3×Dic6semidirect
296. C32D3×Dic6semidirect
297. C32#3797semidirect
298. C32#3797semidirect
334. C3#88957semidirect
335. C3#88958semidirect
336. C3#89040semidirect

Maximal subgroups

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

subgroupindex
C3⋊#62962
#106201542
C3×#62962
C3×#62932
C3×#62892
#890403
#889583
#889573