Finite Group Databasebeta

← #10592438⋊C3 #13141608

Subgroups — #10592438⋊C3 #13141608

Order 1728.

568 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×#634⋊C3C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C2×#3746⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
5#63643⋊C3C212yesmaximal, complemented
6#10592438C311yesmaximal, characteristic, complemented
7C3×#4739—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×#634⋊C3—21no—
11C2×#634⋊C3—24no—
12C2×#634⋊C3C2211yescharacteristic
13C2×#634⋊C3C2212yes—
14C2×C8×H(3,𝔽3)C2212yes—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18#3746⋊C3—24no—
19C22×#634C2×C311yescharacteristic
20C22×C3×Dic6—31no—
21#64133C2×C311yescharacteristic
22C2×#3746C2×C312yescomplemented
23C3×#924—31no—
24C2×C3×Dic12—32no—
25C32×C22⋊C8—21no—
26C32×C22⋊C8D311yescharacteristic
27C32×C22⋊C8C2×C311yescharacteristic
28#63643C2×C312yescomplemented
29C3×#816—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—24no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—21no—
38#634⋊C3—24no—
39#634⋊C3—22no—
40#634⋊C3—22no—
41#634⋊C3D414yes—
42C8×H(3,𝔽3)—22no—
43C3×#451—91no—
44#4739—31no—
45#4739—91no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C22×C311yescharacteristic
49C22×C3⋊Dic3—21no—
50C2×#634—21no—
51C2×#634—24no—
52C2×#634C22×C311yescharacteristic
53C2×#634C22×C312yes—
54C22×C3×Dic3—61no—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6—61no—
59C2×C32×C8—22no—
60C2×C32×C8C2×D312yes—
61C2×C32×C8C22×C312yes—
62#3757—21no—
63C4⋊(C3⋊Dic3)C22×C311yescharacteristic
64#3750—21no—
65#3746—24no—
66C3×#217—61no—
67C3×C4⋊Dic3—31no—
68C3×Dic3⋊C4—61no—
69C3×Dic12—64no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)Q1612yescomplemented
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)C2×D412yes—
76(C3⋊Dic3)⋊C3—24no—
77(C3⋊Dic3)⋊C3—41no—
78(C3⋊Dic3)⋊C3—41no—
79C22×C3×Q8—91no—
80C22×Dic6—31no—
81C22×Dic6—91no—
82C2×C3×Q16—92no—
83C3×C22⋊Q8—91no—
84#924—31no—
85#924—91no—
86C2×Dic12—32no—
87C2×Dic12—92no—
88C3×#103—92no—
89C3×C22⋊C8—31no—
90C3×C22⋊C8—31no—
91C3×C22⋊C8—61no—
92C3×C22⋊C8C3×D311yescharacteristic
93#816—32no—
94#816—92no—
95C23×C32—21no—
96C23×C32D1211yescharacteristic
97C23×C32C3×D411yescharacteristic
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—42no—
103C2×C32×C4D1211yescharacteristic
104C2×C32×C4C3×D411yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C23×C311yescharacteristic
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111#634—24no—
112#634—22no—
113#634—22no—
114#634C3×D414yes—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×Dic6—34no—
120C3×Dic6—64no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C32×C8—22no—
124C32×C8—22no—
125C32×C8—42no—
126#451—91noSylow
127C2×H(3,𝔽3)—22no—
128C2×H(3,𝔽3)#12011yescharacteristic
129C2×H(3,𝔽3)C2×Q1611yescharacteristic
130C2×H(3,𝔽3)#13711yescharacteristic
131C2×C3×Q8—92no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8—184no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—61no—
138C22×C3×C4—181no—
139C22×C3×C4C2×C3×D311yescharacteristic
140C22×Dic3—61no—
141C22×Dic3—181no—
142C2×Dic6—32no—
143C2×Dic6—31no—
144C2×Dic6—64no—
145C2×Dic6—61no—
146C2×Dic6—92no—
147C2×Dic6—91no—
148C2×Dic6—184no—
149C2×Dic6—181no—
150C3×Q16—184no—
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—62no—
154C2×C3×C8C2×C3×D312yes—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C22⋊C4—181no—
158#217—61no—
159#217—181no—
160C4⋊Dic3—31no—
161C4⋊Dic3—91no—
162Dic3⋊C4—61no—
163Dic3⋊C4—181no—
164Dic12—64no—
165Dic12—184no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32Dic1212yescomplemented
175C22×C32C3×Q1612yescomplemented
176C22×C32C2×D1211yescharacteristic
177C22×C32C2×C3×D411yescharacteristic
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—22no—
181C32×C4—41no—
182C32×C4D3×D412yes—
183C32×C4C2×C3×D412yes—
184C3⋊Dic3—24no—
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3×Dic3—64no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C22×Q8—91no—
191C2×Q16—92no—
192C22⋊Q8—91no—
193#103—92no—
194C22⋊C8(C3⋊D3)⋊C311yescharacteristic
195H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—61no—
199C23×C3C3×D1211yescharacteristic
200C3×Q8—94no—
201C3×Q8—182no—
202C3×Q8—182no—
203C3×Q8—184no—
204C2×C3×C4—22no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—122no—
214C2×C3×C4—182no—
215C2×C3×C4—184no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C3×D1211yescharacteristic
219C2×C3×C4C22×C3×D311yescharacteristic
220C2×Dic3—64no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—182no—
225C2×Dic3—184no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228Dic6—34no—
229Dic6—64no—
230Dic6—62no—
231Dic6—62no—
232Dic6—94no—
233Dic6—182no—
234Dic6—184no—
235Dic6—182no—
236C3×C8—22no—
237C3×C8—62no—
238C3×C8—62no—
239C3×C8—122no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—42no—
246C2×C32#88211yescharacteristic
247C2×C32C2×Dic1211yescharacteristic
248C2×C32#90911yescharacteristic
249C2×C32C3×#12011yescharacteristic
250C2×C32C2×C3×Q1611yescharacteristic
251C2×C32C3×#13711yescharacteristic
252C2×Q8—92no—
253C2×Q8—91no—
254C2×Q8—184no—
255C2×Q8—181no—
256C22×C4—181no—
257C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
258Q16—184no—
259C2×C8C2×(C3⋊D3)⋊C312yes—
260C4⋊C4—91no—
261C4⋊C4—181no—
262C22⋊C4—181no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—62no—
275C22×C3—121no—
276C22×C3—121no—
277C22×C3C3×Dic1212yes—
278C22×C3C2×C3×D1211yescharacteristic
279C3×C4—21no—
280C3×C4—32no—
281C3×C4—32no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—62no—
285C3×C4—121no—
286C3×C4—184no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4C3×D3×D412yes—
290Dic3—64no—
291Dic3—121no—
292Dic3—121no—
293Dic3—184no—
294Dic3—361no—
295Dic3—361no—
296C32—21no—
297C32#473911yescharacteristic, complemented
298C32C3×#45111yescharacteristic, complemented
299C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
300Q8—94no—
301Q8—184no—
302Q8—182no—
303Q8—182no—
304C2×C4—22no—
305C2×C4—184no—
306C2×C4—182no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
310C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
311C8—22no—
312C2×C3—22no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—62no—
320C2×C3—62no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—122no—
325C2×C3C3×#88211yescharacteristic
326C2×C3C2×C3×Dic1211yescharacteristic
327C2×C3C3×#90911yescharacteristic
328C22—21no—
329C22—21no—
330C22#3746⋊C312yescomplemented
331C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
332C4—21no—
333C4—184no—
334C4—361no—
335C4—361no—
336C4D4×(C3⋊D3)⋊C312yes—
337C3—31no—
338C3—31no—
339C3—61no—
340C3C3×#473911yescharacteristic
341C2—22no—
342C2#64091⋊C311yescharacteristic
343C2C2×#3746⋊C311yescharacteristic
344C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592438C3semidirect
22. C2×#3746C2×C3semidirect
28. #63643C2×C3semidirect
72. C22×H(3,𝔽3)Q16semidirect
174. C22×C32Dic12semidirect
175. C22×C32C3×Q16semidirect
195. H(3,𝔽3)#451semidirect
297. C32#4739semidirect
298. C32C3×#451semidirect
330. C22#3746⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#64133⋊C32
C2×#3746⋊C32
H(3,𝔽3)×C22⋊C82
#63643⋊C32
#105924383
C3×#47393