Finite Group Databasebeta

← #10592749⋊C3 #13142883

Subgroups — #10592749⋊C3 #13142883

Order 1728.

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

normal subgroups only

This group has 592 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
2H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
3C2×#3756⋊C3C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic
5#63637⋊C3C212yesmaximal, complemented
6#10592749C311yesmaximal, characteristic, complemented
7C3×#5050—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3—21no—
11C2×#634⋊C3—21no—
12C2×#634⋊C3—24no—
13C2×#634⋊C3C2212yes—
14C4⋊C4×H(3,𝔽3)—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)—21no—
17#3756⋊C3—24no—
18C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
19C22×#634C2×C311yescharacteristic
20C22×C3×Dic6—31no—
21C32×C22⋊Q8—21no—
22C32×C22⋊Q8D311yescharacteristic
23C32×C22⋊Q8C2×C311yescharacteristic
24C2×#3756C2×C312yescomplemented
25C2×C3×#216—32no—
26#63657C2×C311yescharacteristic
27#63637C2×C312yescomplemented
28C3×#830—31no—
29C3×#810—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31Q8×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3—22no—
39#634⋊C3—24no—
40#634⋊C3—22no—
41#634⋊C3D414yes—
42(C3⋊(C3⋊C8))⋊C3—22no—
43C3×#451—91no—
44#5050—31no—
45#5050—91no—
46C2×C32×Q8—21no—
47C2×C32×Q8C2×D311yescharacteristic
48C2×C32×Q8C22×C311yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C22×C3⋊Dic3—21no—
53C2×#634—21no—
54C2×#634—24no—
55C2×#634C22×C312yes—
56C22×C3×Dic3—61no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—64no—
59C2×C3×Dic6—61no—
60C32×C4⋊C4—22no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C22⋊C4—22no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4—21no—
69#3756—24no—
70C2×C3⋊(C3⋊C8)C22×C312yes—
71C3×#216—64no—
72C2×C3×C3⋊C8—32no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)Q1612yescomplemented
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×D412yes—
81(C3⋊Dic3)⋊C3—24no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—91no—
85C2×C3×Q16—92no—
86C3×C22⋊Q8—31no—
87C3×C22⋊Q8—31no—
88C3×C22⋊Q8—61no—
89C3×C22⋊Q8C3×D311yescharacteristic
90C2×#216—32no—
91C2×#216—92no—
92C3×#103—92no—
93C3×C22⋊C8—91no—
94#830—31no—
95#830—91no—
96#810—32no—
97#810—92no—
98C23×C32—21no—
99C23×C32#6711yescharacteristic
100C23×C32C3×D411yescharacteristic
101C32×Q8—22no—
102C32×Q8—22no—
103C32×Q8—42no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—42no—
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3—24no—
120C2×C3⋊Dic3—22no—
121#634—24no—
122#634—22no—
123#634C3×D414yes—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—62no—
126C3×Dic6—34no—
127C3×Dic6—64no—
128C3×Dic6—62no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131#451—91noSylow
132C2×H(3,𝔽3)—22no—
133C2×H(3,𝔽3)#12011yescharacteristic
134C2×H(3,𝔽3)C2×Q1611yescharacteristic
135C2×H(3,𝔽3)#13711yescharacteristic
136C2×C3×Q8—31no—
137C2×C3×Q8—31no—
138C2×C3×Q8—61no—
139C2×C3×Q8—92no—
140C2×C3×Q8—181no—
141C2×C3×Q8—184no—
142C2×C3×Q8C2×C3×D311yescharacteristic
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—61no—
146C22×C3×C4—181no—
147C22×C3×C4C2×C3×D311yescharacteristic
148C22×Dic3—61no—
149C22×Dic3—181no—
150C2×Dic6—32no—
151C2×Dic6—64no—
152C2×Dic6—61no—
153C2×Dic6—92no—
154C2×Dic6—184no—
155C2×Dic6—181no—
156C3×Q16—184no—
157C2×C3×C8—92no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—21no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—61no—
169C3×C22⋊C4—62no—
170#216—64no—
171#216—184no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—92no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32#21612yescomplemented
183C22×C32C3×Q1612yescomplemented
184C22×C32C2×#6711yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—22no—
189C32×C4—41no—
190C32×C4—42no—
191C32×C4—42no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4D3×D412yes—
197C32×C4C2×C3×D412yes—
198C3⋊Dic3—24no—
199C3×Dic3—64no—
200C22×Q8—91no—
201C2×Q16—92no—
202C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
203#103—92no—
204C22⋊C8—91no—
205H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—61no—
209C23×C3C3×#6711yescharacteristic
210C3×Q8—22no—
211C3×Q8—62no—
212C3×Q8—62no—
213C3×Q8—94no—
214C3×Q8—122no—
215C3×Q8—182no—
216C3×Q8—184no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—122no—
235C2×C3×C4—182no—
236C2×C3×C4—184no—
237C2×C3×C4C3×#6711yescharacteristic
238C2×C3×C4C22×C3×D311yescharacteristic
239C2×Dic3—64no—
240C2×Dic3—62no—
241C2×Dic3—182no—
242C2×Dic3—184no—
243Dic6—34no—
244Dic6—64no—
245Dic6—62no—
246Dic6—94no—
247Dic6—182no—
248Dic6—184no—
249C3×C8—182no—
250C3⋊C8—62no—
251C3⋊C8—182no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—42no—
258C2×C32#93711yescharacteristic
259C2×C32C2×#21611yescharacteristic
260C2×C32#95111yescharacteristic
261C2×C32C3×#12011yescharacteristic
262C2×C32C2×C3×Q1611yescharacteristic
263C2×C32C3×#13711yescharacteristic
264C2×Q8—92no—
265C2×Q8—184no—
266C2×Q8—181no—
267C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
268C22×C4—181no—
269C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
270Q16—184no—
271C2×C8—92no—
272C4⋊C4—21no—
273C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
274C22⋊C4—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—61no—
282C22×C3—61no—
283C22×C3—61no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—62no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3C3×#21612yes—
290C22×C3C2×C3×#6711yescharacteristic
291C3×C4—21no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—62no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—122no—
305C3×C4—122no—
306C3×C4—184no—
307C3×C4C3×D3×D412yes—
308Dic3—64no—
309Dic3—184no—
310C32—21no—
311C32#505011yescharacteristic, complemented
312C32C3×#45111yescharacteristic, complemented
313C23#638⋊C311yescharacteristic
314Q8—22no—
315Q8—94no—
316Q8—184no—
317Q8—182no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—184no—
322C2×C4—182no—
323C2×C4#638⋊C311yescharacteristic
324C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
325C8—182no—
326C2×C3—22no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—62no—
334C2×C3—62no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—122no—
339C2×C3C3×#93711yescharacteristic
340C2×C3C2×C3×#21611yescharacteristic
341C2×C3C3×#95111yescharacteristic
342C22—21no—
343C22—21no—
344C22#3756⋊C312yescomplemented
345C22C2×#638⋊C311yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—184no—
350C4D4×(C3⋊D3)⋊C312yes—
351C3—31no—
352C3—31no—
353C3—61no—
354C3C3×#505011yescharacteristic
355C2—22no—
356C2#64146⋊C311yescharacteristic
357C2C2×#3756⋊C311yescharacteristic
358C2#64160⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#3756⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #10592749C3semidirect
24. C2×#3756C2×C3semidirect
27. #63637C2×C3semidirect
75. C22×H(3,𝔽3)Q16semidirect
182. C22×C32#216semidirect
183. C22×C32C3×Q16semidirect
205. H(3,𝔽3)#451semidirect
311. C32#5050semidirect
312. C32C3×#451semidirect
344. C22#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
H(3,𝔽3)×C22⋊Q82
C2×#3756⋊C32
#63657⋊C32
#63637⋊C32
#105927493
C3×#50503