Finite Group Databasebeta

← #10592358⋊C3 #13141352

Subgroups — #10592358⋊C3 #13141352

Order 1728.

545 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#95C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#63658⋊C3C212yesmaximal
6#10592358C311yesmaximal, characteristic, complemented
7C3×#4659—31nomaximal
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—22no—
14#3750⋊C3—22no—
15C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
16C2×#3750C2×C311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—31no—
19C32×#95—21no—
20C32×#95D311yescharacteristic
21C32×#95C2×C311yescharacteristic
22#63658C2×C312yes—
23#63658C2×C312yes—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—32no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D412yes—
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3Q812yes—
39C3×#400—91no—
40#4659—31no—
41#4659—91no—
42C22×C32×C4—22no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D312yes—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C22×C312yes—
47C22×C32×C4C22×C311yescharacteristic
48C22×C3⋊Dic3C22×C311yescharacteristic
49C22×C3⋊Dic3C22×C312yes—
50C22×C3⋊Dic3C22×C311yescharacteristic
51C22×C3×Dic3—31no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—32no—
54C4⋊(C3⋊Dic3)—22no—
55#3750—22no—
56C3×C4⋊Dic3—62no—
57C3×Dic3⋊C4—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×Q811yescharacteristic
60C22×H(3,𝔽3)#4111yescharacteristic
61C22×H(3,𝔽3)#4112yes—
62C22×H(3,𝔽3)#4112yes—
63C4×H(3,𝔽3)—22no—
64C4×H(3,𝔽3)—42no—
65(C3⋊Dic3)⋊C3—22no—
66(C3⋊Dic3)⋊C3—41no—
67(C3⋊Dic3)⋊C3—42no—
68C2×C3×C4⋊C4—91no—
69C2×C3×C4⋊C4—91no—
70C2×C4⋊Dic3—31no—
71C2×C4⋊Dic3—91no—
72C2×Dic3⋊C4—31no—
73C2×Dic3⋊C4—91no—
74C3×#95—31no—
75C3×#95—31no—
76C3×#95—61no—
77C3×#95—92no—
78C3×#95—92no—
79C3×#95C3×D311yescharacteristic
80C3⋊#95—32no—
81C3⋊#95—32no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C23×C32—21no—
85C23×C32C22×D311yescharacteristic
86C23×C32C23×C311yescharacteristic
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—24no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—42no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4D1212yes—
101C2×C32×C4C3×D412yes—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—24no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3C3×Q812yes—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—62no—
116#400—91noSylow
117C2×H(3,𝔽3)C22⋊Q811yescharacteristic
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)#12411yescharacteristic
120C2×H(3,𝔽3)#12512yes—
121C2×H(3,𝔽3)#12612yes—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—32no—
125C22×C3×C4—32no—
126C22×C3×C4—62no—
127C22×C3×C4—61no—
128C22×C3×C4—91no—
129C22×C3×C4—91no—
130C22×C3×C4—92no—
131C22×C3×C4C2×C3×D312yes—
132C22×C3×C4C2×C3×D311yescharacteristic
133C22×Dic3—31no—
134C22×Dic3—31no—
135C22×Dic3—32no—
136C22×Dic3—91no—
137C22×Dic3—91no—
138C22×Dic3—92no—
139C3×C4⋊C4—182no—
140C3×C4⋊C4—182no—
141C4⋊Dic3—62no—
142C4⋊Dic3—182no—
143Dic3⋊C4—62no—
144Dic3⋊C4—182no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32C2×D1211yescharacteristic
151C22×C32#23512yes—
152C22×C32#23711yescharacteristic
153C22×C32#23712yes—
154C22×C32D3×Q811yescharacteristic
155C22×C32C2×C3×D411yescharacteristic
156C22×C32C2×C3×Q811yescharacteristic
157C22×C32C3×#4112yes—
158C22×C32C3×#4111yescharacteristic
159C22×C32C3×#4112yes—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—42no—
163C32×C4—44no—
164C32×C4—42no—
165C32×C4—42no—
166C3⋊Dic3—22no—
167C3⋊Dic3—41no—
168C3⋊Dic3—42no—
169C3×Dic3—62no—
170C3×Dic3—122no—
171C3×Dic3—121no—
172C2×C4⋊C4—91no—
173C2×C4⋊C4—91no—
174#95—92no—
175#95—92no—
176#95(C3⋊D3)⋊C311yescharacteristic
177H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
178C23×C3—31no—
179C23×C3—31no—
180C23×C3—61no—
181C23×C3C22×C3×D311yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—64no—
197C2×C3×C4—62no—
198C2×C3×C4—92no—
199C2×C3×C4—122no—
200C2×C3×C4—122no—
201C2×C3×C4—122no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4—184no—
207C2×C3×C4—182no—
208C2×C3×C4C3×D1212yes—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—64no—
215C2×Dic3—62no—
216C2×Dic3—92no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—184no—
222C2×Dic3—182no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32#87511yescharacteristic
229C2×C32#87912yes—
230C2×C32#88611yescharacteristic
231C2×C32#88912yes—
232C2×C32#89711yescharacteristic
233C2×C32C3×C22⋊Q811yescharacteristic
234C2×C32C3×#12311yescharacteristic
235C2×C32C3×#12411yescharacteristic
236C2×C32C3×#12512yes—
237C2×C32C3×#12612yes—
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—91no—
241C22×C4C2×(C3⋊D3)⋊C312yes—
242C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
243C4⋊C4—182no—
244C4⋊C4—182no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—31no—
252C22×C3—32no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—62no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3C2×C3×D1211yescharacteristic
261C22×C3C3×#23512yes—
262C22×C3C3×#23711yescharacteristic
263C22×C3C3×#23712yes—
264C22×C3C3×D3×Q811yescharacteristic
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—62no—
268C3×C4—62no—
269C3×C4—122no—
270C3×C4—122no—
271C3×C4—122no—
272C3×C4—124no—
273C3×C4—182no—
274C3×C4—362no—
275C3×C4—361no—
276Dic3—62no—
277Dic3—121no—
278Dic3—122no—
279Dic3—182no—
280Dic3—362no—
281Dic3—361no—
282C32—21no—
283C32#465911yescharacteristic, complemented
284C32C3×#40011yescharacteristic, complemented
285C23C22×(C3⋊D3)⋊C311yescharacteristic
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—92no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—182no—
294C2×C4—181no—
295C2×C4—184no—
296C2×C4—182no—
297C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—62no—
310C2×C3—61no—
311C2×C3—62no—
312C2×C3—61no—
313C2×C3C3×#87511yescharacteristic
314C2×C3C3×#87912yes—
315C2×C3C3×#88611yescharacteristic
316C2×C3C3×#88912yes—
317C2×C3C3×#89711yescharacteristic
318C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
319C22#3828⋊C312yes—
320C22#3830⋊C311yescharacteristic
321C22#3830⋊C312yes—
322C22Q8×(C3⋊D3)⋊C311yescharacteristic
323C4—22no—
324C4—42no—
325C4—182no—
326C4—362no—
327C4—361no—
328C3—31no—
329C3—31no—
330C3—61no—
331C3C3×#465911yescharacteristic
332C2#64084⋊C311yescharacteristic
333C2#64088⋊C312yes—
334C2#64095⋊C311yescharacteristic
335C2#64098⋊C312yes—
336C2#64106⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×H(3,𝔽3)C23
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#400
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
6. #10592358C3semidirect
177. H(3,𝔽3)#400semidirect
283. C32#4659semidirect
284. C32C3×#400semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923583
C3×#46593