Finite Group Databasebeta

← #13141147

Subgroups — #13141147

Order 1728.

895 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#88552C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4#11720334C214yesmaximal, complemented
5#10591697—31nomaximal
6#10591696—31nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2212yes—
9C22×(C3⋊Dic3)⋊C3—22no—
10C22×(C3⋊Dic3)⋊C3—22no—
11C22×(C3⋊Dic3)⋊C3C414yescomplemented
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14#88552—22no—
15#3757⋊C3—22no—
16C23×C3⋊Dic3D311yescharacteristic
17C23×C3×Dic3—31no—
18C2×#3757D311yescharacteristic
19C2×C3×#217—31no—
20C2×C3×#217—31no—
21C2×C3×#217—31no—
22C2×C3×#217—61no—
23#63579—34no—
24#63579—34no—
25C23×H(3,𝔽3)—24no—
26C23×H(3,𝔽3)C2×C412yes—
27C23×H(3,𝔽3)D412yes—
28C23×H(3,𝔽3)D412yes—
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31C2×#1084—24no—
32C2×(C3⋊Dic3)⋊C3—28no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—24no—
37C2×(C3⋊Dic3)⋊C3C2×C414yes—
38#5211—91no—
39#4943—91no—
40#4943—91no—
41C24×C32—21no—
42C24×C32C2×D311yescharacteristic
43C24×C32C2×D311yescharacteristic
44C22×C3⋊Dic3—22no—
45C22×C3⋊Dic3—22no—
46C22×C3⋊Dic3Dic314yescomplemented
47C22×C3⋊Dic3C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C22×C3×Dic3—34no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—62no—
55C22×C3×Dic3—62no—
56C22×C3×Dic3—62no—
57#3757—22no—
58C3×#217—64no—
59C3×#217—62no—
60C3×#217—62no—
61C3×#217—62no—
62C22×H(3,𝔽3)—24no—
63C22×H(3,𝔽3)—24no—
64C22×H(3,𝔽3)—24no—
65C22×H(3,𝔽3)C22⋊C414yes—
66C22×H(3,𝔽3)C22×C411yescharacteristic
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)#4114yes—
70#1084—42no—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3—42no—
73C23×C3×C4—91no—
74C23×Dic3—31no—
75C23×Dic3—91no—
76C2×C3×C22⋊C4—91no—
77C2×C3×C22⋊C4—91no—
78C2×#217—31no—
79C2×#217—31no—
80C2×#217—31no—
81C2×#217—61no—
82C2×#217—91no—
83C3⋊#95—94no—
84C3⋊#95—94no—
85C3⋊#95—94no—
86C23×C32—24no—
87C23×C32—24no—
88C23×C32—22no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—21no—
92C23×C32—44no—
93C23×C32C4×D312yes—
94C23×C32D1212yes—
95C23×C32C2×Dic312yes—
96C23×C32#6712yes—
97C23×C32#6712yes—
98C23×C32#6712yes—
99C23×C32C22×D311yescharacteristic
100C23×C32C22×D311yescharacteristic
101C2×C3⋊Dic3—28no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—24no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3C2×Dic314yes—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—124no—
119#381—271noSylow
120C2×H(3,𝔽3)—24no—
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)#11712yes—
123C2×H(3,𝔽3)#12312yes—
124C2×H(3,𝔽3)#12312yes—
125C24×C3—31no—
126C24×C3—31no—
127C24×C3—61no—
128C24×C3D3211yescharacteristic
129C22×C3×C4—92no—
130C22×C3×C4—94no—
131C22×C3×C4—92no—
132C22×C3×C4—92no—
133C22×C3×C4—182no—
134C22×C3×C4—182no—
135C22×Dic3—34no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—62no—
141C22×Dic3—62no—
142C22×Dic3—62no—
143C22×Dic3—92no—
144C22×Dic3—94no—
145C22×Dic3—92no—
146C22×Dic3—182no—
147C22×Dic3—182no—
148C3×C22⋊C4—182no—
149C3×C22⋊C4—182no—
150#217—62no—
151#217—62no—
152#217—64no—
153#217—62no—
154#217—182no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—24no—
160C22×C32—24no—
161C22×C32—24no—
162C22×C32—21no—
163C22×C32—24no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—44no—
167C22×C32—44no—
168C22×C32—44no—
169C22×C32(C2×D3)⋊C414yes—
170C22×C32#21714yes—
171C22×C32C2×C4×D311yescharacteristic
172C22×C32C2×D1211yescharacteristic
173C22×C32#23514yes—
174C22×C32#23714yes—
175C22×C32C22×Dic311yescharacteristic
176C22×C32C2×#6711yescharacteristic
177C22×C32C2×#6711yescharacteristic
178C22×C32C2×#6711yescharacteristic
179C3⋊Dic3—24no—
180C3⋊Dic3—42no—
181C3×Dic3—64no—
182C3×Dic3—124no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185C3×Dic3—122no—
186C23×C4—91no—
187C2×C22⋊C4—91no—
188C2×C22⋊C4—91no—
189#95—274no—
190H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
191C23×C3—24no—
192C23×C3—32no—
193C23×C3—32no—
194C23×C3—32no—
195C23×C3—32no—
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—64no—
201C23×C3—64no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—61no—
206C23×C3—124no—
207C23×C3D3×Dic312yes—
208C23×C3#62512yes—
209C23×C3#62612yes—
210C23×C3C2×D3211yescharacteristic
211C2×C3×C4—94no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—188no—
215C2×C3×C4—182no—
216C2×C3×C4—184no—
217C2×C3×C4—184no—
218C2×C3×C4—182no—
219C2×Dic3—34no—
220C2×Dic3—68no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—64no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228C2×Dic3—64no—
229C2×Dic3—62no—
230C2×Dic3—94no—
231C2×Dic3—124no—
232C2×Dic3—182no—
233C2×Dic3—182no—
234C2×Dic3—188no—
235C2×Dic3—182no—
236C2×Dic3—184no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—44no—
244C2×C32#87712yes—
245C2×C32#88612yes—
246C2×C32#92612yes—
247C2×C32C2×(C2×D3)⋊C411yescharacteristic
248C2×C32#92912yes—
249C2×C32#92912yes—
250C2×C32#93512yes—
251C2×C32C2×#21711yescharacteristic
252C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C22×C4—94no—
254C22×C4—92no—
255C22×C4—92no—
256C22×C4—92no—
257C22×C4—182no—
258C22×C4—182no—
259C22⋊C4—182no—
260C22⋊C4—182no—
261C22×C3—24no—
262C22×C3—24no—
263C22×C3—24no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—34no—
267C22×C3—34no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—34no—
273C22×C3—34no—
274C22×C3—64no—
275C22×C3—64no—
276C22×C3—64no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3—64no—
280C22×C3—61no—
281C22×C3—64no—
282C22×C3—61no—
283C22×C3—61no—
284C22×C3—64no—
285C22×C3—124no—
286C22×C3—124no—
287C22×C3—124no—
288C22×C3(C2×D3)⋊Dic314yes—
289C22×C3C2×D3×Dic311yescharacteristic
290C22×C3#380414yes—
291C22×C3C2×#62511yescharacteristic
292C22×C3C2×#62611yescharacteristic
293C3×C4—184no—
294C3×C4—362no—
295C3×C4—362no—
296Dic3—64no—
297Dic3—122no—
298Dic3—122no—
299Dic3—124no—
300Dic3—122no—
301Dic3—184no—
302Dic3—362no—
303C32—21no—
304C32#494311yescharacteristic, complemented
305C32#521111yescharacteristic, complemented
306C23—24no—
307C23((C3⋊D3)⋊C3)⋊C412yes—
308C23#619812yes—
309C23#620012yes—
310C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C2×C4—94no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4—188no—
315C2×C4—184no—
316C2×C4—184no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C3—24no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—32no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—32no—
327C2×C3—32no—
328C2×C3—64no—
329C2×C3—64no—
330C2×C3—62no—
331C2×C3—62no—
332C2×C3—61no—
333C2×C3—62no—
334C2×C3—124no—
335C2×C3#6395512yes—
336C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
337C2×C3#6396112yes—
338C2×C3#6396612yes—
339C22—24no—
340C22—24no—
341C22—24no—
342C22#8845014yes—
343C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
344C22#8867114yes—
345C22C2×#619811yescharacteristic
346C22C2×#620011yescharacteristic
347C4—184no—
348C4—362no—
349C4—362no—
350C3—31no—
351C3—31no—
352C3—61no—
353C3#1059169611yescharacteristic
354C2—24no—
355C2#1172129612yes—
356C2#1172130112yes—
357C2C2×#8845011yescharacteristic
358C2#1172130612yes—
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#381
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. #11720334C2semidirect
11. C22×(C3⋊Dic3)⋊C3C4semidirect
46. C22×C3⋊Dic3Dic3semidirect
190. H(3,𝔽3)#381semidirect
304. C32#4943semidirect
305. C32#5211semidirect

Maximal subgroups

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

subgroupindex
C23×(C3⋊Dic3)⋊C32
C2×#885522
C2×#3757⋊C32
#117203342
#105916973
#105916963