Finite Group Databasebeta

← #13140070

Subgroups — #13140070

Order 1728.

807 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 807 conjugacy classes of subgroups — too many to draw. Only its 139 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×#6229C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3C2×#88447C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#11720334C212yesmaximal
6#10591261—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C2×#6229—22no—
11C2×#6229—22no—
12C2×#6229C414yescomplemented
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15#88449—22no—
16#88447—22no—
17C22×C3×Dic6—32no—
18C22×C3×Dic6—61no—
19C2×Dic3⋊Dic3—32no—
20C2×Dic32—32no—
21#63658D312yes—
22C3×C3⋊#95—32no—
23#63579—32no—
24#63579—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×#1084—21no—
30C2×#1084—21no—
31C2×#1084C2×C412yes—
32C2×#1084C2×C412yes—
33C2×#1084D412yes—
34C2×#1084D412yes—
35#6229—28no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3Q814yescomplemented
42#5229—91no—
43#4646—92no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D312yes—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C2×C3×Dic6—38no—
55C2×C3×Dic6—64no—
56C2×C3×Dic6—64no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6—64no—
59C2×C3×Dic6—64no—
60Dic3⋊Dic3—64no—
61Dic32—64no—
62C22×H(3,𝔽3)C22×C411yescharacteristic
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×Q812yes—
66C22×H(3,𝔽3)#4112yes—
67C4×H(3,𝔽3)—22no—
68#1084—22no—
69#1084C22⋊C414yes—
70(C3⋊Dic3)⋊C3—24no—
71(C3⋊Dic3)⋊C3—42no—
72C22×C3×Q8—91no—
73C22×Dic6—32no—
74C22×Dic6—61no—
75C2×C4⋊Dic3—91no—
76C2×Dic3⋊C4—92no—
77C2×C4×Dic3—91no—
78C2×C4×Dic3—92no—
79C3×#95—92no—
80C3⋊#95—32no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C3⋊#95—92no—
85C23×C32—21no—
86C23×C32C22×D312yes—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4C4×D314yes—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3Dic614yescomplemented
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C3×Dic6—616no—
118C3×Dic6—616no—
119#391—271noSylow
120C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
121C2×H(3,𝔽3)C4×Q812yes—
122C2×H(3,𝔽3)C22⋊Q812yes—
123C2×H(3,𝔽3)#12411yescharacteristic
124C2×H(3,𝔽3)C4⋊Q811yescharacteristic
125C2×C3×Q8—94no—
126C2×C3×Q8—182no—
127C2×C3×Q8—182no—
128C22×C3×C4—32no—
129C22×C3×C4—61no—
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4D3211yescharacteristic
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—61no—
140C22×Dic3—61no—
141C22×Dic3—92no—
142C22×Dic3—92no—
143C2×Dic6—38no—
144C2×Dic6—64no—
145C2×Dic6—64no—
146C2×Dic6—64no—
147C2×Dic6—64no—
148C2×Dic6—64no—
149C4⋊Dic3—182no—
150Dic3⋊C4—184no—
151C4×Dic3—184no—
152C4×Dic3—182no—
153C22×C32—22no—
154C22×C32—22no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32C2×Dic612yes—
159C22×C32C2×C4×D312yes—
160C22×C32#23512yes—
161C22×C32D3×D412yes—
162C22×C32D3×D412yes—
163C22×C32#23712yes—
164C22×C32D3×Q812yes—
165C32×C4—24no—
166C32×C4—24no—
167C3⋊Dic3—24no—
168C3⋊Dic3—42no—
169C3×Dic3—38no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—64no—
173C3×Dic3—64no—
174C3×Dic3—122no—
175C22×Q8—91no—
176C2×C4⋊C4—271no—
177C2×C42—271no—
178#95—92no—
179#95—272no—
180H(3,𝔽3)#39111yescharacteristic, complemented, Sylow
181C23×C3—32no—
182C23×C3—61no—
183C23×C3C2×D3211yescharacteristic
184C3×Q8—188no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—34no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—92no—
194C2×C3×C4—92no—
195C2×C3×C4—92no—
196C2×C3×C4—94no—
197C2×C3×C4—92no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4#62412yes—
206C2×Dic3—34no—
207C2×Dic3—34no—
208C2×Dic3—34no—
209C2×Dic3—34no—
210C2×Dic3—34no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—94no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C2×Dic3—182no—
230Dic6—616no—
231Dic6—616no—
232C2×C32—21no—
233C2×C32—22no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32C4×Dic612yes—
238C2×C32#87812yes—
239C2×C32D3×C22⋊C412yes—
240C2×C32#88512yes—
241C2×C32#88712yes—
242C2×C32#88812yes—
243C2×C32#89612yes—
244C2×Q8—94no—
245C2×Q8—182no—
246C2×Q8—182no—
247C22×C4—91no—
248C22×C4—92no—
249C22×C4—91no—
250C22×C4—92no—
251C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C4⋊C4—542no—
253C42—542no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—32no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3D3×Dic612yes—
267C22×C3#379512yes—
268C22×C3C2×#62411yescharacteristic
269C22×C3#381111yescharacteristic
270C22×C3#381111yescharacteristic
271C3×C4—22no—
272C3×C4—64no—
273C3×C4—64no—
274C3×C4—94no—
275C3×C4—184no—
276C3×C4—182no—
277C3×C4—362no—
278Dic3—38no—
279Dic3—64no—
280Dic3—64no—
281Dic3—64no—
282Dic3—64no—
283Dic3—122no—
284Dic3—184no—
285Dic3—362no—
286C32—21no—
287C32#464612yescomplemented
288C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289Q8—188no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—92no—
293C2×C4—92no—
294C2×C4—92no—
295C2×C4—94no—
296C2×C4—92no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4—181no—
300C2×C4—181no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4#619912yes—
305C2×C3—32no—
306C2×C3—32no—
307C2×C3—32no—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—62no—
315C2×C3—62no—
316C2×C3—61no—
317C2×C3#6384011yescharacteristic
318C2×C3#6384412yes—
319C2×C3#6389612yes—
320C2×C3#6391311yescharacteristic
321C2×C3#6397411yescharacteristic
322C22((C3⋊D3)⋊C3)⋊Q812yes—
323C22#8864812yes—
324C22C2×#619911yescharacteristic
325C22#8867711yescharacteristic
326C22#8867711yescharacteristic
327C4—22no—
328C4—94no—
329C4—182no—
330C4—184no—
331C4—362no—
332C3—32no—
333C3—61no—
334C3#1059127311yescharacteristic
335C2#1172108011yescharacteristic
336C2#1172108312yes—
337C2#1172113512yes—
338C2#1172114311yescharacteristic
339C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#391
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
12. C2×#6229C4semidirect
41. C2×(C3⋊Dic3)⋊C3Q8semidirect
98. C2×C3⋊Dic3Dic6semidirect
180. H(3,𝔽3)#391semidirect
287. C32#4646semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#884492
C2×#884472
#63658⋊C32
#117203342
#105912613