Finite Group Databasebeta

← #13156595

Subgroups — #13156595

Order 1728.

674 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 116 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
1#11723678C211yesmaximal, characteristic
2C2×#88798C212yesmaximal
3C3×#63873C211yesmaximal, characteristic
4C2×C3×#3719C212yesmaximal
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
6#10591626—32nomaximal
7#10590905—31nomaximal
8C2×C3×#634C2212yes—
9C2×C32×Dic6C2212yes—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13#88815—22no—
14#88798—24no—
15C3×#3724—22no—
16C3×#3719—24no—
17C2×#3756—32no—
18C3×#944—32no—
19C2×C3×#216—32no—
20C2×C3×Dic12—32no—
21#63873—31no—
22#63873D311yescharacteristic
23#63866—32no—
24C2×#3719D312yes—
25C2×#3718—32no—
26C2×#3718—32no—
27C2×#3718—62no—
28C8×C3⋊Dic3—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D412yes—
34C3×#634—22no—
35C3×#634—22no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C3⋊(C3⋊(C3⋊C8))D412yes—
40C32×C3⋊C8D412yes—
41#5183—92no—
42#5092—91no—
43C2×C32×Q8—32no—
44C2×#634C2×D312yes—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C2×D312yes—
51C2×C32×C8—31no—
52#3756—64no—
53C4×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)—32no—
55C2×C3⋊(C3⋊C8)—31no—
56C2×C3⋊(C3⋊C8)—61no—
57C2×C3⋊(C3⋊C8)—62no—
58C2×C3⋊(C3⋊C8)—121no—
59C3×#216—64no—
60C3×Dic3⋊C4—64no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8C2×D312yes—
65C3×Dic12—64no—
66#3724—22no—
67#3724—62no—
68Dic3⋊Dic3—64no—
69#3719—24no—
70#3718—64no—
71#3718—64no—
72#3718—68no—
73C22×C33C2×D411yescharacteristic
74C33×C4C2×D411yescharacteristic
75C33×C4C2×D411yescharacteristic
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3Q1614yescomplemented
78C32×Dic3—42no—
79C2×C3×Q16—92no—
80C3×C4⋊Q8—91no—
81#944—32no—
82#944—92no—
83C2×#216—32no—
84C2×#216—92no—
85C2×#216—92no—
86C2×#216—92no—
87C2×#216—182no—
88C2×Dic12—32no—
89#869—91no—
90C8×Dic3—92no—
91C8×Dic3—181no—
92C4×C3⋊C8—91no—
93C32×Q8—62no—
94C32×Q8—62no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—41no—
98C2×C32×C4—62no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D312yes—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3#6712yes—
103#634—22no—
104#634—22no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—42no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C3×Dic6—24no—
114C3×Dic6—22no—
115C3×Dic6—22no—
116C3×Dic6—22no—
117C3×Dic6—22no—
118C3×Dic6—42no—
119C3×Dic6—64no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—62no—
124C3×Dic6—122no—
125C32×C8—32no—
126C3⋊(C3⋊C8)—32no—
127C3⋊(C3⋊C8)—34no—
128C3⋊(C3⋊C8)—64no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—122no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8#6714yes—
133#494—271noSylow
134C2×C33#12711yescharacteristic
135C2×C33C2×Q1612yes—
136C2×C3×Q8—32no—
137C2×C3×Q8—32no—
138C2×C3×Q8—62no—
139C2×C3×Q8—92no—
140C2×Dic6—32no—
141C2×Dic6—32no—
142C2×Dic6—62no—
143C2×Dic6D3212yes—
144C3×Q16—184no—
145C2×C3×C8—32no—
146C2×C3×C8—61no—
147C3×C4⋊C4—182no—
148C3×C42—91no—
149#216—64no—
150#216—184no—
151#216—184no—
152#216—184no—
153#216—188no—
154C4⋊Dic3—182no—
155Dic3⋊C4—64no—
156Dic3⋊C4—184no—
157C4×Dic3—32no—
158C4×Dic3—61no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—92no—
161C2×C3⋊C8—181no—
162C2×C3⋊C8—182no—
163C2×C3⋊C8—361no—
164C2×C3⋊C8D3211yescharacteristic
165Dic12—64no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×#6711yescharacteristic
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—122no—
178C32×C4D3×D412yes—
179C32×C4D3×D411yescharacteristic
180C32×C4D3×D411yescharacteristic
181C32×C4C2×#6712yes—
182C3⋊Dic3—42no—
183C3⋊Dic3#21614yescomplemented
184C3×Dic3—38no—
185C3×Dic3—44no—
186C3×Dic3—42no—
187C3×Dic3—42no—
188C3×Dic3—64no—
189C3×Dic3—124no—
190C3×Dic3—122no—
191C3×Dic3—122no—
192C2×Q16—92no—
193C2×Q16—272no—
194C4⋊Q8—91no—
195C4⋊Q8—271no—
196C4×C8—271no—
197C33#49411yescharacteristic, complemented, Sylow
198C3×Q8—64no—
199C3×Q8—62no—
200C3×Q8—62no—
201C3×Q8—62no—
202C3×Q8—62no—
203C3×Q8—122no—
204C3×Q8—182no—
205C3×Q8—182no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—92no—
212C2×C3×C4—122no—
213C2×C3×C4—182no—
214C2×C3×C4C2×D3211yescharacteristic
215C2×C3×C4C2×D3212yes—
216C2×Dic3—22no—
217C2×Dic3—34no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—122no—
222Dic6—22no—
223Dic6—22no—
224Dic6—64no—
225Dic6—62no—
226Dic6—62no—
227Dic6—62no—
228Dic6—62no—
229Dic6—122no—
230C3×C8—34no—
231C3×C8—62no—
232C3⋊C8—92no—
233C3⋊C8—94no—
234C3⋊C8—182no—
235C3⋊C8—184no—
236C3⋊C8—362no—
237C3⋊C8#62512yes—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—24no—
241C2×C32—22no—
242C2×C32—41no—
243C2×C32—42no—
244C2×C32D3×Q1614yes—
245C2×C32#94012yes—
246C2×C32#94011yescharacteristic
247C2×C32C2×#21612yes—
248C2×Q8—32no—
249C2×Q8—92no—
250Q16—184no—
251Q16—544no—
252C2×C8—31no—
253C2×C8—271no—
254C4⋊C4—182no—
255C4⋊C4—542no—
256C42—91no—
257C22×C3—21no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3#380211yescharacteristic
261C22×C3D3×#6712yes—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—94no—
269C3×C4—124no—
270C3×C4—122no—
271C3×C4—122no—
272C3×C4—362no—
273C3×C4#380211yescharacteristic
274C3×C4C2×#62511yescharacteristic
275C3×C4D3×#6712yes—
276C3×C4#381112yes—
277Dic3—38no—
278Dic3—42no—
279Dic3—64no—
280Dic3—124no—
281Dic3—122no—
282Dic3—122no—
283C32—22no—
284C32—21no—
285C32—41no—
286C32#509211yescharacteristic, complemented
287C32#518312yescomplemented
288Q8—62no—
289Q8—62no—
290Q8—182no—
291Q8—182no—
292C2×C4—62no—
293C2×C4—92no—
294C2×C4—182no—
295C2×C4D3311yescharacteristic
296C8—32no—
297C8—272no—
298C2×C3—22no—
299C2×C3—21no—
300C2×C3—22no—
301C2×C3—24no—
302C2×C3—41no—
303C2×C3—42no—
304C2×C3#6380212yes—
305C2×C3#6391411yescharacteristic
306C2×C3D3×#21614yes—
307C2×C3#6397912yes—
308C22#8895311yescharacteristic
309C4—94no—
310C4—122no—
311C4—362no—
312C4D3×#62511yescharacteristic
313C4#8895411yescharacteristic
314C3—22no—
315C3—21no—
316C3—41no—
317C3#1059090511yescharacteristic, complemented
318C3#1059162612yescomplemented
319C2#1172284712yes—
320C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#494
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
77. C3×C3⋊Dic3Q16semidirect
183. C3⋊Dic3#216semidirect
197. C33#494semidirect
286. C32#5092semidirect
287. C32#5183semidirect
317. C3#10590905semidirect
318. C3#10591626semidirect

Maximal subgroups

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

subgroupindex
#117236782
C2×#887982
C3×#638732
C2×C3×#37192
C3⋊C8×C3⋊Dic32
#105916263
#105909053