Finite Group Databasebeta

← #13156521

Subgroups — #13156521

Order 1728.

542 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C3×#63919C211yesmaximal, characteristic
3#11722039C212yesmaximal
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6#10590901—31nomaximal
7#10590560—32nomaximal
8C2×C3×#634C2212yes—
9C2×C3×#627—21no—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88815—21no—
15#88815—21no—
16C3×#3724—21no—
17C3×#888—32no—
18#63919D311yescharacteristic
19#63873—31no—
20#63866—32no—
21C8×C3⋊Dic3—31no—
22#63637—32no—
23C3×#816—32no—
24C3×#810—32no—
25#63567D312yes—
26#63565—32no—
27#63565—32no—
28#63565—62no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×#634—22no—
36C3×#634—22no—
37C2×C32×Dic3—22no—
38C3×#627—42no—
39C3⋊(C3⋊(C3⋊C8))—21no—
40C32×C3⋊C8—21no—
41#5093—91no—
42#4792—92no—
43C2×#634C2×D312yes—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—62no—
47C2×C3×Dic6—62no—
48C2×#627—21no—
49C2×C32×C8—31no—
50C32×C4⋊C4—32no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—31no—
54C2×C3⋊(C3⋊C8)—61no—
55C2×C3⋊(C3⋊C8)—62no—
56C2×C3⋊(C3⋊C8)—121no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3C2×D312yes—
59C3×C4⋊Dic3C2×D312yes—
60C3×Dic3⋊C4—62no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8C2×D312yes—
65#3724—21no—
66#3724—61no—
67#3724—61no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70C22×C33C2×D411yescharacteristic
71C33×C4#4111yescharacteristic
72C33×C4#4111yescharacteristic
73C3×C3⋊Dic3—42no—
74C3×C3⋊Dic3SD1612yes—
75C3×C3⋊Dic3Q1612yes—
76C32×Dic3—42no—
77C3×C4⋊Q8—91no—
78#944—92no—
79#888—32no—
80#869—91no—
81C3×#103—92no—
82#835—92no—
83#816—32no—
84C8×Dic3—92no—
85C8×Dic3—181no—
86#810—32no—
87#810—92no—
88#810—92no—
89#810—182no—
90C4×C3⋊C8—91no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3#6711yescharacteristic
99C2×C3⋊Dic3#6711yescharacteristic
100#634—22no—
101#634—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—42no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—122no—
112C3×Dic6—64no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117C3×Dic6—122no—
118C3×Dic6—124no—
119#627—42no—
120C32×C8—61no—
121C3⋊(C3⋊C8)—61no—
122C3⋊(C3⋊C8)—62no—
123C3⋊(C3⋊C8)—62no—
124C3⋊(C3⋊C8)—122no—
125C3⋊(C3⋊C8)—122no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128#487—271noSylow
129C2×C33#12411yescharacteristic
130C2×C33C2×SD1611yescharacteristic
131C2×C33C2×Q1611yescharacteristic
132C2×C3×Q8—92no—
133C2×C3×Q8—181no—
134C2×Dic6—32no—
135C2×Dic6—32no—
136C2×Dic6—62no—
137C2×Dic6—62no—
138C2×C3×C8—32no—
139C2×C3×C8—61no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—62no—
143C3×C4⋊C4—181no—
144C3×C42—91no—
145C4⋊Dic3—181no—
146C4⋊Dic3—181no—
147C4⋊Dic3D3212yes—
148Dic3⋊C4—62no—
149Dic3⋊C4—182no—
150Dic3⋊C4—182no—
151C4×Dic3—32no—
152C4×Dic3—61no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—92no—
155C2×C3⋊C8—181no—
156C2×C3⋊C8—182no—
157C2×C3⋊C8—361no—
158C2×C3⋊C8D3211yescharacteristic
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—41no—
162C22×C32D3×D412yes—
163C22×C32C2×#6711yescharacteristic
164C32×C4—22no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4—122no—
171C32×C4#23512yes—
172C32×C4#23712yes—
173C32×C4#23911yescharacteristic
174C32×C4#23911yescharacteristic
175C3⋊Dic3—42no—
176C3⋊Dic3#21412yes—
177C3⋊Dic3#21612yes—
178C3×Dic3—34no—
179C3×Dic3—34no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—82no—
185C3×Dic3—124no—
186C3×Dic3—122no—
187C3×Dic3—122no—
188C4⋊Q8—91no—
189C4⋊Q8—271no—
190#103—92no—
191#103—272no—
192C4×C8—271no—
193C33#48711yescharacteristic, complemented, Sylow
194C3×Q8—182no—
195C3×Q8—182no—
196C3×Q8—362no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—62no—
201C2×C3×C4—62no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—122no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3212yes—
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—32no—
210C2×Dic3—32no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—122no—
216Dic6—64no—
217Dic6—62no—
218Dic6—62no—
219Dic6—62no—
220Dic6—62no—
221Dic6—122no—
222Dic6—124no—
223C3×C8—62no—
224C3×C8—62no—
225C3⋊C8—21no—
226C3⋊C8—182no—
227C3⋊C8—181no—
228C3⋊C8—182no—
229C3⋊C8—362no—
230C3⋊C8—362no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—22no—
236C2×C32—22no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32#88512yes—
241C2×C32D3×SD1612yes—
242C2×C32D3×Q1612yes—
243C2×C32C2×#21411yescharacteristic
244C2×C32C2×#21611yescharacteristic
245C2×C32#94711yescharacteristic
246C2×Q8—92no—
247C2×Q8—181no—
248C2×C8—31no—
249C2×C8—271no—
250C4⋊C4—32no—
251C4⋊C4—181no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C42—91no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3#380211yescharacteristic
259C22×C3D3×#6712yes—
260C3×C4—21no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—92no—
267C3×C4—92no—
268C3×C4—122no—
269C3×C4—122no—
270C3×C4—242no—
271C3×C4—362no—
272C3×C4#379612yes—
273C3×C4#379811yescharacteristic
274C3×C4#379912yes—
275C3×C4#380511yescharacteristic
276Dic3—34no—
277Dic3—34no—
278Dic3—42no—
279Dic3—62no—
280Dic3—62no—
281Dic3—124no—
282Dic3—122no—
283Dic3—122no—
284C32—22no—
285C32—21no—
286C32—41no—
287C32#479212yescomplemented
288C32#509311yescharacteristic, complemented
289Q8—182no—
290Q8—182no—
291Q8—362no—
292C2×C4—62no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—182no—
296C2×C4D3311yescharacteristic
297C8—61no—
298C8—541no—
299C2×C3—21no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3#6379611yescharacteristic
309C2×C3#6380211yescharacteristic
310C2×C3#6390712yes—
311C2×C3#6391511yescharacteristic
312C2×C3D3×#21412yes—
313C2×C3D3×#21612yes—
314C22#8895311yescharacteristic
315C4—92no—
316C4—92no—
317C4—122no—
318C4—362no—
319C4#8896411yescharacteristic
320C4#8896511yescharacteristic
321C3—22no—
322C3—21no—
323C3—41no—
324C3#1059056012yescomplemented
325C3#1059090111yescharacteristic, complemented
326C2#1172283911yescharacteristic
327C2#1172284711yescharacteristic
328C2#1172300711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#487
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
193. C33#487semidirect
287. C32#4792semidirect
288. C32#5093semidirect
324. C3#10590560semidirect
325. C3#10590901semidirect

Maximal subgroups

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

subgroupindex
#117236782
C3×#639192
#117220392
C3⋊C8×C3⋊Dic32
C3×#635672
#105909013
#105905603