Finite Group Databasebeta

← #13156572

Subgroups — #13156572

Order 1728.

526 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723447C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic
4#11722038C212yesmaximal
5C3×#63565C212yesmaximal
6#10591613—32nomaximal
7#10590568—31nomaximal
8C2×C32×Dic6C2212yes—
9C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2211yescharacteristic
13#88806—21no—
14Dic3⋊(C3⋊Dic3)—21no—
15C3×#3724—21no—
16C3×#3724—21no—
17#64099—31no—
18C3×#944—32no—
19#63873D311yescharacteristic
20#63869—32no—
21#63869—61no—
22#63662—32no—
23C3×#835—32no—
24C3×#810—32no—
25#63566—32no—
26#63566—32no—
27#63566—62no—
28#63565D312yes—
29#63555—31no—
30#63554—32no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C3×C3⋊(C3⋊C8)—22no—
40#5082—91no—
41#4797—92no—
42C2×C32×Q8—32no—
43C2×C3×Dic6—22no—
44C2×C3×Dic6C2×D312yes—
45C2×C3×Dic6C2×D312yes—
46C32×C4⋊C4—31no—
47C4⋊(C3⋊Dic3)—31no—
48C4⋊(C3⋊Dic3)—32no—
49C4⋊(C3⋊Dic3)—61no—
50C4⋊(C3⋊Dic3)—61no—
51C4⋊(C3⋊Dic3)—62no—
52C4⋊(C3⋊Dic3)—121no—
53#3750—61no—
54C4×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3C2×D312yes—
58C3×Dic3⋊C4—62no—
59C3×Dic3⋊C4—62no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65#3724—21no—
66#3724—21no—
67#3724—62no—
68#3724—62no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71C22×C33C2×D411yescharacteristic
72C33×C4#4112yes—
73C3⋊(C3⋊Dic3)—41no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—21no—
76C32×Dic3—42no—
77C32×Dic3—41no—
78C3×C4⋊Q8—91no—
79#944—32no—
80#890—92no—
81#890—181no—
82#870—91no—
83C3×#103—92no—
84#835—32no—
85#835—92no—
86#835—92no—
87#835—182no—
88#816—92no—
89#815—92no—
90#810—32no—
91#803—91no—
92C32×Q8—62no—
93C32×Q8—62no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4—62no—
99C2×C32×C4C22×D312yes—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3#6711yescharacteristic
107C2×C3⋊Dic3#6711yescharacteristic
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×Dic6—24no—
118C3×Dic6—22no—
119C3×Dic6—22no—
120C3×Dic6—22no—
121C3×Dic6—22no—
122C3×Dic6—42no—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—122no—
127#491—271noSylow
128C2×C33#12411yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—32no—
132C2×C3×Q8—32no—
133C2×C3×Q8—62no—
134C2×Dic6D3212yes—
135C2×C3×C8—92no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4—181no—
140C3×C42—91no—
141C4⋊Dic3—92no—
142C4⋊Dic3—91no—
143C4⋊Dic3—181no—
144C4⋊Dic3—182no—
145C4⋊Dic3—182no—
146C4⋊Dic3—182no—
147C4⋊Dic3—361no—
148C4⋊Dic3D3211yescharacteristic
149Dic3⋊C4—62no—
150Dic3⋊C4—62no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—182no—
154Dic3⋊C4—182no—
155C4×Dic3—32no—
156C4×Dic3—61no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—62no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—41no—
163C22×C32D3×D412yes—
164C22×C32C2×#6711yescharacteristic
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4—122no—
170C32×C4—121no—
171C32×C4#23512yes—
172C32×C4#23712yes—
173C32×C4#23712yes—
174C3⋊Dic3—21no—
175C3⋊Dic3—21no—
176C3⋊Dic3—121no—
177C3⋊Dic3—122no—
178C3⋊Dic3—122no—
179C3⋊Dic3—242no—
180C3⋊Dic3—242no—
181C3×Dic3—44no—
182C3×Dic3—42no—
183C3×Dic3—42no—
184C3×Dic3—42no—
185C3×Dic3—42no—
186C3×Dic3—62no—
187C3×Dic3—62no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C4⋊Q8—91no—
191#125—271no—
192#103—92no—
193#103—272no—
194#97—271no—
195C33#49111yescharacteristic, complemented, Sylow
196C3×Q8—64no—
197C3×Q8—62no—
198C3×Q8—62no—
199C3×Q8—62no—
200C3×Q8—62no—
201C3×Q8—122no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—122no—
212C2×C3×C4C2×D3211yescharacteristic
213C2×C3×C4C2×D3212yes—
214C2×Dic3—22no—
215C2×Dic3—21no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—181no—
223C2×Dic3—362no—
224C2×Dic3—362no—
225Dic6—22no—
226Dic6—22no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—62no—
230C3⋊C8—122no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—22no—
235C2×C32—22no—
236C2×C32—22no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32#88512yes—
241C2×C32#91512yes—
242C2×C32#91812yes—
243C2×C32#93611yescharacteristic
244C2×C32#94211yescharacteristic
245C2×C32#94211yescharacteristic
246C2×Q8—32no—
247C2×C8—92no—
248C4⋊C4—31no—
249C4⋊C4—181no—
250C4⋊C4—181no—
251C4⋊C4—271no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C42—91no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3D3×#6712yes—
259C22×C3#381111yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—122no—
265C3×C4—124no—
266C3×C4—122no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4—181no—
270C3×C4—181no—
271C3×C4#379512yes—
272C3×C4#380412yes—
273C3×C4#380512yes—
274Dic3—41no—
275Dic3—42no—
276Dic3—62no—
277Dic3—62no—
278Dic3—121no—
279Dic3—121no—
280Dic3—362no—
281Dic3—361no—
282Dic3—362no—
283Dic3—722no—
284Dic3—722no—
285C32—21no—
286C32—22no—
287C32—41no—
288C32#479712yescomplemented
289C32#508211yescharacteristic, complemented
290Q8—62no—
291Q8—62no—
292C2×C4—61no—
293C2×C4—62no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—541no—
297C2×C4D3311yescharacteristic
298C8—182no—
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#6391311yescharacteristic
309C2×C3#6393111yescharacteristic
310C2×C3#6394511yescharacteristic
311C2×C3#6394612yes—
312C2×C3#6394812yes—
313C2×C3#6395912yes—
314C22#8895211yescharacteristic
315C4—121no—
316C4—122no—
317C4—181no—
318C4—181no—
319C4—1081no—
320C4#8896112yes—
321C3—22no—
322C3—21no—
323C3—41no—
324C3#1059056811yescharacteristic, complemented
325C3#1059161312yescomplemented
326C2#1172288911yescharacteristic
327C2#1172289011yescharacteristic
328C2#1172300411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
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
195. C33#491semidirect
288. C32#4797semidirect
289. C32#5082semidirect
324. C3#10590568semidirect
325. C3#10591613semidirect

Maximal subgroups

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

subgroupindex
#117234472
C3×#638732
#117220712
#117220382
C3×#635652
#105916133
#105905683