Finite Group Databasebeta

← #13156577

Subgroups — #13156577

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#11723678C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722040C212yesmaximal
4C3×#63567C212yesmaximal
5C3×#63555C211yesmaximal, characteristic
6#10590898—32nomaximal
7#10590568—31nomaximal
8C2×C3×#634C2212yes—
9C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×C4⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815—21no—
14#88815—21no—
15#88806—21no—
16Dic3⋊(C3⋊Dic3)—21no—
17#64099—31no—
18#63873—31no—
19#63869—32no—
20#63869—61no—
21#63866—32no—
22#63643—32no—
23C3×#816—32no—
24C3×#815—32no—
25C3×#810—32no—
26#63567D312yes—
27#63566—32no—
28#63566—32no—
29#63566—62no—
30#63555D311yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×#634—22no—
37C3×#634—22no—
38C2×C32×Dic3—21no—
39C32×C3⋊C8—22no—
40#4797—92no—
41#4703—91no—
42C2×#634C2×D312yes—
43C2×C3×Dic6—32no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—62no—
46C2×C32×C8—32no—
47C32×C4⋊C4—31no—
48C4⋊(C3⋊Dic3)—31no—
49C4⋊(C3⋊Dic3)—32no—
50C4⋊(C3⋊Dic3)—61no—
51C4⋊(C3⋊Dic3)—61no—
52C4⋊(C3⋊Dic3)—62no—
53C4⋊(C3⋊Dic3)—121no—
54#3750—61no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8C2×D312yes—
62C2×C3×C3⋊C8C2×D312yes—
63#3724—62no—
64#3724—62no—
65#3724—61no—
66#3724—61no—
67Dic3⋊Dic3—62no—
68Dic3⋊Dic3—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—
76C3×C3⋊Dic3—42no—
77C32×Dic3—41no—
78#944—92no—
79#890—92no—
80#890—181no—
81#870—91no—
82#869—91no—
83C3×#103—92no—
84C3×#97—91no—
85#835—92no—
86#816—32no—
87#816—92no—
88#816—92no—
89#816—182no—
90#815—32no—
91#810—32no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D312yes—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3D1211yescharacteristic
105C2×C3⋊Dic3D1211yescharacteristic
106#634—22no—
107#634—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—122no—
117C3×Dic6—64no—
118C3×Dic6—62no—
119C3×Dic6—62no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C3×Dic6—122no—
123C32×C8—62no—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—42no—
127#491—271noSylow
128C2×C33#12411yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—92no—
132C2×Dic6—32no—
133C2×Dic6—32no—
134C2×Dic6—62no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—62no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—61no—
140C3×C42—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—92no—
143C4⋊Dic3—181no—
144C4⋊Dic3—182no—
145C4⋊Dic3—182no—
146C4⋊Dic3—181no—
147C4⋊Dic3—181no—
148C4⋊Dic3—182no—
149C4⋊Dic3—361no—
150C4⋊Dic3D3211yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—182no—
154Dic3⋊C4—182no—
155Dic3⋊C4—182no—
156Dic3⋊C4—182no—
157C4×Dic3—32no—
158C4×Dic3—61no—
159C2×C3⋊C8D3212yes—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—41no—
163C22×C32C2×D1211yescharacteristic
164C22×C32D3×D412yes—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4—121no—
170C32×C4#23512yes—
171C32×C4#23512yes—
172C32×C4#23712yes—
173C3⋊Dic3—21no—
174C3⋊Dic3—21no—
175C3⋊Dic3—42no—
176C3⋊Dic3—121no—
177C3⋊Dic3—122no—
178C3⋊Dic3—122no—
179C3⋊Dic3—242no—
180C3⋊Dic3—242no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—124no—
188C3×Dic3—122no—
189C3×Dic3—122no—
190C4⋊Q8—271no—
191#125—271no—
192#103—92no—
193#103—272no—
194#97—91no—
195C33#49111yescharacteristic, complemented, Sylow
196C3×Q8—182no—
197C3×Q8—182no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3211yescharacteristic
207C2×C3×C4C2×D3212yes—
208C2×Dic3—21no—
209C2×Dic3—32no—
210C2×Dic3—32no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—122no—
216C2×Dic3—182no—
217C2×Dic3—182no—
218C2×Dic3—181no—
219C2×Dic3—362no—
220C2×Dic3—362no—
221Dic6—64no—
222Dic6—62no—
223Dic6—62no—
224Dic6—62no—
225Dic6—62no—
226Dic6—122no—
227C3×C8—62no—
228C3×C8—62no—
229C3×C8—122no—
230C3⋊C8—22no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32#87511yescharacteristic
241C2×C32#88512yes—
242C2×C32#90911yescharacteristic
243C2×C32#90911yescharacteristic
244C2×C32#91512yes—
245C2×C32#91812yes—
246C2×Q8—92no—
247C2×C8—32no—
248C4⋊C4—31no—
249C4⋊C4—271no—
250C4⋊C4—541no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C42—91no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3D3×D1212yes—
259C22×C3#381111yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—122no—
265C3×C4—122no—
266C3×C4—181no—
267C3×C4—181no—
268C3×C4—362no—
269C3×C4#379512yes—
270C3×C4#379812yes—
271C3×C4#380412yes—
272Dic3—41no—
273Dic3—62no—
274Dic3—62no—
275Dic3—121no—
276Dic3—121no—
277Dic3—124no—
278Dic3—122no—
279Dic3—122no—
280Dic3—362no—
281Dic3—362no—
282Dic3—361no—
283Dic3—722no—
284Dic3—722no—
285C32—21no—
286C32—22no—
287C32—41no—
288C32#470311yescharacteristic, complemented
289C32#479712yescomplemented
290Q8—182no—
291Q8—182no—
292C2×C4—61no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—182no—
296C2×C4—541no—
297C2×C4D3311yescharacteristic
298C8—62no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—22no—
303C2×C3—21no—
304C2×C3—22no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3#6380012yes—
309C2×C3#6380112yes—
310C2×C3#6385212yes—
311C2×C3#6391311yescharacteristic
312C2×C3#6393111yescharacteristic
313C2×C3#6394511yescharacteristic
314C22#8895511yescharacteristic
315C4—121no—
316C4—181no—
317C4—181no—
318C4—362no—
319C4—1081no—
320C4#8896012yes—
321C3—21no—
322C3—22no—
323C3—41no—
324C3#1059056811yescharacteristic, complemented
325C3#1059089812yescomplemented
326C2#1172289511yescharacteristic
327C2#1172289711yescharacteristic
328C2#1172301011yescharacteristic
subgroupG/H
centerC22#88955
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#4703semidirect
289. C32#4797semidirect
324. C3#10590568semidirect
325. C3#10590898semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117234472
#117220402
C3×#635672
C3×#635552
#105908983
#105905683