Finite Group Databasebeta

← #13159314

Subgroups — #13159314

Order 1728.

764 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723931C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic
8#10592496—31nomaximal
9#10590568—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21#64153—31no—
22#64099—31no—
23#64099—34no—
24#64099—64no—
25#63873—34no—
26C32×#103—31no—
27#63662—31no—
28#63643—31no—
29#63642—31no—
30#63637D311yescharacteristic
31C3×#810—34no—
32C3×#810D314yes—
33#63566—34no—
34#63566—34no—
35#63555—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4797—94no—
50#4797—91no—
51C2×C32×Q8—31no—
52C2×#634C2×D311yescharacteristic
53C2×C3×Dic6—34no—
54C2×C3×Dic6C2×D314yes—
55C2×C32×C8—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—61no—
62C4⋊(C3⋊Dic3)—64no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)—68no—
65#3750—61no—
66#3750—61no—
67#3750—61no—
68#3750—64no—
69#3750—68no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76#3724—64no—
77#3724—64no—
78C22×C33C2×D411yescharacteristic
79C33×C4—41no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)—21no—
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87#944—94no—
88#944—91no—
89#890—94no—
90#890—91no—
91#890—184no—
92C3×#103—34no—
93C3×#103—91no—
94#835—94no—
95#835—91no—
96#816—94no—
97#816—91no—
98#815—94no—
99#815—91no—
100#810—34no—
101#810C3⋊D311yescharacteristic
102C32×Q8—61no—
103C32×Q8—61no—
104C2×C32×C4—28no—
105C2×C32×C4—24no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—61no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—64no—
125C3×Dic6—24no—
126C3×Dic6—24no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129C32×C8—61no—
130C3⋊(C3⋊C8)—21no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8—64no—
133#491—271noSylow
134C2×C33#12411yescharacteristic
135C2×C33#13711yescharacteristic
136C2×C33#13711yescharacteristic
137C2×C3×Q8—34no—
138C2×C3×Q8—91no—
139C2×Dic6—34no—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4D3214yes—
145C3×C4⋊C4C2×C3⋊D311yescharacteristic
146C4⋊Dic3—94no—
147C4⋊Dic3—91no—
148C4⋊Dic3—184no—
149C4⋊Dic3—184no—
150C4⋊Dic3—181no—
151C4⋊Dic3—188no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—184no—
155Dic3⋊C4—184no—
156Dic3⋊C4—184no—
157Dic3⋊C4—181no—
158Dic3⋊C4—188no—
159C4×Dic3—94no—
160C4×Dic3—91no—
161C4×Dic3—184no—
162C2×C3⋊C8—34no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32D3×D411yescharacteristic
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—48no—
170C32×C4—44no—
171C32×C4—41no—
172C32×C4—121no—
173C32×C4#23514yes—
174C32×C4#23511yescharacteristic
175C32×C4#23714yes—
176C32×C4#23711yescharacteristic
177C3⋊Dic3—41no—
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3⋊Dic3—64no—
181C3⋊Dic3—64no—
182C3⋊Dic3—124no—
183C3⋊Dic3—124no—
184C3⋊Dic3—128no—
185C3⋊Dic3—124no—
186C3⋊Dic3—121no—
187C3×Dic3—44no—
188C3×Dic3—124no—
189C4⋊Q8—271no—
190#125—271no—
191#103—31no—
192#103—91no—
193#103—271no—
194#103—271no—
195#97—271no—
196C33#49111yescharacteristic, complemented, Sylow
197C3×Q8—64no—
198C3×Q8—64no—
199C3×Q8—181no—
200C3×Q8—181no—
201C2×C3×C4—28no—
202C2×C3×C4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—24no—
205C2×C3×C4—64no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—64no—
211C2×Dic3—94no—
212C2×Dic3—94no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3—188no—
218C2×Dic3—184no—
219C2×Dic3—181no—
220Dic6—21no—
221Dic6—21no—
222Dic6—64no—
223Dic6—64no—
224C3×C8—64no—
225C3×C8—181no—
226C3⋊C8—21no—
227C3⋊C8—64no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32#88514yes—
232C2×C32#88511yescharacteristic
233C2×C32#91514yes—
234C2×C32#91511yescharacteristic
235C2×C32#91814yes—
236C2×C32#91811yescharacteristic
237C2×Q8—31no—
238C2×Q8—91no—
239C2×C8—31no—
240C2×C8—91no—
241C4⋊C4—271no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C4⋊C4—541no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C42—271no—
248C22×C3—24no—
249C22×C3#381114yes—
250C22×C3D4×C3⋊D311yescharacteristic
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—48no—
254C3×C4—41no—
255C3×C4—44no—
256C3×C4—124no—
257C3×C4—361no—
258C3×C4#379514yes—
259C3×C4#379514yes—
260C3×C4#382811yescharacteristic
261C3×C4#383011yescharacteristic
262Dic3—41no—
263Dic3—124no—
264Dic3—181no—
265Dic3—181no—
266Dic3—184no—
267Dic3—184no—
268Dic3—364no—
269Dic3—364no—
270Dic3—368no—
271Dic3—361no—
272Dic3—364no—
273C32—24no—
274C32#479714yescomplemented
275C32#479711yescharacteristic, complemented
276Q8—61no—
277Q8—61no—
278Q8—181no—
279Q8—181no—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—181no—
283C2×C4—271no—
284C2×C4—271no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—61no—
288C8—181no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3#6391314yes—
293C2×C3#6393114yes—
294C2×C3#6394514yes—
295C2×C3#6409411yescharacteristic
296C2×C3#6412411yescharacteristic
297C2×C3#6412711yescharacteristic
298C22#8903911yescharacteristic
299C4—41no—
300C4—121no—
301C4—361no—
302C4—541no—
303C4—541no—
304C4—1081no—
305C4#8901411yescharacteristic
306C4#8902111yescharacteristic
307C3—24no—
308C3#1059056814yescomplemented
309C3#1059249611yescharacteristic, complemented
310C2#1172352811yescharacteristic
311C2#1172355411yescharacteristic
312C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
196. C33#491semidirect
274. C32#4797semidirect
275. C32#4797semidirect
308. C3#10590568semidirect
309. C3#10592496semidirect

Maximal subgroups

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

subgroupindex
#117239312
#117234512
C3×#636372
C32×#8102
#117220402
#117220382
#117220212
#105924963
#105905683