Finite Group Databasebeta

← #13161237

Subgroups — #13161237

Order 1728.

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

normal subgroups only

This group has 812 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#11723985C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic
8#10592496—31nomaximal
9#10591616—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12#88857—21no—
13#88857—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64153—31no—
22#64153—34no—
23#64153—64no—
24#64099—31no—
25#63872—34no—
26C32×#103—31no—
27#63662D311yescharacteristic
28#63643—31no—
29#63642—31no—
30#63637—31no—
31C3×#835—34no—
32C3×#835D314yes—
33#63567—34no—
34#63567—34no—
35#63555—34no—
36C33×Q8—21no—
37C33×Q8—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4797—94no—
50#4797—91no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—34no—
56C2×#634—64no—
57C2×C32×C8—31no—
58C32×C4⋊C4—31no—
59C4⋊(C3⋊Dic3)—61no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—61no—
62#3750—61no—
63#3750—61no—
64#3750—64no—
65#3750—64no—
66#3750—68no—
67#3750—68no—
68C4×C3⋊Dic3—31no—
69C4×C3⋊Dic3—34no—
70C4×C3⋊Dic3—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77Dic3⋊Dic3—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#944—184no—
90#890—94no—
91#890—91no—
92C3×#103—34no—
93C3×#103—91no—
94#835—34no—
95#835C3⋊D311yescharacteristic
96#816—94no—
97#816—91no—
98#815—94no—
99#815—91no—
100#810—94no—
101#810—91no—
102C32×Q8—21no—
103C32×Q8—21no—
104C32×Q8—24no—
105C32×Q8—24no—
106C32×Q8—44no—
107C32×Q8—44no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—68no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—61no—
125#634—61no—
126#634—61no—
127#634—64no—
128#634—64no—
129#634—124no—
130#634—124no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C32×C8—61no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8—64no—
137#491—271noSylow
138C2×C33#12411yescharacteristic
139C2×C33#13711yescharacteristic
140C2×C33#13711yescharacteristic
141C2×C3×Q8—24no—
142C2×C3×Q8D3214yes—
143C2×C3×Q8C2×C3⋊D311yescharacteristic
144C2×Dic6—94no—
145C2×Dic6—91no—
146C2×Dic6—184no—
147C2×C3×C8—34no—
148C2×C3×C8—91no—
149C3×C4⋊C4—34no—
150C3×C4⋊C4—91no—
151C4⋊Dic3—34no—
152C4⋊Dic3—181no—
153C4⋊Dic3—184no—
154C4⋊Dic3C2×C3⋊D311yescharacteristic
155Dic3⋊C4—181no—
156Dic3⋊C4—184no—
157Dic3⋊C4—184no—
158Dic3⋊C4—184no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—188no—
162Dic3⋊C4—188no—
163C4×Dic3—94no—
164C4×Dic3—91no—
165C4×Dic3—184no—
166C2×C3⋊C8—34no—
167C2×C3⋊C8C2×C3⋊D311yescharacteristic
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32D3×D411yescharacteristic
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—48no—
174C32×C4—44no—
175C32×C4—41no—
176C32×C4—121no—
177C32×C4#23514yes—
178C32×C4#23511yescharacteristic
179C32×C4#23714yes—
180C32×C4#23711yescharacteristic
181C3⋊Dic3—41no—
182C3⋊Dic3—68no—
183C3⋊Dic3—68no—
184C3⋊Dic3—64no—
185C3⋊Dic3—61no—
186C3⋊Dic3—64no—
187C3⋊Dic3—61no—
188C3⋊Dic3—128no—
189C3⋊Dic3—124no—
190C3⋊Dic3—121no—
191C3×Dic3—44no—
192C3×Dic3—124no—
193C4⋊Q8—271no—
194#125—271no—
195#103—31no—
196#103—91no—
197#103—271no—
198#103—271no—
199#97—271no—
200C33#49111yescharacteristic, complemented, Sylow
201C3×Q8—24no—
202C3×Q8—24no—
203C3×Q8—21no—
204C3×Q8—21no—
205C3×Q8—44no—
206C3×Q8—44no—
207C2×C3×C4—28no—
208C2×C3×C4—24no—
209C2×C3×C4—21no—
210C2×C3×C4—24no—
211C2×C3×C4—64no—
212C2×C3×C4—181no—
213C2×C3×C4C2×D3214yes—
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—64no—
217C2×Dic3—94no—
218C2×Dic3—94no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—184no—
222C2×Dic3—184no—
223C2×Dic3—188no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226Dic6—184no—
227Dic6—184no—
228Dic6—181no—
229Dic6—181no—
230Dic6—364no—
231Dic6—364no—
232C3×C8—64no—
233C3×C8—181no—
234C3⋊C8—21no—
235C3⋊C8—64no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32#88514yes—
240C2×C32#88511yescharacteristic
241C2×C32#91514yes—
242C2×C32#91511yescharacteristic
243C2×C32#91814yes—
244C2×C32#91811yescharacteristic
245C2×Q8—271no—
246C2×Q8D3×C3⋊D311yescharacteristic
247C2×C8—31no—
248C2×C8—91no—
249C4⋊C4—31no—
250C4⋊C4—91no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4—541no—
255C42—271no—
256C22×C3—24no—
257C22×C3#381114yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—48no—
262C3×C4—41no—
263C3×C4—44no—
264C3×C4—124no—
265C3×C4—361no—
266C3×C4#380414yes—
267C3×C4#380414yes—
268C3×C4#382811yescharacteristic
269C3×C4#383011yescharacteristic
270Dic3—41no—
271Dic3—124no—
272Dic3—188no—
273Dic3—188no—
274Dic3—181no—
275Dic3—181no—
276Dic3—184no—
277Dic3—184no—
278Dic3—368no—
279Dic3—361no—
280Dic3—364no—
281C32—24no—
282C32#479714yescomplemented
283C32#479711yescharacteristic, complemented
284Q8—21no—
285Q8—21no—
286Q8—541no—
287Q8—541no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—181no—
291C2×C4—271no—
292C2×C4—271no—
293C2×C4—541no—
294C2×C4C2×D3×C3⋊D311yescharacteristic
295C8—61no—
296C8—181no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—24no—
300C2×C3#6395114yes—
301C2×C3#6395114yes—
302C2×C3#6395314yes—
303C2×C3#6409411yescharacteristic
304C2×C3#6412411yescharacteristic
305C2×C3#6412711yescharacteristic
306C22#8903911yescharacteristic
307C4—41no—
308C4—121no—
309C4—361no—
310C4—541no—
311C4—541no—
312C4—1081no—
313C4#8902811yescharacteristic
314C4#8903111yescharacteristic
315C3—24no—
316C3#1059161614yescomplemented
317C3#1059249611yescharacteristic, complemented
318C2#1172358811yescharacteristic
319C2#1172359011yescharacteristic
320C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
200. C33#491semidirect
282. C32#4797semidirect
283. C32#4797semidirect
316. C3#10591616semidirect
317. C3#10592496semidirect

Maximal subgroups

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

subgroupindex
#117239852
#117234502
C3×#636622
C32×#8352
#117220422
#117220412
#117220212
#105924963
#105916163