Finite Group Databasebeta

← #13159985

Subgroups — #13159985

Order 1728.

816 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 174 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#11723399C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic
4#11722186C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic
6C32×#815C211yesmaximal, characteristic
7#11722052C211yesmaximal, characteristic
8#10592415—31nomaximal
9#10590794—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88807—21no—
18#88806—21no—
19Dic3×C3⋊Dic3—21no—
20Dic3×C3⋊Dic3—21no—
21#88800C414yescomplemented
22C2×#3746—31no—
23C4×#634—31no—
24C4×#634—31no—
25#63844—34no—
26Dic3×Dic6—34no—
27C2×#3719—34no—
28C32×#97—31no—
29#63644—31no—
30#63643—31no—
31#63643—34no—
32#63643—64no—
33#63643D311yescharacteristic
34C3×#816—34no—
35C3×#815D314yes—
36#63573—34no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308C2×C412yes—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634C2×C412yes—
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48C33×C8—21no—
49C32×C3⋊C8C2×C412yes—
50#4789—91no—
51#4716—94no—
52C2×#634—31no—
53C2×#634—34no—
54C2×#634—64no—
55C2×#634C2×D311yescharacteristic
56C2×C3×Dic6—34no—
57C2×C32×C8—24no—
58C2×C32×C8—31no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C2×D311yescharacteristic
61C32×C42—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—61no—
67#3750—61no—
68C4×C3⋊Dic3—61no—
69C4×C3⋊Dic3—61no—
70#3746—34no—
71C3×C4⋊Dic3—34no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8C2×D314yes—
74#3724—64no—
75Dic3⋊Dic3—64no—
76Dic32—64no—
77Dic32—64no—
78#3719—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—22no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—22no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—21no—
87C32×Dic3—21no—
88Q8×Dic3—91no—
89C2×#216—91no—
90C2×Dic12—94no—
91#887—91no—
92C4×Dic6—94no—
93C4×Dic6—94no—
94C3×#103—91no—
95C3×#97—34no—
96#817—94no—
97#816—34no—
98#816—94no—
99#816—91no—
100#816—184no—
101#815C3⋊D311yescharacteristic
102#808—91no—
103C2×C32×C4—24no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C2×C3⋊Dic3—124no—
116#634—21no—
117#634—32no—
118#634—38no—
119#634—68no—
120#634—61no—
121#634—64no—
122#634—124no—
123#634C4×D312yes—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3D1214yes—
127C2×C3×Dic3D1214yes—
128C3×Dic6—38no—
129C3×Dic6—64no—
130C32×C8—21no—
131C32×C8—24no—
132C32×C8—32no—
133C32×C8—44no—
134C3×C3⋊C8C4×D318yes—
135#441—271noSylow
136C2×C33C4×D411yescharacteristic
137C2×C33#13711yescharacteristic
138C2×C33#13711yescharacteristic
139C2×C3×Q8—91no—
140C2×Dic6—34no—
141C2×Dic6—94no—
142C2×Dic6—91no—
143C2×Dic6—184no—
144C2×C3×C8—24no—
145C2×C3×C8—34no—
146C2×C3×C8D3214yes—
147C2×C3×C8C2×C3⋊D311yescharacteristic
148C3×C4⋊C4—91no—
149C3×C42—34no—
150#216—94no—
151C4⋊Dic3—34no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—181no—
156Dic3⋊C4—184no—
157Dic3⋊C4—184no—
158Dic3⋊C4—181no—
159C4×Dic3—184no—
160C4×Dic3—184no—
161C4×Dic3—181no—
162C4×Dic3—181no—
163C4×Dic3C2×C3⋊D311yescharacteristic
164C2×C3⋊C8C2×C3⋊D311yescharacteristic
165Dic12—916no—
166C22×C32—24no—
167C22×C32C2×D1214yes—
168C22×C32D3×D411yescharacteristic
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—61no—
172C32×C4—61no—
173C32×C4C2×C4×D314yes—
174C32×C4C2×C4×D311yescharacteristic
175C32×C4#23514yes—
176C32×C4#23711yescharacteristic
177C3⋊Dic3—22no—
178C3⋊Dic3—41no—
179C3⋊Dic3—62no—
180C3⋊Dic3—68no—
181C3⋊Dic3—128no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3×Dic3—24no—
186C3×Dic3—24no—
187C3×Dic3—68no—
188C3×Dic3—124no—
189C2×Q16—271no—
190C4×Q8—271no—
191C4×Q8—271no—
192#106—271no—
193#103—91no—
194#103—271no—
195#97—31no—
196C33#44111yescharacteristic, complemented, Sylow
197C3×Q8—92no—
198C3×Q8—181no—
199C2×C3×C4—24no—
200C2×C3×C4—34no—
201C2×C3×C4—34no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4C2×D3214yes—
205C2×C3×C4C22×C3⋊D311yescharacteristic
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×Dic3—184no—
209C2×Dic3—181no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212C2×Dic3—364no—
213C2×Dic3—364no—
214C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
215C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
216Dic6—38no—
217Dic6—64no—
218Dic6—98no—
219Dic6—92no—
220Dic6—188no—
221Dic6—184no—
222Dic6—181no—
223Dic6—364no—
224C3×C8—24no—
225C3×C8—21no—
226C3×C8—38no—
227C3×C8—44no—
228C3⋊C8C4×C3⋊D312yes—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32C4×D1214yes—
233C2×C32#67⋊C411yescharacteristic
234C2×C32#90914yes—
235C2×C32#90914yes—
236C2×C32#91511yescharacteristic
237C2×C32#91811yescharacteristic
238C2×Q8—91no—
239C2×Q8—271no—
240Q16—274no—
241C2×C8—31no—
242C2×C8D3×C3⋊D311yescharacteristic
243C4⋊C4—91no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C4⋊C4—541no—
247C42—31no—
248C42—541no—
249C42—541no—
250C22×C3—24no—
251C22×C3D3×D1214yes—
252C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—64no—
256C3×C4—64no—
257C3×C4—182no—
258C3×C4—361no—
259C3×C4C4×D3214yes—
260C3×C4#380414yes—
261C3×C4C2×C4×C3⋊D311yescharacteristic
262C3×C4#382811yescharacteristic
263Dic3—21no—
264Dic3—21no—
265Dic3—68no—
266Dic3—124no—
267Dic3—182no—
268Dic3—188no—
269Dic3—368no—
270Dic3—364no—
271Dic3—361no—
272Dic3—724no—
273C32—24no—
274C32#471614yescomplemented
275C32#478911yescharacteristic, complemented
276Q8—92no—
277Q8—181no—
278Q8—272no—
279Q8—541no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—21no—
288C8—32no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3#6380014yes—
293C2×C3#6380114yes—
294C2×C3#6388014yes—
295C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
296C2×C3#6411811yescharacteristic
297C2×C3#6411811yescharacteristic
298C22D3×C4⋊(C3⋊D3)11yescharacteristic
299C4—61no—
300C4—61no—
301C4—182no—
302C4—361no—
303C4—542no—
304C4—1081no—
305C4C4×D3×C3⋊D311yescharacteristic
306C4#8903111yescharacteristic
307C3—24no—
308C3#1059079414yescomplemented
309C3#1059241511yescharacteristic, complemented
310C2#1172332611yescharacteristic
311C2#1172332711yescharacteristic
312C2#1172347111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
21. #88800C4semidirect
196. C33#441semidirect
274. C32#4716semidirect
275. C32#4789semidirect
308. C3#10590794semidirect
309. C3#10592415semidirect

Maximal subgroups

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

subgroupindex
#117233992
Dic3×#6342
C2×#888002
#117221862
C3×#636432
C32×#8152
#117220522
#105924153
#105907943