Finite Group Databasebeta

← #13159591

Subgroups — #13159591

Order 1728.

720 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 144 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#11723435C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3#11722187C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592514—31nomaximal
9#10590711—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17#88806—21no—
18#88803—21no—
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21#64155—31no—
22#64101—31no—
23#63856—34no—
24#63841—34no—
25C32×#103—31no—
26#63662—31no—
27#63647D311yescharacteristic
28#63644—31no—
29#63644—34no—
30#63644—64no—
31#63634—31no—
32C3×(C2×D3)⋊C8—34no—
33C3×#816D314yescomplemented
34#63575—34no—
35#63566—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3—21no—
42C3×C4×C3⋊D3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C33×C8—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#4815—94no—
50#4723—91no—
51C2×C32×Q8—31no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×C3×Dic6C2×D314yes—
55C2×C32×C8—24no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4—31no—
59#3752—61no—
60#3752—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)—64no—
67#3750—61no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72#3724—64no—
73(C2×D3)⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C32×Dic3—41no—
86C32×Dic3—41no—
87#946—94no—
88#926—91no—
89#924—91no—
90#892—94no—
91C3×#103—34no—
92C3×C22⋊C8—91no—
93#835—94no—
94(C2×D3)⋊C8—34no—
95#818—91no—
96#817—94no—
97#817—91no—
98#817—184no—
99#816—91no—
100#816C3⋊D311yescharacteristic, complemented
101#807—94no—
102C22×C3⋊D3D1211yescharacteristic
103C22×C3×D3—34no—
104C32×Q8—61no—
105C32×Q8—61no—
106C2×C32×C4—24no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3D1211yescharacteristic
118C4×C3⋊D3—21no—
119C4×C3⋊D3—21no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—34no—
123C3×C4×D3—64no—
124C3×C4×D3—64no—
125C3×Dic6—24no—
126C3×Dic6—24no—
127C32×C8—28no—
128C32×C8—24no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—21no—
131C3×C3⋊C8—64no—
132#485—271noSylow
133C2×C33#12311yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—41no—
137C2×C3×Q8—34no—
138C22×C3×C4—91no—
139C2×C4×D3—34no—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—24no—
142C2×C3×C8—91no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—34no—
146#217—181no—
147#217—181no—
148(C2×D3)⋊C4—184no—
149(C2×D3)⋊C4—184no—
150C4⋊Dic3—94no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—184no—
156C4⋊Dic3C2×C3⋊D311yescharacteristic
157Dic3⋊C4—181no—
158Dic3⋊C4—184no—
159C4×Dic3—181no—
160C4×Dic3—184no—
161C2×C3⋊C8—34no—
162C22×C32—24no—
163C22×C32C2×D1211yescharacteristic
164C22×C32D3×D414yes—
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—121no—
174C32×C4—121no—
175C32×C4#23514yes—
176C32×C4#23714yes—
177C32×C4#23711yescharacteristic
178C32×C4#23711yescharacteristic
179C3⋊Dic3—21no—
180C3⋊Dic3—121no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3⋊Dic3—244no—
186C3×Dic3—44no—
187C3×Dic3—44no—
188C3×Dic3—64no—
189C22⋊Q8—271no—
190#117—271no—
191#107—271no—
192#106—271no—
193#103—31no—
194#103—271no—
195C22⋊C8—91no—
196C33#48511yescharacteristic, complemented, Sylow
197C23×C3—91no—
198C22×D3—34no—
199C3×Q8—64no—
200C3×Q8—64no—
201C2×C3×C4—24no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—34no—
212C2×Dic3—184no—
213C2×Dic3—184no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217C2×Dic3—364no—
218C4×D3—64no—
219C4×D3—64no—
220Dic6—21no—
221Dic6—21no—
222C3×C8—28no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—181no—
226C3⋊C8—64no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#88314yes—
231C2×C32#88611yescharacteristic
232C2×C32#90411yescharacteristic
233C2×C32#90911yescharacteristic
234C2×C32#91614yes—
235C2×C32#91814yes—
236C3⋊D3—41no—
237C3×D3—124no—
238C2×Q8—31no—
239C22×C4—91no—
240C2×C8—91no—
241C2×C8D3×C3⋊D311yescharacteristic
242C4⋊C4—31no—
243C4⋊C4—271no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C22⋊C4—541no—
247C22⋊C4—541no—
248C42—541no—
249C22×C3—24no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3D3×D1214yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—64no—
256C2×D3—64no—
257C2×D3—64no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—124no—
261C3×C4—124no—
262C3×C4—181no—
263C3×C4#379514yes—
264C3×C4#380514yes—
265C3×C4#382811yescharacteristic
266C3×C4#383011yescharacteristic
267Dic3—41no—
268Dic3—41no—
269Dic3—64no—
270Dic3—364no—
271Dic3—364no—
272Dic3—361no—
273Dic3—361no—
274Dic3—724no—
275Dic3—724no—
276C32—24no—
277C32#472311yescharacteristic, complemented
278C32#481514yescomplemented
279C23—91no—
280Q8—61no—
281Q8—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—181no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—361no—
296C2×C3#6380114yes—
297C2×C3#6380614yes—
298C2×C3#6389114yes—
299C2×C3#6409211yescharacteristic
300C2×C3#6412511yescharacteristic
301C2×C3#6412711yescharacteristic
302D3—124no—
303C22—181no—
304C22—181no—
305C22—181no—
306C22C3⋊D3×D1211yescharacteristic
307C4—121no—
308C4—121no—
309C4—181no—
310C4—1081no—
311C4—1081no—
312C4#8902111yescharacteristic
313C4#8902911yescharacteristic
314C3—24no—
315C3#1059071114yescomplemented
316C3#1059251411yescharacteristic, complemented
317C2—361no—
318C2#1172332911yescharacteristic
319C2#1172333811yescharacteristic
320C2#1172352311yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×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
3. #11722187C2semidirect
5. C32×#816C2semidirect
6. #11722057C2semidirect
7. #11722038C2semidirect
33. C3×#816D3semidirect
100. #816C3⋊D3semidirect
196. C33#485semidirect
277. C32#4723semidirect
278. C32#4815semidirect
315. C3#10590711semidirect
316. C3#10592514semidirect

Maximal subgroups

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

subgroupindex
#117234352
#117234042
#117221872
C3×#636472
C32×#8162
#117220572
#117220382
#105925143
#105907113