Finite Group Databasebeta

← #13159593

Subgroups — #13159593

Order 1728.

720 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 152 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#11723425C211yesmaximal, characteristic
2#11723424C211yesmaximal, characteristic
3#11722187C211yesmaximal, characteristic, complemented
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic
6#11722055C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592422—31nomaximal
9#10590711—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88806—21no—
18(C2×D3)⋊(C3⋊Dic3)—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21#64135—31no—
22#64133—31no—
23#63856—34no—
24#63841—34no—
25C32×C22⋊C8—31no—
26#63645—31no—
27#63644—31no—
28#63644—34no—
29#63644—64no—
30#63643—31no—
31#63643D311yescharacteristic, complemented
32C3×(C2×D3)⋊C8D314yes—
33C3×#816—34no—
34#63575—34no—
35#63566—34no—
36C22×C32×D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47C33×C8—21no—
48C32×C3⋊C8—21no—
49#4815—91no—
50#4723—94no—
51C22×C32×C4—31no—
52C2×#634C2×D311yescharacteristic
53C2×C3×C4×D3C2×D314yes—
54C2×C3×Dic6—34no—
55C2×C32×C8—24no—
56C2×C32×C8—31no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59#3757—61no—
60#3757—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—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—61no—
69C4×C3⋊Dic3—61no—
70C3×C4⋊Dic3—34no—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—64no—
73(C2×D3)⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C33×C4#4111yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—21no—
87#946—91no—
88#926—94no—
89#924—94no—
90#892—91no—
91C3×#103—91no—
92C3×C22⋊C8—34no—
93#835—91no—
94(C2×D3)⋊C8C3⋊D311yescharacteristic
95#818—94no—
96#817—94no—
97#817—91no—
98#817—184no—
99#816—34no—
100#816—94no—
101#807—91no—
102C23×C32—31no—
103C22×C3×D3D1214yes—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3—124no—
118#634—21no—
119#634—21no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3D1214yes—
123C3×C4×D3—24no—
124C3×C4×D3—24no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C32×C8—28no—
128C32×C8—24no—
129C32×C8—21no—
130C32×C8—61no—
131C3×C3⋊C8—24no—
132#485—271noSylow
133C2×C33#12311yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—41no—
137C2×C3×Q8—91no—
138C22×C3×C4—34no—
139C2×C4×D3C2×C3⋊D311yescharacteristic
140C2×Dic6—34no—
141C2×C3×C8—24no—
142C2×C3×C8—34no—
143C2×C3×C8D3214yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—91no—
146#217—184no—
147#217—184no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—181no—
150C4⋊Dic3—34no—
151C4⋊Dic3—94no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4⋊Dic3—184no—
157Dic3⋊C4—184no—
158Dic3⋊C4—181no—
159C4×Dic3—184no—
160C4×Dic3—181no—
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C22×C32—24no—
163C22×C32—61no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32C2×D1214yes—
167C22×C32D3×D411yescharacteristic
168C2×C3×D3—24no—
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—61no—
174C32×C4#23511yescharacteristic
175C32×C4#23714yes—
176C32×C4#23714yes—
177C32×C4#23711yescharacteristic
178C3⋊Dic3—41no—
179C3⋊Dic3—41no—
180C3⋊Dic3—121no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—124no—
188C3×Dic3—124no—
189C22⋊Q8—271no—
190#117—271no—
191#107—271no—
192#106—271no—
193#103—91no—
194#103—271no—
195C22⋊C8—31no—
196C33#48511yescharacteristic, complemented, Sylow
197C23×C3—34no—
198C22×D3C4⋊(C3⋊D3)11yescharacteristic
199C3×Q8—181no—
200C3×Q8—181no—
201C2×C3×C4—24no—
202C2×C3×C4—34no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—184no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—364no—
216C2×Dic3—364no—
217C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
218C4×D3—21no—
219C4×D3—21no—
220Dic6—64no—
221Dic6—64no—
222C3×C8—28no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—64no—
226C3⋊C8—21no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—121no—
231C2×C32#88311yescharacteristic
232C2×C32#88614yes—
233C2×C32#90414yes—
234C2×C32#90914yes—
235C2×C32#91611yescharacteristic
236C2×C32#91811yescharacteristic
237C3×D3—44no—
238C2×Q8—91no—
239C22×C4—31no—
240C2×C8—31no—
241C2×C8D3×C3⋊D311yescharacteristic
242C4⋊C4—91no—
243C4⋊C4—271no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C22⋊C4—541no—
247C22⋊C4—541no—
248C42—541no—
249C22×C3—24no—
250C22×C3—64no—
251C22×C3—64no—
252C22×C3—64no—
253C22×C3D3×D1214yes—
254C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
255C2×D3—21no—
256C2×D3—21no—
257C2×D3—21no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—64no—
261C3×C4—361no—
262C3×C4—361no—
263C3×C4#379514yes—
264C3×C4#380514yes—
265C3×C4#383011yescharacteristic
266C3×C4#383011yescharacteristic
267Dic3—21no—
268Dic3—124no—
269Dic3—124no—
270Dic3—364no—
271Dic3—364no—
272Dic3—361no—
273Dic3—361no—
274Dic3—724no—
275Dic3—724no—
276C32—24no—
277C32#472314yescomplemented
278C32#481511yescharacteristic, complemented
279C23—31no—
280Q8—181no—
281Q8—181no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—21no—
291C8—61no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—124no—
296C2×C3#6380114yes—
297C2×C3#6380614yes—
298C2×C3#6389114yes—
299C2×C3#6409511yescharacteristic
300C2×C3#6411311yescharacteristic
301C2×C3#6411811yescharacteristic
302D3—41no—
303C22—61no—
304C22—61no—
305C22—61no—
306C22D3×C4⋊(C3⋊D3)11yescharacteristic
307C4—61no—
308C4—361no—
309C4—361no—
310C4—1081no—
311C4—1081no—
312C4#8901411yescharacteristic
313C4#8902911yescharacteristic
314C3—24no—
315C3#1059071114yescomplemented
316C3#1059242211yescharacteristic, complemented
317C2—121no—
318C2#1172332711yescharacteristic
319C2#1172333511yescharacteristic
320C2#1172348011yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×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
4. C3×#63643C2semidirect
6. #11722055C2semidirect
7. #11722040C2semidirect
31. #63643D3semidirect
196. C33#485semidirect
277. C32#4723semidirect
278. C32#4815semidirect
315. C3#10590711semidirect
316. C3#10592422semidirect

Maximal subgroups

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

subgroupindex
#117234252
#117234242
#117221872
C3×#636432
C32×(C2×D3)⋊C82
#117220552
#117220402
#105924223
#105907113