Finite Group Databasebeta

← #13159279

Subgroups — #13159279

Order 1728.

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

normal subgroups only

This group has 792 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
1C3×#64102C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic
7#11722033C211yesmaximal, characteristic
8#10592589—31nomaximal
9#10590555—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×#6293—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64102D311yescharacteristic
22#64097—31no—
23C3×#893—34no—
24#63918—34no—
25C32×#107—31no—
26#63648—31no—
27#63641—31no—
28#63636—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#807D314yescomplemented
33Dic3⋊(C3⋊C8)—34no—
34#63563—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)Q812yescomplemented
44C3×C4×C3⋊D3—42no—
45C2×C32×Dic3—21no—
46#6293—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5007—91no—
50#4890—94no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×#634—61no—
54C2×C3×D12—34no—
55C2×C3×C4×D3—64no—
56C2×#627—64no—
57C2×C32×C8—31no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62#3752—21no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66#3750—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3C2×D314yes—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C33×C4—41no—
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3D812yes—
86C32×Dic3—41no—
87C3×C4×D4—91no—
88#893—34no—
89#888—94no—
90#869—91no—
91C3×#107—34no—
92#832—91no—
93#832—91no—
94#821—94no—
95Dic3⋊C8—94no—
96#809—94no—
97#807—94no—
98#807—91no—
99#807—184no—
100#807C3⋊D311yescharacteristic, complemented
101C4⋊(C3⋊C8)—91no—
102C22×C3⋊D3—21no—
103C22×C3×D3—64no—
104C2×C32×C4—21no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—44no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—124no—
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)Dic612yescomplemented
118C4×C3⋊D3—42no—
119#634—122no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C3×D12—38no—
124C3×D12—64no—
125C3×C4×D3—128no—
126#627—128no—
127C32×C8—61no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—24no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—42no—
137C2×C3×D4—91no—
138C22×C3×C4—181no—
139C2×D12—34no—
140C2×C4×D3—64no—
141C2×Dic6—181no—
142C2×Dic6—184no—
143C2×C3×C8—34no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4D3214yes—
147C3×C4⋊C4C2×C3⋊D311yescharacteristic
148C3×C22⋊C4—181no—
149C3×C42—91no—
150(C2×D3)⋊C4—64no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—184no—
157C4×Dic3—34no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—184no—
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C22×C32—24no—
163C22×C32D3×D414yes—
164C22×C32C2×#6711yescharacteristic
165C2×C3⋊D3—22no—
166C2×C3⋊D3—41no—
167C2×C3⋊D3—41no—
168C2×C3×D3—68no—
169C2×C3×D3—124no—
170C2×C3×D3—124no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—41no—
174C32×C4—44no—
175C32×C4—84no—
176C32×C4—121no—
177C32×C4C2×Dic611yescharacteristic
178C32×C4#23514yes—
179C32×C4#23511yescharacteristic
180C32×C4D3×Q814yes—
181C3⋊Dic3—21no—
182C3⋊Dic3—121no—
183C3⋊Dic3—124no—
184C3⋊Dic3—244no—
185C3⋊Dic3#21312yes—
186C3×Dic3—38no—
187C3×Dic3—44no—
188C3×Dic3—64no—
189C4⋊Q8—271no—
190C4×D4—91no—
191#107—31no—
192#107—271no—
193C4⋊C8—271no—
194#102—271no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—181no—
198C22×D3—64no—
199C3×D4—92no—
200C3×D4—181no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—44no—
205C2×C3×C4—64no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—362no—
209C2×C3×C4C2×D3214yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—21no—
212C2×Dic3—34no—
213C2×Dic3—34no—
214C2×Dic3—184no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217D12—38no—
218D12—64no—
219C4×D3—128no—
220Dic6—368no—
221Dic6—362no—
222C3×C8—64no—
223C3⋊C8—21no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#89614yes—
231C2×C32D3×D814yes—
232C2×C32#91814yes—
233C2×C32#92411yescharacteristic
234C2×C32C2×#21311yescharacteristic
235C2×C32#95111yescharacteristic
236C3⋊D3—42no—
237C3×D3—128no—
238C2×Q8—541no—
239C2×D4—91no—
240C22×C4—181no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C22⋊C4—181no—
248C42—91no—
249C22×C3—24no—
250C22×C3—182no—
251C22×C3—361no—
252C22×C3—361no—
253C22×C3D3×#6714yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—68no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—41no—
261C3×C4—44no—
262C3×C4—84no—
263C3×C4—92no—
264C3×C4—124no—
265C3×C4—181no—
266C3×C4D3×Dic614yes—
267C3×C4#379814yes—
268C3×C4#382811yescharacteristic
269C3×C4Q8×C3⋊D311yescharacteristic
270Dic3—38no—
271Dic3—41no—
272Dic3—64no—
273Dic3—361no—
274Dic3—364no—
275Dic3—724no—
276C32—24no—
277C32#489014yescomplemented
278C32#500711yescharacteristic, complemented
279C23—181no—
280Q8—1082no—
281D4—92no—
282D4—181no—
283C2×C4—21no—
284C2×C4—61no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—362no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—61no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—362no—
296C2×C3#6385114yes—
297C2×C3D3×#21314yes—
298C2×C3#6394314yes—
299C2×C3#6410511yescharacteristic
300C2×C3D8×C3⋊D311yescharacteristic
301C2×C3#6412711yescharacteristic
302D3—128no—
303C22—182no—
304C22—361no—
305C22—361no—
306C22C3⋊D3×#6711yescharacteristic
307C4—41no—
308C4—92no—
309C4—121no—
310C4—181no—
311C4—1081no—
312C4C3⋊D3×Dic611yescharacteristic
313C4#8902011yescharacteristic
314C3—24no—
315C3#1059055514yescomplemented
316C3#1059258911yescharacteristic, complemented
317C2—362no—
318C2#1172341811yescharacteristic
319C2C3⋊D3×#21311yescharacteristic
320C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#474
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
2. #11723535C2semidirect
3. #11722177C2semidirect
4. C32×#807C2semidirect
5. #11722051C2semidirect
32. C3×#807D3semidirect
43. C3×C4⋊(C3⋊D3)Q8semidirect
100. #807C3⋊D3semidirect
117. C4⋊(C3⋊D3)Dic6semidirect
196. C33#474semidirect
277. C32#4890semidirect
278. C32#5007semidirect
315. C3#10590555semidirect
316. C3#10592589semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117235352
#117221772
C32×#8072
#117220512
#117220342
#117220332
#105925893
#105905553