Finite Group Databasebeta

← #13159291

Subgroups — #13159291

Order 1728.

792 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C32×#893C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C3×#63634C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic
7#11722031C211yesmaximal, characteristic
8#10592706—31nomaximal
9#10590555—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×#6293—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4×D4—31no—
22#64078—31no—
23C3×#893D314yes—
24#63918—34no—
25#63659—31no—
26#63659—31no—
27#63634—31no—
28#63634—34no—
29#63634—64no—
30#63634D311yescharacteristic, complemented
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#807—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63563—34no—
35#63562—34no—
36C22×C32×D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C32×D12—21no—
44C32×D12Q812yescomplemented
45C32×C4×D3—42no—
46#6293—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5007—94no—
50#4890—91no—
51C2×C32×D4—31no—
52C22×C32×C4—61no—
53C2×#634—61no—
54C2×C3×D12C2×D314yes—
55C2×C3×C4×D3—24no—
56C2×#627—64no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C22⋊C4—61no—
61C32×C42—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—24no—
72C3×C4⋊Dic3—34no—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—22no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4—41no—
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—21no—
86C32×Dic3D812yes—
87C3×C4×D4—34no—
88#893C3⋊D311yescharacteristic
89#888—91no—
90#869—94no—
91C3×#107—91no—
92#832—94no—
93#832—94no—
94#821—91no—
95Dic3⋊C8—91no—
96#809—91no—
97#807—34no—
98#807—94no—
99#807—91no—
100#807—184no—
101C4⋊(C3⋊C8)—94no—
102C23×C32—61no—
103C22×C3×D3—24no—
104C32×D4—32no—
105C32×D4—61no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4—44no—
112C2×C32×C4—122no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119#634—122no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3#6714yes—
122C2×C3×Dic3#6714yes—
123C3×D12—24no—
124C3×D12Dic618yescomplemented
125C3×C4×D3—48no—
126#627—128no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—68no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—64no—
132#474—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×D811yescharacteristic
135C2×C33#13711yescharacteristic
136C32×D3—42no—
137C2×C3×D4—34no—
138C22×C3×C4—64no—
139C2×D12C2×C3⋊D311yescharacteristic
140C2×C4×D3—21no—
141C2×Dic6—184no—
142C2×Dic6—181no—
143C2×C3×C8—91no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4D3214yes—
147C3×C4⋊C4C2×C3⋊D311yescharacteristic
148C3×C22⋊C4—64no—
149C3×C42—34no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3—34no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—184no—
156Dic3⋊C4—181no—
157C4×Dic3C2×C3⋊D311yescharacteristic
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C22×C32—24no—
163C22×C32—62no—
164C22×C32—121no—
165C22×C32—121no—
166C22×C32D3×D411yescharacteristic
167C22×C32C2×#6714yes—
168C2×C3×D3—28no—
169C2×C3×D3—44no—
170C2×C3×D3—44no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—32no—
174C32×C4—41no—
175C32×C4—44no—
176C32×C4—61no—
177C32×C4—84no—
178C32×C4C2×Dic614yes—
179C32×C4#23514yes—
180C32×C4#23511yescharacteristic
181C32×C4D3×Q811yescharacteristic
182C3⋊Dic3—41no—
183C3⋊Dic3—121no—
184C3⋊Dic3—124no—
185C3⋊Dic3—244no—
186C3×Dic3—24no—
187C3×Dic3—124no—
188C3×Dic3#21318yes—
189C4⋊Q8—271no—
190C4×D4—31no—
191#107—91no—
192#107—271no—
193C4⋊C8—271no—
194#102—271no—
195#102—271no—
196C33#47411yescharacteristic, complemented, Sylow
197C23×C3—64no—
198C22×D3—21no—
199C3×D4—38no—
200C3×D4—64no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—34no—
205C2×C3×C4—34no—
206C2×C3×C4—44no—
207C2×C3×C4—128no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3214yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—64no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—364no—
215C2×Dic3#63811yescharacteristic
216C2×Dic3#63811yescharacteristic
217D12—21no—
218D12#63412yescomplemented
219C4×D3—42no—
220Dic6—368no—
221Dic6—362no—
222C3×C8—181no—
223C3⋊C8—64no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—122no—
231C2×C32#89611yescharacteristic
232C2×C32D3×D811yescharacteristic
233C2×C32#91811yescharacteristic
234C2×C32#92414yes—
235C2×C32C2×#21314yes—
236C2×C32#95114yes—
237C3×D3—48no—
238C2×Q8—541no—
239C2×D4—31no—
240C22×C4—61no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—91no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C22⋊C4—61no—
248C42—31no—
249C22×C3—24no—
250C22×C3—68no—
251C22×C3—124no—
252C22×C3—124no—
253C22×C3D3×#6714yes—
254C22×C3C2×#63811yescharacteristic
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—38no—
261C3×C4—41no—
262C3×C4—44no—
263C3×C4—64no—
264C3×C4—84no—
265C3×C4—361no—
266C3×C4D3×Dic614yes—
267C3×C4#379814yes—
268C3×C4C2×#63411yescharacteristic
269C3×C4#382811yescharacteristic
270Dic3—21no—
271Dic3—124no—
272Dic3—361no—
273Dic3—364no—
274Dic3—724no—
275Dic3#375312yes—
276C32—24no—
277C32#489011yescharacteristic, complemented
278C32#500714yescomplemented
279C23—61no—
280Q8—1082no—
281D4—32no—
282D4—61no—
283C2×C4—21no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—122no—
287C2×C4—181no—
288C2×C4—541no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—181no—
291C8—541no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—128no—
296C2×C3#6385114yes—
297C2×C3D3×#21314yes—
298C2×C3#6394314yes—
299C2×C3#6413311yescharacteristic
300C2×C3C2×#375311yescharacteristic
301C2×C3#6416011yescharacteristic
302D3—42no—
303C22—62no—
304C22—121no—
305C22—121no—
306C22D3×#63811yescharacteristic
307C4—32no—
308C4—41no—
309C4—61no—
310C4—361no—
311C4—1081no—
312C4D3×#63411yescharacteristic
313C4#8902011yescharacteristic
314C3—24no—
315C3#1059055514yescomplemented
316C3#1059270611yescharacteristic, complemented
317C2—122no—
318C2#1172341511yescharacteristic
319C2D3×#375311yescharacteristic
320C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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. #11723534C2semidirect
3. #11722177C2semidirect
4. C3×#63634C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
30. #63634D3semidirect
44. C32×D12Q8semidirect
124. C3×D12Dic6semidirect
196. C33#474semidirect
218. D12#634semidirect
277. C32#4890semidirect
278. C32#5007semidirect
315. C3#10590555semidirect
316. C3#10592706semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117235342
#117221772
C3×#636342
Dic3⋊(C3⋊(C3⋊C8))2
#117220322
#117220312
#105927063
#105905553