Finite Group Databasebeta

← #13160274

Subgroups — #13160274

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2#11722200C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722051C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722020C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592419—31nomaximal
9#10591056—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×#3757C411yescharacteristic
14C3×#3757C411yescharacteristic
15C3×C4⋊(C3⋊Dic3)C411yescharacteristic
16C3×C4⋊(C3⋊Dic3)C411yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64135D311yescharacteristic
22C3×#926—34no—
23C32×C22⋊C8—31no—
24#63657—31no—
25#63657—34no—
26#63657—64no—
27#63642—31no—
28#63641—31no—
29#63641—31no—
30C8×C3⋊Dic3—31no—
31C3×#830D314yes—
32Dic3⋊(C3⋊C8)—34no—
33Dic3⋊(C3⋊C8)—34no—
34#63554—34no—
35Dic3×C3⋊C8—34no—
36C23×C33C2×C411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48C32×C3⋊C8—21no—
49#5000—91no—
50#4720—94no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C2×C32×C8—31no—
55C2×C32×C8—31no—
56#3757Dic311yescharacteristic
57#3757Dic311yescharacteristic
58C4⋊(C3⋊Dic3)Dic311yescharacteristic
59C4⋊(C3⋊Dic3)Dic311yescharacteristic
60C4×C3⋊Dic3C2×D311yescharacteristic
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—64no—
67C2×C3⋊(C3⋊C8)—64no—
68C3×#217—34no—
69C3×#217—34no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3—34no—
72C3×C4×Dic3—34no—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75C2×C3×C3⋊C8C2×D314yes—
76C22×C33—21no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33C22×C411yescharacteristic
80C33×C4—21no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—21no—
87C3×#117—91no—
88#926—34no—
89C3×C22⋊C8—34no—
90#830—94no—
91#830—91no—
92#830—184no—
93#830C3⋊D311yescharacteristic
94#815—94no—
95Dic3⋊C8—94no—
96Dic3⋊C8—94no—
97C8×Dic3—94no—
98C4⋊(C3⋊C8)—91no—
99C4⋊(C3⋊C8)—91no—
100#803—91no—
101C4×C3⋊C8—91no—
102C23×C32—24no—
103C23×C32C4×D314yes—
104C23×C32C2×Dic311yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—44no—
113C2×C32×C4C4×D314yes—
114C2×C32×C4C2×Dic311yescharacteristic
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3C2×Dic311yescharacteristic
118C2×C3⋊Dic3C2×Dic311yescharacteristic
119C2×C3⋊Dic3C2×Dic311yescharacteristic
120C2×C3⋊Dic3C2×Dic311yescharacteristic
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—34no—
125C32×C8—61no—
126C32×C8—61no—
127C3⋊(C3⋊C8)—61no—
128C3⋊(C3⋊C8)—61no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—64no—
131C3⋊(C3⋊C8)—124no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—24no—
135#433—271noSylow
136C2×C33—41no—
137C2×C33#11711yescharacteristic
138C2×C33#13111yescharacteristic
139C2×C33#13111yescharacteristic
140C22×C3×C4—24no—
141C22×C3×C4D3214yes—
142C22×C3×C4C2×C3⋊D311yescharacteristic
143C2×C3×C8—34no—
144C2×C3×C8—34no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C22⋊C4—91no—
148C3×C22⋊C4—91no—
149C3×C42—91no—
150C3×C42—91no—
151#217—34no—
152#217—34no—
153C4⋊Dic3—34no—
154C4⋊Dic3—34no—
155C4×Dic3—34no—
156C4×Dic3—34no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C2×C3⋊C8C2×C3⋊D311yescharacteristic
165C22×C32—28no—
166C22×C32—24no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—24no—
171C22×C32—24no—
172C22×C32—24no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32C2×C4×D314yes—
176C22×C32C22×Dic311yescharacteristic
177C32×C4—21no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4#23511yescharacteristic
183C32×C4#23511yescharacteristic
184C32×C4#23714yes—
185C32×C4#23714yes—
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—21no—
189C3⋊Dic3—21no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C3×Dic3—64no—
193C3×Dic3—64no—
194#117—91no—
195C4⋊C8—271no—
196C4⋊C8—271no—
197C22⋊C8—31no—
198C22⋊C8—271no—
199#97—271no—
200C4×C8—271no—
201C33#43311yescharacteristic, complemented, Sylow
202C23×C3—24no—
203C23×C3D3×Dic314yes—
204C23×C3C4×C3⋊D311yescharacteristic
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4D3×Dic314yes—
218C2×C3×C4C4×C3⋊D311yescharacteristic
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—34no—
222C2×Dic3—34no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C3×C8—64no—
226C3×C8—64no—
227C3⋊C8—21no—
228C3⋊C8—21no—
229C3⋊C8—184no—
230C3⋊C8—184no—
231C3⋊C8—181no—
232C3⋊C8—181no—
233C3⋊C8—364no—
234C3⋊C8—364no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—48no—
239C2×C32—44no—
240C2×C32—41no—
241C2×C32#87714yes—
242C2×C32#90114yes—
243C2×C32#90714yes—
244C2×C32#92611yescharacteristic
245C2×C32#94811yescharacteristic
246C2×C32#94811yescharacteristic
247C22×C4D3×C3⋊D311yescharacteristic
248C2×C8—31no—
249C2×C8—31no—
250C2×C8—271no—
251C2×C8—271no—
252C4⋊C4—91no—
253C4⋊C4—91no—
254C22⋊C4—91no—
255C22⋊C4—91no—
256C42—91no—
257C42—91no—
258C22×C3—28no—
259C22×C3—24no—
260C22×C3—24no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—24no—
265C22×C3—24no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3C2×D3×Dic314yes—
269C22×C3C2×C4×C3⋊D311yescharacteristic
270C3×C4—21no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—44no—
275C3×C4—181no—
276C3×C4—181no—
277C3×C4—181no—
278C3×C4—181no—
279C3×C4#380414yes—
280C3×C4#380414yes—
281C3×C4#383011yescharacteristic
282C3×C4#383011yescharacteristic
283Dic3—64no—
284Dic3—64no—
285Dic3—64no—
286Dic3—64no—
287C32—24no—
288C32#472014yescomplemented
289C32#500011yescharacteristic, complemented
290C23Dic3×C3⋊D311yescharacteristic
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4Dic3×C3⋊D311yescharacteristic
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—61no—
300C8—61no—
301C8—541no—
302C8—541no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—48no—
307C2×C3—44no—
308C2×C3—41no—
309C2×C3#6381414yes—
310C2×C3#6381514yes—
311C2×C3#6395514yes—
312C2×C3#6408611yescharacteristic
313C2×C3#6411011yescharacteristic
314C2×C3#6411611yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22C2×Dic3×C3⋊D311yescharacteristic
319C4—21no—
320C4—181no—
321C4—181no—
322C4—181no—
323C4—181no—
324C4#8902811yescharacteristic
325C4#8902811yescharacteristic
326C3—24no—
327C3#1059105614yescomplemented
328C3#1059241911yescharacteristic, complemented
329C2—41no—
330C2#1172335211yescharacteristic
331C2#1172335411yescharacteristic
332C2#1172359511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722051C2semidirect
5. #11722051C2semidirect
6. #11722020C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
201. C33#433semidirect
288. C32#4720semidirect
289. C32#5000semidirect
327. C3#10591056semidirect
328. C3#10592419semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
#117222002
C32×#8302
#117220512
#117220202
C3⋊C8×C3⋊Dic32
#105924193
#105910563