Finite Group Databasebeta

← #13160285

Subgroups — #13160285

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
1C32×#926C211yesmaximal, characteristic
2#11722200C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722019C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592699—31nomaximal
9#10591056—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×#217C411yescharacteristic
16C32×#217C411yescharacteristic
17C32×C4⋊Dic3C411yescharacteristic
18C32×C4⋊Dic3C411yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C32×#117—31no—
22C3×#926D314yes—
23#63657—31no—
24#63657—34no—
25#63657—64no—
26#63657D311yescharacteristic
27C4⋊(C3⋊(C3⋊C8))—31no—
28C4⋊(C3⋊(C3⋊C8))—31no—
29#63630—31no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×#830—34no—
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×C32×Dic3C2×C411yescharacteristic
42C2×C32×Dic3C2×C411yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5000—94no—
50#4720—91no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D314yes—
54C32×C4⋊C4—31no—
55C32×C4⋊C4—31no—
56C32×C22⋊C4—31no—
57C32×C22⋊C4—31no—
58C32×C42—31no—
59C32×C42—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C2×C3⋊(C3⋊C8)—31no—
62C2×C3⋊(C3⋊C8)—34no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—64no—
65C2×C3⋊(C3⋊C8)—64no—
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×#217Dic314yes—
69C3×#217Dic314yes—
70C3×C4⋊Dic3Dic314yes—
71C3×C4⋊Dic3Dic314yes—
72C3×C4×Dic3C2×D314yes—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8—34no—
76C22×C33—21no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33C22×C411yescharacteristic
80C33×C4—21no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C32×Dic3—21no—
84C32×Dic3—21no—
85C32×Dic3—21no—
86C32×Dic3—21no—
87C3×#117—34no—
88#926C3⋊D311yescharacteristic
89C3×C22⋊C8—91no—
90#830—34no—
91#830—94no—
92#830—91no—
93#830—184no—
94#815—91no—
95Dic3⋊C8—91no—
96Dic3⋊C8—91no—
97C8×Dic3—91no—
98C4⋊(C3⋊C8)—94no—
99C4⋊(C3⋊C8)—94no—
100#803—94no—
101C4×C3⋊C8—94no—
102C23×C32—24no—
103C23×C32C4×D311yescharacteristic
104C23×C32C2×Dic314yes—
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—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—44no—
117C2×C32×C4C4×D311yescharacteristic
118C2×C32×C4C2×Dic314yes—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3×Dic3C2×Dic314yes—
122C2×C3×Dic3C2×Dic314yes—
123C2×C3×Dic3C2×Dic314yes—
124C2×C3×Dic3C2×Dic314yes—
125C3⋊(C3⋊C8)—21no—
126C3⋊(C3⋊C8)—21no—
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—64no—
134C3×C3⋊C8—64no—
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—91no—
144C2×C3×C8—91no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4—34no—
147C3×C22⋊C4—34no—
148C3×C22⋊C4—34no—
149C3×C42—34no—
150C3×C42—34no—
151#217C3⋊Dic311yescharacteristic
152#217C3⋊Dic311yescharacteristic
153C4⋊Dic3C3⋊Dic311yescharacteristic
154C4⋊Dic3C3⋊Dic311yescharacteristic
155C4×Dic3C2×C3⋊D311yescharacteristic
156C4×Dic3C2×C3⋊D311yescharacteristic
157C2×C3⋊C8—34no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C2×C3⋊C8—184no—
165C22×C32—28no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—24no—
170C22×C32—24no—
171C22×C32—24no—
172C22×C32—24no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32C2×C4×D311yescharacteristic
176C22×C32C22×Dic314yes—
177C32×C4—21no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4—61no—
183C32×C4—61no—
184C32×C4—61no—
185C32×C4—61no—
186C32×C4#23514yes—
187C32×C4#23514yes—
188C32×C4#23711yescharacteristic
189C32×C4#23711yescharacteristic
190C3×Dic3—24no—
191C3×Dic3—24no—
192C3×Dic3—24no—
193C3×Dic3—24no—
194#117—31no—
195C4⋊C8—271no—
196C4⋊C8—271no—
197C22⋊C8—91no—
198C22⋊C8—271no—
199#97—271no—
200C4×C8—271no—
201C33#43311yescharacteristic, complemented, Sylow
202C23×C3—24no—
203C23×C3D3×Dic314yes—
204C23×C3C2×C3⋊Dic311yescharacteristic
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—34no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4D3×Dic314yes—
218C2×C3×C4C2×C3⋊Dic311yescharacteristic
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3C2×C3⋊Dic311yescharacteristic
222C2×Dic3C2×C3⋊Dic311yescharacteristic
223C2×Dic3C2×C3⋊Dic311yescharacteristic
224C2×Dic3C2×C3⋊Dic311yescharacteristic
225C3×C8—181no—
226C3×C8—181no—
227C3⋊C8—64no—
228C3⋊C8—64no—
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—41no—
240C2×C32—44no—
241C2×C32#87711yescharacteristic
242C2×C32#90111yescharacteristic
243C2×C32#90711yescharacteristic
244C2×C32#92614yes—
245C2×C32#94814yes—
246C2×C32#94814yes—
247C22×C4D3×C3⋊D311yescharacteristic
248C2×C8—91no—
249C2×C8—91no—
250C2×C8—271no—
251C2×C8—271no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C22⋊C4—31no—
255C22⋊C4—31no—
256C42—31no—
257C42—31no—
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×C3C22×C3⋊Dic311yescharacteristic
270C3×C4—21no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—44no—
275C3×C4—64no—
276C3×C4—64no—
277C3×C4—64no—
278C3×C4—64no—
279C3×C4#380414yes—
280C3×C4#380414yes—
281C3×C4#382811yescharacteristic
282C3×C4#382811yescharacteristic
283Dic3—21no—
284Dic3—21no—
285Dic3—21no—
286Dic3—21no—
287C32—24no—
288C32#472011yescharacteristic, complemented
289C32#500014yescomplemented
290C23D3×C3⋊Dic311yescharacteristic
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4D3×C3⋊Dic311yescharacteristic
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—181no—
300C8—181no—
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#6413511yescharacteristic
313C2×C3#6415711yescharacteristic
314C2×C3#6415711yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22C2×D3×C3⋊Dic311yescharacteristic
319C4—21no—
320C4—61no—
321C4—61no—
322C4—61no—
323C4—61no—
324C4#8903111yescharacteristic
325C4#8903111yescharacteristic
326C3—24no—
327C3#1059105614yescomplemented
328C3#1059269911yescharacteristic, complemented
329C2—41no—
330C2#1172334911yescharacteristic
331C2#1172335011yescharacteristic
332C2#1172359411yescharacteristic
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. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722019C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
201. C33#433semidirect
288. C32#4720semidirect
289. C32#5000semidirect
327. C3#10591056semidirect
328. C3#10592699semidirect

Maximal subgroups

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

subgroupindex
C32×#9262
#117222002
C3×#636572
Dic3⋊(C3⋊(C3⋊C8))2
#117220192
Dic3×C3⋊(C3⋊C8)2
#105926993
#105910563