Finite Group Databasebeta

← #26115233

Subgroups — #26115233

Order 12032.

736 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094021C211yesmaximal, characteristic
2C4×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3C2×#15688852C214yesmaximal, complemented
4C4×C4⋊C8×C47C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×C4×D188—21no—
7C2×C4×D188—21no—
8C2×C4×D188C2212yes—
9C2×C4⋊(C47⋊C8)C2212yes—
10C2×C4×C47⋊C8—21no—
11#15689025—21no—
12#15689025—21no—
13#15689025C412yes—
14#15688993C411yescharacteristic
15#15688878—21no—
16#15688852—28no—
17C2×C4⋊C8×C47—21no—
18C2×C4×C8×C47C2212yes—
19C43×C47C2211yescharacteristic
20C47×#336—21no—
21C22×D188—21no—
22C22×D188—21no—
23C22×D188C2×C412yes—
24C22×C4×D47—22no—
25C22×C47⋊C8—22no—
26#13094021—22no—
27#13094021C814yescomplemented
28C4×D188—24no—
29C4×D188—24no—
30C4×D188—44no—
31C4×D188—42no—
32C4×D188—42no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C2×(C2×D47)⋊C4—22no—
37C2×(C2×D47)⋊C4—22no—
38C2×(C2×D47)⋊C4—22no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)D414yescomplemented
41C4×C47⋊C8—42no—
42C22×C8×C47—22no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47C2311yescharacteristic
50C4⋊C8×C47—24no—
51C4×C8×C47—24no—
52C23×D47—22no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×Dic47—22no—
61C2×D188—24no—
62C2×D188—24no—
63C2×D188—24no—
64C2×D188—24no—
65C2×D188—22no—
66C2×D188—42no—
67C2×D188—42no—
68C2×D188—42no—
69C2×D188—42no—
70C2×D188C2×C814yes—
71C2×C4×D47—44no—
72C2×C4×D47—42no—
73C2×C4×D47—42no—
74C2×C4×D47—44no—
75C2×C8×C47—24no—
76C2×C8×C47—42no—
77C2×C8×C47—42no—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47—22no—
81C42×C47—41no—
82C42×C47C2×C812yes—
83C42×C47M1612yes—
84C42×C47D812yes—
85C42×C47SD1612yes—
86C42×C47C2×D412yes—
87C42×C47#4112yes—
88(C2×D47)⋊C4—44no—
89(C2×D47)⋊C4—44no—
90(C2×D47)⋊C4—42no—
91(C2×D47)⋊C4—42no—
92C4⋊Dic47—24no—
93C4⋊Dic47—42no—
94C4⋊Dic47—42no—
95C2×C47⋊C8—24no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C23×C47C2×C22⋊C411yescharacteristic
99C22×D47—24no—
100C22×D47—42no—
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—42no—
105C22×D47—42no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47C22⋊C812yes—
116C2×C4×C47C22⋊C812yes—
117C2×C4×C47#10212yes—
118C2×C4×C47#10212yes—
119C2×C4×C47#10412yes—
120C2×C4×C47C4×D412yes—
121C2×C4×C47#12112yes—
122C2×C4×C47#12411yescharacteristic
123C2×C4×C47#12711yescharacteristic
124C2×C4×C47C22×C811yescharacteristic
125C2×C4×C47C2×M1611yescharacteristic
126C2×C4×C47#13112yes—
127C2×C4×C47C2×D811yescharacteristic
128C2×C4×C47C2×SD1611yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)12yes—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133D188—28no—
134D188—48no—
135D188—48no—
136D188—44no—
137C4×D47—84no—
138C4×D47—84no—
139C8×C47—44no—
140C47⋊C8—44no—
141#9674—471nomaximal, Sylow
142C22×C47#39011yescharacteristic
143C22×C47C2×C22⋊C811yescharacteristic
144C22×C47#40811yescharacteristic
145C22×C47C2×#10211yescharacteristic
146C22×C47#41811yescharacteristic
147C22×C47C2×#10411yescharacteristic
148C22×C47#42111yescharacteristic
149C2×D47—44no—
150C2×D47—44no—
151C2×D47—44no—
152C2×D47—82no—
153C2×D47—82no—
154C2×D47—82no—
155C2×D47—82no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—41no—
161C4×C47D4⋊C814yescomplemented
162C4×C47C8×D412yes—
163C4×C47#43612yes—
164C4×C47#45912yes—
165C4×C47#46012yes—
166Dic47—82no—
167C4×#127—471no—
168C4×C4⋊C8—471no—
169C4×C4⋊C8D4711yescharacteristic, complemented
170C2×D4⋊C8—474no—
171C2×C47C2×D4⋊C811yescharacteristic
172C2×C47#147811yescharacteristic
173C2×C47#148011yescharacteristic
174C2×C47#148411yescharacteristic
175C2×C47#195411yescharacteristic
176C2×C47#198011yescharacteristic
177C2×C47#198211yescharacteristic
178D47—84no—
179C2×#127—471no—
180C2×C4×D4—472no—
181C2×C4×D4—941no—
182C2×C4×D4—941no—
183C2×C4⋊C8—21no—
184C2×C4⋊C8—472no—
185C2×C4×C8—941no—
186C2×C4×C8C2×D4712yes—
187#390—472no—
188#390—941no—
189#390—941no—
190#384—471no—
191C43C2×D4711yescharacteristic
192#336—21no—
193#336—941no—
194D4⋊C8—948no—
195C47#967411yescharacteristic, complemented, Sylow
196C22×D4—472no—
197C22×D4—941no—
198C22×D4—941no—
199C23×C4—942no—
200C22×C8—22no—
201C22×C8—942no—
202#127—474no—
203#127—942no—
204C4×D4—944no—
205C4×D4—944no—
206C4×D4—1884no—
207C4×D4—1882no—
208C4×D4—1882no—
209C2×C4⋊C4—472no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4—942no—
215C2×C42C4×D4711yescharacteristic
216C2×C42C4×D4711yescharacteristic
217C2×C42D18811yescharacteristic
218C2×C42D18811yescharacteristic
219C2×C42C22⋊D4711yescharacteristic
220C2×C42C22⋊D4711yescharacteristic
221C2×C42C22×D4711yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—474no—
224C4⋊C8—942no—
225C4×C8—24no—
226C4×C8—1882no—
227C24—942no—
228C2×D4—474no—
229C2×D4—944no—
230C2×D4—944no—
231C2×D4—944no—
232C2×D4—942no—
233C2×D4—944no—
234C2×D4—1882no—
235C2×D4—1882no—
236C2×D4—1882no—
237C2×D4—1882no—
238C22×C4—942no—
239C22×C4—1884no—
240C22×C4—1882no—
241C22×C4—1882no—
242C22×C4—1884no—
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4C2×D18811yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—944no—
254C2×C8—1882no—
255C2×C8—1882no—
256C4⋊C4—944no—
257C4⋊C4—1882no—
258C4⋊C4—1882no—
259C22⋊C4—1884no—
260C22⋊C4—1884no—
261C22⋊C4—1882no—
262C22⋊C4—1882no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42C8×D4712yes—
268C42#1062532412yes—
269C42#1062532512yes—
270C42D37612yes—
271C42D4×D4712yes—
272C42#1062535412yes—
273C23—944no—
274C23—1882no—
275C23—1882no—
276C23—1882no—
277C23—1882no—
278C23—1882no—
279C23—1882no—
280C23C2×(C2×D47)⋊C411yescharacteristic
281D4—948no—
282D4—1888no—
283D4—1888no—
284D4—1884no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3764no—
295C2×C4—3764no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4(C2×D47)⋊C812yes—
300C2×C4(C2×D47)⋊C812yes—
301C2×C4#1309394612yes—
302C2×C4#1309394612yes—
303C2×C4#1309395212yes—
304C2×C4#1309398612yes—
305C2×C4#1309399912yes—
306C2×C4#1309401311yescharacteristic
307C2×C4#1309402011yescharacteristic
308C2×C4C2×C8×D4711yescharacteristic
309C2×C4C2×#1062532411yescharacteristic
310C2×C4#1309403212yes—
311C2×C4C2×D37611yescharacteristic
312C2×C4C2×#1062532511yescharacteristic
313C2×C4#1309405212yes—
314C8—44no—
315C8—1884no—
316C22—3762no—
317C22—3762no—
318C22—3762no—
319C22—3762no—
320C22—1884no—
321C22—1884no—
322C22—1884no—
323C22#1568902411yescharacteristic
324C22C2×(C2×D47)⋊C811yescharacteristic
325C22#1568909111yescharacteristic
326C22C2×#1309394611yescharacteristic
327C22#1568912911yescharacteristic
328C22C2×#1309395211yescharacteristic
329C22#1568914111yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—22no—
334C4—41no—
335C4—3762no—
336C4#1568885214yescomplemented
337C4#1568918812yes—
338C4#1568920012yes—
339C4#1568931912yes—
340C4#1568932412yes—
341C2—3764no—
342C2C2×#1568885211yescharacteristic
343C2#2024178011yescharacteristic
344C2#2024178811yescharacteristic
345C2#2024180611yescharacteristic
346C2#2024418111yescharacteristic
347C2#2024430611yescharacteristic
348C2#2024431411yescharacteristic
subgroupG/H
centerC22×C4C2×D188
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC4×C4⋊C8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13094021C4
cosocleC2×C42×C47C23
hypercenterC4×C4⋊C8D47
hypocenterC47#9674
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊(C47⋊C8)C2semidirect
3. C2×#15688852C2semidirect
4. C4×C4⋊C8×C47C2semidirect
27. #13094021C8semidirect
40. C4⋊(C47⋊C8)D4semidirect
161. C4×C47D4⋊C8semidirect
169. C4×C4⋊C8D47semidirect
195. C47#9674semidirect
336. C4#15688852semidirect

Maximal subgroups

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

subgroupindex
C4×#130940212
C4×C4⋊(C47⋊C8)2
C2×#156888522
C4×C4⋊C8×C472
#967447