Finite Group Databasebeta

← #26133308

Subgroups — #26133308

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251999C211yesmaximal, characteristic, complemented
2C2×#15688936C211yesmaximal, characteristic, complemented
3C2×#15688936C211yesmaximal, characteristic, complemented
4#20245016C211yesmaximal, characteristic, complemented
5#20241499C211yesmaximal, characteristic, complemented
6#20241491C211yesmaximal, characteristic, complemented
7C2×C47×#472C211yesmaximal, characteristic
8#15689890—21no—
9C2×#13093964C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11#15689172—22no—
12#15689172—21no—
13C2×#13093960—22no—
14C2×#13093960C2211yescharacteristic
15C2×#13093960C2211yescharacteristic
16#15688936—22no—
17#15688936—22no—
18#15688907—22no—
19C2×AGL(1,ℤ/8)×C47—21no—
20C22×D8×C47C2211yescharacteristic
21C2×#131×C47C2211yescharacteristic
22C47×#472—22no—
23C47×#472C414yes—
24C2×#108×C47C2211yescharacteristic
25C2×#100×C47—21no—
26#13094031—42no—
27#13094031—42no—
28C2×#10625329—24no—
29C2×#10625329—21no—
30C2×#10625329—21no—
31#13093964—21no—
32#13093964—21no—
33#13093964—21no—
34#13093964—21no—
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37#13093960—24no—
38#13093960—21no—
39#13093960—21no—
40#13093960D412yes—
41#13093960D412yes—
42C2×#41×C47D411yescharacteristic
43C22×D4×C47—21no—
44AGL(1,ℤ/8)×C47—24no—
45AGL(1,ℤ/8)×C47—42no—
46C2×SD16×C47—41no—
47C2×D8×C47—21no—
48C2×D8×C47—21no—
49C2×D8×C47—21no—
50C2×D8×C47—21no—
51C2×D8×C47—21no—
52C2×D8×C47—41no—
53C2×D8×C47C2×C412yes—
54#131×C47—21no—
55#131×C47—21no—
56#131×C47C2×C412yes—
57#131×C47D412yes—
58C2×M16×C47—21no—
59C2×M16×C47—21no—
60C2×M16×C47D411yescharacteristic
61C22×C8×C47—21no—
62C22×C8×C47C2311yescharacteristic
63#108×C47—21no—
64#108×C47C2×C412yes—
65#100×C47—24no—
66C24×C47—41no—
67#41×C47—21no—
68#41×C47—21no—
69#41×C47C22⋊C412yes—
70#41×C47D812yes—
71C2×Q8×C47SD1611yescharacteristic
72C2×D4×C47—21no—
73C2×D4×C47—22no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47SD1611yescharacteristic
80C22×C4×C47—21no—
81C22×C4×C47C2×D411yescharacteristic
82SD16×C47—44no—
83D8×C47—22no—
84D8×C47—22no—
85D8×C47—44no—
86D8×C47—41no—
87D8×C47—41no—
88M16×C47—21no—
89M16×C47—22no—
90M16×C47—22no—
91M16×C47—22no—
92M16×C47—41no—
93M16×C47C22⋊C412yes—
94C2×C8×C47—21no—
95C2×C8×C47—22no—
96C2×C8×C47—22no—
97C2×C8×C47—41no—
98C2×C8×C47C22×C411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C2×C8×C47#4111yescharacteristic
104#10625329—22no—
105#10625329—22no—
106#10625329—44no—
107#10625329—41no—
108#10625329—41no—
109C2×C47⋊C8—44no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C47⋊C16—41no—
113C47⋊C16—41no—
114C23×C47—21no—
115C23×C47—42no—
116C23×C47—81no—
117C23×C47—81no—
118C23×C47—162no—
119C23×C47#13711yescharacteristic
120Q8×C47—21no—
121Q8×C47#10212yes—
122D4×C47—22no—
123D4×C47—22no—
124D4×C47—21no—
125D4×C47—42no—
126D4×C47—42no—
127D4×C47—81no—
128D4×C47—81no—
129D4×C47#10212yes—
130C2×C4×C47—21no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47#12011yescharacteristic
135C2×C4×C47#12311yescharacteristic
136C2×C4×C47C2×D811yescharacteristic
137C2×C4×C47C2×SD1611yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C8×C47—22no—
140C8×C47—42no—
141C8×C47C4×D411yescharacteristic
142C8×C47C4×D411yescharacteristic
143C8×C47#12111yescharacteristic
144C8×C47#12111yescharacteristic
145C47⋊C8—42no—
146C47⋊C8—42no—
147#13007—471nomaximal, Sylow
148C22×C47—22no—
149C22×C47—41no—
150C22×C47—81no—
151C22×C47—82no—
152C22×C47—162no—
153C22×C47—162no—
154C22×C47—162no—
155C22×C47#41711yescharacteristic
156C22×C47#44811yescharacteristic
157C22×C47#48011yescharacteristic
158C4×C47—22no—
159C4×C47#38611yescharacteristic
160C4×C47C2×#10211yescharacteristic
161C4×C47#44711yescharacteristic
162C4×C47#48111yescharacteristic
163C2×#472D4711yescharacteristic
164#2916—471no—
165C2×#361—471no—
166C2×#361—471no—
167#2117—471no—
168#1388—471no—
169#1385—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47—162no—
175C2×C47#187411yescharacteristic
176C2×AGL(1,ℤ/8)—21no—
177C22×D8C2×D4711yescharacteristic
178#568—941no—
179C2×#131C2×D4711yescharacteristic
180C2×M32—471no—
181C2×M32—471no—
182#472—22no—
183#472Dic4714yes—
184#430—942no—
185#430—941no—
186C2×#108—471no—
187C2×#108—471no—
188C2×#108—942no—
189C2×#108C2×D4711yescharacteristic
190C2×#100—21no—
191#361—942no—
192#361—942no—
193#350—942no—
194C47#1300711yescharacteristic, complemented, Sylow
195C2×#41C22⋊D4711yescharacteristic
196C22×D4—21no—
197AGL(1,ℤ/8)—24no—
198AGL(1,ℤ/8)—42no—
199C2×SD16—41no—
200C2×D8—21no—
201C2×D8—21no—
202C2×D8—21no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—41no—
206C2×D8C2×Dic4712yes—
207#131—21no—
208#131—21no—
209#131—1882no—
210#131—1882no—
211#131C2×Dic4712yes—
212#131C22⋊D4712yes—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—944no—
216C2×M16—941no—
217C2×M16—941no—
218C2×M16C22⋊D4711yescharacteristic
219C22×C8—21no—
220C22×C8C22×D4711yescharacteristic
221M32—941no—
222M32—941no—
223M32—941no—
224M32—941no—
225C2×C16—941no—
226C2×C16—941no—
227#108—21no—
228#108—472no—
229#108—472no—
230#108—944no—
231#108—941no—
232#108—941no—
233#108C2×Dic4712yes—
234#100—24no—
235C24—41no—
236#41—21no—
237#41—21no—
238#41#1062533412yes—
239#41C22⋊Dic4712yes—
240C2×Q8#1062533511yescharacteristic
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4#1062533511yescharacteristic
249C22×C4—21no—
250C22×C4C2×C22⋊D4711yescharacteristic
251SD16—44no—
252D8—22no—
253D8—22no—
254D8—44no—
255D8—41no—
256D8—41no—
257M16—21no—
258M16—22no—
259M16—22no—
260M16—22no—
261M16—41no—
262M16—942no—
263M16—942no—
264M16—1884no—
265M16—1881no—
266M16—1881no—
267M16C22⋊Dic4712yes—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—1884no—
273C2×C8—1881no—
274C2×C8—1881no—
275C2×C8D4×D4711yescharacteristic
276C2×C8D4×D4711yescharacteristic
277C2×C8#1062535211yescharacteristic
278C2×C8#1062535211yescharacteristic
279C2×C8C22×Dic4711yescharacteristic
280C2×C8C2×C22⋊D4711yescharacteristic
281C16—1881no—
282C16—1881no—
283C23—21no—
284C23—42no—
285C23—81no—
286C23—81no—
287C23—162no—
288C23#1309405411yescharacteristic
289Q8—21no—
290Q8#1309394512yes—
291D4—22no—
292D4—22no—
293D4—21no—
294D4—42no—
295D4—42no—
296D4—81no—
297D4—81no—
298D4#1309394512yes—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4C2×C22⋊Dic4711yescharacteristic
303C2×C4#1309399211yescharacteristic
304C2×C4#1309400511yescharacteristic
305C2×C4C2×#1062533411yescharacteristic
306C2×C4C2×#1062533511yescharacteristic
307C2×C4#1309404911yescharacteristic
308C8—22no—
309C8—42no—
310C8—1882no—
311C8—1882no—
312C8D4×Dic4711yescharacteristic
313C8D4×Dic4711yescharacteristic
314C8#1309399611yescharacteristic
315C8#1309399611yescharacteristic
316C22—22no—
317C22—41no—
318C22—81no—
319C22—82no—
320C22—162no—
321C22—162no—
322C22—162no—
323C22#1568912411yescharacteristic
324C22#1568926111yescharacteristic
325C22#1568943411yescharacteristic
326C4—22no—
327C4#1568900311yescharacteristic
328C4C2×#1309394511yescharacteristic
329C4#1568925611yescharacteristic
330C4#1568943911yescharacteristic
331C2—21no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—162no—
336C2#2024370611yescharacteristic
subgroupG/H
centerC2#20243706
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC2×C47×#472C2
socleC2×C47#1874
cosocleC22×C8×C47C23
hypercenterC2×#472D47
hypocenterC47#13007
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #20251999C2semidirect
2. C2×#15688936C2semidirect
3. C2×#15688936C2semidirect
4. #20245016C2semidirect
5. #20241499C2semidirect
6. #20241491C2semidirect
194. C47#13007semidirect

Maximal subgroups

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

subgroupindex
#202519992
C2×#156889362
#202450162
#202414992
#202414912
C2×C47×#4722
#1300747