Finite Group Databasebeta

← #26115365

Subgroups — #26115365

Order 12032.

610 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251746C211yesmaximal, characteristic
2#20243539C211yesmaximal, characteristic, complemented
3#20243243C211yesmaximal, characteristic, complemented
4C2×#15688869C211yesmaximal, characteristic, complemented
5C2×#15688866C211yesmaximal, characteristic
6C2×#15688866C211yesmaximal, characteristic
7C2×C47×#332C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9C2×#13094013—21no—
10C2×#13094013—21no—
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×Dic47⋊C8—22no—
14C2×#13093934—21no—
15#15689024—22no—
16#15689024C414yes—
17C42×Dic47C2211yescharacteristic
18#15688877—21no—
19#15688869—22no—
20#15688866—22no—
21#15688866—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C47×#332—22no—
26C22×D188—21no—
27C22×D188—21no—
28C22×D188C2×C412yes—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31#13094021—21no—
32#13094021—21no—
33#13094021—21no—
34#13094021—21no—
35#13094013—42no—
36#13094013—42no—
37#13094013—42no—
38#13094013—42no—
39C2×(C2×D47)⋊C4—24no—
40C2×(C2×D47)⋊C4—24no—
41C2×C4×Dic47C2×C412yes—
42C2×C4×Dic47D412yes—
43C2×C4×Dic47D411yescharacteristic
44C2×C4×Dic47D411yescharacteristic
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)D412yes—
49Dic47⋊C8—44no—
50#13093934—42no—
51C22×Q8×C47—21no—
52C22×C8×C47—21no—
53C22×C8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47C2×C412yes—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47D412yes—
65C23×D47—22no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C22×C4×C47—22no—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47C22⋊C412yes—
77C22×Dic47C22⋊C412yes—
78C2×D188—22no—
79C2×D188—22no—
80C2×D188—44no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—42no—
90C2×D188—42no—
91C2×C8×C47—22no—
92C2×C8×C47—41no—
93C2×C8×C47—41no—
94C2×C8×C47—41no—
95C2×C8×C47—42no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47#4111yescharacteristic
103C42×C47#4111yescharacteristic
104(C2×D47)⋊C4—48no—
105(C2×D47)⋊C4—48no—
106C4×Dic47—22no—
107C4×Dic47—21no—
108C4×Dic47—21no—
109C4×Dic47—22no—
110C4×Dic47—42no—
111C4×Dic47D812yes—
112C4×Dic47SD1612yes—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—42no—
118C23×C47C2×C22⋊C411yescharacteristic
119C22×D47—42no—
120C22×D47—44no—
121C22×D47—42no—
122C22×D47—44no—
123C22×D47—42no—
124Q8×C47—81no—
125Q8×C47—81no—
126Q8×C47—81no—
127Q8×C47—81no—
128C2×C4×C47—22no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12411yescharacteristic
137C2×C4×C47#12711yescharacteristic
138C2×C4×C47C2×D811yescharacteristic
139C2×C4×C47C2×SD1611yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)12yes—
143C2×C4×C47#13712yes—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47#10214yes—
150D188—82no—
151D188—82no—
152D188—81no—
153D188—81no—
154D188—81no—
155D188—81no—
156D188—82no—
157D188—81no—
158D188—81no—
159C8×C47—42no—
160C8×C47—81no—
161C47⋊C8—42no—
162C47⋊C8—81no—
163#9704—471nomaximal, Sylow
164C22×C47#39011yescharacteristic
165C22×C47#41312yes—
166C22×C47C2×#10211yescharacteristic
167C22×C47#41712yes—
168C22×C47#41811yescharacteristic
169C2×D47—84no—
170C2×D47—84no—
171C2×D47—82no—
172C2×D47—82no—
173C2×D47—82no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—82no—
177C4×C47#45911yescharacteristic
178C4×C47#45911yescharacteristic
179C4×C47#46011yescharacteristic
180C4×C47#46011yescharacteristic
181C4×C47#48912yes—
182C4×C47#49611yescharacteristic
183C4×C47#49711yescharacteristic
184Dic47—24no—
185Dic47—42no—
186#2858—471no—
187#1842—471no—
188#1767—471no—
189C2×#332—471no—
190C2×#332D4711yescharacteristic, complemented
191C2×#331—471no—
192C2×#331—471no—
193C2×C47#154712yes—
194C2×C47#155712yes—
195C2×C47#195911yescharacteristic
196C2×C47#196711yescharacteristic
197C2×C47#198211yescharacteristic
198D47—162no—
199C2×C4⋊Q8C2×D4711yescharacteristic
200C2×#127—471no—
201C2×#124—941no—
202C2×#124—941no—
203C2×C4⋊C8—471no—
204C2×C4⋊C8—471no—
205C2×C4⋊C8—942no—
206C2×C4⋊C8C2×D4711yescharacteristic
207C2×C4⋊C8C2×D4711yescharacteristic
208C2×#97—941no—
209#390—474no—
210#390—942no—
211C43—471no—
212#336—941no—
213#332—22no—
214#332—942no—
215#331—942no—
216#331—942no—
217C47#970411yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C22×D4—472no—
220C22×D4—941no—
221C22×D4—941no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—941no—
225C22×C8—941no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230#127—941no—
231#127—941no—
232#127—941no—
233#127—941no—
234#124—1882no—
235#124—1882no—
236#124—1882no—
237#124—1882no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C4×D4712yes—
240C2×C22⋊C4—944no—
241C2×C22⋊C4—944no—
242C2×C42—471no—
243C2×C42—471no—
244C2×C42—472no—
245C2×C42—472no—
246C2×C42C22×D4711yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—472no—
251C4⋊C8—941no—
252C4⋊C8—941no—
253C4⋊C8—941no—
254C4⋊C8—1884no—
255C4⋊C8C22⋊D4712yes—
256#97—1882no—
257C24—942no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—942no—
265C2×D4—942no—
266C2×D4—1884no—
267C2×D4—1881no—
268C2×D4—1881no—
269C2×D4—1881no—
270C2×D4—1881no—
271C2×D4—1881no—
272C2×D4—1881no—
273C2×D4—1881no—
274C2×D4—1881no—
275C2×D4—1882no—
276C2×D4—1882no—
277C22×C4—22no—
278C22×C4—472no—
279C22×C4—472no—
280C22×C4C2×C4×D4711yescharacteristic
281C22×C4D4×D4711yescharacteristic
282C22×C4D4×D4711yescharacteristic
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—942no—
289C2×C8—1881no—
290C2×C8—1882no—
291C2×C8—1881no—
292C2×C8—1881no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—1888no—
298C22⋊C4—1888no—
299C42—472no—
300C42—472no—
301C42—941no—
302C42—942no—
303C42—941no—
304C42—942no—
305C42—1882no—
306C42#1062535011yescharacteristic
307C42D4×D4711yescharacteristic
308C42#1062535211yescharacteristic
309C42C2×C22⋊D4711yescharacteristic
310C23—1882no—
311C23—1884no—
312C23—1882no—
313C23—1884no—
314C23—1882no—
315C23C22⋊C4×D4711yescharacteristic
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—81no—
320D4—3762no—
321D4—3762no—
322D4—3762no—
323D4—3761no—
324D4—3761no—
325D4—3761no—
326D4—3761no—
327D4—3761no—
328D4—3761no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—474no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—942no—
337C2×C4—942no—
338C2×C4—942no—
339C2×C4(C22⋊D47)⋊C411yescharacteristic
340C2×C4C4×C22⋊D4711yescharacteristic
341C2×C4#1309399611yescharacteristic
342C2×C4#1309399711yescharacteristic
343C2×C4#1309401011yescharacteristic
344C2×C4#1309402011yescharacteristic
345C2×C4D8×D4711yescharacteristic
346C2×C4SD16×D4711yescharacteristic
347C2×C4#1309404811yescharacteristic
348C2×C4#1309405112yes—
349C2×C4#1309405111yescharacteristic
350C2×C4#1309405612yes—
351C8—42no—
352C8—81no—
353C8—3761no—
354C8—1882no—
355C22—3762no—
356C22—3762no—
357C22—3764no—
358C22—3764no—
359C22—3762no—
360C22#1568902311yescharacteristic
361C22#1568910612yes—
362C22#102×D4711yescharacteristic
363C22#1568912312yes—
364C22#1568912611yescharacteristic
365C4—21no—
366C4—21no—
367C4—82no—
368C4—944no—
369C4—1882no—
370C4#1568931611yescharacteristic
371C4#1568931711yescharacteristic
372C4#1568932111yescharacteristic
373C4#1568932211yescharacteristic
374C4#1568948412yes—
375C4#1568950911yescharacteristic
376C4#1568951611yescharacteristic
377C2—7522no—
378C2#2024206412yes—
379C2#2024210912yes—
380C2#2024420111yescharacteristic
381C2#2024423611yescharacteristic
382C2#2024431311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#332C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenterC2×#332D47
hypocenterC47#9704
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20243539C2semidirect
3. #20243243C2semidirect
4. C2×#15688869C2semidirect
7. C2×C47×#332C2semidirect
190. C2×#332D47semidirect
217. C47#9704semidirect

Maximal subgroups

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

subgroupindex
#202517462
#202435392
#202432432
C2×#156888692
C2×#156888662
C2×C47×#3322
#970447