Finite Group Databasebeta

← #26112522

Subgroups — #26112522

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2#20244194C211yesmaximal, characteristic
3#20243393C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241223C211yesmaximal, characteristic, complemented
7C2×C47×#446C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—22no—
11#15689323C414yescomplemented
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093948—21no—
15C2×#13093946—21no—
16C2×#13093946—21no—
17C2×#13093944—21no—
18#15688993C2211yescharacteristic
19#15688885—21no—
20#15688866—22no—
21C2×C4×Q8×C47C2211yescharacteristic
22C47×#446—22no—
23C47×#446—22no—
24C2×C4⋊C8×C47—21no—
25C2×C4⋊C8×C47C2211yescharacteristic
26C2×#97×C47—21no—
27C2×C4×C8×C47C2211yescharacteristic
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C2×#10625336—24no—
32C2×#10625336—42no—
33C2×#10625336—42no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021C2×C412yes—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)C2×C412yes—
47#13093948—42no—
48#13093946—42no—
49#13093946—42no—
50#13093944—24no—
51C22×Q8×C47D411yescharacteristic
52C22×C8×C47—21no—
53C22×C8×C47D411yescharacteristic
54C22×C8×C47D411yescharacteristic
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47C2×C412yes—
58C4×Q8×C47D412yes—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47D411yescharacteristic
61C2×C42×C47—21no—
62C2×C42×C47C2311yescharacteristic
63C4⋊C8×C47—24no—
64C4⋊C8×C47—21no—
65C4⋊C8×C47D412yes—
66#97×C47—24no—
67C4×C8×C47—21no—
68C4×C8×C47—21no—
69C23×D47—41no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47C22⋊C412yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—41no—
82C2×D188—22no—
83C2×D188—22no—
84C2×D188—22no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—81no—
97C2×D188—81no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C2×C8×C47—21no—
101C2×C8×C47—22no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C2×C8×C47SD1612yescomplemented
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47C22⋊C412yes—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47C22×C411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47#4111yescharacteristic
117#10625336—48no—
118C4⋊Dic47—41no—
119C4⋊Dic47—41no—
120C4⋊Dic47—41no—
121C4⋊Dic47—41no—
122C4⋊Dic47—81no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C23×C47#12111yescharacteristic
130C22×D47—42no—
131C22×D47—81no—
132C22×D47—81no—
133C22×D47—81no—
134C22×D47—81no—
135C22×D47—81no—
136C22×D47—81no—
137Q8×C47—22no—
138Q8×C47—22no—
139Q8×C47—41no—
140Q8×C47—41no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47C4×D411yescharacteristic
152C2×C4×C47#12011yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47#12311yescharacteristic
155C2×C4×C47C2×SD1611yescharacteristic
156C2×C4×C47#13511yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C47#13711yescharacteristic
162C2×C4×C47#13711yescharacteristic
163C2×Dic47—81no—
164C2×Dic47—81no—
165C2×Dic47—81no—
166D188—42no—
167D188—42no—
168D188—42no—
169D188—42no—
170D188—82no—
171D188—82no—
172D188—81no—
173D188—81no—
174D188—81no—
175D188—81no—
176C8×C47—22no—
177C8×C47—41no—
178C8×C47—42no—
179C47⋊C8—42no—
180C47⋊C8—81no—
181#9230—471nomaximal, Sylow
182C22×C47#38611yescharacteristic
183C22×C47#46011yescharacteristic
184C22×C47#46411yescharacteristic
185C22×C47#46811yescharacteristic
186C22×C47#46911yescharacteristic
187C22×C47#47111yescharacteristic
188C22×C47#47211yescharacteristic
189C2×D47—82no—
190C2×D47—82no—
191C2×D47—82no—
192C2×D47—161no—
193C2×D47—161no—
194C2×D47—161no—
195C2×D47—161no—
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—42no—
199C4×C47—42no—
200C4×C47#41711yescharacteristic
201C4×C47#41811yescharacteristic
202C4×C47C4×SD1611yescharacteristic
203C4×C47#44011yescharacteristic
204C4×C47#44811yescharacteristic
205C4×C47#44911yescharacteristic
206C4×C47#46811yescharacteristic
207C4×C47#46911yescharacteristic
208Dic47—161no—
209C2×#460—471no—
210C2×#446D4711yescharacteristic, complemented
211#1957—471no—
212#1803—471no—
213C2×#331—471no—
214#1296—471no—
215#1277—471no—
216C2×C47#168611yescharacteristic
217C2×C47#168811yescharacteristic
218C2×C47#169811yescharacteristic
219C2×C47#170211yescharacteristic
220C2×C47#188011yescharacteristic
221C2×C47#189511yescharacteristic
222C2×C47#190711yescharacteristic
223D47—162no—
224C22×SD16—941no—
225C2×#127—471no—
226C2×C4×Q8C2×D4711yescharacteristic
227#460—474no—
228#460—942no—
229#446—22no—
230#446—22no—
231C2×C4⋊C8—21no—
232C2×C4⋊C8—471no—
233C2×C4⋊C8—471no—
234C2×C4⋊C8C2×D4711yescharacteristic
235C2×#103—941no—
236C2×#102—941no—
237C2×#102—941no—
238C2×#102—941no—
239C2×#97—21no—
240C2×C4×C8C2×D4711yescharacteristic
241#384—471no—
242#340—941no—
243#331—942no—
244C47#923011yescharacteristic, complemented, Sylow
245C22×Q8D18811yescharacteristic
246C22×D4—941no—
247C22×D4—941no—
248C22×D4—941no—
249C2×SD16—944no—
250C2×SD16—1882no—
251C2×SD16—1882no—
252C22×C8—21no—
253C22×C8—941no—
254C22×C8—941no—
255C22×C8C22⋊D4711yescharacteristic
256C22×C8C22⋊D4711yescharacteristic
257#127—472no—
258#127—941no—
259#127—941no—
260#127—941no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D4712yes—
264C4×Q8C22⋊D4712yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—941no—
267C2×C4⋊C4—941no—
268C2×C4⋊C4—941no—
269C2×C4⋊C4D18811yescharacteristic
270C2×C42—21no—
271C2×C42C22×D4711yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—21no—
274C4⋊C8—472no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278C4⋊C8C22⋊D4712yes—
279#103—1882no—
280#102—944no—
281#102—1882no—
282#102—1882no—
283#97—24no—
284C4×C8—21no—
285C4×C8—21no—
286C24—1881no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8(C2×D47)⋊C412yes—
293C2×D4—942no—
294C2×D4—942no—
295C2×D4—942no—
296C2×D4—3761no—
297C2×D4—3761no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1881no—
302C2×D4—1881no—
303C2×D4—1881no—
304C2×D4—1881no—
305C2×D4—1881no—
306C2×D4—1881no—
307C2×D4—1881no—
308C2×D4—1881no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—1881no—
312C22×C4C2×D18811yescharacteristic
313C22×C4#1062535011yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—1888no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—942no—
325C2×C8—1881no—
326C2×C8—1881no—
327C2×C8—1881no—
328C2×C8—1881no—
329C2×C8—1881no—
330C2×C8#1062533612yescomplemented
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—3761no—
335C4⋊C4—1881no—
336C4⋊C4—1881no—
337C4⋊C4—1881no—
338C4⋊C4—1881no—
339C4⋊C4(C2×D47)⋊C412yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D4711yescharacteristic
343C42#1062535011yescharacteristic
344C42C2×C22⋊D4711yescharacteristic
345C42C2×C22⋊D4711yescharacteristic
346C23—3761no—
347C23—3761no—
348C23—3761no—
349C23—3761no—
350C23—3761no—
351C23—3761no—
352C23—1882no—
353C23#1309399811yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358D4—3762no—
359D4—3762no—
360D4—3761no—
361D4—3761no—
362D4—3761no—
363D4—3761no—
364D4—1882no—
365D4—1882no—
366D4—1882no—
367D4—1882no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—3761no—
377C2×C4—3761no—
378C2×C4—3761no—
379C2×C4C2×(C2×D47)⋊C411yescharacteristic
380C2×C4C4×C22⋊D4711yescharacteristic
381C2×C4C4×C22⋊D4711yescharacteristic
382C2×C4#1309399311yescharacteristic
383C2×C4#1309399811yescharacteristic
384C2×C4#1309400711yescharacteristic
385C2×C4C2×#1062533611yescharacteristic
386C2×C4#1309404411yescharacteristic
387C2×C4#1309404911yescharacteristic
388C2×C4#1309405211yescharacteristic
389C2×C4#1309405211yescharacteristic
390C2×C4#1309405211yescharacteristic
391C2×C4#1309405511yescharacteristic
392C2×C4#1309405711yescharacteristic
393C8—22no—
394C8—41no—
395C8—42no—
396C8—3761no—
397C8—1882no—
398C22—3762no—
399C22—3762no—
400C22—3762no—
401C22—7521no—
402C22—7521no—
403C22—7521no—
404C22—7521no—
405C22#1568900211yescharacteristic
406C22#1568932311yescharacteristic
407C22#1568934511yescharacteristic
408C22#1568936911yescharacteristic
409C22#1568937611yescharacteristic
410C22#1568938611yescharacteristic
411C22#1568939311yescharacteristic
412C4—21no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—7521no—
417C4#1568912511yescharacteristic
418C4#1568912911yescharacteristic
419C4C4×#1062533611yescharacteristic
420C4#1568922111yescharacteristic
421C4#1568926211yescharacteristic
422C4#1568926811yescharacteristic
423C4#1568936911yescharacteristic
424C4#1568937611yescharacteristic
425C2—7522no—
426C2#2024286011yescharacteristic
427C2#2024287211yescharacteristic
428C2#2024292811yescharacteristic
429C2#2024295611yescharacteristic
430C2#2024374311yescharacteristic
431C2#2024382611yescharacteristic
432C2#2024389611yescharacteristic
subgroupG/H
centerC23#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#446C2
socleC23×C47#121
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenterC2×#446D47
hypocenterC47#9230
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
3. #20243393C2semidirect
5. #20241272C2semidirect
6. #20241223C2semidirect
7. C2×C47×#446C2semidirect
11. #15689323C4semidirect
106. C2×C8×C47SD16semidirect
210. C2×#446D47semidirect
244. C47#9230semidirect
330. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
#202441942
#202433932
C2×#156888662
#202412722
#202412232
C2×C47×#4462
#923047