Finite Group Databasebeta

← #26110904

Subgroups — #26110904

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252499C211yesmaximal, characteristic, complemented
2#20244183C211yesmaximal, characteristic, complemented
3#20241779C211yesmaximal, characteristic, complemented
4#20241770C211yesmaximal, characteristic, complemented
5#20241239C211yesmaximal, characteristic, complemented
6#20241226C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C47C211yesmaximal, characteristic
8Q8×C47⋊C8—21no—
9D4×C47⋊C8—22no—
10D4×C47⋊C8—21no—
11#15689176—21no—
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14C2×C4⋊(C47⋊C8)—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C22⋊(C47⋊C8)—22no—
17C2×C4×C47⋊C8C2211yescharacteristic
18Q8⋊(C47⋊C8)—21no—
19D4⋊(C47⋊C8)—21no—
20D4⋊(C47⋊C8)—21no—
21D4⋊(C47⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C47C411yescharacteristic
23C4×#41×C47C2211yescharacteristic
24C2×C4×D4×C47C2211yescharacteristic
25C47×#442—21no—
26C47×#440—21no—
27C4×SD16×C47—21no—
28C4×D8×C47—21no—
29C47×#428C411yescharacteristic
30C47×#418C411yescharacteristic
31C47×#417C411yescharacteristic
32C4×M16×C47C2211yescharacteristic
33C22×C47⋊C8—21no—
34C22×C47⋊C8—22no—
35C22×C47⋊C8D412yes—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)D411yescharacteristic
41C4⋊(C47⋊C8)D411yescharacteristic
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—44no—
46C4×C47⋊C8—22no—
47C4×C47⋊C8—21no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C2×#41×C47C2×C411yescharacteristic
52C22×D4×C47C2×C411yescharacteristic
53C23×C4×C47—21no—
54AGL(1,ℤ/8)×C47—22no—
55AGL(1,ℤ/8)×C47C814yescomplemented
56C2×SD16×C47—21no—
57C2×D8×C47—21no—
58C2×M16×C47C2×C411yescharacteristic
59C2×M16×C47C2×C411yescharacteristic
60C4×Q8×C47D411yescharacteristic
61C4×Q8×C47D411yescharacteristic
62C4×D4×C47—21no—
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47—21no—
68C4×D4×C47D411yescharacteristic
69C4×D4×C47D411yescharacteristic
70#117×C47—21no—
71#117×C47C2×C411yescharacteristic
72C2×C4⋊C4×C47C2×C411yescharacteristic
73C2×C22⋊C4×C47—21no—
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76#107×C47—21no—
77#106×C47—21no—
78#103×C47—21no—
79#102×C47—21no—
80#102×C47—21no—
81#102×C47—21no—
82#97×C47—21no—
83C4×C8×C47—21no—
84C24×C47—21no—
85#41×C47—21no—
86#41×C47—21no—
87#41×C47C2×C812yes—
88#41×C47M1612yes—
89C2×Q8×C47C22⋊C411yescharacteristic
90C2×D4×C47—21no—
91C2×D4×C47—21no—
92C2×D4×C47—21no—
93C2×D4×C47—21no—
94C2×D4×C47—21no—
95C2×D4×C47—21no—
96C2×D4×C47C22⋊C411yescharacteristic
97C2×D4×C47C2×C812yescomplemented
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—42no—
103C22×C4×C47—42no—
104C22×C4×C47—41no—
105C22×C4×C47—41no—
106C22×C4×C47C22×C411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47#4111yescharacteristic
109SD16×C47—24no—
110D8×C47—24no—
111M16×C47—21no—
112M16×C47—21no—
113M16×C47—21no—
114M16×C47C2×C812yescomplemented
115C2×C8×C47—21no—
116C2×C8×C47—21no—
117C4⋊C4×C47—21no—
118C4⋊C4×C47—21no—
119C4⋊C4×C47—21no—
120C4⋊C4×C47—21no—
121C4⋊C4×C47C22⋊C411yescharacteristic
122C4⋊C4×C47C22⋊C411yescharacteristic
123C22⋊C4×C47—21no—
124C22⋊C4×C47—21no—
125C22⋊C4×C47—21no—
126C22⋊C4×C47—41no—
127C22⋊C4×C47—41no—
128C42×C47—21no—
129C42×C47—21no—
130C42×C47—21no—
131C42×C47C2×D411yescharacteristic
132C42×C47C2×D411yescharacteristic
133C42×C47C2×D411yescharacteristic
134C42×C47#4111yescharacteristic
135C2×C47⋊C8—22no—
136C2×C47⋊C8—21no—
137C2×C47⋊C8—22no—
138C2×C47⋊C8—21no—
139C2×C47⋊C8—41no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—42no—
142C2×C47⋊C8—41no—
143C2×C47⋊C8—82no—
144C23×C47—21no—
145C23×C47—22no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—41no—
149C23×C47—41no—
150C23×C47C4×D411yescharacteristic
151Q8×C47—21no—
152Q8×C47C22⋊C812yescomplemented
153D4×C47—22no—
154D4×C47—22no—
155D4×C47—22no—
156D4×C47—21no—
157D4×C47—22no—
158D4×C47—41no—
159D4×C47—41no—
160D4×C47C22⋊C812yescomplemented
161C2×C4×C47—21no—
162C2×C4×C47—22no—
163C2×C4×C47—22no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—42no—
172C2×C4×C47—41no—
173C2×C4×C47—81no—
174C2×C4×C47—82no—
175C2×C4×C47—81no—
176C2×C4×C47—81no—
177C2×C4×C47C2×C22⋊C411yescharacteristic
178C2×C4×C47C4×D411yescharacteristic
179C2×C4×C47#12011yescharacteristic
180C2×C4×C47#12111yescharacteristic
181C2×C4×C47#12111yescharacteristic
182C2×C4×C47#12311yescharacteristic
183C2×C4×C47C22×C811yescharacteristic
184C2×C4×C47C2×M1611yescharacteristic
185C2×C4×C47#13111yescharacteristic
186C2×C4×C47#13111yescharacteristic
187C8×C47—22no—
188C8×C47—41no—
189C47⋊C8—42no—
190C47⋊C8—42no—
191C47⋊C8—41no—
192C47⋊C8—41no—
193C47⋊C8—81no—
194#8954—471nomaximal, Sylow
195C22×C47—21no—
196C22×C47—21no—
197C22×C47—22no—
198C22×C47—44no—
199C22×C47—41no—
200C22×C47—41no—
201C22×C47—42no—
202C22×C47—82no—
203C22×C47—81no—
204C22×C47—81no—
205C22×C47#38611yescharacteristic
206C22×C47C8×D411yescharacteristic
207C22×C47#43511yescharacteristic
208C4×C47—22no—
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—41no—
213C4×C47—41no—
214C4×C47—81no—
215C4×C47C2×C22⋊C811yescharacteristic
216C4×C47#40811yescharacteristic
217C4×C47C8×D411yescharacteristic
218C4×C47#43511yescharacteristic
219C4×C47#45311yescharacteristic
220C4×C47#45411yescharacteristic
221#2988—471no—
222C4×AGL(1,ℤ/8)D4711yescharacteristic
223#1954—471no—
224#1478—471no—
225#1476—471no—
226#1283—471no—
227#1278—471no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47—42no—
231C2×C47—82no—
232C2×C47#186911yescharacteristic
233C2×C47#188111yescharacteristic
234C2×C47#188311yescharacteristic
235C2×AGL(1,ℤ/8)Dic4711yescharacteristic
236C4×#41C2×D4711yescharacteristic
237C2×C4×D4C2×D4711yescharacteristic
238C8×Q8—941no—
239#442—21no—
240#440—21no—
241C4×SD16—21no—
242C4×D8—21no—
243C8×D4—942no—
244C8×D4—941no—
245#431—471no—
246#431—471no—
247#431—941no—
248#428Dic4711yescharacteristic
249C2×C4⋊C8—471no—
250C2×C4⋊C8—941no—
251#418Dic4711yescharacteristic
252#417Dic4711yescharacteristic
253C2×C22⋊C8—942no—
254C4×M16C2×D4711yescharacteristic
255C2×C4×C8—471no—
256Q8⋊C8—941no—
257D4⋊C8—941no—
258D4⋊C8—941no—
259D4⋊C8—941no—
260C47#895411yescharacteristic, complemented, Sylow
261C2×#41C2×Dic4711yescharacteristic
262C22×D4C2×Dic4711yescharacteristic
263C23×C4—21no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)C47⋊C814yescomplemented
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16C2×Dic4711yescharacteristic
269C2×M16C2×Dic4711yescharacteristic
270C22×C8—472no—
271C22×C8—942no—
272C22×C8—941no—
273C4×Q8C22⋊D4711yescharacteristic
274C4×Q8C22⋊D4711yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22⋊D4711yescharacteristic
282C4×D4C22⋊D4711yescharacteristic
283#117—21no—
284#117C2×Dic4711yescharacteristic
285C2×C4⋊C4C2×Dic4711yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D4711yescharacteristic
289#107—21no—
290#106—21no—
291C4⋊C8—471no—
292C4⋊C8—471no—
293C4⋊C8—942no—
294C4⋊C8—941no—
295C4⋊C8—941no—
296C4⋊C8—941no—
297#103—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—942no—
302C22⋊C8—941no—
303C22⋊C8—941no—
304C22⋊C8—1884no—
305#97—21no—
306C4×C8—21no—
307C4×C8—942no—
308C4×C8—941no—
309C4×C8—941no—
310C4×C8—941no—
311C4×C8—941no—
312C24—21no—
313#41—21no—
314#41—21no—
315#41C2×C47⋊C812yes—
316#41#1062532912yes—
317C2×Q8C22⋊Dic4711yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C2×C47⋊C812yescomplemented
325C2×D4C22⋊Dic4711yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4D4×D4711yescharacteristic
335C22×C4#1062535211yescharacteristic
336C22×C4C22×Dic4711yescharacteristic
337SD16—24no—
338D8—24no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16C2×C47⋊C812yescomplemented
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—942no—
346C2×C8—941no—
347C2×C8—941no—
348C2×C8—942no—
349C2×C8—3762no—
350C2×C8—1881no—
351C2×C8—1881no—
352C2×C8—1881no—
353C2×C8—1882no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4C22⋊Dic4711yescharacteristic
359C4⋊C4C22⋊Dic4711yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42D4×D4711yescharacteristic
369C42#1062535211yescharacteristic
370C42C2×C22⋊D4711yescharacteristic
371C42C2×C22⋊D4711yescharacteristic
372C23—21no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23D4×Dic4711yescharacteristic
379Q8—21no—
380Q8C22⋊(C47⋊C8)12yescomplemented
381D4—22no—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—22no—
386D4—41no—
387D4—41no—
388D4C22⋊(C47⋊C8)12yescomplemented
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊Dic4711yescharacteristic
406C2×C4D4×Dic4711yescharacteristic
407C2×C4#1309399211yescharacteristic
408C2×C4#1309399611yescharacteristic
409C2×C4#1309399611yescharacteristic
410C2×C4#1309400511yescharacteristic
411C2×C4C22×C47⋊C811yescharacteristic
412C2×C4C2×#1062532911yescharacteristic
413C2×C4#1309403111yescharacteristic
414C2×C4#1309403111yescharacteristic
415C8—22no—
416C8—41no—
417C8—3761no—
418C8—1882no—
419C8—1881no—
420C8—1882no—
421C8—1881no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—44no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22#1568900311yescharacteristic
433C22D4×C47⋊C811yescharacteristic
434C22#1568919511yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4C2×C22⋊(C47⋊C8)11yescharacteristic
443C4#1568909011yescharacteristic
444C4D4×C47⋊C811yescharacteristic
445C4#1568919511yescharacteristic
446C4#1568928911yescharacteristic
447C4#1568929411yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2#2024368111yescharacteristic
453C2#2024375211yescharacteristic
454C2#2024376611yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC4×AGL(1,ℤ/8)×C47C2
socleC22×C47#386
prime radicalC2×AGL(1,ℤ/8)×C47C4
cosocleC2×C42×C47C23
hypercenterC4×AGL(1,ℤ/8)D47
hypocenterC47#8954
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20252499C2semidirect
2. #20244183C2semidirect
3. #20241779C2semidirect
4. #20241770C2semidirect
5. #20241239C2semidirect
6. #20241226C2semidirect
55. AGL(1,ℤ/8)×C47C8semidirect
97. C2×D4×C47C2×C8semidirect
114. M16×C47C2×C8semidirect
152. Q8×C47C22⋊C8semidirect
160. D4×C47C22⋊C8semidirect
260. C47#8954semidirect
265. AGL(1,ℤ/8)C47⋊C8semidirect
324. C2×D4C2×C47⋊C8semidirect
342. M16C2×C47⋊C8semidirect
380. Q8C22⋊(C47⋊C8)semidirect
388. D4C22⋊(C47⋊C8)semidirect

Maximal subgroups

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

subgroupindex
#202524992
#202441832
#202417792
#202417702
#202412392
#202412262
C4×AGL(1,ℤ/8)×C472
#895447