Finite Group Databasebeta

← #26112173

Subgroups — #26112173

Order 12032.

562 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
1#20251956C211yesmaximal, characteristic, complemented
2#20242056C211yesmaximal, characteristic, complemented
3C2×#15688867C211yesmaximal, characteristic, complemented
4#20241775C211yesmaximal, characteristic, complemented
5#20241283C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#3038C211yesmaximal, characteristic
8C2×#13094031—21no—
9#15689176C2211yescharacteristic
10#15689150—22no—
11#15689150—21no—
12#15689147—21no—
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17#15688867—21no—
18#15688867—21no—
19#15688867—22no—
20Q8⋊(C47⋊C8)—21no—
21D4⋊(C47⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C47—21no—
23C2×#127×C47C2211yescharacteristic
24C4×#41×C47C2211yescharacteristic
25C47×#460—21no—
26C47×#460C412yes—
27C47×#459—21no—
28C47×#459C412yes—
29C47×#423C2211yescharacteristic
30C47×#418—21no—
31#13094031—42no—
32#13094031—42no—
33C2×#10625329—22no—
34C2×#10625329—21no—
35C22×C47⋊C8—22no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—21no—
38C4⋊(C47⋊C8)—22no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)D411yescharacteristic
46C4⋊(C47⋊C8)D411yescharacteristic
47C4⋊(C47⋊C8)D412yes—
48C22⋊(C47⋊C8)—21no—
49C4×C47⋊C8—21no—
50C2×#41×C47D411yescharacteristic
51C22×D4×C47—21no—
52C22×D4×C47—21no—
53C22×D4×C47—21no—
54AGL(1,ℤ/8)×C47—42no—
55AGL(1,ℤ/8)×C47—42no—
56C2×SD16×C47—22no—
57C2×D8×C47—22no—
58C2×M16×C47—21no—
59#127×C47—21no—
60#127×C47—21no—
61#127×C47—21no—
62#127×C47C2×C412yes—
63C4×Q8×C47C2×C411yescharacteristic
64C4×Q8×C47D411yescharacteristic
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47C2×C411yescharacteristic
68C4×D4×C47D411yescharacteristic
69#117×C47—21no—
70#117×C47D411yescharacteristic
71C2×C42×C47—21no—
72C2×C42×C47C2311yescharacteristic
73C4⋊C8×C47—21no—
74C4⋊C8×C47C2×C412yes—
75#102×C47—22no—
76#102×C47—22no—
77C24×C47—41no—
78#41×C47—21no—
79#41×C47—21no—
80#41×C47D812yes—
81#41×C47SD1612yes—
82C2×Q8×C47C22⋊C411yescharacteristic
83C2×D4×C47—21no—
84C2×D4×C47—22no—
85C2×D4×C47—22no—
86C2×D4×C47—22no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—41no—
91C2×D4×C47—41no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—81no—
95C2×D4×C47—81no—
96C2×D4×C47—82no—
97C2×D4×C47C22⋊C411yescharacteristic
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47#4111yescharacteristic
103C22×C4×C47#4111yescharacteristic
104SD16×C47—44no—
105D8×C47—44no—
106M16×C47—41no—
107M16×C47—41no—
108C2×C8×C47—22no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47C22⋊C411yescharacteristic
112C4⋊C4×C47C22⋊C411yescharacteristic
113C22⋊C4×C47—21no—
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47C22×C411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47C2×D411yescharacteristic
123#10625329—44no—
124#10625329—41no—
125#10625329—41no—
126C2×C47⋊C8—21no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—44no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—81no—
135C23×C47—21no—
136C23×C47—42no—
137C23×C47—82no—
138C23×C47—82no—
139C23×C47—81no—
140C23×C47—81no—
141C23×C47#12311yescharacteristic
142Q8×C47—21no—
143Q8×C47#10212yes—
144D4×C47—22no—
145D4×C47—22no—
146D4×C47—21no—
147D4×C47—42no—
148D4×C47—42no—
149D4×C47—82no—
150D4×C47—82no—
151D4×C47—82no—
152D4×C47—81no—
153D4×C47—81no—
154D4×C47—81no—
155D4×C47—81no—
156D4×C47#10212yes—
157C2×C4×C47—21no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—81no—
168C2×C4×C47C2×C22⋊C411yescharacteristic
169C2×C4×C47C4×D411yescharacteristic
170C2×C4×C47C4×D411yescharacteristic
171C2×C4×C47#12011yescharacteristic
172C2×C4×C47#12111yescharacteristic
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47C2×D811yescharacteristic
175C2×C4×C47C2×SD1611yescharacteristic
176C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C47#13711yescharacteristic
178C8×C47—42no—
179C47⋊C8—41no—
180C47⋊C8—42no—
181C47⋊C8—41no—
182C47⋊C8—81no—
183C47⋊C8—81no—
184#9173—471nomaximal, Sylow
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—22no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—82no—
191C22×C47—82no—
192C22×C47—82no—
193C22×C47—162no—
194C22×C47—162no—
195C22×C47#38611yescharacteristic
196C22×C47#48011yescharacteristic
197C22×C47#48111yescharacteristic
198C4×C47—22no—
199C4×C47—21no—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—41no—
203C4×C47—41no—
204C4×C47C2×#10211yescharacteristic
205C4×C47#41711yescharacteristic
206C4×C47#44711yescharacteristic
207C4×C47#44811yescharacteristic
208C4×C47#47111yescharacteristic
209C4×C47#47211yescharacteristic
210#3038D4711yescharacteristic
211#2906—471no—
212#1545—471no—
213C2×#331—471no—
214#1477—471no—
215#1301—471no—
216#1292—471no—
217C2×C47—21no—
218C2×C47—21no—
219C2×C47—42no—
220C2×C47—162no—
221C2×C47#187411yescharacteristic
222C2×C47#190311yescharacteristic
223C2×C47#190711yescharacteristic
224C2×AGL(1,ℤ/8)—21no—
225C2×#131—941no—
226C2×#127C2×D4711yescharacteristic
227C4×#41C2×D4711yescharacteristic
228#460—21no—
229#460Dic4712yes—
230#459—21no—
231#459Dic4712yes—
232#431—471no—
233#424—942no—
234#424—941no—
235#423—471no—
236#423—941no—
237#423C2×D4711yescharacteristic
238C2×C4⋊C8—471no—
239C2×C4⋊C8—471no—
240C2×C4⋊C8—941no—
241#418—21no—
242#331—941no—
243#331—941no—
244#331—942no—
245Q8⋊C8—941no—
246D4⋊C8—941no—
247C47#917311yescharacteristic, complemented, Sylow
248C2×#41C22⋊D4711yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252AGL(1,ℤ/8)—42no—
253AGL(1,ℤ/8)—42no—
254C2×SD16—22no—
255C2×D8—22no—
256#131—1882no—
257#131—1882no—
258C2×M16—21no—
259C2×M16—942no—
260C2×M16—941no—
261C22×C8—942no—
262C22×C8—941no—
263C22×C8—941no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic4712yes—
268C4×Q8C2×Dic4711yescharacteristic
269C4×Q8C22⋊D4711yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic4711yescharacteristic
273C4×D4C22⋊D4711yescharacteristic
274#117—21no—
275#117C22⋊D4711yescharacteristic
276C2×C42—21no—
277C2×C42C22×D4711yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—471no—
280C4⋊C8—471no—
281C4⋊C8—472no—
282C4⋊C8—942no—
283C4⋊C8—942no—
284C4⋊C8—941no—
285C4⋊C8—941no—
286C4⋊C8—941no—
287C4⋊C8—941no—
288C4⋊C8—941no—
289C4⋊C8C2×Dic4712yes—
290#102—22no—
291#102—22no—
292C22⋊C8—941no—
293C4×C8—941no—
294C24—41no—
295#41—21no—
296#41—21no—
297#41#1062533412yes—
298#41#1062533512yes—
299C2×Q8C22⋊Dic4711yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊Dic4711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#1062535211yescharacteristic
319C22×C4#1062535211yescharacteristic
320C22×C4C2×C22⋊D4711yescharacteristic
321SD16—44no—
322D8—44no—
323M16—41no—
324M16—41no—
325M16—1884no—
326M16—1881no—
327M16—1881no—
328C2×C8—22no—
329C2×C8—941no—
330C2×C8—941no—
331C2×C8—942no—
332C2×C8—3761no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1884no—
336C2×C8—1881no—
337C2×C8—1881no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4C22⋊Dic4711yescharacteristic
341C4⋊C4C22⋊Dic4711yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42D4×D4711yescharacteristic
349C42D4×D4711yescharacteristic
350C42C22×Dic4711yescharacteristic
351C42C2×C22⋊D4711yescharacteristic
352C23—21no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23#1309400511yescharacteristic
359Q8—21no—
360Q8#1309394512yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4#1309394512yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×C22⋊Dic4711yescharacteristic
386C2×C4D4×Dic4711yescharacteristic
387C2×C4D4×Dic4711yescharacteristic
388C2×C4#1309399211yescharacteristic
389C2×C4#1309399611yescharacteristic
390C2×C4#1309399611yescharacteristic
391C2×C4C2×#1062533411yescharacteristic
392C2×C4C2×#1062533511yescharacteristic
393C2×C4#1309404911yescharacteristic
394C2×C4#1309405411yescharacteristic
395C8—42no—
396C8—3761no—
397C8—3761no—
398C8—1881no—
399C8—1882no—
400C8—1881no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22#1568900311yescharacteristic
412C22#1568943411yescharacteristic
413C22#1568943911yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4C2×#1309394511yescharacteristic
421C4#1568912411yescharacteristic
422C4#1568925611yescharacteristic
423C4#1568926111yescharacteristic
424C4#1568938411yescharacteristic
425C4#1568939111yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2#2024370611yescharacteristic
431C2#2024387011yescharacteristic
432C2#2024389811yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3038C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3038D47
hypocenterC47#9173
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20251956C2semidirect
2. #20242056C2semidirect
3. C2×#15688867C2semidirect
4. #20241775C2semidirect
5. #20241283C2semidirect
6. #20241262C2semidirect
247. C47#9173semidirect

Maximal subgroups

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

subgroupindex
#202519562
#202420562
C2×#156888672
#202417752
#202412832
#202412622
C47×#30382
#917347