Finite Group Databasebeta

← #26112178

Subgroups — #26112178

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252868C211yesmaximal, characteristic
2#20242057C211yesmaximal, characteristic, complemented
3C2×#15688866C211yesmaximal, characteristic, complemented
4#20241775C211yesmaximal, characteristic, complemented
5#20241285C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#2925C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—21no—
11#15689323C412yes—
12#15689318—21no—
13#15689318C412yes—
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17#15689127—21no—
18#15688866—21no—
19#15688866—21no—
20#15688866—22no—
21Q8⋊(C47⋊C8)—21no—
22D4⋊(C47⋊C8)—21no—
23C4×#41×C47C2211yescharacteristic
24C47×#433—22no—
25C47×#432—21no—
26C47×#431—21no—
27C47×#423C2211yescharacteristic
28C2×C4⋊C8×C47C2211yescharacteristic
29C47×#408—21no—
30C47×#405—21no—
31C22×D188—21no—
32C22×D188—21no—
33C22×D188—21no—
34#13094049—42no—
35#13094049—42no—
36C2×#10625336—22no—
37C2×#10625334—22no—
38C2×#10625329—21no—
39C22×C47⋊C8—21no—
40#13094021—21no—
41#13094021—21no—
42#13094021—21no—
43#13094021C2×C412yes—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)C2×C412yes—
49#13093944—22no—
50#13093944—22no—
51C22⋊(C47⋊C8)—21no—
52C4×C47⋊C8—21no—
53C2×#41×C47D411yescharacteristic
54C2×M16×C47—21no—
55C22×C8×C47—21no—
56C4×Q8×C47C2×C411yescharacteristic
57C4×Q8×C47D411yescharacteristic
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47C2×C411yescharacteristic
61C4×D4×C47D411yescharacteristic
62#117×C47—21no—
63#117×C47D411yescharacteristic
64C2×C42×C47—21no—
65C2×C42×C47C2311yescharacteristic
66C4⋊C8×C47—21no—
67C4⋊C8×C47—21no—
68C4⋊C8×C47D411yescharacteristic
69C4⋊C8×C47D411yescharacteristic
70C4⋊C8×C47D412yes—
71C22⋊C8×C47—22no—
72C22⋊C8×C47—22no—
73#97×C47—22no—
74C4×C8×C47—22no—
75C23×D47—41no—
76#41×C47—21no—
77#41×C47—21no—
78#41×C47—21no—
79#41×C47—21no—
80C2×Q8×C47C22⋊C411yescharacteristic
81C2×D4×C47—21no—
82C2×D4×C47C22⋊C411yescharacteristic
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4111yescharacteristic
88C22×C4×C47#4111yescharacteristic
89C2×D188—22no—
90C2×D188—22no—
91C2×D188—22no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—82no—
100C2×D188—81no—
101C2×D188—81no—
102M16×C47—41no—
103M16×C47—41no—
104C2×C8×C47—21no—
105C2×C8×C47—21no—
106C2×C8×C47—22no—
107C2×C8×C47—41no—
108C2×C8×C47—41no—
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#10625336—44no—
124#10625334—44no—
125#10625329—41no—
126#10625329—41no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—81no—
131C23×C47—21no—
132C23×C47#12311yescharacteristic
133C22×D47—42no—
134C22×D47—82no—
135C22×D47—82no—
136C22×D47—81no—
137C22×D47—81no—
138Q8×C47—21no—
139Q8×C47—21no—
140D4×C47—21no—
141D4×C47—21no—
142D4×C47—41no—
143D4×C47—41no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—81no—
155C2×C4×C47C2×C22⋊C411yescharacteristic
156C2×C4×C47C4×D411yescharacteristic
157C2×C4×C47C4×D411yescharacteristic
158C2×C4×C47#12011yescharacteristic
159C2×C4×C47#12111yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#13511yescharacteristic
162C2×C4×C47#13511yescharacteristic
163C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
165D188—42no—
166D188—42no—
167D188—82no—
168D188—82no—
169D188—82no—
170D188—81no—
171D188—81no—
172D188—81no—
173D188—81no—
174C8×C47—41no—
175C8×C47—41no—
176C8×C47—42no—
177C47⋊C8—42no—
178C47⋊C8—81no—
179C47⋊C8—81no—
180#9174—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47#38611yescharacteristic
187C22×C47#48211yescharacteristic
188C22×C47#48211yescharacteristic
189C2×D47—82no—
190C2×D47—82no—
191C2×D47—82no—
192C2×D47—162no—
193C2×D47—162no—
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47#41611yescharacteristic
201C4×C47#41811yescharacteristic
202C4×C47#44911yescharacteristic
203C4×C47#44911yescharacteristic
204C4×C47#47111yescharacteristic
205C4×C47#47211yescharacteristic
206#3038—471no—
207#2925D4711yescharacteristic, complemented
208#1545—471no—
209C2×#331—471no—
210#1477—471no—
211#1301—471no—
212#1292—471no—
213C2×C47—21no—
214C2×C47—21no—
215C2×C47—81no—
216C2×C47#187511yescharacteristic
217C2×C47#190311yescharacteristic
218C2×C47#190711yescharacteristic
219D47—162no—
220C2×AGL(1,ℤ/8)—941no—
221C2×#127—471no—
222C4×#41C2×D4711yescharacteristic
223#460—472no—
224#460—941no—
225#459—472no—
226#459—941no—
227#433—22no—
228#432—21no—
229#431—21no—
230#431—471no—
231#423—471no—
232#423C2×D4711yescharacteristic
233C2×C4⋊C8—471no—
234C2×C4⋊C8C2×D4711yescharacteristic
235#418—941no—
236#408—21no—
237#405—21no—
238#331—941no—
239#331—941no—
240#331—942no—
241Q8⋊C8—941no—
242D4⋊C8—941no—
243C47#917411yescharacteristic, complemented, Sylow
244C2×#41C22⋊D4711yescharacteristic
245C22×D4—941no—
246C22×D4—941no—
247C22×D4—941no—
248AGL(1,ℤ/8)—1882no—
249AGL(1,ℤ/8)—1882no—
250C2×SD16—942no—
251C2×D8—942no—
252C2×M16—21no—
253C2×M16—941no—
254C22×C8—21no—
255C22×C8—941no—
256#127—472no—
257#127—941no—
258#127—941no—
259#127—941no—
260C4×Q8C4×D4711yescharacteristic
261C4×Q8D18811yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C4×D4711yescharacteristic
265C4×D4D18811yescharacteristic
266#117—21no—
267#117C22⋊D4711yescharacteristic
268C2×C42—21no—
269C2×C42C22×D4711yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—472no—
273C4⋊C8—941no—
274C4⋊C8—941no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8C22⋊D4711yescharacteristic
278C4⋊C8C22⋊D4711yescharacteristic
279C4⋊C8C22⋊D4712yes—
280#102—942no—
281#102—942no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—941no—
285#97—22no—
286C4×C8—22no—
287C4×C8—941no—
288C24—1881no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8(C2×D47)⋊C411yescharacteristic
294C2×D4—21no—
295C2×D4—942no—
296C2×D4—942no—
297C2×D4—942no—
298C2×D4—3761no—
299C2×D4—3761no—
300C2×D4—3762no—
301C2×D4—1881no—
302C2×D4—1881no—
303C2×D4—1881no—
304C2×D4—1881no—
305C2×D4—1881no—
306C2×D4—1881no—
307C2×D4—1881no—
308C2×D4(C2×D47)⋊C411yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#1062535011yescharacteristic
313C22×C4#1062535011yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—1884no—
316D8—1884no—
317M16—41no—
318M16—41no—
319M16—1881no—
320M16—1881no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—942no—
327C2×C8—3761no—
328C2×C8—1881no—
329C2×C8—1881no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4(C2×D47)⋊C411yescharacteristic
333C4⋊C4(C2×D47)⋊C411yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C2×C4×D4711yescharacteristic
341C42C2×D18811yescharacteristic
342C42C2×C22⋊D4711yescharacteristic
343C42C2×C22⋊D4711yescharacteristic
344C23—21no—
345C23—3762no—
346C23—3762no—
347C23—3761no—
348C23—3761no—
349C23—1882no—
350C23#1309400711yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—3762no—
358D4—3762no—
359D4—3762no—
360D4—3761no—
361D4—3761no—
362D4—3761no—
363D4—3761no—
364D4—1882no—
365D4—1882no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×(C2×D47)⋊C411yescharacteristic
378C2×C4C4×C22⋊D4711yescharacteristic
379C2×C4C4×C22⋊D4711yescharacteristic
380C2×C4#1309399311yescharacteristic
381C2×C4#1309399811yescharacteristic
382C2×C4#1309399811yescharacteristic
383C2×C4#1309404411yescharacteristic
384C2×C4#1309404411yescharacteristic
385C2×C4#1309404911yescharacteristic
386C2×C4#1309404911yescharacteristic
387C8—41no—
388C8—41no—
389C8—42no—
390C8—3761no—
391C8—3761no—
392C8—1882no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—3762no—
399C22—3762no—
400C22—3762no—
401C22—7522no—
402C22—7522no—
403C22#1568900211yescharacteristic
404C22#1568944711yescharacteristic
405C22#1568944711yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1568911711yescharacteristic
413C4#1568912711yescharacteristic
414C4#1568926811yescharacteristic
415C4#1568926811yescharacteristic
416C4#1568938611yescharacteristic
417C4#1568939311yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—7522no—
422C2#2024371011yescharacteristic
423C2#2024386811yescharacteristic
424C2#2024389611yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2925C2
socleC22×C47#386
prime radical#20252868C2
cosocleC2×C42×C47C23
hypercenter#2925D47
hypocenterC47#9174
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #20242057C2semidirect
3. C2×#15688866C2semidirect
4. #20241775C2semidirect
5. #20241285C2semidirect
6. #20241262C2semidirect
7. C47×#2925C2semidirect
207. #2925D47semidirect
243. C47#9174semidirect

Maximal subgroups

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

subgroupindex
#202528682
#202420572
C2×#156888662
#202417752
#202412852
#202412622
C47×#29252
#917447