Finite Group Databasebeta

← #26133294

Subgroups — #26133294

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251983C211yesmaximal, characteristic, complemented
2#20245124C211yesmaximal, characteristic, complemented
3C2×#15688933C211yesmaximal, characteristic, complemented
4#20245016C211yesmaximal, characteristic, complemented
5#20241502C211yesmaximal, characteristic, complemented
6#20241488C211yesmaximal, characteristic, complemented
7C47×#3067C211yesmaximal, characteristic
8#15689606—21no—
9C2×#13093964C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11#15689165—22no—
12#15689165—21no—
13#15689161—21no—
14#15689161C2211yescharacteristic
15C2×#13093956—21no—
16C2×#13093956C2211yescharacteristic
17#15688933—22no—
18#15688933—21no—
19#15688933—21no—
20#15688907—22no—
21C47×#575—21no—
22C2×#135×C47C2211yescharacteristic
23C2×#131×C47C2211yescharacteristic
24C47×#473C412yescomplemented
25C47×#472C412yescomplemented
26C47×#471—22no—
27C2×#108×C47C2211yescharacteristic
28C47×#413—21no—
29Q8×Dic47—41no—
30D4×Dic47—42no—
31D4×Dic47—41no—
32#13093983—22no—
33#13093983—21no—
34C2×C4⋊Dic47—22no—
35C2×C4⋊Dic47—21no—
36#13093964—21no—
37#13093964—21no—
38#13093964—21no—
39#13093964—21no—
40C2×C47⋊C16—21no—
41C2×C47⋊C16—21no—
42#13093958—22no—
43#13093958—21no—
44#13093956—22no—
45#13093956—21no—
46#13093956D411yescharacteristic
47#13093956D412yes—
48#13093956D411yescharacteristic
49C2×#41×C47—21no—
50C2×#41×C47D411yescharacteristic
51#137×C47—22no—
52#137×C47—41no—
53AGL(1,ℤ/8)×C47—22no—
54AGL(1,ℤ/8)×C47—41no—
55#135×C47—21no—
56#135×C47—21no—
57#135×C47—21no—
58#135×C47—21no—
59#135×C47—42no—
60C2×Q16×C47C2×C411yescharacteristic, complemented
61C2×SD16×C47—21no—
62C2×D8×C47C2×C411yescharacteristic, complemented
63#131×C47—21no—
64#131×C47—21no—
65#131×C47C2×C412yes—
66#131×C47D412yes—
67C2×M16×C47—21no—
68C2×M16×C47—21no—
69C2×M16×C47D411yescharacteristic
70C22×C8×C47—21no—
71C22×C8×C47C2311yescharacteristic
72#108×C47—21no—
73#108×C47C2×C412yescomplemented
74#101×C47—22no—
75#100×C47—22no—
76#41×C47—21no—
77#41×C47—21no—
78#41×C47—41no—
79#41×C47—41no—
80#41×C47—41no—
81#41×C47—41no—
82#41×C47C22⋊C412yes—
83#41×C47SD1612yes—
84C2×Q8×C47—21no—
85C2×Q8×C47D811yescharacteristic
86C2×D4×C47—21no—
87C2×D4×C47—21no—
88C2×D4×C47—41no—
89C2×D4×C47D811yescharacteristic
90C22×C4×C47—21no—
91C22×C4×C47—41no—
92C22×C4×C47C2×D411yescharacteristic
93C22×Dic47—42no—
94C22×Dic47—41no—
95Q16×C47—21no—
96Q16×C47—21no—
97Q16×C47—42no—
98SD16×C47—42no—
99SD16×C47—42no—
100SD16×C47—41no—
101SD16×C47—41no—
102D8×C47—21no—
103D8×C47—21no—
104D8×C47—42no—
105M16×C47—21no—
106M16×C47—22no—
107M16×C47—22no—
108M16×C47—22no—
109M16×C47—41no—
110M16×C47C22⋊C412yes—
111C2×C8×C47—21no—
112C2×C8×C47—22no—
113C2×C8×C47—22no—
114C2×C8×C47—41no—
115C2×C8×C47C22×C411yescharacteristic
116C2×C8×C47C2×D411yescharacteristic
117C2×C8×C47C2×D411yescharacteristic
118C2×C8×C47C2×D411yescharacteristic
119C2×C8×C47#4111yescharacteristic
120C2×C8×C47#4111yescharacteristic
121C22⋊Dic47—42no—
122C22⋊Dic47—41no—
123C4⋊Dic47—21no—
124C4⋊Dic47—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47—42no—
127C4⋊Dic47—41no—
128C4×Dic47—42no—
129C4×Dic47—41no—
130C47⋊C16—41no—
131C47⋊C16—41no—
132C23×C47—21no—
133C23×C47—41no—
134C23×C47AGL(1,ℤ/8)11yescharacteristic
135Q8×C47—21no—
136Q8×C47—41no—
137Q8×C47—41no—
138Q8×C47#10212yes—
139D4×C47—22no—
140D4×C47—22no—
141D4×C47—21no—
142D4×C47—41no—
143D4×C47—41no—
144D4×C47—81no—
145D4×C47—81no—
146D4×C47#10212yes—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—41no—
151C2×C4×C47—81no—
152C2×C4×C47—81no—
153C2×C4×C47C2×C22⋊C411yescharacteristic
154C2×C4×C47#12011yescharacteristic
155C2×C4×C47#12311yescharacteristic
156C2×C4×C47C2×D811yescharacteristic
157C2×C4×C47C2×SD1611yescharacteristic
158C2×C4×C47#13711yescharacteristic
159C2×Dic47—41no—
160C2×Dic47—42no—
161C2×Dic47—41no—
162C2×Dic47—161no—
163C2×Dic47—162no—
164C8×C47—22no—
165C8×C47—42no—
166C8×C47C4×D411yescharacteristic
167C8×C47C4×D411yescharacteristic
168C8×C47#12111yescharacteristic
169C8×C47#12111yescharacteristic
170#13005—471nomaximal, Sylow
171C22×C47—22no—
172C22×C47—41no—
173C22×C47—42no—
174C22×C47—81no—
175C22×C47—161no—
176C22×C47#41711yescharacteristic
177C22×C47#44711yescharacteristic
178C22×C47#48111yescharacteristic
179C4×C47—22no—
180C4×C47—81no—
181C4×C47#38611yescharacteristic
182C4×C47C2×#10211yescharacteristic
183C4×C47#44811yescharacteristic
184C4×C47#48011yescharacteristic
185Dic47—81no—
186Dic47—82no—
187Dic47—81no—
188#3067D4711yescharacteristic
189#2912—471no—
190#2145—471no—
191C2×#360—471no—
192#2117—471no—
193#1389—471no—
194#1383—471no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—42no—
199C2×C47—161no—
200C2×C47#187411yescharacteristic
201#575—21no—
202C2×#135C2×D4711yescharacteristic
203C2×#131C2×D4711yescharacteristic
204#520—941no—
205C2×M32—471no—
206C2×M32—471no—
207#473Dic4712yescomplemented
208#472Dic4712yescomplemented
209#471—22no—
210C2×#108C2×D4711yescharacteristic
211#428—942no—
212#428—941no—
213#427—471no—
214#427—941no—
215C2×#106—471no—
216C2×#106—941no—
217#413—21no—
218#360—942no—
219#360—941no—
220#360—941no—
221#350—942no—
222C47#1300511yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C2×#41C22⋊D4711yescharacteristic
225#137—22no—
226#137—41no—
227AGL(1,ℤ/8)—22no—
228AGL(1,ℤ/8)—41no—
229#135—21no—
230#135—21no—
231#135—21no—
232#135—21no—
233#135—42no—
234C2×Q16C2×Dic4711yescharacteristic, complemented
235C2×SD16—21no—
236C2×D8C2×Dic4711yescharacteristic, complemented
237#131—21no—
238#131—21no—
239#131C2×Dic4712yes—
240#131C22⋊D4712yes—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16C22⋊D4711yescharacteristic
244C22×C8—21no—
245C22×C8C22×D4711yescharacteristic
246C4×Q8—1881no—
247C4×D4—1882no—
248C4×D4—1881no—
249#117—942no—
250#117—941no—
251C2×C4⋊C4—942no—
252C2×C4⋊C4—941no—
253M32—941no—
254M32—941no—
255M32—941no—
256M32—941no—
257C2×C16—941no—
258C2×C16—941no—
259#108—21no—
260#108C2×Dic4712yescomplemented
261#107—942no—
262#107—941no—
263#106—472no—
264#106—471no—
265#106—471no—
266#106—942no—
267#106—941no—
268#101—22no—
269#100—22no—
270#41—21no—
271#41—21no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41#1062533512yes—
277#41C22⋊Dic4712yes—
278C2×Q8—21no—
279C2×Q8#1062533411yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4#1062533411yescharacteristic
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—1882no—
287C22×C4—1881no—
288C22×C4C2×C22⋊D4711yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—42no—
292SD16—42no—
293SD16—42no—
294SD16—41no—
295SD16—41no—
296D8—21no—
297D8—21no—
298D8—42no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—22no—
303M16—41no—
304M16C22⋊Dic4712yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8D4×D4711yescharacteristic
310C2×C8D4×D4711yescharacteristic
311C2×C8#1062535211yescharacteristic
312C2×C8#1062535211yescharacteristic
313C2×C8C22×Dic4711yescharacteristic
314C2×C8C2×C22⋊D4711yescharacteristic
315C4⋊C4—942no—
316C4⋊C4—941no—
317C4⋊C4—941no—
318C4⋊C4—1882no—
319C4⋊C4—1881no—
320C22⋊C4—1882no—
321C22⋊C4—1881no—
322C42—1882no—
323C42—1881no—
324C16—1881no—
325C16—1881no—
326C23—21no—
327C23—41no—
328C23#1309404911yescharacteristic
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8#1309394512yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4#1309394512yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—1881no—
348C2×C4—1882no—
349C2×C4—1881no—
350C2×C4—7521no—
351C2×C4—7522no—
352C2×C4C2×C22⋊Dic4711yescharacteristic
353C2×C4#1309399211yescharacteristic
354C2×C4#1309400511yescharacteristic
355C2×C4C2×#1062533411yescharacteristic
356C2×C4C2×#1062533511yescharacteristic
357C2×C4#1309405411yescharacteristic
358C8—22no—
359C8—42no—
360C8D4×Dic4711yescharacteristic
361C8D4×Dic4711yescharacteristic
362C8#1309399611yescharacteristic
363C8#1309399611yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—81no—
368C22—161no—
369C22#1568912411yescharacteristic
370C22#1568925611yescharacteristic
371C22#1568943911yescharacteristic
372C4—22no—
373C4—81no—
374C4—3762no—
375C4—3761no—
376C4—3761no—
377C4#1568900311yescharacteristic
378C4C2×#1309394511yescharacteristic
379C4#1568926111yescharacteristic
380C4#1568943411yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—161no—
386C2#2024370611yescharacteristic
subgroupG/H
centerC2#20243706
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC47×#3067C2
socleC2×C47#1874
cosocleC22×C8×C47C23
hypercenter#3067D47
hypocenterC47#13005
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20251983C2semidirect
2. #20245124C2semidirect
3. C2×#15688933C2semidirect
4. #20245016C2semidirect
5. #20241502C2semidirect
6. #20241488C2semidirect
24. C47×#473C4semidirect
25. C47×#472C4semidirect
60. C2×Q16×C47C2×C4semidirect
62. C2×D8×C47C2×C4semidirect
73. #108×C47C2×C4semidirect
207. #473Dic47semidirect
208. #472Dic47semidirect
222. C47#13005semidirect
234. C2×Q16C2×Dic47semidirect
236. C2×D8C2×Dic47semidirect
260. #108C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202519832
#202451242
C2×#156889332
#202450162
#202415022
#202414882
C47×#30672
#1300547