Finite Group Databasebeta

← #26220646

Subgroups — #26220646

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689234C211yesmaximal, characteristic
2#20246027C212yesmaximal, complemented
3#20244264C211yesmaximal, characteristic
4#20244252C211yesmaximal, characteristic
5#20244117C211yesmaximal, characteristic, complemented
6#20244086C212yesmaximal, complemented
7#20244070C212yesmaximal, complemented
8#20243287C211yesmaximal, characteristic, complemented
9#20243202C212yesmaximal, complemented
10#20243076C211yesmaximal, characteristic, complemented
11C47×#1944C211yesmaximal, characteristic, complemented
12C22×D376C2211yescharacteristic
13C2×#13093986C2212yes—
14#15689234C418yescomplemented
15C2×#13093956C2211yescharacteristic
16C2×#13093946C2212yes—
17C2×#13093944C2212yes—
18C2×#13093944C2212yes—
19C2×#13093934C2211yescharacteristic
20C2×#13093934C2211yescharacteristic
21C2×C4×C47⋊C8C2211yescharacteristic
22C2×C8×Dic47C2211yescharacteristic
23C2×C8×Dic47C2211yescharacteristic
24#15689025C2212yes—
25#15689024C2212yes—
26#15689022C2212yes—
27#15689015—22no—
28#15689013—22no—
29#15688985—22no—
30#15688980—22no—
31C4⋊C4×Dic47C2212yes—
32#15688969C2211yescharacteristic
33#15688885C2212yes—
34#15688885C2212yes—
35C2×#107×C47C2211yescharacteristic
36C2×#106×C47C2211yescharacteristic
37C2×#97×C47C2211yescharacteristic
38C47×#389C2212yes—
39C47×#340C2212yes—
40C22×D188C2312yes—
41C22×C4×D47—22no—
42C2×D376—22no—
43C2×D376—22no—
44C2×D376C2×C414yes—
45C22×C47⋊C8C2311yescharacteristic
46C22×C47⋊C8C2311yescharacteristic
47#13093986—24no—
48#13093986C2×C418yes—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47C2312yes—
51C2×(C2×D47)⋊C4—22no—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—22no—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56C2×C4×Dic47—22no—
57C2×C4×Dic47C2311yescharacteristic
58C2×C4×Dic47C2311yescharacteristic
59#13093956—22no—
60#13093946—24no—
61#13093944—24no—
62#13093944C2×C418yes—
63#13093934—22no—
64#13093934C2×C414yes—
65C4×C47⋊C8—22no—
66C8×Dic47—22no—
67C8×Dic47—22no—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C47⋊#95D412yes—
71C47⋊#95D412yes—
72C22×C8×C47C2311yescharacteristic
73C22×C8×C47C2311yescharacteristic
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47C2312yes—
76C2×C4⋊C4×C47C2312yes—
77C2×C42×C47C2311yescharacteristic
78#107×C47—22no—
79#106×C47C2×C414yes—
80#97×C47—22no—
81#95×C47—22no—
82C23×D47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—22no—
89C22×Dic47C2×D412yes—
90C22×Dic47C2×D412yes—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—22no—
95C2×D188—22no—
96C2×D188C22×C414yes—
97C2×C4×D47—28no—
98C2×C4×D47—82no—
99C2×C4×D47—82no—
100C2×C8×C47—21no—
101C2×C8×C47—21no—
102C2×C8×C47C22×C412yes—
103C2×C8×C47#4112yes—
104C2×C8×C47#4112yes—
105C2×C8×C47#4112yes—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—82no—
110C4⋊C4×C47C22×C414yes—
111C42×C47—21no—
112C42×C47—21no—
113(C2×D47)⋊C4—28no—
114(C2×D47)⋊C4—82no—
115(C2×D47)⋊C4—82no—
116(C2×D47)⋊C4—82no—
117(C2×D47)⋊C4—82no—
118C4⋊Dic47—22no—
119C4⋊Dic47—22no—
120C4⋊Dic47—82no—
121C4×Dic47—21no—
122C4×Dic47—21no—
123C4×Dic47—82no—
124C4×Dic47C22×C412yes—
125C4×Dic47#4112yes—
126C2×C47⋊C8—21no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8C22×C412yes—
131C2×C47⋊C8#4112yes—
132D376—28no—
133C23×C47C22×D411yescharacteristic
134C22×D47—24no—
135C22×D47—42no—
136C22×D47—42no—
137C22×D47—42no—
138C22×D47—42no—
139C22×D47—42no—
140C22×D47—42no—
141C2×C4×C47—24no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47C23×C411yescharacteristic
152C2×C4×C47C2×#4111yescharacteristic
153C2×C4×C47C2×#4111yescharacteristic
154C2×C4×C47C2×#4111yescharacteristic
155C2×C4×C47C2×#4111yescharacteristic
156C2×C4×C47C2×#4111yescharacteristic
157C2×Dic47—22no—
158C2×Dic47—22no—
159C2×Dic47—22no—
160C2×Dic47—22no—
161C2×Dic47—22no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47C4×D414yes—
166D188—24no—
167D188—24no—
168D188—42no—
169D188—42no—
170C4×D47—88no—
171C8×C47—22no—
172C8×C47#11714yes—
173C47⋊C8—22no—
174C47⋊C8—22no—
175#21290—471nomaximal, Sylow
176C22×C47C2×C4×D411yescharacteristic
177C22×C47#53211yescharacteristic
178C22×C47#53211yescharacteristic
179C22×C47#57612yes—
180C22×C47#57712yes—
181C2×D47—44no—
182C2×D47—44no—
183C2×D47—44no—
184C2×D47—82no—
185C2×D47—82no—
186C2×D47—82no—
187C2×D47—82no—
188C4×C47—41no—
189C4×C47—41no—
190C4×C47—44no—
191C4×C47—82no—
192C4×C47C2×#11711yescharacteristic
193C4×C47C4×#4111yescharacteristic
194C4×C47#52911yescharacteristic
195C4×C47#52911yescharacteristic
196Dic47—24no—
197Dic47—42no—
198Dic47—82no—
199C2×#442—471no—
200#2319—472no—
201#1972—471no—
202#1970—471no—
203#1944D4711yescharacteristic, complemented
204#1941—471no—
205#1936—472no—
206#1934—472no—
207#1776—471no—
208C4×#102—472no—
209#1726—471no—
210C2×C47#229311yescharacteristic
211C2×C47#294612yes—
212C2×C47#317311yescharacteristic
213C2×C47#317611yescharacteristic
214C2×C47#317912yes—
215D47—84no—
216C22×D8—471no—
217C2×C4×D4—472no—
218#442—478no—
219C2×#107C2×D4711yescharacteristic
220C2×#106—471no—
221C2×#106C2×D4711yescharacteristic
222C2×#102—472no—
223C2×#102—472no—
224C2×#102—472no—
225C2×#97—471no—
226C2×#97—471no—
227C2×#97C2×D4711yescharacteristic
228C2×C4×C8—471no—
229C2×C4×C8—471no—
230C2×C4×C8—471no—
231#390—472no—
232#390—472no—
233#389—472no—
234#389C2×D4712yes—
235#388—942no—
236#388—942no—
237#381—942no—
238#380—942no—
239C4×C4⋊C4—472no—
240#376—471no—
241#340—472no—
242#340—472no—
243#340C2×D4712yes—
244C47#2129011yescharacteristic, complemented, Sylow
245C22×D4—472no—
246C23×C4—942no—
247C2×D8—474no—
248C2×D8—942no—
249C2×D8—942no—
250C22×C8—471no—
251C22×C8—471no—
252C22×C8C22×D4711yescharacteristic
253C22×C8C22×D4711yescharacteristic
254C4×D4—478no—
255C4×D4—944no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—472no—
258C2×C4⋊C4—942no—
259C2×C4⋊C4C22×D4712yes—
260C2×C4⋊C4C22×D4712yes—
261C2×C22⋊C4—942no—
262C2×C22⋊C4—942no—
263C2×C22⋊C4—942no—
264C2×C22⋊C4—942no—
265C2×C22⋊C4—942no—
266C2×C42—471no—
267C2×C42—471no—
268C2×C42—942no—
269C2×C42C22×D4711yescharacteristic
270#107—22no—
271#106—942no—
272#106C4×D4714yes—
273#102—478no—
274#102—944no—
275#102—944no—
276#97—22no—
277#97—474no—
278#97—942no—
279C4×C8—942no—
280C4×C8—942no—
281C4×C8—942no—
282#95—22no—
283#95—472no—
284#95—472no—
285#95—942no—
286#95—942no—
287C24—942no—
288C2×D4—474no—
289C2×D4—942no—
290C2×D4—942no—
291C2×D4—942no—
292C2×D4—942no—
293C2×D4—942no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—472no—
297C22×C4—472no—
298C22×C4—948no—
299C22×C4—942no—
300C22×C4—3762no—
301C22×C4—3762no—
302C22×C4D4×D4711yescharacteristic
303C22×C4D4×D4711yescharacteristic
304C22×C4C23×D4711yescharacteristic
305D8—948no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—472no—
309C2×C8—472no—
310C2×C8—941no—
311C2×C8—941no—
312C2×C8—941no—
313C2×C8—941no—
314C2×C8C2×C4×D4712yes—
315C2×C8#1062535212yes—
316C2×C8#1062535412yes—
317C2×C8#1062535412yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—82no—
322C4⋊C4—942no—
323C4⋊C4—942no—
324C4⋊C4—3762no—
325C4⋊C4C2×C4×D4714yes—
326C22⋊C4—948no—
327C22⋊C4—3762no—
328C22⋊C4—3762no—
329C22⋊C4—3762no—
330C22⋊C4—3762no—
331C42—21no—
332C42—21no—
333C42—472no—
334C42—472no—
335C42—941no—
336C42—941no—
337C42—3762no—
338C23—944no—
339C23—1882no—
340C23—1882no—
341C23—1882no—
342C23—1882no—
343C23—1882no—
344C23—1882no—
345C23C2×D4×D4711yescharacteristic
346D4—944no—
347D4—944no—
348D4—1882no—
349D4—1882no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—474no—
361C2×C4—942no—
362C2×C4—942no—
363C2×C4—942no—
364C2×C4—942no—
365C2×C4—942no—
366C2×C4—3768no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4—1882no—
370C2×C4C22×C4×D4711yescharacteristic
371C2×C4#41×D4711yescharacteristic
372C2×C4#41×D4711yescharacteristic
373C2×C4C2×#1062535211yescharacteristic
374C2×C4C2×#1062535411yescharacteristic
375C2×C4C2×#1062535411yescharacteristic
376C8—22no—
377C8—942no—
378C8—942no—
379C8#1309398214yes—
380C22—3762no—
381C22—3762no—
382C22—3762no—
383C22—3762no—
384C22—1884no—
385C22—1884no—
386C22—1884no—
387C22C4×D4×D4711yescharacteristic
388C22#1568966811yescharacteristic
389C22#1568967411yescharacteristic
390C22#1568992712yes—
391C22#1568993212yes—
392C4—41no—
393C4—44no—
394C4—41no—
395C4—82no—
396C4—944no—
397C4—3762no—
398C4—1882no—
399C4C2×#1309398211yescharacteristic
400C4C4×#1062535411yescharacteristic
401C4#1568965811yescharacteristic
402C4#1568965911yescharacteristic
403C2—3764no—
404C2#2024584611yescharacteristic
405C2#2025221912yes—
406C2#2025400611yescharacteristic
407C2#2025403311yescharacteristic
408C2#2025407612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1944C2
socleC23×C47C22×D4
prime radicalC22×D376C22
cosocleC22×C4×C47C24
hypercenter#1944D47
hypocenterC47#21290
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
2. #20246027C2semidirect
5. #20244117C2semidirect
6. #20244086C2semidirect
7. #20244070C2semidirect
8. #20243287C2semidirect
9. #20243202C2semidirect
10. #20243076C2semidirect
11. C47×#1944C2semidirect
14. #15689234C4semidirect
203. #1944D47semidirect
244. C47#21290semidirect

Maximal subgroups

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

subgroupindex
C2×#156892342
#202460272
#202442642
#202442522
#202441172
#202440862
#202440702
#202432872
#202432022
#202430762
C47×#19442
#2129047