Finite Group Databasebeta

← #26112490

Subgroups — #26112490

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689325C211yesmaximal, characteristic
2C2×#15689202C211yesmaximal, characteristic
3#20243385C211yesmaximal, characteristic
4C2×#15688869C211yesmaximal, characteristic, complemented
5#20241271C211yesmaximal, characteristic, complemented
6#20241223C211yesmaximal, characteristic, complemented
7C47×#1990C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C22×#10625324—21no—
10C2×C4×D188C2211yescharacteristic
11#15689325—22no—
12#15689325C414yescomplemented
13#15689202—22no—
14#15689202—22no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×#13093948—21no—
19C2×#13093944—21no—
20C2×(C2×D47)⋊C8—21no—
21#15688885—21no—
22#15688869—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4×C8×C47C2211yescharacteristic
25C47×#384C2211yescharacteristic
26C47×#340—21no—
27C47×#340—21no—
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C2×#10625336—24no—
31C2×#10625336—42no—
32C2×#10625336—42no—
33C2×#10625324—24no—
34C2×#10625324—42no—
35C2×#10625324—42no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—21no—
38C22×C47⋊C8—21no—
39C4×D188—21no—
40C4×D188—21no—
41C4×D188C2×C412yes—
42C4×D188D412yes—
43C2×C4⋊Dic47D411yescharacteristic
44C2×(C2×D47)⋊C4—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)C2×C412yes—
50C4⋊(C47⋊C8)D412yes—
51#13093948—24no—
52#13093944—42no—
53(C2×D47)⋊C8—24no—
54C22×Q8×C47—21no—
55C22×C8×C47D411yescharacteristic
56C22×C8×C47D411yescharacteristic
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47C2×C412yes—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C42×C47C2311yescharacteristic
67C4×C8×C47—21no—
68C4×C8×C47—21no—
69C23×D47—21no—
70C2×Q8×C47—22no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—21no—
82C2×D188—21no—
83C2×D188—21no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188C22⋊C412yes—
88C2×C4×D47—22no—
89C2×C4×D47—22no—
90C2×C4×D47—41no—
91C2×C4×D47—41no—
92C2×C4×D47—41no—
93C2×C4×D47—41no—
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47SD1612yescomplemented
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—81no—
109C42×C47C22×C411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47#4111yescharacteristic
113#10625336—48no—
114(C2×D47)⋊C4—22no—
115(C2×D47)⋊C4—22no—
116C4⋊Dic47—21no—
117C4⋊Dic47C22⋊C412yes—
118C2×C47⋊C8—22no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127#10625324—48no—
128C23×C47#12111yescharacteristic
129C22×D47—22no—
130C22×D47—41no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—41no—
136Q8×C47—42no—
137Q8×C47—42no—
138Q8×C47—81no—
139Q8×C47—81no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—42no—
143C2×C4×C47—81no—
144C2×C4×C47—81no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47C2×C22⋊C411yescharacteristic
149C2×C4×C47C4×D411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47#12011yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12311yescharacteristic
154C2×C4×C47C2×SD1611yescharacteristic
155C2×C4×C47#13511yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C47#13711yescharacteristic
161C2×C4×C47#13711yescharacteristic
162C2×Dic47—22no—
163C2×Dic47—41no—
164C2×Dic47—41no—
165D188—22no—
166D188—22no—
167D188—41no—
168D188—41no—
169C4×D47—42no—
170C4×D47—42no—
171C4×D47—42no—
172C4×D47—42no—
173C8×C47—22no—
174C8×C47—41no—
175C47⋊C8—42no—
176C47⋊C8—42no—
177C47⋊C8—81no—
178#9225—471nomaximal, Sylow
179C22×C47#38611yescharacteristic
180C22×C47#46111yescharacteristic
181C22×C47#46311yescharacteristic
182C22×C47#46811yescharacteristic
183C22×C47#46911yescharacteristic
184C22×C47#47111yescharacteristic
185C22×C47#47311yescharacteristic
186C2×D47—42no—
187C2×D47—42no—
188C2×D47—42no—
189C2×D47—81no—
190C2×D47—81no—
191C2×D47—81no—
192C2×D47—81no—
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—82no—
196C4×C47—161no—
197C4×C47#41711yescharacteristic
198C4×C47#41811yescharacteristic
199C4×C47C4×SD1611yescharacteristic
200C4×C47#44011yescharacteristic
201C4×C47#45011yescharacteristic
202C4×C47#45211yescharacteristic
203C4×C47#46811yescharacteristic
204C4×C47#46911yescharacteristic
205Dic47—42no—
206C2×#461—471no—
207C2×#436—471no—
208#1990D4711yescharacteristic, complemented
209#1802—471no—
210C2×#332—471no—
211#1296—471no—
212#1277—471no—
213C2×C47#169111yescharacteristic
214C2×C47#169311yescharacteristic
215C2×C47#169911yescharacteristic
216C2×C47#170111yescharacteristic
217C2×C47#187711yescharacteristic
218C2×C47#189511yescharacteristic
219C2×C47#190811yescharacteristic
220D47—82no—
221C22×SD16—941no—
222C22×M16—941no—
223C2×C4⋊Q8C2×D4711yescharacteristic
224C2×C4×D4—471no—
225#461—474no—
226#461—942no—
227#436—942no—
228#436—942no—
229C2×C4⋊C8—471no—
230C2×C4⋊C8—471no—
231C2×C4⋊C8—471no—
232C2×#103—941no—
233C2×#102—941no—
234C2×C22⋊C8—941no—
235C2×C4×C8C2×D4711yescharacteristic
236#384C2×D4711yescharacteristic
237#340—21no—
238#340—21no—
239#340—941no—
240#332—942no—
241C47#922511yescharacteristic, complemented, Sylow
242C22×Q8—21no—
243C22×D4—471no—
244C23×C4—941no—
245C2×SD16—944no—
246C2×SD16—1882no—
247C2×SD16—1882no—
248C2×M16—944no—
249C2×M16—1882no—
250C2×M16—1882no—
251C22×C8—941no—
252C22×C8—941no—
253C22×C8—941no—
254C22×C8C22⋊D4711yescharacteristic
255C22×C8C22⋊D4711yescharacteristic
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C4×D4712yes—
260C4×D4—472no—
261C4×D4—472no—
262C4×D4—941no—
263C4×D4—941no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—471no—
270C2×C22⋊C4—941no—
271C2×C42C22×D4711yescharacteristic
272C4⋊C8—472no—
273C4⋊C8—472no—
274C4⋊C8—941no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278#103—944no—
279#102—1882no—
280C22⋊C8—944no—
281C4×C8—21no—
282C4×C8—21no—
283C24—941no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—472no—
291C2×D4—941no—
292C2×D4—941no—
293C2×D4—941no—
294C2×D4—941no—
295C2×D4—941no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—941no—
299C22×C4—942no—
300C22×C4—942no—
301C22×C4—1881no—
302C22×C4—1881no—
303C22×C4—1881no—
304C22×C4—1881no—
305C22×C4D4×D4711yescharacteristic
306C22×C4#1062535211yescharacteristic
307C22×C4C2×C22⋊D4711yescharacteristic
308SD16—1888no—
309M16—1888no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—942no—
316C2×C8—942no—
317C2×C8—1881no—
318C2×C8—1881no—
319C2×C8—1881no—
320C2×C8—1881no—
321C2×C8—1881no—
322C2×C8—1881no—
323C2×C8—1881no—
324C2×C8#1062533612yescomplemented
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—472no—
335C4⋊C4—941no—
336C22⋊C4—942no—
337C22⋊C4—942no—
338C42C2×C4×D4711yescharacteristic
339C42#1062535011yescharacteristic
340C42D4×D4711yescharacteristic
341C42D4×D4711yescharacteristic
342C23—942no—
343C23—1881no—
344C23—1881no—
345C23—1881no—
346C23—1881no—
347C23—1881no—
348C23—1881no—
349C23#1309399511yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—942no—
355D4—942no—
356D4—1881no—
357D4—1881no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—942no—
367C2×C4—1881no—
368C2×C4—1881no—
369C2×C4—1882no—
370C2×C4—1882no—
371C2×C4—1882no—
372C2×C4—1882no—
373C2×C4C22⋊C4×D4711yescharacteristic
374C2×C4(C22⋊D47)⋊C411yescharacteristic
375C2×C4C4×C22⋊D4711yescharacteristic
376C2×C4#1309399211yescharacteristic
377C2×C4#1309399711yescharacteristic
378C2×C4#1309400611yescharacteristic
379C2×C4C2×#1062533611yescharacteristic
380C2×C4#1309404411yescharacteristic
381C2×C4#1309404811yescharacteristic
382C2×C4#1309404811yescharacteristic
383C2×C4#1309404911yescharacteristic
384C2×C4#1309405111yescharacteristic
385C2×C4#1309405311yescharacteristic
386C2×C4#1309405511yescharacteristic
387C8—22no—
388C8—41no—
389C8—3761no—
390C8—1882no—
391C8—1882no—
392C22—3761no—
393C22—3761no—
394C22—3761no—
395C22—3761no—
396C22—1882no—
397C22—1882no—
398C22—1882no—
399C22#1568899911yescharacteristic
400C22#1568932511yescharacteristic
401C22#1568933511yescharacteristic
402C22#1568936611yescharacteristic
403C22#1568937311yescharacteristic
404C22#1568938311yescharacteristic
405C22#1568939511yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—161no—
410C4—1882no—
411C4#1568912111yescharacteristic
412C4#1568912611yescharacteristic
413C4C4×#1062533611yescharacteristic
414C4#1568922111yescharacteristic
415C4#1568927311yescharacteristic
416C4#1568928311yescharacteristic
417C4#1568936811yescharacteristic
418C4#1568937511yescharacteristic
419C2—3762no—
420C2#2024288411yescharacteristic
421C2#2024289811yescharacteristic
422C2#2024293211yescharacteristic
423C2#2024294611yescharacteristic
424C2#2024372511yescharacteristic
425C2#2024382311yescharacteristic
426C2#2024390611yescharacteristic
subgroupG/H
centerC23#13093995
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1990C2
socleC23×C47#121
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1990D47
hypocenterC47#9225
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#15688869C2semidirect
5. #20241271C2semidirect
6. #20241223C2semidirect
7. C47×#1990C2semidirect
12. #15689325C4semidirect
99. C2×C8×C47SD16semidirect
208. #1990D47semidirect
241. C47#9225semidirect
324. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893252
C2×#156892022
#202433852
C2×#156888692
#202412712
#202412232
C47×#19902
#922547