Finite Group Databasebeta

← #26112550

Subgroups — #26112550

Order 12032.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253419C211yesmaximal, characteristic
2#20244216C211yesmaximal, characteristic
3#20241882C211yesmaximal, characteristic
4#20241817C211yesmaximal, characteristic, complemented
5#20241291C211yesmaximal, characteristic, complemented
6#20241221C211yesmaximal, characteristic, complemented
7C42×Q8×C47C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689441—21no—
10#15689408C412yescomplemented
11#15689323C412yescomplemented
12#15689262—21no—
13C4×#10625336—22no—
14#15689176C2211yescharacteristic
15#15689176C2211yescharacteristic
16#15689147C2211yescharacteristic
17C2×#13093952—21no—
18C2×#13093952—21no—
19#15688856—22no—
20Q8⋊(C47⋊C8)—22no—
21C2×C4×Q8×C47—21no—
22C2×C4×Q8×C47C2211yescharacteristic
23C4×C4⋊C4×C47—22no—
24C4×C4⋊C4×C47—21no—
25C4×C4⋊C4×C47—21no—
26C4×C4⋊C4×C47C2211yescharacteristic
27C43×C47—21no—
28C43×C47C2211yescharacteristic
29C2×#10625350—21no—
30C2×#10625336—22no—
31C2×#10625329—21no—
32#13094024C2×C411yescharacteristic
33#13094021C2×C411yescharacteristic
34#13094012—21no—
35#13093998—21no—
36#13093998—21no—
37C4×D188—21no—
38C4×D188—21no—
39#13093955—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)C2×C412yes—
44#13093952—42no—
45#13093952—42no—
46#13093948—22no—
47#13093944—22no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8—21no—
52C22×Q8×C47D411yescharacteristic
53C4×Q8×C47—28no—
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47C2×C412yes—
57C4×Q8×C47D412yes—
58C2×C4⋊C4×C47—22no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47D411yescharacteristic
62C2×C42×C47—22no—
63C2×C42×C47—21no—
64C2×C42×C47—21no—
65C2×C42×C47—22no—
66C2×C42×C47—21no—
67C2×C42×C47—21no—
68C2×C42×C47D411yescharacteristic
69C2×C42×C47D411yescharacteristic
70C2×C42×C47C2311yescharacteristic
71#95×C47—22no—
72#95×C47D412yes—
73C2×Q8×C47—24no—
74C2×Q8×C47—21no—
75C2×Q8×C47C22⋊C412yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C2×C22⋊D47—41no—
85#10625350—41no—
86#10625350—41no—
87#10625350—41no—
88#10625350—41no—
89C2×D188—21no—
90C2×D188—21no—
91C2×D188—21no—
92C2×C4×D47—41no—
93C2×Dic94—21no—
94C4⋊C4×C47—28no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47C22⋊C412yes—
100C42×C47—28no—
101C42×C47—24no—
102C42×C47—22no—
103C42×C47—22no—
104C42×C47—21no—
105C42×C47—21no—
106C42×C47—21no—
107C42×C47—42no—
108C42×C47—44no—
109C42×C47—41no—
110C42×C47—41no—
111C42×C47SD1612yescomplemented
112C42×C47C22×C411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47#4111yescharacteristic
115C42×C47#4111yescharacteristic
116#10625336—44no—
117(C2×D47)⋊C4—41no—
118(C2×D47)⋊C4—41no—
119C4⋊Dic47—21no—
120C4⋊Dic47—21no—
121#10625329—41no—
122#10625329—41no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C23×C47#12011yescharacteristic
127C22×D47—41no—
128Q8×C47—24no—
129Q8×C47—22no—
130Q8×C47#10414yescomplemented
131C2×C4×C47—22no—
132C2×C4×C47—24no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—24no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—42no—
146C2×C4×C47C2×C22⋊C411yescharacteristic
147C2×C4×C47C4×D411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47#12311yescharacteristic
152C2×C4×C47C2×SD1611yescharacteristic
153C2×C4×C47#13511yescharacteristic
154C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C47#13711yescharacteristic
156C22⋊D47—81no—
157C22⋊D47—81no—
158C2×Dic47—41no—
159D188—42no—
160D188—41no—
161D188—41no—
162D188—81no—
163C4×D47—81no—
164C4×D47—81no—
165Dic94—41no—
166Dic94—41no—
167C47⋊C8—42no—
168C47⋊C8—81no—
169C47⋊C8—81no—
170#9234—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47#38611yescharacteristic
174C22×C47#44811yescharacteristic
175C22×C47#44911yescharacteristic
176C2×D47—81no—
177C2×D47—81no—
178C2×D47—81no—
179C4×C47—24no—
180C4×C47—24no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—42no—
185C4×C47—44no—
186C4×C47—42no—
187C4×C47C2×#10412yes—
188C4×C47C4×SD1611yescharacteristic
189C4×C47#44011yescharacteristic
190C4×C47#46011yescharacteristic
191C4×C47#46411yescharacteristic
192Dic47—81no—
193#3103—471no—
194C42×Q8D4711yescharacteristic, complemented
195#1962—471no—
196#1502—471no—
197#1487—471no—
198#1303—471no—
199#1276—471no—
200C2×C47—21no—
201C2×C47—21no—
202C2×C47#188011yescharacteristic
203C2×C47#189611yescharacteristic
204C2×C47#190611yescharacteristic
205D47—161no—
206#532—471no—
207C2×C4×Q8—21no—
208C2×C4×Q8C2×D4711yescharacteristic
209#481—941no—
210#475—472no—
211#460—472no—
212#448—941no—
213C4×SD16—942no—
214#431—471no—
215#431—471no—
216#423—471no—
217C2×#104—941no—
218C2×#104—941no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2×D4711yescharacteristic
223C43—21no—
224C43C2×D4711yescharacteristic
225#327—942no—
226Q8⋊C8—942no—
227C47#923411yescharacteristic, complemented, Sylow
228C2×#41—941no—
229C22×Q8C22⋊D4711yescharacteristic
230C2×SD16—942no—
231C2×M16—941no—
232C4⋊Q8—471no—
233#127—471no—
234#124—941no—
235#121—941no—
236#121—941no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D4712yes—
241C4×Q8D18812yes—
242C4×D4—941no—
243C4×D4—941no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D4711yescharacteristic
255C2×C42C22⋊D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257#106—942no—
258C4⋊C8—472no—
259C4⋊C8—941no—
260C4⋊C8—941no—
261C4⋊C8—941no—
262#104—1882no—
263#104—1882no—
264#103—942no—
265#102—942no—
266C22⋊C8—941no—
267C22⋊C8—941no—
268C4×C8—941no—
269C4×C8—941no—
270#95—22no—
271#95C22⋊D4712yes—
272#41—1881no—
273#41—1881no—
274#41—1881no—
275#41—1881no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—941no—
279C2×Q8(C2×D47)⋊C412yes—
280C2×D4—941no—
281C2×D4—941no—
282C2×D4—941no—
283C2×D4—1881no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—1881no—
290C22×C4C2×C22⋊D4711yescharacteristic
291C22×C4C2×C22⋊D4711yescharacteristic
292C22×C4C2×C22⋊D4711yescharacteristic
293SD16—1884no—
294M16—1881no—
295M16—1881no—
296C2×C8—942no—
297C2×C8—1881no—
298C2×C8—1881no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—941no—
305C4⋊C4—941no—
306C4⋊C4(C2×D47)⋊C412yes—
307C22⋊C4—1881no—
308C22⋊C4—1881no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1062533612yescomplemented
321C42C2×C4×D4711yescharacteristic
322C42C2×D18811yescharacteristic
323C42#1062535011yescharacteristic
324C42#1062535011yescharacteristic
325C23—1881no—
326C23#1309399311yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—1881no—
330Q8—1881no—
331Q8#1309395214yescomplemented
332D4—3761no—
333D4—3761no—
334D4—3761no—
335D4—1881no—
336D4—1882no—
337D4—1881no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—3761no—
354C2×C4—3761no—
355C2×C4—1881no—
356C2×C4C2×(C2×D47)⋊C411yescharacteristic
357C2×C4C4×C22⋊D4711yescharacteristic
358C2×C4C4×C22⋊D4711yescharacteristic
359C2×C4#1309399811yescharacteristic
360C2×C4#1309399811yescharacteristic
361C2×C4#1309400711yescharacteristic
362C2×C4C2×#1062533611yescharacteristic
363C2×C4#1309404411yescharacteristic
364C2×C4#1309404911yescharacteristic
365C2×C4#1309405511yescharacteristic
366C8—3761no—
367C8—3761no—
368C8—1882no—
369C22—21no—
370C22—21no—
371C22—3761no—
372C22—3761no—
373C22—3761no—
374C22#1568900211yescharacteristic
375C22#1568926211yescharacteristic
376C22#1568926811yescharacteristic
377C4—24no—
378C4—24no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—3761no—
386C4C2×#1309395212yes—
387C4C4×#1062533611yescharacteristic
388C4#1568922111yescharacteristic
389C4#1568932311yescharacteristic
390C4#1568934511yescharacteristic
391C2—21no—
392C2—21no—
393C2—7521no—
394C2#2024374311yescharacteristic
395C2#2024383111yescharacteristic
396C2#2024388911yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC42×Q8×C47C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenterC42×Q8D47
hypocenterC47#9234
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #20241817C2semidirect
5. #20241291C2semidirect
6. #20241221C2semidirect
7. C42×Q8×C47C2semidirect
10. #15689408C4semidirect
11. #15689323C4semidirect
111. C42×C47SD16semidirect
130. Q8×C47#104semidirect
194. C42×Q8D47semidirect
227. C47#9234semidirect
320. C42#10625336semidirect
331. Q8#13093952semidirect

Maximal subgroups

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

subgroupindex
#202534192
#202442162
#202418822
#202418172
#202412912
#202412212
C42×Q8×C472
#923447