Finite Group Databasebeta

← #26111144

Subgroups — #26111144

Order 12032.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253419C211yesmaximal, characteristic
2#20244307C211yesmaximal, characteristic
3C4×#13093948C211yesmaximal, characteristic, complemented
4#20241838C211yesmaximal, characteristic
5#20241289C211yesmaximal, characteristic, complemented
6#20241226C211yesmaximal, characteristic, complemented
7Q8×M16×C47C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689441—21no—
10#15689408C411yescharacteristic, complemented
11#15689408C411yescharacteristic, complemented
12#15689323C411yescharacteristic, complemented
13#15689323C411yescharacteristic, complemented
14#15689262—21no—
15C4×#10625336—21no—
16C4×#10625336—21no—
17#15689176C2211yescharacteristic
18C2×#13093950—21no—
19C2×#13093950—21no—
20C2×#13093948—21no—
21C2×C4×C47⋊C8C2211yescharacteristic
22C4×C4⋊Dic47C2211yescharacteristic
23#15688885—21no—
24#15688850—22no—
25C2×C4×Q8×C47C2211yescharacteristic
26C47×#446—21no—
27C47×#446—22no—
28C8×Q8×C47—21no—
29C47×#423—21no—
30C47×#423C2211yescharacteristic
31C4×M16×C47—21no—
32C4×M16×C47C2211yescharacteristic
33C2×#10625350—21no—
34C2×#10625336—21no—
35C2×#10625336—21no—
36C22×C47⋊C8—21no—
37#13094024C2×C411yescharacteristic
38#13094021C2×C411yescharacteristic
39#13094012—21no—
40#13093998—21no—
41#13093998—21no—
42C4×D188—21no—
43C4×D188—21no—
44C2×C4⋊Dic47—21no—
45C2×C4×Dic47—21no—
46#13093955—21no—
47#13093955—21no—
48C4⋊(C47⋊C8)C2×C411yescharacteristic
49C4⋊(C47⋊C8)C2×C411yescharacteristic
50#13093950—42no—
51#13093950—42no—
52#13093948—24no—
53#13093948—21no—
54#13093948—21no—
55#13093944—21no—
56#13093944—21no—
57C22⋊(C47⋊C8)—21no—
58C4×C47⋊C8—21no—
59C4×C47⋊C8—21no—
60C4×C47⋊C8—21no—
61C47⋊#95—21no—
62C22×Q8×C47D411yescharacteristic
63C2×M16×C47—21no—
64C2×M16×C47D411yescharacteristic
65C2×M16×C47D411yescharacteristic
66C4×Q8×C47—21no—
67C4×Q8×C47—21no—
68C4×Q8×C47C2×C411yescharacteristic
69C4×Q8×C47C2×C411yescharacteristic
70C4×Q8×C47D411yescharacteristic
71C4×Q8×C47D411yescharacteristic
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47D411yescharacteristic
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C4⋊C8×C47—22no—
77C4⋊C8×C47—22no—
78C4⋊C8×C47—21no—
79C4⋊C8×C47D412yes—
80#97×C47—22no—
81#97×C47—21no—
82C4×C8×C47—22no—
83C4×C8×C47—21no—
84C2×Q8×C47—21no—
85C2×Q8×C47—22no—
86C2×Q8×C47—41no—
87C2×Q8×C47C22⋊C411yescharacteristic
88C2×Q8×C47C22⋊C411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C2×C22⋊D47—41no—
95C22×Dic47—41no—
96#10625350—41no—
97#10625350—41no—
98#10625350—41no—
99#10625350—41no—
100C2×D188—21no—
101C2×D188—21no—
102C2×D188—21no—
103C2×C4×D47—41no—
104C2×Dic94—21no—
105M16×C47—21no—
106M16×C47—21no—
107M16×C47—21no—
108M16×C47—41no—
109M16×C47—41no—
110M16×C47SD1612yescomplemented
111C2×C8×C47—21no—
112C2×C8×C47—21no—
113C2×C8×C47—22no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—21no—
116C4⋊C4×C47—21no—
117C4⋊C4×C47—21no—
118C4⋊C4×C47—21no—
119C4⋊C4×C47C22⋊C411yescharacteristic
120C4⋊C4×C47C22⋊C411yescharacteristic
121C42×C47—21no—
122C42×C47—21no—
123C42×C47—21no—
124C42×C47—21no—
125C42×C47C22×C411yescharacteristic
126C42×C47C2×D411yescharacteristic
127C42×C47C2×D411yescharacteristic
128C42×C47#4111yescharacteristic
129#10625336—42no—
130#10625336—42no—
131(C2×D47)⋊C4—41no—
132(C2×D47)⋊C4—41no—
133C4⋊Dic47—22no—
134C4⋊Dic47—21no—
135C4⋊Dic47—21no—
136C4⋊Dic47—41no—
137C4×Dic47—41no—
138C4×Dic47—41no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—21no—
141C2×C47⋊C8—21no—
142C2×C47⋊C8—41no—
143C2×C47⋊C8—41no—
144C23×C47#12111yescharacteristic
145C22×D47—41no—
146Q8×C47—21no—
147Q8×C47—21no—
148Q8×C47—21no—
149Q8×C47—42no—
150Q8×C47#10412yescomplemented
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47—81no—
160C2×C4×C47C2×C22⋊C411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47C4×D411yescharacteristic
163C2×C4×C47#12011yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47#12311yescharacteristic
166C2×C4×C47C2×SD1611yescharacteristic
167C2×C4×C47#13511yescharacteristic
168C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C47#13711yescharacteristic
170C22⋊D47—81no—
171C22⋊D47—81no—
172C2×Dic47—42no—
173C2×Dic47—41no—
174C2×Dic47—81no—
175C2×Dic47—81no—
176D188—42no—
177D188—41no—
178D188—41no—
179D188—41no—
180D188—41no—
181C4×D47—81no—
182C4×D47—81no—
183Dic94—41no—
184Dic94—41no—
185C8×C47—22no—
186C8×C47—41no—
187C8×C47—42no—
188C47⋊C8—42no—
189C47⋊C8—41no—
190C47⋊C8—41no—
191#8992—471nomaximal, Sylow
192C22×C47—21no—
193C22×C47—21no—
194C22×C47#38611yescharacteristic
195C22×C47#46011yescharacteristic
196C22×C47#46411yescharacteristic
197C2×D47—81no—
198C2×D47—81no—
199C2×D47—81no—
200C4×C47—21no—
201C4×C47—22no—
202C4×C47—21no—
203C4×C47—21no—
204C4×C47—41no—
205C4×C47—41no—
206C4×C47—41no—
207C4×C47C2×#10411yescharacteristic
208C4×C47#42111yescharacteristic
209C4×C47C4×SD1611yescharacteristic
210C4×C47#44011yescharacteristic
211C4×C47#44811yescharacteristic
212C4×C47#44911yescharacteristic
213Dic47—82no—
214Dic47—81no—
215#3103—471no—
216Q8×M16D4711yescharacteristic, complemented
217#1980—471no—
218C4×#103—471no—
219#1492—471no—
220#1303—471no—
221#1278—471no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47#187111yescharacteristic
225C2×C47#188011yescharacteristic
226C2×C47#188111yescharacteristic
227D47—161no—
228#532—471no—
229C2×C4×Q8C2×D4711yescharacteristic
230#481—941no—
231#475—471no—
232#475—471no—
233#460—471no—
234#460—471no—
235#448—941no—
236#446—21no—
237#446—22no—
238C8×Q8—21no—
239C4×SD16—941no—
240C4×SD16—941no—
241#431—471no—
242#423—21no—
243#423C2×D4711yescharacteristic
244C2×#104—941no—
245C2×#104—941no—
246C2×#103—941no—
247C4×M16—21no—
248C4×M16C2×D4711yescharacteristic
249C2×C4×C8—471no—
250C4×C4⋊C4—471no—
251#340—941no—
252D4⋊C8—942no—
253C47#899211yescharacteristic, complemented, Sylow
254C2×#41—941no—
255C22×Q8D18811yescharacteristic
256C2×SD16—941no—
257C2×SD16—941no—
258C2×M16—21no—
259C2×M16C22⋊D4711yescharacteristic
260C2×M16C22⋊D4711yescharacteristic
261C22×C8—941no—
262C4⋊Q8—471no—
263#127—471no—
264#124—941no—
265#121—941no—
266#121—941no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D4711yescharacteristic
270C4×Q8C4×D4711yescharacteristic
271C4×Q8C22⋊D4711yescharacteristic
272C4×Q8C22⋊D4711yescharacteristic
273C4×D4—941no—
274C4×D4—941no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4D18811yescharacteristic
278C2×C42—21no—
279C2×C42—941no—
280C2×C42C22×D4711yescharacteristic
281#106—941no—
282#106—941no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—471no—
287C4⋊C8—471no—
288C4⋊C8C22⋊D4712yes—
289#104—1882no—
290#104—1882no—
291#103—944no—
292#103—941no—
293#103—941no—
294#102—941no—
295#102—941no—
296C22⋊C8—941no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—941no—
302C4×C8—941no—
303C4×C8—941no—
304#95—941no—
305#41—1881no—
306#41—1881no—
307#41—1881no—
308#41—1881no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—941no—
313C2×Q8(C2×D47)⋊C411yescharacteristic
314C2×Q8(C2×D47)⋊C411yescharacteristic
315C2×D4—941no—
316C2×D4—941no—
317C2×D4—941no—
318C2×D4—1881no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—1881no—
322C22×C4—1881no—
323C22×C4C2×D18811yescharacteristic
324C22×C4#1062535011yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326SD16—1882no—
327SD16—1882no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16#1062533612yescomplemented
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—942no—
338C2×C8—941no—
339C2×C8—941no—
340C2×C8—1881no—
341C2×C8—1881no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—942no—
348C4⋊C4—941no—
349C4⋊C4—941no—
350C4⋊C4—1881no—
351C4⋊C4(C2×D47)⋊C411yescharacteristic
352C4⋊C4(C2×D47)⋊C411yescharacteristic
353C22⋊C4—1881no—
354C22⋊C4—1881no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—1881no—
360C42—1881no—
361C42C2×C4×D4711yescharacteristic
362C42#1062535011yescharacteristic
363C42C2×C22⋊D4711yescharacteristic
364C42C2×C22⋊D4711yescharacteristic
365C23—1881no—
366C23#1309399811yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—1881no—
372Q8—1881no—
373Q8#1309395012yescomplemented
374D4—3761no—
375D4—3761no—
376D4—1882no—
377D4—1881no—
378D4—1881no—
379D4—1881no—
380D4—1881no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—3761no—
391C2×C4—3761no—
392C2×C4—3761no—
393C2×C4—3761no—
394C2×C4—1882no—
395C2×C4—1881no—
396C2×C4C2×(C2×D47)⋊C411yescharacteristic
397C2×C4C4×C22⋊D4711yescharacteristic
398C2×C4C4×C22⋊D4711yescharacteristic
399C2×C4#1309399311yescharacteristic
400C2×C4#1309399811yescharacteristic
401C2×C4#1309400711yescharacteristic
402C2×C4C2×#1062533611yescharacteristic
403C2×C4#1309404411yescharacteristic
404C2×C4#1309404911yescharacteristic
405C2×C4#1309405511yescharacteristic
406C8—22no—
407C8—41no—
408C8—42no—
409C8—1882no—
410C8—1881no—
411C8—1881no—
412C22—21no—
413C22—21no—
414C22—3761no—
415C22—3761no—
416C22—3761no—
417C22#1568900211yescharacteristic
418C22#1568932311yescharacteristic
419C22#1568934511yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—3762no—
428C4—3761no—
429C4C2×#1309395011yescharacteristic
430C4#1568913911yescharacteristic
431C4C4×#1062533611yescharacteristic
432C4#1568922111yescharacteristic
433C4#1568926211yescharacteristic
434C4#1568926811yescharacteristic
435C2—21no—
436C2—21no—
437C2—7521no—
438C2#2024369011yescharacteristic
439C2#2024374311yescharacteristic
440C2#2024375011yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingQ8×M16×C47C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenterQ8×M16D47
hypocenterC47#8992
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C4×#13093948C2semidirect
5. #20241289C2semidirect
6. #20241226C2semidirect
7. Q8×M16×C47C2semidirect
10. #15689408C4semidirect
11. #15689408C4semidirect
12. #15689323C4semidirect
13. #15689323C4semidirect
110. M16×C47SD16semidirect
150. Q8×C47#104semidirect
216. Q8×M16D47semidirect
253. C47#8992semidirect
333. M16#10625336semidirect
373. Q8#13093950semidirect

Maximal subgroups

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

subgroupindex
#202534192
#202443072
C4×#130939482
#202418382
#202412892
#202412262
Q8×M16×C472
#899247