Finite Group Databasebeta

← #26131222

Subgroups — #26131222

Order 12032.

838 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689097C211yesmaximal, characteristic
2#20245879C211yesmaximal, characteristic
3#20241507C214yesmaximal, complemented
4C2×C47×#413C211yesmaximal, characteristic
5C2×#13093983—21no—
6C2×#13093983C411yescharacteristic, complemented
7C22×C4×Dic47C411yescharacteristic, complemented
8#15689097—22no—
9#15689097C414yescomplemented
10C2×#13093939—21no—
11C2×#13093939C2212yes—
12#15689030—21no—
13#15689026—21no—
14#15689003C412yescomplemented
15#15689003C2212yes—
16C4×C22⋊Dic47—21no—
17C4×C22⋊Dic47—21no—
18C22×#41×C47C2211yescharacteristic
19C22×M16×C47—21no—
20C47×#413—22no—
21C47×#413C414yescomplemented
22C2×#101×C47—21no—
23C2×#100×C47C2212yes—
24C23×Dic47C2×C412yes—
25#13093983—24no—
26#13093983—42no—
27#13093983—42no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—41no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47C2×C412yes—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—41no—
38C2×C4×Dic47—41no—
39C2×C4×Dic47—41no—
40#13093939—24no—
41#13093939—22no—
42#13093939C2×C414yes—
43C47⋊#95—22no—
44C47⋊#95—22no—
45C2×#41×C47—41no—
46C2×#41×C47—41no—
47C2×#41×C47—42no—
48C2×#41×C47C2×C412yes—
49C2×#41×C47C2×C412yes—
50C2×#41×C47D412yes—
51C2×#41×C47Q812yes—
52C22×Q8×C47D411yescharacteristic
53C22×D4×C47C2×C411yescharacteristic
54C22×D4×C47D411yescharacteristic
55C22×D4×C47C2311yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D411yescharacteristic
59C2×M16×C47—24no—
60C2×M16×C47—22no—
61C2×M16×C47—41no—
62C2×M16×C47—42no—
63C22×C8×C47—41no—
64#101×C47—24no—
65#100×C47—22no—
66#100×C47C2×C414yes—
67C24×C47C22⋊C411yescharacteristic
68C24×C47C22×C411yescharacteristic
69C24×C47C2×D411yescharacteristic
70#41×C47—42no—
71#41×C47—42no—
72#41×C47—42no—
73#41×C47—42no—
74#41×C47—42no—
75#41×C47—42no—
76#41×C47—42no—
77#41×C47—42no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47C22⋊C412yes—
81C2×Q8×C47C4⋊C412yes—
82C2×D4×C47—41no—
83C2×D4×C47—41no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47C4212yes—
89C2×D4×C47C22⋊C412yes—
90C2×D4×C47C4⋊C412yes—
91C2×D4×C47C4⋊C412yes—
92C2×D4×C47C22×C411yescharacteristic
93C2×D4×C47C22×C411yescharacteristic
94C2×D4×C47C2×D411yescharacteristic
95C2×D4×C47C2×Q811yescharacteristic
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47C4212yes—
106C22×C4×C47C22⋊C411yescharacteristic
107C22×C4×C47C22⋊C411yescharacteristic
108C22×C4×C47C22⋊C412yes—
109C22×C4×C47C22⋊C412yes—
110C22×C4×C47C22⋊C411yescharacteristic
111C22×C4×C47C4⋊C412yes—
112C22×C4×C47C4⋊C412yes—
113C22×C4×C47C4⋊C412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—44no—
118C22×Dic47—42no—
119C22×Dic47—42no—
120M16×C47—24no—
121M16×C47—42no—
122M16×C47—44no—
123C2×C8×C47—42no—
124C2×C8×C47—44no—
125C22⋊Dic47—24no—
126C22⋊Dic47—44no—
127C22⋊Dic47—44no—
128C22⋊Dic47—42no—
129C22⋊Dic47—42no—
130C4⋊Dic47—42no—
131C4⋊Dic47—44no—
132C4×Dic47—42no—
133C4×Dic47—44no—
134C4×Dic47—41no—
135C4×Dic47—41no—
136C4×Dic47—42no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—21no—
140C23×C47—21no—
141C23×C47—21no—
142C23×C47—21no—
143C23×C47—42no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47#9512yes—
147C23×C47C2×C4211yescharacteristic
148C23×C47C2×C22⋊C411yescharacteristic
149C23×C47C2×C22⋊C411yescharacteristic
150C23×C47C2×C4⋊C411yescharacteristic
151C23×C47C2×C4⋊C411yescharacteristic
152Q8×C47—42no—
153Q8×C47—42no—
154D4×C47—42no—
155D4×C47—42no—
156D4×C47—42no—
157D4×C47—42no—
158D4×C47—41no—
159D4×C47—41no—
160D4×C47—41no—
161D4×C47—41no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—22no—
173C2×C4×C47—22no—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47#9512yes—
181C2×C4×C47#9512yes—
182C2×C4×C47#9512yes—
183C2×C4×C47#9912yes—
184C2×C4×C47#9911yescharacteristic
185C2×C4×C47#9911yescharacteristic
186C2×C4×C47C2×C22⋊C411yescharacteristic
187C2×C4×C47C2×C4⋊C411yescharacteristic
188C2×Dic47—24no—
189C2×Dic47—42no—
190C2×Dic47—44no—
191C2×Dic47—42no—
192C2×Dic47—44no—
193C2×Dic47—44no—
194C2×Dic47—42no—
195C2×Dic47—42no—
196C2×Dic47—42no—
197C2×Dic47—84no—
198C8×C47—44no—
199#12573—471nomaximal, Sylow
200C22×C47—22no—
201C22×C47—21no—
202C22×C47—22no—
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—21no—
206C22×C47—42no—
207C22×C47—41no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47—41no—
211C22×C47—42no—
212C22×C47—42no—
213C22×C47—42no—
214C22×C47—42no—
215C22×C47—81no—
216C22×C47—81no—
217C22×C47C2×#9511yescharacteristic
218C22×C47C2×#9911yescharacteristic
219C22×C47C2×#9911yescharacteristic
220C4×C47—21no—
221C4×C47—22no—
222C4×C47—22no—
223C4×C47—21no—
224C4×C47#34212yes—
225C4×C47#34212yes—
226Dic47—44no—
227Dic47—82no—
228Dic47—84no—
229C2×#413D4711yescharacteristic
230C2×#410—471no—
231#2297—471no—
232#1390—474no—
233C2×C47—21no—
234C2×C47—21no—
235C2×C47—41no—
236C2×C47—41no—
237C2×C47—41no—
238C2×C47—42no—
239C2×C47—41no—
240C2×C47C2×#34211yescharacteristic
241C2×C47#212012yes—
242C22×#41C2×D4711yescharacteristic
243C22×M16—21no—
244C2×#117—471no—
245C2×#117—941no—
246C22×C42—471no—
247#413—22no—
248#413Dic4714yescomplemented
249C2×#101—21no—
250C2×#100C2×D4712yes—
251#410—474no—
252#410—942no—
253C2×#99—472no—
254C2×#99—941no—
255#391—941no—
256#390—941no—
257#386—472no—
258#386—472no—
259C4×C22⋊C4—941no—
260C4×C22⋊C4—941no—
261C47#1257311yescharacteristic, complemented, Sylow
262C2×#41—41no—
263C2×#41—42no—
264C2×#41—41no—
265C2×#41Dic9412yes—
266C2×#41C4×D4712yes—
267C2×#41C2×Dic4712yes—
268C2×#41C22⋊D4712yes—
269C22×Q8C22⋊D4711yescharacteristic
270C22×D4C4×D4711yescharacteristic
271C22×D4D18811yescharacteristic
272C22×D4C22×D4711yescharacteristic
273C23×C4—472no—
274C23×C4C4×D4711yescharacteristic
275C23×C4D18811yescharacteristic
276C23×C4C22⋊D4711yescharacteristic
277C2×M16—24no—
278C2×M16—22no—
279C2×M16—41no—
280C2×M16—42no—
281C22×C8—41no—
282#117—944no—
283#117—1882no—
284#117—1882no—
285C2×C4⋊C4—941no—
286C2×C4⋊C4—1881no—
287C2×C22⋊C4—472no—
288C2×C22⋊C4—942no—
289C2×C22⋊C4—942no—
290C2×C22⋊C4—942no—
291C2×C42—941no—
292C2×C42—941no—
293C2×C42—941no—
294C2×C42—1881no—
295C2×C42—1881no—
296C2×C42—1881no—
297#101—24no—
298#100—22no—
299#100C2×Dic4714yes—
300#99—474no—
301#99—944no—
302#99—942no—
303#95—942no—
304#95—942no—
305C24(C2×D47)⋊C411yescharacteristic
306C24C2×C4×D4711yescharacteristic
307C24C2×D18811yescharacteristic
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314#41—42no—
315#41—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8Dic47⋊C412yes—
319C2×Q8C22⋊Dic4712yes—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4C4×Dic4712yes—
327C2×D4Dic47⋊C412yes—
328C2×D4C4⋊Dic4712yes—
329C2×D4(C2×D47)⋊C412yes—
330C2×D4C2×Dic9411yescharacteristic
331C2×D4C2×C4×D4711yescharacteristic
332C2×D4C22×Dic4711yescharacteristic
333C2×D4C2×C22⋊D4711yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—942no—
344C22×C4—942no—
345C22×C4—1884no—
346C22×C4—1882no—
347C22×C4—1882no—
348C22×C4C4×Dic4712yes—
349C22×C4Dic47⋊C412yes—
350C22×C4Dic47⋊C412yes—
351C22×C4C4⋊Dic4712yes—
352C22×C4(C2×D47)⋊C411yescharacteristic
353C22×C4(C2×D47)⋊C411yescharacteristic
354C22×C4(C2×D47)⋊C412yes—
355C22×C4(C2×D47)⋊C411yescharacteristic
356C22×C4C22⋊Dic4712yes—
357C22×C4C2×C22⋊D4711yescharacteristic
358M16—24no—
359M16—42no—
360M16—44no—
361C2×C8—42no—
362C2×C8—44no—
363C4⋊C4—1882no—
364C4⋊C4—1884no—
365C22⋊C4—944no—
366C22⋊C4—1884no—
367C22⋊C4—1884no—
368C22⋊C4—1882no—
369C22⋊C4—1882no—
370C42—1882no—
371C42—1884no—
372C42—1881no—
373C42—1882no—
374C42—1881no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23C47⋊#9512yes—
385C23C2×C4×Dic4711yescharacteristic
386C23C2×(C2×D47)⋊C411yescharacteristic
387C23C2×(C2×D47)⋊C411yescharacteristic
388C23C2×Dic47⋊C411yescharacteristic
389C23C2×C4⋊Dic4711yescharacteristic
390Q8—42no—
391Q8—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—944no—
419C2×C4—3764no—
420C2×C4—1884no—
421C2×C4—1882no—
422C2×C4—1882no—
423C2×C4—1882no—
424C2×C4—1882no—
425C2×C4—1884no—
426C2×C4—1882no—
427C2×C4—1884no—
428C2×C4C47⋊#9512yes—
429C2×C4C47⋊#9512yes—
430C2×C4C47⋊#9512yes—
431C2×C4#1309393811yescharacteristic
432C2×C4#1309393812yes—
433C2×C4#1309393811yescharacteristic
434C2×C4C2×C22⋊Dic4711yescharacteristic
435C2×C4C2×Dic47⋊C411yescharacteristic
436C8—44no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22C2×C47⋊#9511yescharacteristic
455C22C2×#1309393811yescharacteristic
456C22C2×#1309393811yescharacteristic
457C4—21no—
458C4—22no—
459C4—22no—
460C4—21no—
461C4—3762no—
462C4—3764no—
463C4—1884no—
464C4#1568888912yes—
465C4#1568888912yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—42no—
472C2—41no—
473C2C2×#1568888911yescharacteristic
474C2#2024503012yes—
subgroupG/H
centerC2×C4#13093938
commutatorC23×C47C2×C42
FrattiniC22×D4C22×D47
FittingC2×C47×#413C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×D4×C47C23
hypercenterC2×#413D47
hypocenterC47#12573
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #20241507C2semidirect
6. C2×#13093983C4semidirect
7. C22×C4×Dic47C4semidirect
9. #15689097C4semidirect
14. #15689003C4semidirect
21. C47×#413C4semidirect
248. #413Dic47semidirect
261. C47#12573semidirect

Maximal subgroups

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

subgroupindex
C2×#156890972
#202458792
#202415072
C2×C47×#4132
#1257347