Finite Group Databasebeta

← #26131265

Subgroups — #26131265

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689107C211yesmaximal, characteristic
2#20245980C211yesmaximal, characteristic
3#20241514C214yesmaximal, complemented
4C2×C47×#413C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×#13093983C411yescharacteristic
7C22×C4⋊Dic47C411yescharacteristic
8#15689107—22no—
9#15689107C414yescomplemented
10C2×#13093943C2212yes—
11C2×#13093941—21no—
12#15689030C2212yes—
13#15689026C412yes—
14#15689003—21no—
15#15689003—21no—
16#15688983—21no—
17#15688983—21no—
18C22×#41×C47C2211yescharacteristic
19C22×M16×C47—21no—
20C47×#413—22no—
21C47×#413C414yescomplemented
22C2×#101×C47C2212yes—
23C2×#100×C47—21no—
24C23×Dic47—21no—
25C2×#10625329—24no—
26C2×#10625329—22no—
27C2×#10625329—41no—
28C2×#10625329—42no—
29C22×C47⋊C8—41no—
30#13093983—42no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—41no—
34C2×C4⋊Dic47—41no—
35C2×C4⋊Dic47C2×C412yes—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—22no—
38C2×C22⋊Dic47—22no—
39C2×C4×Dic47C2×C412yes—
40#13093943—22no—
41#13093943C2×C414yescomplemented
42#13093941—24no—
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×C47C2311yescharacteristic
53C22×D4×C47C2×C411yescharacteristic
54C22×D4×C47D411yescharacteristic
55C22×D4×C47D411yescharacteristic
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—22no—
65#101×C47C2×C414yescomplemented
66#100×C47—24no—
67C24×C47C22⋊C411yescharacteristic
68C24×C47C22⋊C411yescharacteristic
69C24×C47C22⋊C411yescharacteristic
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×C411yescharacteristic
81C2×Q8×C47C22×C411yescharacteristic
82C2×Q8×C47C2×D411yescharacteristic
83C2×Q8×C47C2×Q811yescharacteristic
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47C4212yes—
91C2×D4×C47C22⋊C412yes—
92C2×D4×C47C22⋊C412yes—
93C2×D4×C47C4⋊C412yes—
94C2×D4×C47C4⋊C412yes—
95C2×D4×C47C4⋊C412yes—
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⋊C412yes—
107C22×C4×C47C22⋊C412yes—
108C22×C4×C47C22⋊C411yescharacteristic
109C22×C4×C47C4⋊C412yes—
110C22×C4×C47C4⋊C412yes—
111C22×C4×C47C4⋊C412yes—
112C22×C4×C47C22×C411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—42no—
118C22×Dic47—82no—
119M16×C47—24no—
120M16×C47—42no—
121M16×C47—44no—
122C2×C8×C47—42no—
123C2×C8×C47—44no—
124C22⋊Dic47—42no—
125C22⋊Dic47—82no—
126C22⋊Dic47—82no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129C4⋊Dic47—42no—
130C4⋊Dic47—42no—
131C4×Dic47—42no—
132#10625329—24no—
133#10625329—42no—
134#10625329—44no—
135C2×C47⋊C8—42no—
136C2×C47⋊C8—44no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—21no—
140C23×C47—21no—
141C23×C47—21no—
142C23×C47—21no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—41no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47—41no—
149C23×C47#9512yes—
150C23×C47#9512yes—
151C23×C47#9512yes—
152C23×C47C2×C22⋊C411yescharacteristic
153Q8×C47—41no—
154Q8×C47—41no—
155Q8×C47—41no—
156Q8×C47—41no—
157D4×C47—42no—
158D4×C47—42no—
159D4×C47—42no—
160D4×C47—42no—
161D4×C47—42no—
162D4×C47—42no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—21no—
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—22no—
181C2×C4×C47#9512yes—
182C2×C4×C47#9912yes—
183C2×C4×C47#9911yescharacteristic
184C2×C4×C47#9911yescharacteristic
185C2×C4×C47C2×C4211yescharacteristic
186C2×C4×C47C2×C22⋊C411yescharacteristic
187C2×C4×C47C2×C22⋊C411yescharacteristic
188C2×C4×C47C2×C4⋊C411yescharacteristic
189C2×C4×C47C2×C4⋊C411yescharacteristic
190C2×C4×C47C2×C4⋊C411yescharacteristic
191C2×Dic47—42no—
192C2×Dic47—42no—
193C2×Dic47—42no—
194C2×Dic47—42no—
195C2×Dic47—42no—
196C2×Dic47—42no—
197C2×Dic47—84no—
198C8×C47—44no—
199C47⋊C8—44no—
200#12587—471nomaximal, Sylow
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—22no—
204C22×C47—22no—
205C22×C47—22no—
206C22×C47—42no—
207C22×C47—42no—
208C22×C47—42no—
209C22×C47—42no—
210C22×C47—41no—
211C22×C47—41no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—41no—
215C22×C47—41no—
216C22×C47—81no—
217C22×C47—81no—
218C22×C47C2×#9511yescharacteristic
219C22×C47C2×#9911yescharacteristic
220C22×C47C2×#9911yescharacteristic
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—21no—
226C4×C47—21no—
227C4×C47#34212yes—
228C4×C47#34212yes—
229Dic47—82no—
230Dic47—82no—
231C2×#413—471no—
232C2×#413D4711yescharacteristic
233#2311—471no—
234#1394—474no—
235C2×C47—21no—
236C2×C47—21no—
237C2×C47—42no—
238C2×C47—42no—
239C2×C47—42no—
240C2×C47C2×#34211yescharacteristic
241C2×C47#213411yescharacteristic
242C2×C47#213511yescharacteristic
243C22×#41C2×D4711yescharacteristic
244C22×M16—21no—
245C22×M16—941no—
246C2×#117—471no—
247C22×C4⋊C4—471no—
248#413—22no—
249#413—474no—
250#413—942no—
251#413Dic4714yescomplemented
252C2×#101—472no—
253C2×#101C2×D4712yes—
254C2×#100—21no—
255C2×#100—941no—
256#391—472no—
257#390—472no—
258#386—941no—
259#386—941no—
260#380—941no—
261#380—941no—
262C47#1258711yescharacteristic, complemented, Sylow
263C2×#41—41no—
264C2×#41—41no—
265C2×#41—42no—
266C2×#41Dic9412yes—
267C2×#41C4×D4712yes—
268C2×#41C2×Dic4712yes—
269C2×#41C22⋊D4712yes—
270C22×Q8C22×D4711yescharacteristic
271C22×D4C4×D4711yescharacteristic
272C22×D4D18811yescharacteristic
273C22×D4C22⋊D4711yescharacteristic
274C23×C4—941no—
275C23×C4C4×D4711yescharacteristic
276C23×C4D18811yescharacteristic
277C23×C4C22⋊D4711yescharacteristic
278C2×M16—24no—
279C2×M16—22no—
280C2×M16—41no—
281C2×M16—42no—
282C2×M16—944no—
283C2×M16—942no—
284C2×M16—1881no—
285C2×M16—1882no—
286C22×C8—41no—
287C22×C8—1881no—
288#117—1882no—
289C2×C4⋊C4—472no—
290C2×C4⋊C4—941no—
291C2×C4⋊C4—941no—
292C2×C4⋊C4—1881no—
293C2×C4⋊C4—1881no—
294C2×C22⋊C4—942no—
295C2×C22⋊C4—942no—
296C2×C22⋊C4—941no—
297C2×C42—472no—
298#101—22no—
299#101—474no—
300#101—942no—
301#101C2×Dic4714yescomplemented
302#100—24no—
303#100—944no—
304#95—942no—
305#95—942no—
306C24(C2×D47)⋊C411yescharacteristic
307C24(C2×D47)⋊C411yescharacteristic
308C24(C2×D47)⋊C411yescharacteristic
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8C2×Dic9411yescharacteristic
320C2×Q8C2×C4×D4711yescharacteristic
321C2×Q8C22×Dic4711yescharacteristic
322C2×Q8C2×C22⋊D4711yescharacteristic
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4C4×Dic4712yes—
330C2×D4Dic47⋊C412yes—
331C2×D4Dic47⋊C412yes—
332C2×D4C4⋊Dic4712yes—
333C2×D4(C2×D47)⋊C412yes—
334C2×D4C22⋊Dic4712yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—942no—
345C22×C4—942no—
346C22×C4—3762no—
347C22×C4—1882no—
348C22×C4C4×Dic4712yes—
349C22×C4Dic47⋊C412yes—
350C22×C4Dic47⋊C412yes—
351C22×C4C4⋊Dic4712yes—
352C22×C4(C2×D47)⋊C412yes—
353C22×C4(C2×D47)⋊C411yescharacteristic
354C22×C4C22⋊Dic4712yes—
355C22×C4C2×C4×D4711yescharacteristic
356C22×C4C2×D18811yescharacteristic
357C22×C4C2×C22⋊D4711yescharacteristic
358M16—24no—
359M16—42no—
360M16—44no—
361M16—944no—
362M16—1882no—
363M16—1884no—
364C2×C8—42no—
365C2×C8—44no—
366C2×C8—1882no—
367C2×C8—1884no—
368C4⋊C4—1882no—
369C4⋊C4—1881no—
370C4⋊C4—1881no—
371C4⋊C4—1882no—
372C22⋊C4—3762no—
373C22⋊C4—3762no—
374C22⋊C4—1882no—
375C42—1882no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C47⋊#9512yes—
389C23C47⋊#9512yes—
390C23C47⋊#9512yes—
391C23C2×(C2×D47)⋊C411yescharacteristic
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396D4—42no—
397D4—42no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—42no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
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—22no—
419C2×C4—22no—
420C2×C4—3764no—
421C2×C4—1882no—
422C2×C4—1882no—
423C2×C4—1882no—
424C2×C4—1882no—
425C2×C4—1882no—
426C2×C4—1882no—
427C2×C4C47⋊#9512yes—
428C2×C4#1309393812yes—
429C2×C4#1309393811yescharacteristic
430C2×C4#1309393811yescharacteristic
431C2×C4C2×C4×Dic4711yescharacteristic
432C2×C4C2×(C2×D47)⋊C411yescharacteristic
433C2×C4C2×C22⋊Dic4711yescharacteristic
434C2×C4C2×Dic47⋊C411yescharacteristic
435C2×C4C2×Dic47⋊C411yescharacteristic
436C2×C4C2×C4⋊Dic4711yescharacteristic
437C8—44no—
438C8—1884no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—81no—
455C22—81no—
456C22C2×C47⋊#9511yescharacteristic
457C22C2×#1309393811yescharacteristic
458C22C2×#1309393811yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—3762no—
466C4—3762no—
467C4#1568888912yes—
468C4#1568888912yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—42no—
474C2C2×#1568888911yescharacteristic
475C2#2024507911yescharacteristic
476C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC2×C47×#413C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenterC2×#413D47
hypocenterC47#12587
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #20241514C2semidirect
9. #15689107C4semidirect
21. C47×#413C4semidirect
41. #13093943C2×C4semidirect
65. #101×C47C2×C4semidirect
251. #413Dic47semidirect
262. C47#12587semidirect
301. #101C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891072
#202459802
#202415142
C2×C47×#4132
#1258747