Finite Group Databasebeta

← #26131259

Subgroups — #26131259

Order 12032.

822 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689107C211yesmaximal, characteristic
2#20245879C211yesmaximal, 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—
16C4×C22⋊Dic47—21no—
17C4×C22⋊Dic47—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—41no—
31#13093983—41no—
32C2×C4⋊Dic47C2×C412yes—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—42no—
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—42no—
47C2×#41×C47—41no—
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—22no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
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—44no—
118C22×Dic47—42no—
119M16×C47—24no—
120M16×C47—42no—
121M16×C47—44no—
122C2×C8×C47—42no—
123C2×C8×C47—44no—
124C22⋊Dic47—44no—
125C22⋊Dic47—44no—
126C22⋊Dic47—42no—
127C4⋊Dic47—42no—
128C4×Dic47—42no—
129C4×Dic47—41no—
130C4×Dic47—41no—
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—42no—
144C23×C47—42no—
145C23×C47—42no—
146C23×C47#9512yes—
147C23×C47#9512yes—
148C23×C47#9512yes—
149C23×C47C2×C22⋊C411yescharacteristic
150Q8×C47—41no—
151Q8×C47—41no—
152Q8×C47—41no—
153Q8×C47—41no—
154D4×C47—42no—
155D4×C47—42no—
156D4×C47—42no—
157D4×C47—42no—
158D4×C47—42no—
159D4×C47—42no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—22no—
167C2×C4×C47—22no—
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#9512yes—
179C2×C4×C47#9912yes—
180C2×C4×C47#9911yescharacteristic
181C2×C4×C47#9911yescharacteristic
182C2×C4×C47C2×C4211yescharacteristic
183C2×C4×C47C2×C22⋊C411yescharacteristic
184C2×C4×C47C2×C22⋊C411yescharacteristic
185C2×C4×C47C2×C4⋊C411yescharacteristic
186C2×C4×C47C2×C4⋊C411yescharacteristic
187C2×C4×C47C2×C4⋊C411yescharacteristic
188C2×Dic47—24no—
189C2×Dic47—42no—
190C2×Dic47—48no—
191C2×Dic47—42no—
192C2×Dic47—42no—
193C2×Dic47—42no—
194C2×Dic47—42no—
195C8×C47—44no—
196C47⋊C8—44no—
197#12585—471nomaximal, Sylow
198C22×C47—22no—
199C22×C47—22no—
200C22×C47—22no—
201C22×C47—22no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—42no—
205C22×C47—42no—
206C22×C47—42no—
207C22×C47—42no—
208C22×C47—42no—
209C22×C47—81no—
210C22×C47—81no—
211C22×C47C2×#9511yescharacteristic
212C22×C47C2×#9911yescharacteristic
213C22×C47C2×#9911yescharacteristic
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—21no—
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47#34212yes—
221C4×C47#34212yes—
222Dic47—44no—
223Dic47—82no—
224C2×#413—471no—
225C2×#413D4711yescharacteristic
226#2297—471no—
227#1394—474no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47—42no—
231C2×C47—42no—
232C2×C47—42no—
233C2×C47C2×#34211yescharacteristic
234C2×C47#213212yes—
235C22×#41C2×D4711yescharacteristic
236C22×M16—21no—
237C22×M16—941no—
238C2×#117—471no—
239C22×C42—471no—
240#413—22no—
241#413—474no—
242#413—942no—
243#413Dic4714yescomplemented
244C2×#101—472no—
245C2×#101C2×D4712yes—
246C2×#100—21no—
247C2×#100—941no—
248#391—472no—
249#390—472no—
250#386—941no—
251#386—941no—
252C4×C22⋊C4—941no—
253C4×C22⋊C4—941no—
254C47#1258511yescharacteristic, complemented, Sylow
255C2×#41—42no—
256C2×#41—41no—
257C2×#41—41no—
258C2×#41Dic9412yes—
259C2×#41C4×D4712yes—
260C2×#41C2×Dic4712yes—
261C2×#41C22⋊D4712yes—
262C22×Q8C22×D4711yescharacteristic
263C22×D4C4×D4711yescharacteristic
264C22×D4D18811yescharacteristic
265C22×D4C22⋊D4711yescharacteristic
266C23×C4—941no—
267C23×C4C4×D4711yescharacteristic
268C23×C4D18811yescharacteristic
269C23×C4C22⋊D4711yescharacteristic
270C2×M16—24no—
271C2×M16—22no—
272C2×M16—41no—
273C2×M16—42no—
274C2×M16—944no—
275C2×M16—942no—
276C2×M16—1881no—
277C2×M16—1882no—
278C22×C8—41no—
279C22×C8—1881no—
280#117—1881no—
281#117—1881no—
282C2×C4⋊C4—472no—
283C2×C22⋊C4—941no—
284C2×C22⋊C4—942no—
285C2×C22⋊C4—942no—
286C2×C42—472no—
287C2×C42—941no—
288C2×C42—941no—
289C2×C42—1882no—
290#101—22no—
291#101—474no—
292#101—942no—
293#101C2×Dic4714yescomplemented
294#100—24no—
295#100—944no—
296#95—942no—
297#95—942no—
298C24(C2×D47)⋊C411yescharacteristic
299C24(C2×D47)⋊C411yescharacteristic
300C24(C2×D47)⋊C411yescharacteristic
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—42no—
308#41—42no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8C2×Dic9411yescharacteristic
312C2×Q8C2×C4×D4711yescharacteristic
313C2×Q8C22×Dic4711yescharacteristic
314C2×Q8C2×C22⋊D4711yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4C4×Dic4712yes—
322C2×D4Dic47⋊C412yes—
323C2×D4Dic47⋊C412yes—
324C2×D4C4⋊Dic4712yes—
325C2×D4(C2×D47)⋊C412yes—
326C2×D4C22⋊Dic4712yes—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—942no—
337C22×C4—942no—
338C22×C4—1884no—
339C22×C4—1882no—
340C22×C4C4×Dic4712yes—
341C22×C4Dic47⋊C412yes—
342C22×C4Dic47⋊C412yes—
343C22×C4C4⋊Dic4712yes—
344C22×C4(C2×D47)⋊C412yes—
345C22×C4(C2×D47)⋊C411yescharacteristic
346C22×C4C22⋊Dic4712yes—
347C22×C4C2×C4×D4711yescharacteristic
348C22×C4C2×D18811yescharacteristic
349C22×C4C2×C22⋊D4711yescharacteristic
350M16—24no—
351M16—42no—
352M16—44no—
353M16—944no—
354M16—1882no—
355M16—1884no—
356C2×C8—42no—
357C2×C8—44no—
358C2×C8—1882no—
359C2×C8—1884no—
360C4⋊C4—1882no—
361C22⋊C4—1884no—
362C22⋊C4—1884no—
363C22⋊C4—1882no—
364C42—1882no—
365C42—1882no—
366C42—1881no—
367C42—1881no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23C47⋊#9512yes—
378C23C47⋊#9512yes—
379C23C47⋊#9512yes—
380C23C2×(C2×D47)⋊C411yescharacteristic
381Q8—41no—
382Q8—41no—
383Q8—41no—
384Q8—41no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—42no—
390D4—42no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—944no—
410C2×C4—1882no—
411C2×C4—1882no—
412C2×C4—1882no—
413C2×C4—1882no—
414C2×C4—1888no—
415C2×C4—1882no—
416C2×C4C47⋊#9512yes—
417C2×C4#1309393811yescharacteristic
418C2×C4#1309393812yes—
419C2×C4#1309393811yescharacteristic
420C2×C4C2×C4×Dic4711yescharacteristic
421C2×C4C2×(C2×D47)⋊C411yescharacteristic
422C2×C4C2×C22⋊Dic4711yescharacteristic
423C2×C4C2×Dic47⋊C411yescharacteristic
424C2×C4C2×Dic47⋊C411yescharacteristic
425C2×C4C2×C4⋊Dic4711yescharacteristic
426C8—44no—
427C8—1884no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22C2×C47⋊#9511yescharacteristic
442C22C2×#1309393811yescharacteristic
443C22C2×#1309393811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—3762no—
451C4—1884no—
452C4#1568888912yes—
453C4#1568888912yes—
454C2—21no—
455C2—21no—
456C2—42no—
457C2—42no—
458C2—42no—
459C2C2×#1568888911yescharacteristic
460C2#2024507112yes—
subgroupG/H
centerC2×C4#13093938
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC2×C47×#413C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenterC2×#413D47
hypocenterC47#12585
LCSG ▷ C2×C4×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. #20241514C2semidirect
9. #15689107C4semidirect
21. C47×#413C4semidirect
41. #13093943C2×C4semidirect
65. #101×C47C2×C4semidirect
243. #413Dic47semidirect
254. C47#12585semidirect
293. #101C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891072
#202458792
#202415142
C2×C47×#4132
#1258547