Finite Group Databasebeta

← #26132251

Subgroups — #26132251

Order 12032.

600 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×#15688910C212yesmaximal
2#20243380C211yesmaximal, characteristic, complemented
3#20243376C211yesmaximal, characteristic, complemented
4#20241506C212yesmaximal, complemented
5C47×#2321C211yesmaximal, characteristic
6#15689147—21no—
7C2×#13093941C2212yes—
8C2×#13093939C2212yes—
9#15688983—21no—
10#15688910—24no—
11#15688892—21no—
12#15688890—21no—
13C2×#120×C47C2211yescharacteristic
14C2×C4×D4×C47C2211yescharacteristic
15C22×C22⋊C4×C47C411yescharacteristic, complemented
16C47×#390—21no—
17C47×#388—21no—
18C47×#388—21no—
19C47×#386—21no—
20C47×#386C2211yescharacteristic
21C47×#385C411yescharacteristic, complemented
22C47×#379C411yescharacteristic, complemented
23C4×C22⋊C4×C47C411yescharacteristic, complemented
24C2×#10625329—22no—
25C2×C4⋊Dic47—41no—
26C2×C22⋊Dic47—22no—
27C4⋊(C47⋊C8)—41no—
28C4⋊(C47⋊C8)—41no—
29#13093941—22no—
30#13093941—22no—
31#13093939—22no—
32#13093939—22no—
33C47⋊#95—41no—
34C25×C47C2×C411yescharacteristic
35C22×D4×C47—21no—
36C22×D4×C47C2311yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40#120×C47—21no—
41#120×C47—21no—
42#120×C47D412yes—
43#120×C47Q812yes—
44C4×D4×C47—41no—
45C4×D4×C47—41no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—41no—
55C2×C22⋊C4×C47—41no—
56C2×C22⋊C4×C47C2×C411yescharacteristic
57C2×C22⋊C4×C47C2×C411yescharacteristic
58C2×C22⋊C4×C47C2×C411yescharacteristic
59C2×C22⋊C4×C47D411yescharacteristic
60C2×C42×C47—21no—
61C2×C42×C47D411yescharacteristic
62#95×C47—21no—
63#95×C47—21no—
64#95×C47—21no—
65C24×C47—21no—
66C24×C47—21no—
67C24×C47—41no—
68C24×C47—41no—
69C24×C47—81no—
70C24×C47—81no—
71C24×C47C4⋊C412yescomplemented
72C24×C47C22×C411yescharacteristic
73C24×C47C2×D411yescharacteristic
74C2×D4×C47—22no—
75C2×D4×C47—22no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—81no—
79C2×D4×C47—81no—
80C2×D4×C47C2×D411yescharacteristic
81C2×D4×C47C2×Q811yescharacteristic
82C2×D4×C47#4111yescharacteristic
83C2×D4×C47#4111yescharacteristic
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—41no—
92C22×C4×C47—81no—
93C22×C4×C47—81no—
94C22×C4×C47—81no—
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C22⋊C411yescharacteristic
98C22×C4×C47C22⋊C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×Dic47—42no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—81no—
103C22⋊C4×C47—21no—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—22no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—81no—
113C22⋊C4×C47—81no—
114C22⋊C4×C47—81no—
115C22⋊C4×C47C4⋊C412yescomplemented
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—81no—
119C22⋊Dic47—42no—
120C22⋊Dic47—42no—
121C4⋊Dic47—82no—
122#10625329—42no—
123#10625329—42no—
124C2×C47⋊C8—42no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—21no—
128C23×C47—21no—
129C23×C47—22no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—41no—
135C23×C47—41no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47—42no—
139C23×C47—42no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—82no—
143C23×C47—82no—
144C23×C47—82no—
145C23×C47—82no—
146C23×C47—84no—
147C23×C47C2×C22⋊C411yescharacteristic
148C23×C47C2×C4⋊C411yescharacteristic
149C23×C47#11711yescharacteristic
150C23×C47#12111yescharacteristic
151C23×C47C22⋊Q811yescharacteristic
152D4×C47—41no—
153D4×C47—41no—
154D4×C47—41no—
155D4×C47—41no—
156D4×C47—82no—
157D4×C47—82no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—22no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—81no—
178C2×C4×C47—81no—
179C2×C4×C47—81no—
180C2×C4×C47—82no—
181C2×C4×C47#9911yescharacteristic
182C2×C4×C47#9911yescharacteristic
183C2×C4×C47#12111yescharacteristic
184C2×C4×C47C22⋊Q811yescharacteristic
185C2×Dic47—82no—
186C2×Dic47—82no—
187C2×Dic47—82no—
188C47⋊C8—82no—
189#12795—471nomaximal, Sylow
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—41no—
197C22×C47—41no—
198C22×C47—41no—
199C22×C47—41no—
200C22×C47—42no—
201C22×C47—42no—
202C22×C47—42no—
203C22×C47—41no—
204C22×C47—41no—
205C22×C47—41no—
206C22×C47—41no—
207C22×C47—42no—
208C22×C47—81no—
209C22×C47—81no—
210C22×C47—81no—
211C22×C47—84no—
212C22×C47—84no—
213C22×C47—81no—
214C22×C47—81no—
215C22×C47#38011yescharacteristic
216C22×C47C2×#9911yescharacteristic
217C22×C47#41011yescharacteristic
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—21no—
221C4×C47—21no—
222C4×C47—42no—
223C4×C47—41no—
224C4×C47—41no—
225C4×C47—41no—
226C4×C47—81no—
227C4×C47—81no—
228Dic47—162no—
229#2321D4711yescharacteristic
230C2×AΣL(1,𝔽16)—472no—
231#1800—471no—
232#1799—471no—
233#1390—472no—
234C2×C47—21no—
235C2×C47—21no—
236C2×C47—41no—
237C2×C47—41no—
238C2×C47—41no—
239C2×C47—41no—
240C2×C47—82no—
241C2×C47#179911yescharacteristic
242C2×C47#212111yescharacteristic
243C2×C47#212211yescharacteristic
244C2×#120C2×D4711yescharacteristic
245C2×C4×D4C2×D4711yescharacteristic
246C22×C22⋊C4Dic4711yescharacteristic, complemented
247#423—941no—
248C2×#100—472no—
249C2×#99—472no—
250#390—21no—
251#388—21no—
252#388—21no—
253#386—21no—
254#386C2×D4711yescharacteristic
255#385Dic4711yescharacteristic, complemented
256#380—941no—
257#379Dic4711yescharacteristic, complemented
258C4×C22⋊C4Dic4711yescharacteristic, complemented
259AΣL(1,𝔽16)—944no—
260#343—941no—
261#342—941no—
262C47#1279511yescharacteristic, complemented, Sylow
263C25C2×Dic4711yescharacteristic
264C22×D4—21no—
265C22×D4C22×D4711yescharacteristic
266C23×C4C2×Dic4711yescharacteristic
267C23×C4C2×Dic4711yescharacteristic
268C23×C4C22⋊D4711yescharacteristic
269C2×M16—942no—
270#120—21no—
271#120—21no—
272#120Dic9412yes—
273#120D18812yes—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—1881no—
278C2×C4⋊C4C22⋊D4711yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—942no—
288C2×C22⋊C4C2×Dic4711yescharacteristic
289C2×C22⋊C4C2×Dic4711yescharacteristic
290C2×C22⋊C4C2×Dic4711yescharacteristic
291C2×C22⋊C4C22⋊D4711yescharacteristic
292C2×C42—21no—
293C2×C42C22⋊D4711yescharacteristic
294C4⋊C8—1881no—
295C4⋊C8—1881no—
296#100—942no—
297#100—942no—
298#99—942no—
299#99—942no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—1881no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—81no—
309C24—81no—
310C24C4⋊Dic4712yescomplemented
311C24C22×Dic4711yescharacteristic
312C24C2×C22⋊D4711yescharacteristic
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4C2×Dic9411yescharacteristic
320C2×D4C2×D18811yescharacteristic
321C2×D4#1062535011yescharacteristic
322C2×D4#1062535011yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—1882no—
335C22×C4C22⋊Dic4711yescharacteristic
336C22×C4C22⋊Dic4711yescharacteristic
337C22×C4C22⋊Dic4711yescharacteristic
338C22×C4C22⋊Dic4711yescharacteristic
339C22×C4C2×C22⋊D4711yescharacteristic
340M16—1882no—
341M16—1882no—
342C2×C8—1882no—
343C4⋊C4—41no—
344C4⋊C4—81no—
345C4⋊C4—3762no—
346C22⋊C4—21no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4—1882no—
359C22⋊C4—1882no—
360C22⋊C4C4⋊Dic4712yescomplemented
361C42—21no—
362C42—21no—
363C42—81no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—84no—
386C23C2×C22⋊Dic4711yescharacteristic
387C23C2×C4⋊Dic4711yescharacteristic
388C23#1309398311yescharacteristic
389C23#1309399811yescharacteristic
390C23#1309400311yescharacteristic
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—82no—
396D4—82no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—3762no—
421C2×C4—3762no—
422C2×C4—3762no—
423C2×C4#1309393911yescharacteristic
424C2×C4#1309393911yescharacteristic
425C2×C4#1309399811yescharacteristic
426C2×C4#1309400311yescharacteristic
427C8—3762no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—84no—
450C22—84no—
451C22—81no—
452C22—81no—
453C22#1568898311yescharacteristic
454C22C2×#1309393911yescharacteristic
455C22#1568909711yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—81no—
465C4—81no—
466C4—7522no—
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—82no—
474C2#2024337611yescharacteristic
475C2#2024503411yescharacteristic
476C2#2024503811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#2321C2
socleC22×C47C2×#99
prime radicalC2×#120×C47C22
cosocleC22×D4×C47C23
hypercenter#2321D47
hypocenterC47#12795
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. #20243380C2semidirect
3. #20243376C2semidirect
4. #20241506C2semidirect
15. C22×C22⋊C4×C47C4semidirect
21. C47×#385C4semidirect
22. C47×#379C4semidirect
23. C4×C22⋊C4×C47C4semidirect
71. C24×C47C4⋊C4semidirect
115. C22⋊C4×C47C4⋊C4semidirect
246. C22×C22⋊C4Dic47semidirect
255. #385Dic47semidirect
257. #379Dic47semidirect
258. C4×C22⋊C4Dic47semidirect
262. C47#12795semidirect
310. C24C4⋊Dic47semidirect
360. C22⋊C4C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889102
#202433802
#202433762
#202415062
C47×#23212
#1279547