Finite Group Databasebeta

← #26109012

Subgroups — #26109012

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093998C211yesmaximal, characteristic
2C2×#15688858C212yesmaximal
3#20241615C211yesmaximal, characteristic, complemented
4#20241268C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7C2×C4×D188C411yescharacteristic, complemented
8C2×C4×C22⋊D47—21no—
9#15689025C411yescharacteristic, complemented
10#15689021C411yescharacteristic, complemented
11#15689014—21no—
12#15689002—21no—
13#15688983C2211yescharacteristic
14C4×C4⋊Dic47C411yescharacteristic, complemented
15C4×(C2×D47)⋊C4—21no—
16C2×C47⋊#95—21no—
17C2×C47⋊#95—21no—
18C2×C47⋊#95C2212yes—
19#15688858—24no—
20C22×C42×C47C2211yescharacteristic
21C2×C4⋊C8×C47—21no—
22C2×C22⋊C8×C47C2212yes—
23C47×#336—21no—
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—21no—
26C23×Dic47—22no—
27C22×C4×D47—21no—
28#13093998—21no—
29#13093998—21no—
30#13093998D412yes—
31#13093998Q812yes—
32C4×D188—42no—
33C4×C22⋊D47—44no—
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×C4⋊Dic47C2×C411yescharacteristic
36C2×C4⋊Dic47C2×C411yescharacteristic
37C2×Dic47⋊C4—21no—
38C2×C22⋊Dic47—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×(C2×D47)⋊C4—21no—
41C2×(C2×D47)⋊C4—21no—
42C2×(C2×D47)⋊C4—21no—
43C2×C4×Dic47—21no—
44C47⋊#95—24no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—22no—
49C47⋊#95—22no—
50C47⋊#95—42no—
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C23×C4×C47C2311yescharacteristic
54C22×C8×C47—22no—
55C2×C42×C47—42no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C4⋊C8×C47—42no—
61C22⋊C8×C47—22no—
62C22⋊C8×C47—22no—
63C24×C47C2×D411yescharacteristic
64C23×D47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×Q811yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C2×C22⋊D47—22no—
84C2×C22⋊D47—22no—
85C2×C22⋊D47—42no—
86C2×C22⋊D47—42no—
87C22×Dic47—21no—
88C22×Dic47—24no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C2×D188—21no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188C4⋊C412yes—
101C2×C4×D47—42no—
102C2×C4×D47—42no—
103C2×C4×D47—42no—
104C2×C8×C47—42no—
105C2×C8×C47—42no—
106C2×C8×C47—42no—
107C42×C47—22no—
108C42×C47—22no—
109C42×C47—41no—
110C42×C47—41no—
111C22⋊Dic47—42no—
112(C2×D47)⋊C4—22no—
113(C2×D47)⋊C4—22no—
114(C2×D47)⋊C4—42no—
115(C2×D47)⋊C4—42no—
116(C2×D47)⋊C4—42no—
117C4⋊Dic47—21no—
118C4⋊Dic47—41no—
119C4⋊Dic47—41no—
120C4⋊Dic47C4⋊C412yes—
121Dic47⋊C4—42no—
122C4×Dic47—42no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47C2×C22⋊C411yescharacteristic
128C23×C47#12111yescharacteristic
129C23×C47C22⋊Q811yescharacteristic
130C23×C47#13512yes—
131C23×C47AGL(1,ℤ/8)12yes—
132C22×D47—22no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—41no—
136C22×D47—41no—
137C22×D47—41no—
138C22×D47—41no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—44no—
158C2×C4×C47—44no—
159C2×C4×C47#9912yes—
160C2×C4×C47#10412yes—
161C2×C4×C47C2×C4⋊C411yescharacteristic
162C2×C4×C47#11711yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47C22⋊Q811yescharacteristic
165C22⋊D47—44no—
166C22⋊D47—44no—
167C2×Dic47—22no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—44no—
171C2×Dic47—44no—
172C2×Dic47—44no—
173C2×Dic47—41no—
174C2×Dic47—41no—
175C2×Dic47—41no—
176C2×Dic47—41no—
177C2×Dic47—41no—
178C2×Dic47—41no—
179C2×Dic47—82no—
180C2×Dic47—82no—
181C2×Dic47—82no—
182C2×Dic47—82no—
183D188—42no—
184D188—42no—
185C4×D47—84no—
186C8×C47—82no—
187#8573—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—22no—
193C22×C47—22no—
194C22×C47—22no—
195C22×C47—22no—
196C22×C47#38011yescharacteristic
197C22×C47C2×#9911yescharacteristic
198C22×C47#41011yescharacteristic
199C22×C47#41611yescharacteristic
200C22×C47#41811yescharacteristic
201C22×C47C2×#10411yescharacteristic
202C22×C47#42111yescharacteristic
203C22×C47#46312yes—
204C22×C47#47812yes—
205C2×D47—42no—
206C2×D47—42no—
207C2×D47—42no—
208C2×D47—81no—
209C2×D47—81no—
210C2×D47—81no—
211C2×D47—81no—
212C4×C47—21no—
213C4×C47—21no—
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—41no—
219C4×C47—41no—
220Dic47—42no—
221Dic47—84no—
222Dic47—81no—
223Dic47—81no—
224C4×#121—471no—
225#1797D4711yescharacteristic, complemented
226C2×#327—472no—
227#1434—471no—
228#1294—472no—
229C2×C47—22no—
230C2×C47—22no—
231C2×C47#150011yescharacteristic
232C2×C47#150711yescharacteristic
233C2×C47#151411yescharacteristic
234C2×C47#151611yescharacteristic
235C2×C47#179911yescharacteristic
236C2×C47#180411yescharacteristic
237C2×C47#180611yescharacteristic
238D47—82no—
239C2×#121—471no—
240C2×C4×D4—471no—
241C2×C4×D4—941no—
242C22×C42C2×D4711yescharacteristic
243C2×C4⋊C8—21no—
244C2×C22⋊C8C2×D4712yes—
245#390—471no—
246#389—471no—
247#388—941no—
248#386—941no—
249#380—471no—
250C4×C4⋊C4—471no—
251C4×C22⋊C4—941no—
252C2×#95—472no—
253C2×#95—941no—
254C2×#95—941no—
255#336—21no—
256#327—944no—
257C47#857311yescharacteristic, complemented, Sylow
258C22×D4—471no—
259C22×D4—941no—
260C23×C4—942no—
261C23×C4—941no—
262C23×C4D18811yescharacteristic
263C23×C4D18811yescharacteristic
264C23×C4C22×D4711yescharacteristic
265C22×C8—22no—
266#121—472no—
267#121—472no—
268#121—941no—
269#121—941no—
270C4×D4—1882no—
271C4×D4—1884no—
272C2×C4⋊C4—471no—
273C2×C4⋊C4—471no—
274C2×C4⋊C4—471no—
275C2×C4⋊C4—941no—
276C2×C22⋊C4—941no—
277C2×C22⋊C4—941no—
278C2×C22⋊C4—941no—
279C2×C22⋊C4—941no—
280C2×C22⋊C4—941no—
281C2×C42—42no—
282C2×C42—941no—
283C2×C42C4×D4711yescharacteristic
284C2×C42C4×D4711yescharacteristic
285C2×C42C22⋊D4711yescharacteristic
286C2×C42C22⋊D4711yescharacteristic
287C4⋊C8—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290#95—944no—
291#95—941no—
292#95—941no—
293#95—941no—
294#95—942no—
295#95—942no—
296#95—1882no—
297C24—941no—
298C24C2×D18811yescharacteristic
299C2×D4—472no—
300C2×D4—942no—
301C2×D4—942no—
302C2×D4—941no—
303C2×D4—1881no—
304C2×D4—1881no—
305C2×D4—1882no—
306C2×D4—1882no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—941no—
316C22×C4—944no—
317C22×C4—941no—
318C22×C4—941no—
319C22×C4—1882no—
320C22×C4—1882no—
321C22×C4—1882no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4—1882no—
326C22×C4—1882no—
327C22×C4—1882no—
328C22×C4(C2×D47)⋊C411yescharacteristic
329C22×C4(C2×D47)⋊C411yescharacteristic
330C22×C4(C2×D47)⋊C411yescharacteristic
331C22×C4(C2×D47)⋊C411yescharacteristic
332C22×C4C2×C4×D4711yescharacteristic
333C22×C4D4×D4711yescharacteristic
334C22×C4#1062535211yescharacteristic
335C22×C4Q8×D4711yescharacteristic
336C22×C4#1062535411yescharacteristic
337C22×C4C2×C22⋊D4711yescharacteristic
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—472no—
342C4⋊C4—941no—
343C4⋊C4—1881no—
344C4⋊C4—1881no—
345C4⋊C4—1882no—
346C22⋊C4—942no—
347C22⋊C4—942no—
348C22⋊C4—1882no—
349C22⋊C4—1882no—
350C22⋊C4—1882no—
351C22⋊C4—1882no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—1882no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—942no—
362C23—1881no—
363C23—1881no—
364C23—1881no—
365C23—1881no—
366C23—1881no—
367C23—1881no—
368C23C2×(C2×D47)⋊C411yescharacteristic
369C23#1309399911yescharacteristic
370C23#1309400411yescharacteristic
371C23#1309404612yes—
372C23#1309405212yes—
373D4—1882no—
374D4—1882no—
375D4—1884no—
376D4—1884no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—942no—
398C2×C4—3764no—
399C2×C4—3762no—
400C2×C4—3762no—
401C2×C4—3762no—
402C2×C4—3762no—
403C2×C4—1884no—
404C2×C4—1884no—
405C2×C4—1884no—
406C2×C4—1881no—
407C2×C4—1881no—
408C2×C4—1881no—
409C2×C4—1881no—
410C2×C4—1881no—
411C2×C4—1881no—
412C2×C4—1881no—
413C2×C4—1881no—
414C2×C4#1309393812yes—
415C2×C4#1309395012yes—
416C2×C4C4⋊C4×D4711yescharacteristic
417C2×C4#1309398211yescharacteristic
418C2×C4#1309399511yescharacteristic
419C2×C4#1309400011yescharacteristic
420C8—82no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—22no—
429C22—3761no—
430C22—3761no—
431C22—3761no—
432C22—3761no—
433C22—1882no—
434C22—1882no—
435C22—1882no—
436C22#1568898011yescharacteristic
437C22C2×#1309393811yescharacteristic
438C22#1568909611yescharacteristic
439C22#1568912011yescharacteristic
440C22#1568912911yescharacteristic
441C22C2×#1309395011yescharacteristic
442C22#1568913911yescharacteristic
443C22#1568933912yes—
444C22#1568942412yes—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—3764no—
454C4—3761no—
455C4—3761no—
456C4—1882no—
457C2—22no—
458C2—22no—
459C2—3762no—
460C2#2024187511yescharacteristic
461C2#2024190511yescharacteristic
462C2#2024193611yescharacteristic
463C2#2024194511yescharacteristic
464C2#2024337311yescharacteristic
465C2#2024339911yescharacteristic
466C2#2024340511yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1797D47
hypocenterC47#8573
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20241615C2semidirect
4. #20241268C2semidirect
5. C47×#1797C2semidirect
7. C2×C4×D188C4semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
14. C4×C4⋊Dic47C4semidirect
225. #1797D47semidirect
257. C47#8573semidirect

Maximal subgroups

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

subgroupindex
C4×#130939982
C2×#156888582
#202416152
#202412682
C47×#17972
#857347