Finite Group Databasebeta

← #26111070

Subgroups — #26111070

Order 12032.

664 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic
2C4⋊C4×D188C211yesmaximal, characteristic
3#20244350C211yesmaximal, characteristic, complemented
4C2×#15688850C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1981C211yesmaximal, characteristic, complemented
8C2×C4×D188—21no—
9C2×C4×D188C2211yescharacteristic
10C2×C4⋊C4×D47—21no—
11#15689400—22no—
12#15689400C414yescomplemented
13C2×#13093957—21no—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093944—21no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15689021—21no—
18#15688997—22no—
19#15688993C2211yescharacteristic
20#15688980—21no—
21#15688885—22no—
22#15688850—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×#103×C47—21no—
26C2×#103×C47—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C22×C4×D47—22no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—42no—
36C4×D188—42no—
37C4×D188C2×C412yes—
38C4×D188D412yes—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×C4⋊Dic47D412yes—
43C2×Dic47⋊C4—22no—
44C4⋊C4×D47—42no—
45C4⋊C4×D47—42no—
46C2×(C2×D47)⋊C4—21no—
47C2×(C2×D47)⋊C4—22no—
48#13093957—24no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)C2×C412yescomplemented
51#13093944—24no—
52C4×C47⋊C8—21no—
53C4×C47⋊C8—21no—
54C47⋊#95—22no—
55C22×Q8×C47—21no—
56C22×C8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47C2×C412yescomplemented
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C42×C47—21no—
66C2×C42×C47C2311yescharacteristic
67C4⋊C8×C47—21no—
68C4⋊C8×C47—21no—
69#103×C47—42no—
70#103×C47—42no—
71#95×C47—21no—
72C23×D47—21no—
73C2×Q8×C47—22no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—41no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—21no—
86C22×Dic47—22no—
87C2×D188—21no—
88C2×D188—21no—
89C2×D188—21no—
90C2×D188C22⋊C412yes—
91C2×D188SD1612yes—
92C2×D188Q1612yes—
93C2×C4×D47—22no—
94C2×C4×D47—22no—
95C2×C4×D47—41no—
96C2×C4×D47—41no—
97C2×C4×D47—41no—
98C2×C4×D47—41no—
99C2×C4×D47—44no—
100C2×C4×D47—44no—
101C2×C4×D47—82no—
102C2×C8×C47—41no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47D812yes—
113C42×C47—41no—
114C42×C47—41no—
115C42×C47C22×C411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47#4111yescharacteristic
118C42×C47#4111yescharacteristic
119(C2×D47)⋊C4—22no—
120(C2×D47)⋊C4—22no—
121(C2×D47)⋊C4—44no—
122C4⋊Dic47—21no—
123C4⋊Dic47—41no—
124C4⋊Dic47—41no—
125C4⋊Dic47—41no—
126C4⋊Dic47—41no—
127C4⋊Dic47—42no—
128C4⋊Dic47C22⋊C412yes—
129Dic47⋊C4—44no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C23×C47#12011yescharacteristic
137C22×D47—22no—
138C22×D47—22no—
139C22×D47—22no—
140C22×D47—41no—
141C22×D47—41no—
142C22×D47—41no—
143C22×D47—41no—
144Q8×C47—42no—
145Q8×C47—42no—
146Q8×C47—81no—
147Q8×C47—81no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—22no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—81no—
158C2×C4×C47—81no—
159C2×C4×C47C2×C22⋊C411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12311yescharacteristic
165C2×C4×C47C2×D811yescharacteristic
166C2×C4×C47C2×SD1611yescharacteristic
167C2×C4×C47C2×Q1611yescharacteristic
168C2×C4×C47#13511yescharacteristic
169C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C47#13711yescharacteristic
173C2×Dic47—22no—
174C2×Dic47—41no—
175C2×Dic47—41no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C2×Dic47—42no—
179D188—22no—
180D188—22no—
181D188—22no—
182D188#10314yescomplemented
183C4×D47—42no—
184C4×D47—42no—
185C4×D47—42no—
186C4×D47—42no—
187C4×D47—88no—
188C8×C47—81no—
189C47⋊C8—42no—
190C47⋊C8—42no—
191#8981—471nomaximal, Sylow
192C22×C47#38611yescharacteristic
193C22×C47#44711yescharacteristic
194C22×C47#44911yescharacteristic
195C22×C47#45011yescharacteristic
196C22×C47#45111yescharacteristic
197C22×C47#45511yescharacteristic
198C22×C47#45611yescharacteristic
199C2×D47—24no—
200C2×D47—42no—
201C2×D47—42no—
202C2×D47—42no—
203C2×D47—42no—
204C2×D47—42no—
205C2×D47—42no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—42no—
209C4×C47—82no—
210C4×C47—81no—
211C4×C47C2×#10311yescharacteristic
212C4×C47#41711yescharacteristic
213C4×C47C4×D811yescharacteristic
214C4×C47#44211yescharacteristic
215C4×C47#45911yescharacteristic
216C4×C47#46311yescharacteristic
217C4×C47#48311yescharacteristic
218C4×C47#48411yescharacteristic
219Dic47—42no—
220Dic47—82no—
221C2×#474—471no—
222D4×C4⋊C4—471no—
223#1990—471no—
224#1981D4711yescharacteristic, complemented
225C2×D4⋊C8—471no—
226#1296—471no—
227#1279—471no—
228C2×C47#163911yescharacteristic
229C2×C47#164411yescharacteristic
230C2×C47#164911yescharacteristic
231C2×C47#165411yescharacteristic
232C2×C47#187211yescharacteristic
233C2×C47#187911yescharacteristic
234C2×C47#188211yescharacteristic
235D47—44no—
236C2×C4⋊Q8C2×D4711yescharacteristic
237C2×C4×D4—471no—
238C2×C4×D4—941no—
239C22×C4⋊C4—941no—
240#474—474no—
241#474—942no—
242C2×#107—941no—
243C2×C4⋊C8—471no—
244C2×C4⋊C8C2×D4711yescharacteristic
245C2×#103—21no—
246C2×#103—21no—
247C2×#102—941no—
248C2×C4×C8—471no—
249#389—941no—
250#385—942no—
251#384—471no—
252#380—941no—
253C4×C4⋊C4C2×D4711yescharacteristic
254#340—942no—
255D4⋊C8—942no—
256C47#898111yescharacteristic, complemented, Sylow
257C22×Q8—21no—
258C22×D4—471no—
259C23×C4—942no—
260C23×C4—941no—
261C22×C8—21no—
262C22×C8—941no—
263C22×C8—941no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C4×D4712yescomplemented
268C4×D4—472no—
269C4×D4—472no—
270C4×D4—941no—
271C4×D4—941no—
272C4×D4—1882no—
273C4×D4—1882no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—471no—
277C2×C4⋊C4—472no—
278C2×C4⋊C4—942no—
279C2×C4⋊C4—941no—
280C2×C4⋊C4—941no—
281C2×C4⋊C4—1882no—
282C2×C4⋊C4—1882no—
283C2×C4⋊C4C22⋊D4711yescharacteristic
284C2×C4⋊C4C22⋊D4711yescharacteristic
285C2×C22⋊C4—941no—
286C2×C22⋊C4—942no—
287C2×C42—21no—
288C2×C42C22×D4711yescharacteristic
289#107—944no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—472no—
293C4⋊C8—941no—
294#103—42no—
295#103—42no—
296#102—944no—
297C4×C8—941no—
298C4×C8—941no—
299#95—21no—
300#95—942no—
301C24—941no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—472no—
309C2×D4—472no—
310C2×D4—472no—
311C2×D4—941no—
312C2×D4—941no—
313C2×D4—941no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—941no—
318C22×C4—942no—
319C22×C4—942no—
320C22×C4—942no—
321C22×C4—3762no—
322C22×C4—1884no—
323C22×C4—1884no—
324C22×C4—1881no—
325C22×C4—1881no—
326C22×C4—1881no—
327C22×C4—1881no—
328C22×C4D4×D4711yescharacteristic
329C22×C4D4×D4711yescharacteristic
330C22×C4C2×C22⋊D4711yescharacteristic
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—942no—
335C2×C8—942no—
336C2×C8—1881no—
337C2×C8—1881no—
338C2×C8—1881no—
339C2×C8—1881no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—472no—
348C4⋊C4—941no—
349C4⋊C4—1882no—
350C4⋊C4—1884no—
351C4⋊C4—1881no—
352C4⋊C4—1881no—
353C4⋊C4—1881no—
354C4⋊C4—1881no—
355C4⋊C4#1062533412yes—
356C22⋊C4—942no—
357C22⋊C4—942no—
358C22⋊C4—1884no—
359C42—41no—
360C42—41no—
361C42C2×C4×D4711yescharacteristic
362C42#1062535011yescharacteristic
363C42D4×D4711yescharacteristic
364C42#1062535211yescharacteristic
365C23—942no—
366C23—942no—
367C23—942no—
368C23—1881no—
369C23—1881no—
370C23—1881no—
371C23—1881no—
372C23#1309399211yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—81no—
376Q8—81no—
377D4—474no—
378D4—942no—
379D4—942no—
380D4—942no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—942no—
393C2×C4—3768no—
394C2×C4—1882no—
395C2×C4—1882no—
396C2×C4—1882no—
397C2×C4—1881no—
398C2×C4—1881no—
399C2×C4—1882no—
400C2×C4—1882no—
401C2×C4—1882no—
402C2×C4—1882no—
403C2×C4C22⋊C4×D4711yescharacteristic
404C2×C4(C22⋊D47)⋊C411yescharacteristic
405C2×C4C4×C22⋊D4711yescharacteristic
406C2×C4#1309399511yescharacteristic
407C2×C4#1309399711yescharacteristic
408C2×C4#1309400611yescharacteristic
409C2×C4C2×#1062533411yescharacteristic
410C2×C4SD16×D4711yescharacteristic
411C2×C4Q16×D4711yescharacteristic
412C2×C4#1309404411yescharacteristic
413C2×C4#1309404911yescharacteristic
414C2×C4#1309405011yescharacteristic
415C2×C4#1309405111yescharacteristic
416C2×C4#1309405611yescharacteristic
417C8—81no—
418C8—1882no—
419C8—1882no—
420C22—944no—
421C22—1882no—
422C22—1882no—
423C22—1882no—
424C22—1882no—
425C22—1882no—
426C22—1882no—
427C22#1568899911yescharacteristic
428C22#1568925611yescharacteristic
429C22#1568926611yescharacteristic
430C22#1568927211yescharacteristic
431C22#1568927711yescharacteristic
432C22#1568929911yescharacteristic
433C22#1568930411yescharacteristic
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4—3762no—
440C4—1882no—
441C4#103×D4711yescharacteristic
442C4#1568912311yescharacteristic
443C4C4×#1062533411yescharacteristic
444C4#1568923311yescharacteristic
445C4#1568931511yescharacteristic
446C4#1568933511yescharacteristic
447C4#1568945211yescharacteristic
448C4#1568945711yescharacteristic
449C2—1884no—
450C2#2024256511yescharacteristic
451C2#2024260011yescharacteristic
452C2#2024263511yescharacteristic
453C2#2024267011yescharacteristic
454C2#2024369211yescharacteristic
455C2#2024373911yescharacteristic
456C2#2024375411yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1981C2
socleC23×C47#120
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1981D47
hypocenterC47#8981
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. #20244350C2semidirect
5. #20241272C2semidirect
6. #20241229C2semidirect
7. C47×#1981C2semidirect
12. #15689400C4semidirect
50. C4⋊(C47⋊C8)C2×C4semidirect
60. C4⋊Q8×C47C2×C4semidirect
182. D188#103semidirect
224. #1981D47semidirect
256. C47#8981semidirect
267. C4⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C4⋊C4×D1882
#202443502
C2×#156888502
#202412722
#202412292
C47×#19812
#898147