Finite Group Databasebeta

← #26200232

Subgroups — #26200232

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689470C211yesmaximal, characteristic, complemented
2#20253203C211yesmaximal, characteristic, complemented
3#20252868C211yesmaximal, characteristic
4#20242804C214yesmaximal, complemented
5#20242797C214yesmaximal, complemented
6#20242144C211yesmaximal, characteristic, complemented
7#20242068C211yesmaximal, characteristic, complemented
8#20242055C211yesmaximal, characteristic, complemented
9C8×#41×C47C211yesmaximal, characteristic, complemented
10C2×#13094049—21no—
11C2×#13094024C2211yescharacteristic
12C2×#13094021C2211yescharacteristic
13#15689470—21no—
14#15689470—21no—
15#15689470C2214yes—
16#15689418C2212yescomplemented
17#15689413C2212yescomplemented
18#15689408—21no—
19#15689403—21no—
20#15689323—21no—
21#15689323C2212yes—
22#15689318—21no—
23#15689318C2212yes—
24#15689166—21no—
25#15689147C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689127—21no—
28#15689124—21no—
29C2×#13093946—22no—
30#15688874C2214yescomplemented
31#15688871C2214yescomplemented
32#15688866C2214yes—
33C4×#41×C47C2211yescharacteristic
34C8×Q8×C47C2212yescomplemented
35C8×D4×C47—22no—
36C8×D4×C47C2212yescomplemented
37C47×#431—21no—
38C47×#431C2211yescharacteristic
39C2×C4×C8×C47—21no—
40C2×C4×C8×C47C2211yescharacteristic
41C22×Dic94—21no—
42C22×D188—21no—
43C22×D188—21no—
44C22×D188—21no—
45#13094049—44no—
46C2×#10625336—22no—
47C2×#10625334—22no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50#13094024—21no—
51#13094024—21no—
52#13094024—21no—
53#13094024C2312yes—
54#13094021—21no—
55#13094021—21no—
56#13094021—21no—
57#13094021C2312yes—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—21no—
60#13093957—22no—
61#13093955—22no—
62C4⋊(C47⋊C8)—21no—
63C4⋊(C47⋊C8)—21no—
64C4⋊(C47⋊C8)C2312yes—
65C4⋊(C47⋊C8)C2312yes—
66#13093948—22no—
67#13093946—24no—
68#13093946—24no—
69#13093945—22no—
70#13093944—22no—
71#13093944—22no—
72C2×#41×C47D411yescharacteristic
73C22×C8×C47—22no—
74C22×C8×C47D412yes—
75C4×Q8×C47D411yescharacteristic
76C4×Q8×C47C2311yescharacteristic
77C4×D4×C47—21no—
78C4×D4×C47—21no—
79C4×D4×C47D411yescharacteristic
80C4×D4×C47C2311yescharacteristic
81#117×C47—21no—
82#117×C47D411yescharacteristic
83C2×C42×C47—21no—
84C2×C42×C47C2311yescharacteristic
85C4⋊C8×C47—22no—
86C4⋊C8×C47C2312yes—
87C22⋊C8×C47—21no—
88C22⋊C8×C47—22no—
89C4×C8×C47—22no—
90C4×C8×C47—21no—
91C4×C8×C47D411yescharacteristic
92C4×C8×C47D411yescharacteristic
93C4×C8×C47C2312yes—
94C23×D47—41no—
95#41×C47—22no—
96#41×C47SD1614yes—
97C2×Q8×C47C2×D411yescharacteristic
98C2×D4×C47—21no—
99C2×D4×C47C2×D411yescharacteristic
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47#4111yescharacteristic
106C22×Dic47—41no—
107C2×D188—22no—
108C2×D188—22no—
109C2×D188—22no—
110C2×D188—44no—
111C2×D188—42no—
112C2×D188—42no—
113C2×D188—41no—
114C2×D188—41no—
115C2×D188—41no—
116C2×D188—82no—
117C2×Dic94—22no—
118C2×Dic94—41no—
119C2×Dic94—82no—
120C2×C8×C47—22no—
121C2×C8×C47—42no—
122C2×C8×C47—44no—
123C2×C8×C47C2×D412yes—
124C2×C8×C47C2×D412yes—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47C2×D411yescharacteristic
128C4⋊C4×C47C2×D411yescharacteristic
129C22⋊C4×C47—21no—
130C22⋊C4×C47—21no—
131C22⋊C4×C47—21no—
132C42×C47—21no—
133C42×C47—21no—
134C42×C47—21no—
135C42×C47C2×D411yescharacteristic
136C42×C47C2×D411yescharacteristic
137C42×C47#4111yescharacteristic
138C42×C47C2411yescharacteristic
139#10625336—44no—
140#10625334—44no—
141C4⋊Dic47—22no—
142C4⋊Dic47—22no—
143C4⋊Dic47—41no—
144C4⋊Dic47—41no—
145#10625329—42no—
146#10625329—42no—
147C2×C47⋊C8—22no—
148C2×C47⋊C8—22no—
149C23×C47—21no—
150C23×C47#12111yescharacteristic
151C22×D47—42no—
152C22×D47—82no—
153C22×D47—82no—
154C22×D47—82no—
155Q8×C47—21no—
156Q8×C47C2×SD1612yes—
157D4×C47—24no—
158D4×C47—21no—
159D4×C47C2×SD1612yes—
160C2×C4×C47—21no—
161C2×C4×C47—22no—
162C2×C4×C47—24no—
163C2×C4×C47—22no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47C2×SD1612yes—
172C2×C4×C47#13512yes—
173C2×C4×C47C22×D411yescharacteristic
174C2×C4×C47C22×D411yescharacteristic
175C2×C4×C47C2×#4111yescharacteristic
176C2×Dic47—42no—
177C2×Dic47—82no—
178D188—42no—
179D188—42no—
180D188—44no—
181D188—82no—
182D188—82no—
183D188—82no—
184D188—82no—
185Dic94—82no—
186Dic94—82no—
187C8×C47—22no—
188C8×C47—22no—
189C8×C47—42no—
190C47⋊C8—42no—
191C47⋊C8—42no—
192#19274—471nomaximal, Sylow
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—22no—
196C22×C47—41no—
197C22×C47—41no—
198C22×C47#46512yes—
199C22×C47C2×#12111yescharacteristic
200C2×D47—82no—
201C2×D47—82no—
202C2×D47—82no—
203C2×D47—164no—
204C4×C47—21no—
205C4×C47—22no—
206C4×C47—21no—
207C4×C47—41no—
208C4×C47—41no—
209C4×C47#46512yes—
210C4×C47#47212yes—
211C4×C47C22×SD1611yescharacteristic
212C4×C47C2×#13511yescharacteristic
213C4×C47#57512yes—
214Dic47—82no—
215C2×#486—471no—
216#3079—471no—
217#3038—471no—
218C8×#41D4711yescharacteristic, complemented
219#1678—474no—
220#1677—474no—
221#1566—471no—
222#1548—471no—
223#1545—471no—
224C2×C47—22no—
225C2×C47—42no—
226C2×C47#304011yescharacteristic
227C2×C47C2×#46511yescharacteristic
228C2×C47C2×#47211yescharacteristic
229D47—162no—
230C2×AGL(1,ℤ/8)—941no—
231C2×C4⋊Q8—471no—
232C2×#127—471no—
233C4×#41C2×D4711yescharacteristic
234#486—474no—
235#486—941no—
236#486—941no—
237#477—472no—
238#476—472no—
239#475—941no—
240#474—941no—
241#460—472no—
242#460—941no—
243#459—472no—
244#459—941no—
245C8×Q8C2×D4712yescomplemented
246C8×D4—22no—
247C8×D4C2×D4712yescomplemented
248#431—21no—
249#431C2×D4711yescharacteristic
250#428—941no—
251#423—471no—
252#423—471no—
253#418—941no—
254#417—941no—
255C2×#102—942no—
256C2×C4×C8—21no—
257C2×C4×C8C2×D4711yescharacteristic
258#334—474no—
259#333—474no—
260#331—474no—
261C47#1927411yescharacteristic, complemented, Sylow
262C2×#41D18811yescharacteristic
263C22×Q8—941no—
264C22×D4—941no—
265C22×D4—941no—
266C22×D4—941no—
267AGL(1,ℤ/8)—1884no—
268C2×SD16—942no—
269C2×D8—942no—
270C2×M16—941no—
271C2×M16—941no—
272C22×C8—22no—
273C22×C8C22⋊D4712yes—
274C4⋊Q8—472no—
275C4⋊Q8—941no—
276C4⋊Q8—941no—
277C4⋊Q8—941no—
278#127—472no—
279#127—941no—
280#127—941no—
281#127—941no—
282C4×Q8D18811yescharacteristic
283C4×Q8C22×D4711yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4D18811yescharacteristic
287C4×D4C22×D4711yescharacteristic
288#117—21no—
289#117D18811yescharacteristic
290C2×C4⋊C4—941no—
291C2×C4⋊C4—941no—
292C2×C42—21no—
293C2×C42C22×D4711yescharacteristic
294#107—942no—
295#106—942no—
296C4⋊C8—22no—
297C4⋊C8—472no—
298C4⋊C8—472no—
299C4⋊C8—941no—
300C4⋊C8—941no—
301C4⋊C8C22×D4712yes—
302#103—942no—
303#102—942no—
304#102—944no—
305#102—944no—
306#102—942no—
307#102—942no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8C22⋊D4711yescharacteristic
313C4×C8C22⋊D4711yescharacteristic
314C4×C8C22×D4712yes—
315C24—1881no—
316#41—22no—
317#41#1062532514yes—
318C2×Q8—942no—
319C2×Q8—3762no—
320C2×Q8—1881no—
321C2×Q8C2×D18811yescharacteristic
322C2×D4—21no—
323C2×D4—942no—
324C2×D4—942no—
325C2×D4—942no—
326C2×D4—3762no—
327C2×D4—1884no—
328C2×D4—1881no—
329C2×D4—1881no—
330C2×D4—1881no—
331C2×D4—1882no—
332C2×D4—1882no—
333C2×D4C2×D18811yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—1881no—
338C22×C4C2×D18811yescharacteristic
339C22×C4#1062535011yescharacteristic
340C22×C4C2×C22⋊D4711yescharacteristic
341SD16—1884no—
342D8—1884no—
343M16—1882no—
344M16—1882no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—942no—
349C2×C8—942no—
350C2×C8C2×C22⋊D4712yes—
351C2×C8C2×C22⋊D4712yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—942no—
355C4⋊C4—942no—
356C4⋊C4—1881no—
357C4⋊C4—1881no—
358C4⋊C4C2×D18811yescharacteristic
359C4⋊C4C2×D18811yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×D18811yescharacteristic
367C42#1062535011yescharacteristic
368C42C2×C22⋊D4711yescharacteristic
369C42C23×D4711yescharacteristic
370C23—21no—
371C23—3762no—
372C23—3762no—
373C23—3762no—
374C23—1882no—
375C23#1309399811yescharacteristic
376Q8—21no—
377Q8—3762no—
378Q8—3762no—
379Q8C2×#1062532512yes—
380D4—24no—
381D4—21no—
382D4—3762no—
383D4—3762no—
384D4—3762no—
385D4—3762no—
386D4—1882no—
387D4—1882no—
388D4—1884no—
389D4C2×#1062532512yes—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—3762no—
399C2×C4—1882no—
400C2×C4#1309399811yescharacteristic
401C2×C4#1309399811yescharacteristic
402C2×C4#1309399811yescharacteristic
403C2×C4C2×#1062532512yes—
404C2×C4#1309404612yes—
405C2×C4C22×C22⋊D4711yescharacteristic
406C2×C4C22×D18811yescharacteristic
407C2×C4C2×#1062535011yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—1882no—
412C8—1882no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—3762no—
419C22—3762no—
420C22—3762no—
421C22—7524no—
422C22#1568935212yes—
423C22C2×#1309399811yescharacteristic
424C4—22no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—3762no—
430C4#1568935212yes—
431C4#1568939312yes—
432C4C22×#1062532511yescharacteristic
433C4C2×#1309404611yescharacteristic
434C4#1568992112yes—
435C2—22no—
436C2—42no—
437C2—7522no—
438C2#2025288211yescharacteristic
439C2C2×#1568935211yescharacteristic
440C2C2×#1568939311yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC8×#41×C47C2
socleC22×C47C2×#121
prime radical#20252868C2
hypercenterC8×#41D47
hypocenterC47#19274
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15689470C2semidirect
2. #20253203C2semidirect
4. #20242804C2semidirect
5. #20242797C2semidirect
6. #20242144C2semidirect
7. #20242068C2semidirect
8. #20242055C2semidirect
9. C8×#41×C47C2semidirect
16. #15689418C22semidirect
17. #15689413C22semidirect
30. #15688874C22semidirect
31. #15688871C22semidirect
34. C8×Q8×C47C22semidirect
36. C8×D4×C47C22semidirect
218. C8×#41D47semidirect
245. C8×Q8C2×D47semidirect
247. C8×D4C2×D47semidirect
261. C47#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#156894702
#202532032
#202528682
#202428042
#202427972
#202421442
#202420682
#202420552
C8×#41×C472
#1927447