Finite Group Databasebeta

← #26111745

Subgroups — #26111745

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094049C211yesmaximal, characteristic
2#20241775C211yesmaximal, characteristic, complemented
3C2×#15688852C211yesmaximal, characteristic, complemented
4#20241708C211yesmaximal, characteristic, complemented
5#20241237C211yesmaximal, characteristic, complemented
6#20241225C211yesmaximal, characteristic, complemented
7C8×#41×C47C211yesmaximal, characteristic, complemented
8C2×#13094049C411yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689233—21no—
11#15689221—21no—
12C4×#10625336—21no—
13C4×#10625334—21no—
14#15689176C2211yescharacteristic
15#15689166C411yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17#15689127C411yescharacteristic
18#15689124C411yescharacteristic
19C4×#10625329C2211yescharacteristic
20Q8⋊(C47⋊C8)—21no—
21#15688852—21no—
22#15688852—21no—
23D4⋊(C47⋊C8)—21no—
24#15688846—22no—
25C4×#41×C47C2211yescharacteristic
26C8×Q8×C47—21no—
27C8×D4×C47—22no—
28C8×D4×C47—21no—
29C47×#431—21no—
30C47×#431C2211yescharacteristic
31C2×C4×C8×C47—21no—
32C2×C4×C8×C47C2211yescharacteristic
33C22×D188C2×C411yescharacteristic
34C22×C4×D47—21no—
35#13094049—22no—
36#13094049C814yescomplemented
37C2×#10625336—21no—
38C2×#10625334—21no—
39C2×#10625329C2×C411yescharacteristic
40C2×#10625329C2×C411yescharacteristic
41C22×C47⋊C8—21no—
42C4×D188—21no—
43C4×D188—21no—
44C4×D188—21no—
45C4×D188—21no—
46C2×C4⋊Dic47C2×C411yescharacteristic
47C2×(C2×D47)⋊C4—21no—
48#13093957—21no—
49#13093955—21no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)—21no—
53#13093948—21no—
54#13093945—21no—
55#13093944—21no—
56#13093944—21no—
57C22⋊(C47⋊C8)—21no—
58#13093935—21no—
59C4×C47⋊C8—21no—
60C4×C47⋊C8—21no—
61C2×#41×C47C2×C411yescharacteristic
62C22×C8×C47—22no—
63C22×C8×C47—21no—
64C4×Q8×C47D411yescharacteristic
65C4×Q8×C47D411yescharacteristic
66C4×D4×C47—21no—
67C4×D4×C47—21no—
68C4×D4×C47D411yescharacteristic
69C4×D4×C47D411yescharacteristic
70#117×C47—21no—
71#117×C47C2×C411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47C2311yescharacteristic
74C4⋊C8×C47—22no—
75C4⋊C8×C47—21no—
76C22⋊C8×C47—22no—
77C22⋊C8×C47D412yes—
78C4×C8×C47—22no—
79C4×C8×C47—21no—
80C4×C8×C47—21no—
81C4×C8×C47D411yescharacteristic
82C4×C8×C47D411yescharacteristic
83C23×D47—21no—
84#41×C47—21no—
85#41×C47—21no—
86#41×C47C2×C812yes—
87#41×C47M1612yes—
88C2×Q8×C47C22⋊C411yescharacteristic
89C2×D4×C47—21no—
90C2×D4×C47C22⋊C411yescharacteristic
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47C22×C411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47#4111yescharacteristic
97C22×Dic47—21no—
98C2×D188—21no—
99C2×D188—21no—
100C2×D188—21no—
101C2×D188—21no—
102C2×D188—21no—
103C2×D188C2×C812yescomplemented
104C2×C4×D47—41no—
105C2×C4×D47—41no—
106C2×C4×D47—81no—
107C2×C4×D47—81no—
108C2×C8×C47—21no—
109C2×C8×C47—22no—
110C2×C8×C47—21no—
111C2×C8×C47—42no—
112C2×C8×C47—44no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47C22⋊C411yescharacteristic
116C4⋊C4×C47C22⋊C411yescharacteristic
117C22⋊C4×C47—21no—
118C22⋊C4×C47—21no—
119C22⋊C4×C47—21no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—21no—
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C42×C47#4111yescharacteristic
127#10625336—24no—
128#10625334—24no—
129(C2×D47)⋊C4—41no—
130(C2×D47)⋊C4—41no—
131C4⋊Dic47—21no—
132C4⋊Dic47—21no—
133#10625329—21no—
134#10625329—21no—
135#10625329—21no—
136#10625329C2×C812yescomplemented
137C2×C47⋊C8—21no—
138C2×C47⋊C8—21no—
139C2×C47⋊C8—41no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—81no—
142C23×C47—21no—
143C23×C47C4×D411yescharacteristic
144C22×D47—22no—
145C22×D47—41no—
146C22×D47—41no—
147C22×D47—81no—
148C22×D47—81no—
149Q8×C47—21no—
150Q8×C47C22⋊C812yes—
151D4×C47—22no—
152D4×C47—22no—
153D4×C47—21no—
154D4×C47C22⋊C812yes—
155C2×C4×C47—21no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47C2×C22⋊C411yescharacteristic
167C2×C4×C47C4×D411yescharacteristic
168C2×C4×C47#12011yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47#12311yescharacteristic
172C2×C4×C47C22×C811yescharacteristic
173C2×C4×C47C2×M1611yescharacteristic
174C2×C4×C47#13111yescharacteristic
175C2×C4×C47#13111yescharacteristic
176C2×Dic47—41no—
177C2×Dic47—81no—
178D188—22no—
179D188—22no—
180D188—41no—
181D188—41no—
182C4×D47—82no—
183C4×D47—81no—
184C4×D47—81no—
185C8×C47—22no—
186C8×C47—42no—
187C8×C47—41no—
188C47⋊C8—22no—
189C47⋊C8—41no—
190C47⋊C8—81no—
191C47⋊C8—81no—
192#9095—471nomaximal, Sylow
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—22no—
196C22×C47—41no—
197C22×C47—41no—
198C22×C47#38611yescharacteristic
199C22×C47C8×D411yescharacteristic
200C22×C47#43511yescharacteristic
201C2×D47—42no—
202C2×D47—82no—
203C2×D47—81no—
204C2×D47—81no—
205C2×D47—82no—
206C4×C47—22no—
207C4×C47—21no—
208C4×C47—21no—
209C4×C47—21no—
210C4×C47—41no—
211C4×C47—41no—
212C4×C47C2×C22⋊C811yescharacteristic
213C4×C47#40811yescharacteristic
214C4×C47C8×D411yescharacteristic
215C4×C47#43511yescharacteristic
216C4×C47#47111yescharacteristic
217C4×C47#47211yescharacteristic
218Dic47—81no—
219C8×#41D4711yescharacteristic, complemented
220C4×AGL(1,ℤ/8)—471no—
221#1477—471no—
222C2×D4⋊C8—471no—
223#1460—471no—
224#1283—471no—
225#1278—471no—
226C2×C47—21no—
227C2×C47—21no—
228C2×C47—42no—
229C2×C47#186911yescharacteristic
230C2×C47#190611yescharacteristic
231C2×C47#190711yescharacteristic
232D47—82no—
233C2×AGL(1,ℤ/8)—471no—
234C4×#41C2×D4711yescharacteristic
235C2×C4×D4—471no—
236C8×Q8—21no—
237#442—941no—
238#440—941no—
239C4×SD16—941no—
240C4×D8—941no—
241C8×D4—21no—
242C8×D4—22no—
243#431—21no—
244#431—471no—
245#431C2×D4711yescharacteristic
246#428—471no—
247C2×C4⋊C8—471no—
248#418—471no—
249#417—471no—
250C4×M16—471no—
251C2×C4×C8—21no—
252C2×C4×C8C2×D4711yescharacteristic
253Q8⋊C8—941no—
254D4⋊C8—941no—
255D4⋊C8—941no—
256D4⋊C8—941no—
257#323—942no—
258C47#909511yescharacteristic, complemented, Sylow
259C2×#41C4×D4711yescharacteristic
260C22×D4—471no—
261C23×C4—941no—
262AGL(1,ℤ/8)—474no—
263AGL(1,ℤ/8)—942no—
264C2×SD16—941no—
265C2×D8—941no—
266C2×M16—471no—
267C2×M16—471no—
268C22×C8—22no—
269C22×C8—21no—
270C22×C8—941no—
271C4×Q8D18811yescharacteristic
272C4×Q8C22⋊D4711yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—941no—
276C4×D4—941no—
277C4×D4—941no—
278C4×D4—941no—
279C4×D4D18811yescharacteristic
280C4×D4C22⋊D4711yescharacteristic
281#117—21no—
282#117C4×D4711yescharacteristic
283C2×C4⋊C4—471no—
284C2×C22⋊C4—941no—
285C2×C42—21no—
286C2×C42C22×D4711yescharacteristic
287#107—941no—
288#106—941no—
289C4⋊C8—22no—
290C4⋊C8—21no—
291C4⋊C8—941no—
292C4⋊C8—941no—
293C4⋊C8—941no—
294#103—941no—
295#102—941no—
296#102—941no—
297#102—941no—
298C22⋊C8—22no—
299C22⋊C8—941no—
300C22⋊C8C22⋊D4712yes—
301#97—941no—
302C4×C8—22no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—941no—
306C4×C8—941no—
307C4×C8C22⋊D4711yescharacteristic
308C4×C8C22⋊D4711yescharacteristic
309C24—941no—
310#41—21no—
311#41—21no—
312#41C8×D4712yes—
313#41#1062532412yes—
314C2×Q8(C2×D47)⋊C411yescharacteristic
315C2×D4—21no—
316C2×D4—472no—
317C2×D4—941no—
318C2×D4—941no—
319C2×D4—941no—
320C2×D4—941no—
321C2×D4—941no—
322C2×D4(C2×D47)⋊C411yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—941no—
327C22×C4—3761no—
328C22×C4—3761no—
329C22×C4—1881no—
330C22×C4—1881no—
331C22×C4C2×C4×D4711yescharacteristic
332C22×C4#1062535011yescharacteristic
333C22×C4C2×C22⋊D4711yescharacteristic
334SD16—944no—
335D8—944no—
336M16—472no—
337M16—941no—
338M16—941no—
339M16—941no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—42no—
344C2×C8—44no—
345C2×C8—941no—
346C2×C8—941no—
347C2×C8—3761no—
348C2×C8—1881no—
349C2×C8—1881no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—941no—
353C4⋊C4—941no—
354C4⋊C4(C2×D47)⋊C411yescharacteristic
355C4⋊C4(C2×D47)⋊C411yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—1881no—
360C22⋊C4—1881no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42C2×D18811yescharacteristic
365C42#1062535011yescharacteristic
366C42C2×C22⋊D4711yescharacteristic
367C42C2×C22⋊D4711yescharacteristic
368C23—21no—
369C23—942no—
370C23—3761no—
371C23—3761no—
372C23—1881no—
373C23—1881no—
374C23C4×C22⋊D4711yescharacteristic
375Q8—21no—
376Q8(C2×D47)⋊C812yes—
377D4—22no—
378D4—22no—
379D4—21no—
380D4—942no—
381D4—942no—
382D4—1881no—
383D4—1881no—
384D4(C2×D47)⋊C812yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—3762no—
397C2×C4—3761no—
398C2×C4—3761no—
399C2×C4—3761no—
400C2×C4—1881no—
401C2×C4C2×(C2×D47)⋊C411yescharacteristic
402C2×C4C4×C22⋊D4711yescharacteristic
403C2×C4#1309399311yescharacteristic
404C2×C4#1309399811yescharacteristic
405C2×C4#1309399811yescharacteristic
406C2×C4#1309400711yescharacteristic
407C2×C4C2×C8×D4711yescharacteristic
408C2×C4C2×#1062532411yescharacteristic
409C2×C4#1309403011yescharacteristic
410C2×C4#1309403011yescharacteristic
411C8—22no—
412C8—42no—
413C8—41no—
414C8—942no—
415C8—3761no—
416C8—3761no—
417C8—1881no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—3762no—
424C22—3762no—
425C22—3761no—
426C22—3761no—
427C22—1882no—
428C22#1568900211yescharacteristic
429C22C8×C22⋊D4711yescharacteristic
430C22#1568919411yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—3761no—
438C4C2×(C2×D47)⋊C811yescharacteristic
439C4#1568908911yescharacteristic
440C4C8×C22⋊D4711yescharacteristic
441C4#1568919411yescharacteristic
442C4#1568938611yescharacteristic
443C4#1568939311yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—3762no—
448C2#2024368011yescharacteristic
449C2#2024388911yescharacteristic
450C2#2024389611yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC8×#41×C47C2
socleC22×C47#386
prime radicalC2×#13094049C4
cosocleC2×C42×C47C23
hypercenterC8×#41D47
hypocenterC47#9095
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #20241775C2semidirect
3. C2×#15688852C2semidirect
4. #20241708C2semidirect
5. #20241237C2semidirect
6. #20241225C2semidirect
7. C8×#41×C47C2semidirect
36. #13094049C8semidirect
103. C2×D188C2×C8semidirect
136. #10625329C2×C8semidirect
219. C8×#41D47semidirect
258. C47#9095semidirect

Maximal subgroups

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

subgroupindex
C4×#130940492
#202417752
C2×#156888522
#202417082
#202412372
#202412252
C8×#41×C472
#909547