Finite Group Databasebeta

← #26318771

Subgroups — #26318771

Order 12032.

774 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245852C211yesmaximal, characteristic
2#20244860C211yesmaximal, characteristic, complemented
3#20244792C211yesmaximal, characteristic
4#20244713C211yesmaximal, characteristic, complemented
5#20244521C211yesmaximal, characteristic
6#20244008C212yesmaximal
7#20243959C212yesmaximal, complemented
8#20243956C212yesmaximal, complemented
9#20243634C212yesmaximal
10#20243531C211yesmaximal, characteristic, complemented
11C4×C4⋊Q8×C47C211yesmaximal, characteristic, complemented
12C2×C4×D188C2211yescharacteristic
13C2×#13093981—21no—
14C2×#13093957C2212yes—
15C2×#13093957C2212yes—
16C2×#13093944C2212yes—
17C2×#13093944C2212yes—
18#15689025C2211yescharacteristic
19#15689025C2211yescharacteristic
20#15689021C2211yescharacteristic
21#15689021C2211yescharacteristic
22#15689014—21no—
23#15689014—21no—
24C4×C4⋊Dic47C2211yescharacteristic
25C4×(C2×D47)⋊C4—21no—
26#15688885C2212yes—
27#15688885C2212yes—
28#15688885C2212yes—
29#15688885C2212yes—
30#15688878C2212yes—
31#15688878C2211yescharacteristic
32#15688878C2211yescharacteristic
33#15688878C2212yes—
34C2×C4⋊Q8×C47C2211yescharacteristic
35C2×C4×Q8×C47—21no—
36C47×#391—21no—
37C47×#389—21no—
38C47×#389C2212yes—
39C47×#384C2211yescharacteristic
40C4×C4⋊C4×C47—21no—
41C4×C4⋊C4×C47C2212yes—
42C43×C47C2211yescharacteristic
43C22×D188C2311yescharacteristic
44C22×C4×D47—21no—
45C22×C47⋊C8C2312yes—
46C22×C47⋊C8C2312yes—
47C4×D188—22no—
48C4×D188—22no—
49#13093981—44no—
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2311yescharacteristic
52C2×C4⋊Dic47C2311yescharacteristic
53C2×Dic47⋊C4—21no—
54C2×Dic47⋊C4—21no—
55C2×(C2×D47)⋊C4—21no—
56C2×(C2×D47)⋊C4—21no—
57C2×(C2×D47)⋊C4—21no—
58C2×(C2×D47)⋊C4—21no—
59C2×(C2×D47)⋊C4—21no—
60C2×C4×Dic47—21no—
61#13093957—24no—
62#13093957—24no—
63#13093944—24no—
64#13093944—24no—
65C47⋊#95—21no—
66C47⋊#95—21no—
67C47⋊#95—21no—
68C22×Q8×C47—21no—
69C4⋊Q8×C47—24no—
70C4×Q8×C47—44no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47C2312yes—
76C2×C4⋊C4×C47C2312yes—
77C2×C42×C47—22no—
78C2×C42×C47D411yescharacteristic
79C2×C42×C47D411yescharacteristic
80C2×C42×C47D411yescharacteristic
81C2×C42×C47D411yescharacteristic
82C2×C42×C47C2311yescharacteristic
83C2×C42×C47C2311yescharacteristic
84C2×C42×C47C2311yescharacteristic
85#95×C47—22no—
86#95×C47—22no—
87C23×D47—21no—
88C2×Q8×C47—24no—
89C2×Q8×C47—42no—
90C2×Q8×C47—42no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2411yescharacteristic
100C22×Dic47—21no—
101C22×Dic47—21no—
102C22×Dic47—21no—
103C2×D188—22no—
104C2×D188—22no—
105C2×D188—21no—
106C2×D188#4112yes—
107C2×C4×D47—24no—
108C2×C4×D47—42no—
109C2×C4×D47—42no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—42no—
115C4⋊C4×C47—44no—
116C4⋊C4×C47—42no—
117C4⋊C4×C47#4114yes—
118C42×C47—22no—
119C42×C47—22no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—22no—
123C42×C47—41no—
124C42×C47—41no—
125C42×C47—44no—
126C42×C47D814yes—
127(C2×D47)⋊C4—24no—
128(C2×D47)⋊C4—42no—
129(C2×D47)⋊C4—42no—
130(C2×D47)⋊C4—42no—
131(C2×D47)⋊C4—42no—
132C4⋊Dic47—21no—
133C4⋊Dic47—22no—
134C4⋊Dic47—22no—
135C4⋊Dic47#4112yes—
136Dic47⋊C4—42no—
137Dic47⋊C4—42no—
138C4×Dic47—42no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—22no—
141C2×C47⋊C8—22no—
142C2×C47⋊C8—22no—
143C2×C47⋊C8—22no—
144C2×C47⋊C8—22no—
145C23×C47C22×D411yescharacteristic
146C22×D47—22no—
147C22×D47—41no—
148C22×D47—41no—
149C22×D47—41no—
150C22×D47—41no—
151C22×D47—41no—
152C22×D47—41no—
153Q8×C47—48no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—24no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—21no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47C2×D812yes—
175C2×C4×C47C2×D812yes—
176C2×C4×C47C2×D812yes—
177C2×C4×C47#13712yes—
178C2×C4×C47C2×#4111yescharacteristic
179C2×C4×C47C2×#4111yescharacteristic
180C2×C4×C47C2×#4112yes—
181C2×C4×C472−1+412yes—
182C2×Dic47—22no—
183C2×Dic47—41no—
184C2×Dic47—41no—
185C2×Dic47—41no—
186C2×Dic47—41no—
187C2×Dic47—41no—
188C2×Dic47—41no—
189C2×Dic47—41no—
190C2×Dic47—41no—
191D188—24no—
192D188—42no—
193C4×D47—48no—
194C47⋊C8—42no—
195C47⋊C8—42no—
196#29388—471nomaximal, Sylow
197C22×C47#52511yescharacteristic
198C22×C47#52511yescharacteristic
199C22×C47#53611yescharacteristic
200C22×C47C22×D811yescharacteristic
201C22×C47C2×#13711yescharacteristic
202C22×C47#57511yescharacteristic
203C22×C47#57511yescharacteristic
204C2×D47—42no—
205C2×D47—42no—
206C2×D47—42no—
207C2×D47—81no—
208C2×D47—81no—
209C2×D47—81no—
210C2×D47—81no—
211C4×C47—22no—
212C4×C47—22no—
213C4×C47—22no—
214C4×C47—22no—
215C4×C47—44no—
216C4×C47—41no—
217C4×C47—41no—
218C4×C47—82no—
219C4×C47#55212yes—
220C4×C47#55612yes—
221Dic47—42no—
222Dic47—81no—
223Dic47—81no—
224C4×C4⋊Q8D4711yescharacteristic, complemented
225#2293—471no—
226#2083—471no—
227#2071—471no—
228#2057—471no—
229#2023—471no—
230#1925—472no—
231#1918—472no—
232#1918—472no—
233#1862—472no—
234#1839—471no—
235C2×C47#259911yescharacteristic
236C2×C47#310011yescharacteristic
237C2×C47#310011yescharacteristic
238C2×C47#311611yescharacteristic
239C2×C47#311711yescharacteristic
240C2×C47#322611yescharacteristic
241C2×C47#324011yescharacteristic
242D47—82no—
243C2×C4⋊Q8C2×D4711yescharacteristic
244C2×C4×Q8—21no—
245C2×C4×D4—471no—
246C2×#117—941no—
247C2×#107—472no—
248C2×#107—472no—
249C2×#102—472no—
250C2×#102—472no—
251#391—21no—
252#390—471no—
253#390—471no—
254#389—21no—
255#389—471no—
256#389—471no—
257#389C2×D4712yes—
258#388—941no—
259#388—941no—
260#384C2×D4711yescharacteristic
261C4×C4⋊C4—21no—
262C4×C4⋊C4—471no—
263C4×C4⋊C4C2×D4712yes—
264C4×C22⋊C4—941no—
265C43C2×D4711yescharacteristic
266#340—472no—
267#340—472no—
268#340—472no—
269#340—472no—
270#336—471no—
271#336—471no—
272#336—472no—
273#336—472no—
274C47#2938811yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×D4—471no—
277C23×C4—941no—
278C22×C8—472no—
279C22×C8—472no—
280C4⋊Q8—24no—
281C4×Q8—44no—
282C4×D4—942no—
283C4×D4—942no—
284#117—1884no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—471no—
290C2×C4⋊C4—471no—
291C2×C4⋊C4—471no—
292C2×C4⋊C4—941no—
293C2×C4⋊C4—941no—
294C2×C4⋊C4C22×D4712yes—
295C2×C4⋊C4C22×D4712yes—
296C2×C22⋊C4—941no—
297C2×C22⋊C4—941no—
298C2×C22⋊C4—941no—
299C2×C22⋊C4—941no—
300C2×C22⋊C4—941no—
301C2×C42—22no—
302C2×C42—941no—
303C2×C42C22⋊D4711yescharacteristic
304C2×C42C22⋊D4711yescharacteristic
305C2×C42C22⋊D4711yescharacteristic
306C2×C42C22⋊D4711yescharacteristic
307C2×C42C22×D4711yescharacteristic
308C2×C42C22×D4711yescharacteristic
309C2×C42C22×D4711yescharacteristic
310#107—944no—
311#107—944no—
312#102—944no—
313#102—944no—
314#95—22no—
315#95—22no—
316#95—941no—
317#95—941no—
318#95—941no—
319C24—941no—
320C2×Q8—24no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×D4—472no—
324C2×D4—941no—
325C2×D4—942no—
326C2×D4—942no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—944no—
330C22×C4—941no—
331C22×C4—941no—
332C22×C4—941no—
333C22×C4—1882no—
334C22×C4—1882no—
335C22×C4C2×C22⋊D4711yescharacteristic
336C22×C4C2×C22⋊D4711yescharacteristic
337C22×C4C2×C22⋊D4711yescharacteristic
338C22×C4C2×C22⋊D4711yescharacteristic
339C22×C4C2×C22⋊D4711yescharacteristic
340C22×C4C2×C22⋊D4711yescharacteristic
341C22×C4C23×D4711yescharacteristic
342C2×C8—942no—
343C2×C8—942no—
344C2×C8—942no—
345C2×C8—942no—
346C2×C8—942no—
347C2×C8—942no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—472no—
356C4⋊C4—941no—
357C4⋊C4—942no—
358C4⋊C4—942no—
359C4⋊C4—1882no—
360C4⋊C4—1882no—
361C4⋊C4#1062535014yes—
362C22⋊C4—944no—
363C22⋊C4—1882no—
364C22⋊C4—1882no—
365C22⋊C4—1882no—
366C22⋊C4—1882no—
367C42—22no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—41no—
373C42—41no—
374C42—44no—
375C42—1882no—
376C42#1062533414yes—
377C23—942no—
378C23—1881no—
379C23—1881no—
380C23—1881no—
381C23—1881no—
382C23—1881no—
383C23—1881no—
384C23C22×C22⋊D4711yescharacteristic
385Q8—48no—
386D4—944no—
387D4—1882no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—942no—
409C2×C4—1888no—
410C2×C4—1881no—
411C2×C4—1881no—
412C2×C4—1881no—
413C2×C4—1881no—
414C2×C4—1881no—
415C2×C4—1881no—
416C2×C4—1881no—
417C2×C4—1881no—
418C2×C4C2×#1062533412yes—
419C2×C4C2×#1062533412yes—
420C2×C4C2×#1062533412yes—
421C2×C4#1309405512yes—
422C2×C4#41×D4711yescharacteristic
423C2×C4#41×D4711yescharacteristic
424C2×C4C2×#1062535012yes—
425C2×C4#1309407212yes—
426C8—1882no—
427C8—1882no—
428C22—3761no—
429C22—3761no—
430C22—3761no—
431C22—3761no—
432C22—1882no—
433C22—1882no—
434C22—1882no—
435C22#1568963011yescharacteristic
436C22#1568963411yescharacteristic
437C22#1568969711yescharacteristic
438C22C22×#1062533411yescharacteristic
439C22C2×#1309405511yescharacteristic
440C22#1568992111yescharacteristic
441C22#1568992111yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—44no—
449C4—82no—
450C4—3761no—
451C4—3761no—
452C4—1882no—
453C4#1568981412yes—
454C4#1568984512yes—
455C2—3762no—
456C2#2024895911yescharacteristic
457C2#2025338411yescharacteristic
458C2#2025338811yescharacteristic
459C2#2025354411yescharacteristic
460C2#2025355911yescharacteristic
461C2#2025450911yescharacteristic
462C2#2025462811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×C4⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenterC4×C4⋊Q8D47
hypocenterC47#29388
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20244860C2semidirect
4. #20244713C2semidirect
7. #20243959C2semidirect
8. #20243956C2semidirect
10. #20243531C2semidirect
11. C4×C4⋊Q8×C47C2semidirect
224. C4×C4⋊Q8D47semidirect
274. C47#29388semidirect

Maximal subgroups

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

subgroupindex
#202458522
#202448602
#202447922
#202447132
#202445212
#202440082
#202439592
#202439562
#202436342
#202435312
C4×C4⋊Q8×C472
#2938847