Finite Group Databasebeta

← #26275183

Subgroups — #26275183

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253246C212yesmaximal
2#20251985C211yesmaximal, characteristic
3#20244901C212yesmaximal, complemented
4#20244735C212yesmaximal, complemented
5#20244173C211yesmaximal, characteristic, complemented
6#20244137C211yesmaximal, characteristic, complemented
7#20243618C212yesmaximal
8#20243551C211yesmaximal, characteristic, complemented
9#20243453C211yesmaximal, characteristic
10#20243227C211yesmaximal, characteristic
11C47×#3164C211yesmaximal, characteristic, complemented
12C2×#13094030C2211yescharacteristic
13#15689603C2212yes—
14#15689427—22no—
15#15689420—22no—
16#15689410—22no—
17#15689405—22no—
18C2×#13093960C2211yescharacteristic
19#15689166C2212yes—
20#15689166C2212yes—
21#15689162—22no—
22#15689160—21no—
23C2×#13093958C2211yescharacteristic
24#107×D47—21no—
25C2×#13093955C2212yes—
26#15689150C2212yes—
27C2×#13093950C2211yescharacteristic
28#15689122C2212yes—
29#15689117C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31#15688988C2212yes—
32#15688892C2212yes—
33#15688891C2212yes—
34#15688887C2211yescharacteristic
35#15688882C2211yescharacteristic
36#15688882C2212yes—
37C47×#544C2212yes—
38C47×#501—22no—
39C47×#500—21no—
40C47×#499—21no—
41C47×#428C2212yes—
42C47×#427C2211yescharacteristic
43C2×#107×C47C2211yescharacteristic
44C47×#405C2211yescharacteristic
45C2×#10625350C2311yescharacteristic
46#13094030—22no—
47#13094030Q814yescomplemented
48C2×#10625329C2312yes—
49C2×#10625329C2311yescharacteristic
50C2×#10625324—21no—
51C22×C47⋊C8C2311yescharacteristic
52C2×C8×D47—21no—
53C4×Dic94—22no—
54C4×D188—22no—
55(C22⋊D47)⋊C4—42no—
56#13093982—22no—
57C2×C4⋊Dic47C2312yes—
58C4⋊C4×D47—22no—
59C2×C4×Dic47C2311yescharacteristic
60#13093960—22no—
61#13093958—21no—
62#13093958—21no—
63#13093957—22no—
64#13093957—22no—
65#13093955—22no—
66#13093955—22no—
67#13093955—22no—
68#13093955—22no—
69C4⋊(C47⋊C8)—22no—
70C4⋊(C47⋊C8)—22no—
71#13093950Q814yescomplemented
72#13093947—22no—
73#13093947D411yescharacteristic
74#13093947D411yescharacteristic
75#13093944—22no—
76#13093944D411yescharacteristic
77#13093944D411yescharacteristic
78C8×Dic47—22no—
79C47⋊#95—42no—
80C2×M16×C47C2311yescharacteristic
81C22×C8×C47C2311yescharacteristic
82C4⋊Q8×C47—22no—
83C4⋊Q8×C47—22no—
84#125×C47—22no—
85#125×C47—22no—
86C22⋊Q8×C47—42no—
87#117×C47C2311yescharacteristic
88#117×C47C2312yes—
89C2×C4⋊C4×C47C2312yes—
90#107×C47—21no—
91#107×C47—21no—
92#107×C47—22no—
93#107×C47D411yescharacteristic
94#107×C47D411yescharacteristic
95#106×C47—22no—
96#106×C47D412yes—
97#97×C47—21no—
98C4×C8×C47—21no—
99C2×Q8×C47—42no—
100C22×C4×C47—22no—
101C22×C4×C47C2411yescharacteristic
102C2×C22⋊D47—21no—
103C22×Dic47—21no—
104C22×Dic47—22no—
105#10625350—21no—
106#10625350—21no—
107#10625350C2×Q812yes—
108#10625350C2×Q812yes—
109C2×D188C2×D411yescharacteristic
110C2×C4×D47—21no—
111C2×C4×D47—42no—
112C2×Dic94C2×D411yescharacteristic
113M16×C47—21no—
114M16×C47C2×Q812yes—
115C2×C8×C47—21no—
116C2×C8×C47C2×D411yescharacteristic
117C2×C8×C47C2×D411yescharacteristic
118C2×C8×C47C2×Q812yes—
119C2×C8×C47#4112yes—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—42no—
123C4⋊C4×C47—42no—
124C4⋊C4×C47—42no—
125C4⋊C4×C47C2×D411yescharacteristic
126C4⋊C4×C47C2×D411yescharacteristic
127C4⋊C4×C47C2×D412yes—
128C4⋊C4×C47C2×D412yes—
129C22⋊C4×C47—21no—
130C22⋊C4×C47—22no—
131C42×C47—21no—
132C42×C47—22no—
133(C2×D47)⋊C4—42no—
134C4⋊Dic47—22no—
135C4⋊Dic47—22no—
136Dic47⋊C4—42no—
137C4×Dic47—21no—
138C4×Dic47—42no—
139C4×Dic47C2×Q812yes—
140#10625329—21no—
141#10625329—22no—
142#10625329—22no—
143#10625329C2×Q812yes—
144C2×C47⋊C8—21no—
145C2×C47⋊C8—21no—
146C2×C47⋊C8—21no—
147C2×C47⋊C8—22no—
148C2×C47⋊C8C2×D411yescharacteristic
149C2×C47⋊C8C2×D411yescharacteristic
150#10625324—24no—
151C8×D47—24no—
152C23×C47H(5,𝔽2)11yescharacteristic
153C22×D47—21no—
154Q8×C47—162no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—42no—
160C2×C4×C47—82no—
161C2×C4×C47C22×D411yescharacteristic
162C2×C4×C47C22×D411yescharacteristic
163C2×C4×C47C22×Q811yescharacteristic
164C2×C4×C47C22×Q811yescharacteristic
165C2×C4×C47C2×#4111yescharacteristic
166C2×C4×C472−1+411yescharacteristic
167C22⋊D47—22no—
168C22⋊D47—22no—
169C2×Dic47—21no—
170C2×Dic47—22no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—42no—
174C2×Dic47—82no—
175D188—21no—
176D188C4⋊Q812yes—
177C4×D47—22no—
178C4×D47—22no—
179C4×D47—162no—
180Dic94—21no—
181Dic94C4⋊Q812yes—
182C8×C47—22no—
183C8×C47C22⋊Q812yes—
184C8×C47C22⋊Q812yes—
185C47⋊C8—22no—
186C47⋊C8—21no—
187C47⋊C8—21no—
188C47⋊C8—42no—
189#26082—471nomaximal, Sylow
190C22×C47—21no—
191C22×C47—21no—
192C22×C47#54411yescharacteristic
193C22×C47D4211yescharacteristic
194C22×C47#55411yescharacteristic
195C2×D47—22no—
196C2×D47—41no—
197C2×D47—41no—
198C4×C47—42no—
199C4×C47—41no—
200C4×C47—42no—
201C4×C47—41no—
202C4×C47—82no—
203C4×C47C2×C22⋊Q811yescharacteristic
204C4×C47C2×C4⋊Q811yescharacteristic
205C4×C47#54711yescharacteristic
206C4×C47D4×Q811yescharacteristic
207Dic47—22no—
208Dic47—42no—
209Dic47—82no—
210#3164D4711yescharacteristic, complemented
211#3084—472no—
212#2913—471no—
213#2090—472no—
214#2061—472no—
215#1952—471no—
216#1945—471no—
217#1860—472no—
218#1846—471no—
219#1816—471no—
220#1764—471no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47#244011yescharacteristic
224C2×C47#329011yescharacteristic
225C2×C47#329011yescharacteristic
226D47—42no—
227C2×#131—471no—
228#544C2×D4712yes—
229#520—472no—
230#501—22no—
231#500—21no—
232#499—21no—
233#479—942no—
234#478—942no—
235#476—942no—
236#475—942no—
237C2×#108—471no—
238#428—472no—
239#428—472no—
240#428—942no—
241#428C2×D4712yes—
242#427—941no—
243#427C2×D4711yescharacteristic
244C2×#107—471no—
245C2×#107—941no—
246C2×#107C2×D4711yescharacteristic
247C2×#106—472no—
248#424—472no—
249C2×#104—471no—
250#417—472no—
251#416—471no—
252#405C2×D4711yescharacteristic
253C2×C4×C8—471no—
254#382—472no—
255#343—472no—
256#343—472no—
257#341—471no—
258#339—471no—
259#339—472no—
260C47#2608211yescharacteristic, complemented, Sylow
261C2×#41—471no—
262#131—474no—
263#131—942no—
264C2×M16—471no—
265C2×M16—472no—
266C2×M16—941no—
267C2×M16C22×D4711yescharacteristic
268C22×C8—471no—
269C22×C8—941no—
270C22×C8C22×D4711yescharacteristic
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273#125—22no—
274#125—22no—
275C22⋊Q8—42no—
276C4×Q8—942no—
277C4×D4—942no—
278C4×D4—1882no—
279#117—942no—
280#117C22×D4711yescharacteristic
281#117C22×D4712yes—
282C2×C4⋊C4—472no—
283C2×C4⋊C4—942no—
284C2×C4⋊C4C22×D4712yes—
285C2×C42—471no—
286#108—942no—
287#107—21no—
288#107—21no—
289#107—22no—
290#107—941no—
291#107—941no—
292#107—942no—
293#107—942no—
294#107C22⋊D4711yescharacteristic
295#107C22⋊D4711yescharacteristic
296#106—22no—
297#106—942no—
298#106—942no—
299#106—942no—
300#106—942no—
301#106C22⋊D4712yes—
302C4⋊C8—942no—
303C4⋊C8—942no—
304#104—474no—
305#103—471no—
306#103—471no—
307#103—942no—
308#102—471no—
309#102—471no—
310#102—942no—
311#97—21no—
312C4×C8—21no—
313C4×C8—942no—
314#95—1882no—
315#41—472no—
316#41—472no—
317#41—941no—
318#41—941no—
319C2×Q8—42no—
320C2×Q8—471no—
321C2×D4—471no—
322C2×D4—941no—
323C22×C4—22no—
324C22×C4—941no—
325C22×C4—941no—
326C22×C4—942no—
327C22×C4—1882no—
328C22×C4C23×D4711yescharacteristic
329M16—21no—
330M16—472no—
331M16—941no—
332M16—944no—
333M16—942no—
334M16—942no—
335M16Q8×D4712yes—
336C2×C8—21no—
337C2×C8—471no—
338C2×C8—471no—
339C2×C8—941no—
340C2×C8—941no—
341C2×C8—941no—
342C2×C8—944no—
343C2×C8—942no—
344C2×C8Q8×D4712yes—
345C2×C8#1062535412yes—
346C2×C8C2×C22⋊D4711yescharacteristic
347C2×C8C2×C22⋊D4711yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—942no—
354C4⋊C4—942no—
355C4⋊C4—1882no—
356C4⋊C4D4×D4711yescharacteristic
357C4⋊C4D4×D4711yescharacteristic
358C4⋊C4C2×C22⋊D4712yes—
359C4⋊C4C2×C22⋊D4712yes—
360C22⋊C4—22no—
361C22⋊C4—21no—
362C22⋊C4—1882no—
363C42—22no—
364C42—21no—
365C42—472no—
366C42—941no—
367C42—1882no—
368C23—941no—
369C23#1309406911yescharacteristic
370Q8—162no—
371Q8—472no—
372Q8—941no—
373D4—472no—
374D4—942no—
375D4—942no—
376D4—941no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—941no—
384C2×C4—942no—
385C2×C4—942no—
386C2×C4—942no—
387C2×C4—3762no—
388C2×C4—1882no—
389C2×C4—1881no—
390C2×C4—1881no—
391C2×C4—7522no—
392C2×C4C2×D4×D4711yescharacteristic
393C2×C4C22×C22⋊D4711yescharacteristic
394C2×C4C2×Q8×D4711yescharacteristic
395C2×C4C2×Q8×D4711yescharacteristic
396C2×C4C2×#1062535411yescharacteristic
397C2×C4#1309407211yescharacteristic
398C8—22no—
399C8—941no—
400C8—941no—
401C8—942no—
402C8—1882no—
403C8#1309400012yes—
404C8#1309400012yes—
405C22—21no—
406C22—21no—
407C22—942no—
408C22—1881no—
409C22—1881no—
410C22#1568975011yescharacteristic
411C22D4×C22⋊D4711yescharacteristic
412C22#1568983411yescharacteristic
413C4—41no—
414C4—42no—
415C4—42no—
416C4—41no—
417C4—82no—
418C4—942no—
419C4—3762no—
420C4—1882no—
421C4C2×#1309400011yescharacteristic
422C4C4⋊Q8×D4711yescharacteristic
423C4#1568977611yescharacteristic
424C4Q8×C22⋊D4711yescharacteristic
425C2—21no—
426C2—21no—
427C2—1882no—
428C2#2024708611yescharacteristic
429C2#2025517011yescharacteristic
430C2#2025517411yescharacteristic
subgroupG/H
centerC22D4×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#3164C2
socleC22×C47D42
prime radicalC2×#10625350C23
hypercenter#3164D47
hypocenterC47#26082
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3164

Complements

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

NH ≅ G/Ntype
3. #20244901C2semidirect
4. #20244735C2semidirect
5. #20244173C2semidirect
6. #20244137C2semidirect
8. #20243551C2semidirect
11. C47×#3164C2semidirect
47. #13094030Q8semidirect
71. #13093950Q8semidirect
210. #3164D47semidirect
260. C47#26082semidirect

Maximal subgroups

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

subgroupindex
#202532462
#202519852
#202449012
#202447352
#202441732
#202441372
#202436182
#202435512
#202434532
#202432272
C47×#31642
#2608247