Finite Group Databasebeta

← #26199789

Subgroups — #26199789

Order 12032.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254700C211yesmaximal, characteristic
2#20254516C211yesmaximal, characteristic
3#20252134C211yesmaximal, characteristic
4#20242908C212yesmaximal, complemented
5#20242884C212yesmaximal, complemented
6#20242784C212yesmaximal, complemented
7#20242760C212yesmaximal, complemented
8#20242140C211yesmaximal, characteristic, complemented
9#20242045C211yesmaximal, characteristic, complemented
10C2×#15688872C211yesmaximal, characteristic, complemented
11C47×#3165C211yesmaximal, characteristic, complemented
12C2×#13094032—21no—
13#15689673C2211yescharacteristic
14#15689519—21no—
15#15689471—21no—
16#15689441—22no—
17#15689410C2212yes—
18#15689410C2212yes—
19#15689410C2212yes—
20#15689408—21no—
21#15689354—22no—
22#15689325C2212yes—
23#15689323—21no—
24#15689202C2212yes—
25#15689196—22no—
26C8×D188C2212yes—
27#15689176C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30#15689091—21no—
31#15688876C2212yes—
32#15688874C2212yes—
33#15688872C2212yes—
34#15688872C2211yescharacteristic
35#15688872C2212yes—
36#15688872C2211yescharacteristic
37#15688869C2211yescharacteristic
38#15688869C2211yescharacteristic
39C47×#533C2211yescharacteristic
40C2×C4⋊Q8×C47C2211yescharacteristic
41C47×#501—22no—
42C47×#500C2212yes—
43C47×#498C2212yes—
44C47×#428—21no—
45C47×#428—21no—
46C4×M16×C47C2211yescharacteristic
47C2×#10625350—21no—
48C2×#10625336—22no—
49#13094032—44no—
50C2×#10625324—22no—
51C22×C47⋊C8—21no—
52C22×C47⋊C8—21no—
53C2×C8×D47—22no—
54#13094024D411yescharacteristic
55#13094021D411yescharacteristic
56#13094012—21no—
57#13093998—21no—
58#13093998D412yes—
59C4×D188—21no—
60C4×D188C2312yes—
61#13093955—22no—
62#13093955—22no—
63#13093955—22no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)C2312yes—
67C4⋊(C47⋊C8)C2311yescharacteristic
68C4⋊(C47⋊C8)C2311yescharacteristic
69C4⋊(C47⋊C8)C2311yescharacteristic
70C4⋊(C47⋊C8)C2311yescharacteristic
71#13093948—22no—
72#13093944—22no—
73#13093944—22no—
74#13093944—22no—
75C22⋊(C47⋊C8)—21no—
76(C2×D47)⋊C8—22no—
77(C2×D47)⋊C8—22no—
78C4×C47⋊C8—21no—
79C22×Q8×C47—21no—
80C2×M16×C47D411yescharacteristic
81C2×M16×C47D411yescharacteristic
82C4⋊Q8×C47—21no—
83C4⋊Q8×C47—21no—
84C4⋊Q8×C47—21no—
85C4⋊Q8×C47C2311yescharacteristic
86C4⋊Q8×C47C2312yes—
87C4⋊Q8×C47C2311yescharacteristic
88#125×C47—21no—
89C22⋊Q8×C47—21no—
90C22⋊Q8×C47—21no—
91C4×Q8×C47—21no—
92C4×Q8×C47—21no—
93#117×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C42×C47C2311yescharacteristic
97#107×C47—22no—
98#107×C47—22no—
99#106×C47—22no—
100#106×C47—22no—
101#97×C47D412yes—
102C4×C8×C47C2312yes—
103C2×Q8×C47—21no—
104C2×Q8×C47—21no—
105C2×Q8×C47—22no—
106C2×Q8×C47—41no—
107C2×Q8×C47—82no—
108C22×C4×C47—41no—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C2×C22⋊D47—22no—
113#10625350—42no—
114#10625350—42no—
115C2×D188—21no—
116C2×D188—21no—
117C2×D188C2×D412yes—
118C2×C4×D47—22no—
119C2×Dic94—21no—
120M16×C47—22no—
121M16×C47SD1614yescomplemented
122C2×C8×C47C2×D412yes—
123C2×C8×C47C2×D412yes—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47—22no—
128C4⋊C4×C47—21no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—41no—
131C4⋊C4×C47—41no—
132C4⋊C4×C47—41no—
133C4⋊C4×C47—41no—
134C22⋊C4×C47—41no—
135C42×C47—41no—
136C42×C47C2×D411yescharacteristic
137C42×C47C2×D411yescharacteristic
138C42×C47#4111yescharacteristic
139C42×C47C2411yescharacteristic
140#10625336—82no—
141(C2×D47)⋊C4—22no—
142(C2×D47)⋊C4—22no—
143C4⋊Dic47—21no—
144C4⋊Dic47C2×D412yes—
145C2×C47⋊C8—21no—
146C2×C47⋊C8—21no—
147C2×C47⋊C8—21no—
148C2×C47⋊C8—21no—
149C2×C47⋊C8—22no—
150C2×C47⋊C8—42no—
151C2×C47⋊C8—81no—
152#10625324—44no—
153C8×D47—44no—
154C23×C47#12111yescharacteristic
155C22×D47—22no—
156Q8×C47—41no—
157Q8×C47—41no—
158Q8×C47—81no—
159Q8×C47—82no—
160Q8×C47—82no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—42no—
167C2×C4×C47—82no—
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47C2×SD1612yes—
172C2×C4×C47#13712yes—
173C2×C4×C47C22×D411yescharacteristic
174C2×C4×C47C22×D411yescharacteristic
175C2×C4×C47C2×#4111yescharacteristic
176C22⋊D47—44no—
177C2×Dic47—22no—
178D188—22no—
179D188—22no—
180D188—42no—
181D188—81no—
182C4×D47—42no—
183C4×D47—42no—
184Dic94—42no—
185C8×C47—22no—
186C8×C47C2×SD1614yes—
187C47⋊C8—41no—
188C47⋊C8—41no—
189C47⋊C8—41no—
190C47⋊C8—42no—
191C47⋊C8—41no—
192#19233—471nomaximal, Sylow
193C22×C47—21no—
194C22×C47—21no—
195C22×C47#46112yes—
196C22×C47C2×#12111yescharacteristic
197C2×D47—42no—
198C2×D47—42no—
199C2×D47—42no—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—21no—
203C4×C47—81no—
204C4×C47—82no—
205C4×C47—81no—
206C4×C47#46112yes—
207C4×C47C22×SD1611yescharacteristic
208C4×C47C2×#13711yescharacteristic
209C4×C47#57711yescharacteristic
210C4×C47#57811yescharacteristic
211Dic47—42no—
212#3246—471no—
213#3227—471no—
214#3165D4711yescharacteristic, complemented
215#2935—471no—
216#1695—472no—
217#1691—472no—
218#1675—472no—
219#1671—472no—
220#1565—471no—
221#1542—471no—
222C2×#333—471no—
223C2×C47—22no—
224C2×C47C2×#46111yescharacteristic
225C2×C47#306311yescharacteristic
226C2×C47#306811yescharacteristic
227D47—82no—
228C2×#131—941no—
229#533C2×D4711yescharacteristic
230#532—471no—
231C2×C4⋊Q8C2×D4711yescharacteristic
232#501—22no—
233#500C2×D4712yes—
234#498—941no—
235#498C2×D4712yes—
236#486—941no—
237#481—942no—
238#476—472no—
239#476—472no—
240#476—472no—
241#475—941no—
242#465—942no—
243#461—472no—
244#460—941no—
245#436—472no—
246#435—942no—
247C8×D4—472no—
248#431—471no—
249#428—21no—
250#428—21no—
251C2×C4⋊C8—471no—
252C2×C4⋊C8—471no—
253#408—941no—
254C4×M16C2×D4711yescharacteristic
255#335—472no—
256#334—472no—
257#333—472no—
258#333—472no—
259#333—471no—
260#333—471no—
261#332—471no—
262#332—471no—
263C47#1923311yescharacteristic, complemented, Sylow
264C2×#41—941no—
265C22×Q8—21no—
266C2×SD16—942no—
267#131—1884no—
268C2×M16—942no—
269C2×M16C22⋊D4711yescharacteristic
270C2×M16C22⋊D4711yescharacteristic
271C22×C8—941no—
272C22×C8—941no—
273C22×C8—942no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—471no—
278C4⋊Q8C22×D4712yes—
279C4⋊Q8C22×D4711yescharacteristic
280C4⋊Q8C22×D4711yescharacteristic
281#127—471no—
282#125—21no—
283#124—941no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286#121—472no—
287#121—941no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×D4—472no—
291C4×D4—941no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C22×D4711yescharacteristic
296#107—22no—
297#107—22no—
298#106—22no—
299#106—22no—
300#106—942no—
301#106—942no—
302#106—942no—
303C4⋊C8—471no—
304C4⋊C8—471no—
305C4⋊C8—471no—
306C4⋊C8—471no—
307C4⋊C8—472no—
308C4⋊C8—941no—
309C4⋊C8—941no—
310#103—942no—
311#102—942no—
312#102—942no—
313#102—942no—
314C22⋊C8—941no—
315C22⋊C8—942no—
316C22⋊C8—942no—
317#97C22⋊D4712yes—
318C4×C8—941no—
319C4×C8C22×D4712yes—
320#41—1882no—
321#41—1882no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—941no—
328C2×D4—472no—
329C2×D4—942no—
330C2×D4—941no—
331C2×D4—941no—
332C22×C4—41no—
333C22×C4—942no—
334C22×C4D4×D4711yescharacteristic
335C22×C4#1062535211yescharacteristic
336C22×C4C2×C22⋊D4711yescharacteristic
337SD16—3762no—
338M16—22no—
339M16—1884no—
340M16#1062533614yescomplemented
341C2×C8—941no—
342C2×C8—941no—
343C2×C8—942no—
344C2×C8—941no—
345C2×C8—941no—
346C2×C8—3761no—
347C2×C8—1882no—
348C2×C8—1884no—
349C2×C8C2×C22⋊D4712yes—
350C2×C8C2×C22⋊D4712yes—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—472no—
362C4⋊C4—941no—
363C22⋊C4—41no—
364C22⋊C4—942no—
365C22⋊C4—942no—
366C42—41no—
367C42D4×D4711yescharacteristic
368C42#1062535211yescharacteristic
369C42C2×C22⋊D4711yescharacteristic
370C42C23×D4711yescharacteristic
371C23—942no—
372C23#1309399511yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—81no—
376Q8—82no—
377Q8—82no—
378Q8—1882no—
379D4—942no—
380D4—942no—
381D4—3761no—
382D4—1884no—
383D4—1882no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—942no—
392C2×C4—1882no—
393C2×C4—1882no—
394C2×C4#1309399511yescharacteristic
395C2×C4#1309399511yescharacteristic
396C2×C4#1309399511yescharacteristic
397C2×C4C2×#1062533612yes—
398C2×C4#1309405512yes—
399C2×C4C2×D4×D4711yescharacteristic
400C2×C4C22×C22⋊D4711yescharacteristic
401C2×C4C2×#1062535211yescharacteristic
402C8—22no—
403C8—1881no—
404C8—1881no—
405C8—1881no—
406C8—1882no—
407C8—1881no—
408C8C2×#1062533614yes—
409C22—21no—
410C22—21no—
411C22—1882no—
412C22—1882no—
413C22—1882no—
414C22#1568932512yes—
415C22C2×#1309399511yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4—1882no—
423C4#1568932512yes—
424C4C22×#1062533611yescharacteristic
425C4C2×#1309405511yescharacteristic
426C4#1568993211yescharacteristic
427C4#1568993911yescharacteristic
428C2—22no—
429C2—3762no—
430C2C2×#1568932511yescharacteristic
431C2#2025307311yescharacteristic
432C2#2025311211yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3165C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenter#3165D47
hypocenterC47#19233
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20242908C2semidirect
5. #20242884C2semidirect
6. #20242784C2semidirect
7. #20242760C2semidirect
8. #20242140C2semidirect
9. #20242045C2semidirect
10. C2×#15688872C2semidirect
11. C47×#3165C2semidirect
121. M16×C47SD16semidirect
214. #3165D47semidirect
263. C47#19233semidirect
340. M16#10625336semidirect

Maximal subgroups

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

subgroupindex
#202547002
#202545162
#202521342
#202429082
#202428842
#202427842
#202427602
#202421402
#202420452
C2×#156888722
C47×#31652
#1923347