Finite Group Databasebeta

← #26112359

Subgroups — #26112359

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253176C211yesmaximal, characteristic
2#41×C47⋊C8C211yesmaximal, characteristic
3#20241886C211yesmaximal, characteristic, complemented
4#20241842C211yesmaximal, characteristic, complemented
5#20241283C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1806C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689430—21no—
10#15689423—21no—
11#15689418C411yescharacteristic, complemented
12#15689413C411yescharacteristic, complemented
13#15689408C411yescharacteristic, complemented
14#15689403C411yescharacteristic, complemented
15Q8×C47⋊C8—21no—
16D4×C47⋊C8—22no—
17D4×C47⋊C8—21no—
18#15689176—21no—
19#15689176C2211yescharacteristic
20#15689159—21no—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22#15689119—21no—
23C2×C4×C47⋊C8—21no—
24C2×C4×C47⋊C8C2211yescharacteristic
25#15688859—22no—
26#15688853—21no—
27#15688853—21no—
28C4×#41×C47C2211yescharacteristic
29C47×#423C2211yescharacteristic
30C2×#104×C47—21no—
31C4×C4⋊C4×C47C2211yescharacteristic
32C47×#339—21no—
33C22×C47⋊C8—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36#13094024C2×C411yescharacteristic
37#13094024C2×C411yescharacteristic
38#13094016—21no—
39#13094003—21no—
40#13094003—21no—
41C4×Dic94—21no—
42C4×Dic94—21no—
43#13093983—21no—
44#13093957—21no—
45#13093957—21no—
46#13093955—21no—
47#13093955—21no—
48C4⋊(C47⋊C8)—22no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)C2×C411yescharacteristic
52C4⋊(C47⋊C8)C2×C411yescharacteristic
53#13093948—21no—
54#13093948—21no—
55#13093945—21no—
56#13093945—21no—
57C22⋊(C47⋊C8)—22no—
58C22⋊(C47⋊C8)D412yes—
59C4×C47⋊C8—22no—
60C4×C47⋊C8—21no—
61C4×C47⋊C8—21no—
62C4×C47⋊C8D411yescharacteristic
63C4×C47⋊C8D411yescharacteristic
64C2×#41×C47D411yescharacteristic
65C2×M16×C47—21no—
66C4×Q8×C47C2×C411yescharacteristic
67C4×Q8×C47D411yescharacteristic
68C4×D4×C47—21no—
69C4×D4×C47—21no—
70C4×D4×C47C2×C411yescharacteristic
71C4×D4×C47D411yescharacteristic
72#117×C47—21no—
73#117×C47D411yescharacteristic
74C2×C4⋊C4×C47—21no—
75C2×C42×C47—21no—
76C2×C42×C47—21no—
77C2×C42×C47C2311yescharacteristic
78C4⋊C8×C47—21no—
79C4⋊C8×C47—21no—
80#104×C47—42no—
81#95×C47—21no—
82#41×C47—21no—
83#41×C47—21no—
84#41×C47SD1612yes—
85#41×C47Q1612yes—
86C2×Q8×C47C22⋊C411yescharacteristic
87C2×D4×C47—21no—
88C2×D4×C47C22⋊C411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—41no—
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47#4111yescharacteristic
96C2×Dic94—21no—
97C2×Dic94—21no—
98M16×C47—41no—
99M16×C47—41no—
100C2×C8×C47—41no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47C22⋊C411yescharacteristic
106C4⋊C4×C47C22⋊C411yescharacteristic
107C22⋊C4×C47—21no—
108C22⋊C4×C47—21no—
109C22⋊C4×C47—21no—
110C42×C47—21no—
111C42×C47—21no—
112C42×C47—21no—
113C42×C47—41no—
114C42×C47—41no—
115C42×C47C22×C411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47C2×D411yescharacteristic
118C42×C47#4111yescharacteristic
119C22⋊Dic47—41no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122C4⋊Dic47—21no—
123C4⋊Dic47—21no—
124Dic47⋊C4—41no—
125Dic47⋊C4—41no—
126C4×Dic47—41no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—42no—
133C2×C47⋊C8—44no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C23×C47—21no—
137C23×C47#12111yescharacteristic
138Q8×C47—21no—
139Q8×C47#10312yes—
140D4×C47—22no—
141D4×C47—22no—
142D4×C47—21no—
143D4×C47#10312yes—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—42no—
156C2×C4×C47—81no—
157C2×C4×C47—81no—
158C2×C4×C47C2×C22⋊C411yescharacteristic
159C2×C4×C47C4×D411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47#12011yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47#12311yescharacteristic
164C2×C4×C47C2×SD1611yescharacteristic
165C2×C4×C47C2×Q1611yescharacteristic
166C2×C4×C47#13511yescharacteristic
167C2×C4×C47#13511yescharacteristic
168C2×Dic47—41no—
169C2×Dic47—41no—
170Dic94—41no—
171Dic94—41no—
172Dic94—81no—
173C8×C47—81no—
174C47⋊C8—22no—
175C47⋊C8—42no—
176C47⋊C8—41no—
177C47⋊C8—41no—
178C47⋊C8—41no—
179#9207—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—22no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47#38611yescharacteristic
186C22×C47#46511yescharacteristic
187C22×C47#46711yescharacteristic
188C4×C47—21no—
189C4×C47—22no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47—82no—
195C4×C47C2×#10311yescharacteristic
196C4×C47#41611yescharacteristic
197C4×C47#44311yescharacteristic
198C4×C47#44411yescharacteristic
199C4×C47#46511yescharacteristic
200C4×C47#46711yescharacteristic
201Dic47—81no—
202Dic47—81no—
203#3076—471no—
204C8×#41—471no—
205#1806D4711yescharacteristic
206#1503—471no—
207#1493—471no—
208#1301—471no—
209#1279—471no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—42no—
213C2×C47#187611yescharacteristic
214C2×C47#189211yescharacteristic
215C2×C47#190611yescharacteristic
216#533—471no—
217C4×#41C2×D4711yescharacteristic
218#479—941no—
219#478—941no—
220#477—471no—
221#476—471no—
222#475—471no—
223#474—471no—
224C8×Q8—941no—
225C8×D4—942no—
226C8×D4—941no—
227#431—471no—
228#431—941no—
229#427—941no—
230#423C2×D4711yescharacteristic
231C2×C4⋊C8—471no—
232C2×#104—21no—
233#416—941no—
234C2×C4×C8—471no—
235C2×C4×C8—941no—
236C4×C4⋊C4C2×D4711yescharacteristic
237#339—21no—
238#328—942no—
239Q8⋊C8—941no—
240Q8⋊C8—941no—
241C47#920711yescharacteristic, complemented, Sylow
242C2×#41C22⋊D4711yescharacteristic
243C2×M16—21no—
244C22×C8—942no—
245C22×C8—941no—
246C22×C8—941no—
247C4⋊Q8—471no—
248C4⋊Q8—471no—
249#125—941no—
250C22⋊Q8—941no—
251C22⋊Q8—941no—
252C4×Q8—941no—
253C4×Q8—941no—
254C4×Q8C4×D4711yescharacteristic
255C4×Q8D18811yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C4×D4711yescharacteristic
259C4×D4D18811yescharacteristic
260#117—21no—
261#117—941no—
262#117C22⋊D4711yescharacteristic
263C2×C4⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×D4711yescharacteristic
267#107—941no—
268#107—941no—
269#106—941no—
270#106—941no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—471no—
274C4⋊C8—471no—
275C4⋊C8—942no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278#104—42no—
279#103—941no—
280#103—941no—
281#102—941no—
282#102—941no—
283C22⋊C8—472no—
284C22⋊C8—942no—
285C4×C8—471no—
286C4×C8—471no—
287C4×C8—942no—
288C4×C8—941no—
289C4×C8—941no—
290#95—21no—
291#41—21no—
292#41—21no—
293#41#1062533512yes—
294#41#1062533712yes—
295C2×Q8—941no—
296C2×Q8—941no—
297C2×Q8(C2×D47)⋊C411yescharacteristic
298C2×D4—21no—
299C2×D4(C2×D47)⋊C411yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4D4×D4711yescharacteristic
305C22×C4#1062535211yescharacteristic
306C22×C4C2×C22⋊D4711yescharacteristic
307M16—41no—
308M16—41no—
309C2×C8—41no—
310C2×C8—941no—
311C2×C8—941no—
312C2×C8—941no—
313C2×C8—941no—
314C2×C8—942no—
315C2×C8—1882no—
316C2×C8—1884no—
317C2×C8—1881no—
318C2×C8—1881no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—941no—
324C4⋊C4—941no—
325C4⋊C4—941no—
326C4⋊C4—941no—
327C4⋊C4—1881no—
328C4⋊C4—1881no—
329C4⋊C4(C2×D47)⋊C411yescharacteristic
330C4⋊C4(C2×D47)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—1881no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—41no—
340C42—1881no—
341C42C2×C4×D4711yescharacteristic
342C42C2×D18811yescharacteristic
343C42D4×D4711yescharacteristic
344C42#1062535211yescharacteristic
345C23—21no—
346C23#1309399611yescharacteristic
347Q8—21no—
348Q8—3761no—
349Q8—1881no—
350Q8—1881no—
351Q8#1309394712yes—
352D4—22no—
353D4—22no—
354D4—21no—
355D4#1309394712yes—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—1881no—
371C2×C4—1881no—
372C2×C4C2×(C2×D47)⋊C411yescharacteristic
373C2×C4(C22⋊D47)⋊C411yescharacteristic
374C2×C4(C22⋊D47)⋊C411yescharacteristic
375C2×C4#1309399411yescharacteristic
376C2×C4#1309399611yescharacteristic
377C2×C4#1309400811yescharacteristic
378C2×C4C2×#1062533511yescharacteristic
379C2×C4C2×#1062533711yescharacteristic
380C2×C4#1309404411yescharacteristic
381C2×C4#1309404411yescharacteristic
382C8—81no—
383C8—942no—
384C8—1881no—
385C8—1882no—
386C8—1881no—
387C8—1881no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22#1568900011yescharacteristic
394C22#1568935011yescharacteristic
395C22#1568936211yescharacteristic
396C4—21no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—3761no—
404C4—3761no—
405C4C2×#1309394711yescharacteristic
406C4#1568911711yescharacteristic
407C4#1568923711yescharacteristic
408C4#1568924211yescharacteristic
409C4#1568935011yescharacteristic
410C4#1568936211yescharacteristic
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2024371311yescharacteristic
415C2#2024380911yescharacteristic
416C2#2024388711yescharacteristic
subgroupG/H
centerC2×C4#13093996
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1806C2
socleC22×C47#386
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#1806D47
hypocenterC47#9207
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. #20241886C2semidirect
4. #20241842C2semidirect
5. #20241283C2semidirect
6. #20241229C2semidirect
11. #15689418C4semidirect
12. #15689413C4semidirect
13. #15689408C4semidirect
14. #15689403C4semidirect
241. C47#9207semidirect

Maximal subgroups

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

subgroupindex
#202531762
#41×C47⋊C82
#202418862
#202418422
#202412832
#202412292
C47×#18062
#920747