Finite Group Databasebeta

← #26111132

Subgroups — #26111132

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10625329C211yesmaximal, characteristic
2#20244333C211yesmaximal, characteristic
3C4×#13093948C211yesmaximal, characteristic
4#20241843C211yesmaximal, characteristic
5#20241290C211yesmaximal, characteristic
6#20241224C211yesmaximal, characteristic
7C47×#3104C211yesmaximal, characteristic
8#15689254—22no—
9#15689254—21no—
10Q8×C47⋊C8—21no—
11#15689147—21no—
12#15689147C2211yescharacteristic
13C2×#13093948—21no—
14C4×#10625329—21no—
15C4×#10625329C2211yescharacteristic
16C2×C4×C47⋊C8C2211yescharacteristic
17C4×C4⋊Dic47C2211yescharacteristic
18#15688885—21no—
19#15688882—21no—
20#15688882—21no—
21Q8⋊(C47⋊C8)—22no—
22C47×#533C2211yescharacteristic
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#484—21no—
25C47×#477C411yescharacteristic, complemented
26C47×#477C411yescharacteristic, complemented
27C47×#462C411yescharacteristic, complemented
28C47×#462C411yescharacteristic, complemented
29C47×#451—21no—
30C4×Q16×C47—21no—
31C4×Q16×C47—21no—
32C47×#431C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329D411yescharacteristic
35C2×#10625329D411yescharacteristic
36C22×C47⋊C8—21no—
37C2×C4⋊Dic47—21no—
38C2×C4×Dic47—21no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)—22no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D412yes—
43#13093948—24no—
44#13093935—22no—
45#13093935—21no—
46C4×C47⋊C8—22no—
47C4×C47⋊C8—21no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C47⋊#95—21no—
51C22×Q8×C47D411yescharacteristic
52C2×Q16×C47—21no—
53C2×Q16×C47—21no—
54C4⋊Q8×C47C2×C411yescharacteristic
55C4⋊Q8×C47C2×C411yescharacteristic
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47—21no—
62C4×Q8×C47—21no—
63C4×Q8×C47C2×C411yescharacteristic
64C4×Q8×C47C2×C411yescharacteristic
65C4×Q8×C47D411yescharacteristic
66C4×Q8×C47D411yescharacteristic
67#117×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C42×C47—21no—
71C2×C42×C47C2311yescharacteristic
72#107×C47—21no—
73#107×C47—21no—
74C4⋊C8×C47C2×C411yescharacteristic
75C4⋊C8×C47C2×C411yescharacteristic
76#103×C47—21no—
77#103×C47—21no—
78#103×C47—21no—
79#103×C47—21no—
80C22⋊C8×C47—21no—
81C4×C8×C47—21no—
82C2×Q8×C47—21no—
83C2×Q8×C47—22no—
84C2×Q8×C47—21no—
85C2×Q8×C47—21no—
86C2×Q8×C47—41no—
87C2×Q8×C47C22⋊C411yescharacteristic
88C2×Q8×C47C22⋊C411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—41no—
95Q16×C47—42no—
96Q16×C47—42no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47C22⋊C411yescharacteristic
111C4⋊C4×C47C22⋊C411yescharacteristic
112C22⋊C4×C47—41no—
113C42×C47—21no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47—41no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47#4111yescharacteristic
122C4⋊Dic47—22no—
123C4⋊Dic47—41no—
124C4×Dic47—41no—
125C4×Dic47—41no—
126#10625329—21no—
127#10625329—21no—
128#10625329—21no—
129#10625329—41no—
130#10625329—41no—
131#10625329Q1612yescomplemented
132C2×C47⋊C8—21no—
133C2×C47⋊C8—22no—
134C2×C47⋊C8—21no—
135C2×C47⋊C8—22no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C23×C47#12111yescharacteristic
139Q8×C47—21no—
140Q8×C47—21no—
141Q8×C47—21no—
142Q8×C47—42no—
143Q8×C47—42no—
144Q8×C47—41no—
145Q8×C47—41no—
146Q8×C47#10412yes—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—41no—
154C2×C4×C47—42no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
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×Q1611yescharacteristic
165C2×C4×C47#13511yescharacteristic
166C2×C4×C47#13711yescharacteristic
167C2×C4×C47#13711yescharacteristic
168C2×Dic47—42no—
169C2×Dic47—81no—
170C2×Dic47—81no—
171C8×C47—41no—
172C8×C47—41no—
173C47⋊C8—22no—
174C47⋊C8—42no—
175C47⋊C8—41no—
176C47⋊C8—42no—
177#8990—471nomaximal, Sylow
178C22×C47—21no—
179C22×C47—21no—
180C22×C47#38611yescharacteristic
181C22×C47#46211yescharacteristic
182C22×C47#46411yescharacteristic
183C4×C47—21no—
184C4×C47—22no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—41no—
188C4×C47—41no—
189C4×C47—41no—
190C4×C47—81no—
191C4×C47—81no—
192C4×C47C2×#10411yescharacteristic
193C4×C47#42111yescharacteristic
194C4×C47C4×Q1611yescharacteristic
195C4×C47#44111yescharacteristic
196C4×C47#45111yescharacteristic
197C4×C47#45211yescharacteristic
198Dic47—82no—
199#3104D4711yescharacteristic
200Q8×M16—471no—
201#1986—471no—
202C4×#103—471no—
203#1493—471no—
204#1303—471no—
205#1278—471no—
206C2×C47—21no—
207C2×C47—21no—
208C2×C47#187111yescharacteristic
209C2×C47#187811yescharacteristic
210C2×C47#188111yescharacteristic
211#533C2×D4711yescharacteristic
212C2×C4×Q8C2×D4711yescharacteristic
213#484—21no—
214#477Dic4711yescharacteristic, complemented
215#477Dic4711yescharacteristic, complemented
216#462Dic4711yescharacteristic, complemented
217#462Dic4711yescharacteristic, complemented
218#451—21no—
219#446—942no—
220#446—941no—
221C8×Q8—941no—
222C4×Q16—21no—
223C4×Q16—21no—
224#431C2×D4711yescharacteristic
225#423—471no—
226#423—941no—
227C2×#103—941no—
228C4×M16—471no—
229C4×M16—941no—
230C2×C4×C8—471no—
231C4×C4⋊C4—471no—
232#340—941no—
233#339—941no—
234#339—941no—
235Q8⋊C8—942no—
236C47#899011yescharacteristic, complemented, Sylow
237C22×Q8C22⋊D4711yescharacteristic
238C2×Q16—21no—
239C2×Q16—21no—
240C2×M16—471no—
241C2×M16—471no—
242C2×M16—941no—
243C22×C8—941no—
244C4⋊Q8C2×Dic4711yescharacteristic
245C4⋊Q8C2×Dic4711yescharacteristic
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C2×Dic4711yescharacteristic
254C4×Q8C2×Dic4711yescharacteristic
255C4×Q8C22⋊D4711yescharacteristic
256C4×Q8C22⋊D4711yescharacteristic
257#117—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—941no—
260C2×C4⋊C4C22⋊D4711yescharacteristic
261C2×C42—21no—
262C2×C42—941no—
263C2×C42C22×D4711yescharacteristic
264#107—21no—
265#107—21no—
266C4⋊C8—472no—
267C4⋊C8—942no—
268C4⋊C8—942no—
269C4⋊C8—941no—
270C4⋊C8C2×Dic4711yescharacteristic
271C4⋊C8C2×Dic4711yescharacteristic
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#103—944no—
277C22⋊C8—21no—
278#97—942no—
279#97—941no—
280C4×C8—21no—
281C4×C8—942no—
282C4×C8—941no—
283C4×C8—941no—
284C4×C8—941no—
285#95—941no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8C22⋊Dic4711yescharacteristic
292C2×Q8C22⋊Dic4711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—1881no—
296C22×C4D4×D4711yescharacteristic
297C22×C4#1062535211yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299Q16—42no—
300Q16—42no—
301M16—472no—
302M16—941no—
303M16—941no—
304M16—941no—
305M16—1881no—
306M16—1881no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—941no—
310C2×C8—942no—
311C2×C8—942no—
312C2×C8—941no—
313C2×C8—1881no—
314C2×C8—1881no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—942no—
327C4⋊C4—1881no—
328C4⋊C4C22⋊Dic4711yescharacteristic
329C4⋊C4C22⋊Dic4711yescharacteristic
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—41no—
336C42—1881no—
337C42—1881no—
338C42D4×D4711yescharacteristic
339C42#1062535211yescharacteristic
340C42C22×Dic4711yescharacteristic
341C42C2×C22⋊D4711yescharacteristic
342C23#1309399611yescharacteristic
343Q8—21no—
344Q8—21no—
345Q8—21no—
346Q8—42no—
347Q8—42no—
348Q8—41no—
349Q8—41no—
350Q8#1309395112yes—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—3761no—
363C2×C4—3761no—
364C2×C4—1882no—
365C2×C4C2×C22⋊Dic4711yescharacteristic
366C2×C4D4×Dic4711yescharacteristic
367C2×C4D4×Dic4711yescharacteristic
368C2×C4#1309399211yescharacteristic
369C2×C4#1309399611yescharacteristic
370C2×C4#1309400511yescharacteristic
371C2×C4Q16×D4711yescharacteristic
372C2×C4#1309404711yescharacteristic
373C2×C4#1309405611yescharacteristic
374C2×C4#1309405611yescharacteristic
375C8—41no—
376C8—41no—
377C8—942no—
378C8—1882no—
379C8—1881no—
380C8—1882no—
381C22—21no—
382C22—21no—
383C22#1568900311yescharacteristic
384C22#1568933111yescharacteristic
385C22#1568934711yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—3762no—
396C4C2×#1309395111yescharacteristic
397C4#1568914011yescharacteristic
398C4Q16×Dic4711yescharacteristic
399C4#1568923011yescharacteristic
400C4#1568927711yescharacteristic
401C4#1568928411yescharacteristic
402C2—21no—
403C2—21no—
404C2#2024369111yescharacteristic
405C2#2024373111yescharacteristic
406C2#2024375211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3104C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#3104D47
hypocenterC47#8990
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
25. C47×#477C4semidirect
26. C47×#477C4semidirect
27. C47×#462C4semidirect
28. C47×#462C4semidirect
131. #10625329Q16semidirect
214. #477Dic47semidirect
215. #477Dic47semidirect
216. #462Dic47semidirect
217. #462Dic47semidirect
236. C47#8990semidirect

Maximal subgroups

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

subgroupindex
Q8×#106253292
#202443332
C4×#130939482
#202418432
#202412902
#202412242
C47×#31042
#899047