Finite Group Databasebeta

← #26107906

Subgroups — #26107906

Order 12032.

744 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093943C211yesmaximal, characteristic, complemented
2#20241986C212yesmaximal
3#20241835C212yesmaximal, complemented
4#20241732C211yesmaximal, characteristic, complemented
5C2×C47×#556C211yesmaximal, characteristic
6#15689180C2212yes—
7C2×#13093943—21no—
8C4×#10625329C2212yes—
9#15688892—21no—
10#15688863—24no—
11Q8⋊(C47⋊C8)—24no—
12#15688849—21no—
13#15688849—21no—
14C47×#556—24no—
15C47×#556—24no—
16C2×C4⋊Q8×C47C411yescharacteristic
17C2×#126×C47—21no—
18C2×#126×C47—21no—
19C2×#124×C47C411yescharacteristic
20C2×#124×C47C2211yescharacteristic
21C2×C22⋊Q8×C47—21no—
22C2×C4×Q8×C47C411yescharacteristic
23C2×C4×Q8×C47C2211yescharacteristic
24C2×C4×Q8×C47C2211yescharacteristic
25C2×C4×D4×C47C411yescharacteristic
26C2×#117×C47—21no—
27C2×#10625329—22no—
28C4⋊(C47⋊C8)—22no—
29#13093943—24no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C4×C47⋊C8—22no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×Q8×C47C2×C411yescharacteristic
36C22×Q8×C47D411yescharacteristic
37C22×D4×C47C2×C411yescharacteristic
38C23×C4×C47—21no—
39C4⋊Q8×C47—22no—
40C4⋊Q8×C47—22no—
41#126×C47—28no—
42#126×C47—28no—
43#124×C47—22no—
44#124×C47—22no—
45#124×C47—22no—
46#124×C47D414yes—
47C22⋊Q8×C47—28no—
48C4×Q8×C47—22no—
49C4×Q8×C47—22no—
50C4×Q8×C47—22no—
51C4×Q8×C47—22no—
52C4×Q8×C47—22no—
53C4×Q8×C47D414yes—
54C4×D4×C47—22no—
55C4×D4×C47—22no—
56#117×C47—28no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47C2×C411yescharacteristic
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47—21no—
70C2×C42×C47—21no—
71C2×C42×C47—21no—
72C2×C42×C47C2×C411yescharacteristic
73C2×C42×C47C2×C411yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C24×C47—21no—
76C2×Q8×C47—22no—
77C2×Q8×C47—21no—
78C2×Q8×C47—21no—
79C2×Q8×C47—22no—
80C2×Q8×C47—41no—
81C2×Q8×C47—42no—
82C2×Q8×C47—42no—
83C2×Q8×C47C22⋊C412yes—
84C2×Q8×C47C22⋊C412yes—
85C2×D4×C47—21no—
86C2×D4×C47—42no—
87C2×D4×C47C22⋊C412yes—
88C22×C4×C47—24no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—42no—
93C22×C4×C47—42no—
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C22⋊C411yescharacteristic
98C22×C4×C47C22×C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C4⋊C4×C47—24no—
102C4⋊C4×C47—24no—
103C4⋊C4×C47—24no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47C22⋊C412yes—
110C22⋊C4×C47—24no—
111C22⋊C4×C47—24no—
112C22⋊C4×C47—24no—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—24no—
115C42×C47—24no—
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47—21no—
120C42×C47—24no—
121C42×C47C2×D412yes—
122C42×C47C2×D412yes—
123#10625329—24no—
124#10625329—42no—
125C2×C47⋊C8—42no—
126C2×C47⋊C8—42no—
127C23×C47—22no—
128C23×C47—42no—
129C23×C47—42no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47C2×C22⋊C411yescharacteristic
133Q8×C47—24no—
134Q8×C47—44no—
135Q8×C47—42no—
136Q8×C47—42no—
137Q8×C47—42no—
138D4×C47—42no—
139D4×C47—42no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—22no—
145C2×C4×C47—21no—
146C2×C4×C47—22no—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—41no—
154C2×C4×C47—48no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47#10412yes—
162C2×C4×C47C2×C22⋊C411yescharacteristic
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47#12012yes—
165C2×C4×C47#12012yes—
166C47⋊C8—44no—
167C47⋊C8—82no—
168#8368—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—42no—
172C22×C47—42no—
173C22×C47—42no—
174C22×C47—82no—
175C22×C47—82no—
176C22×C47#37911yescharacteristic
177C22×C47C2×#10411yescharacteristic
178C22×C47#42111yescharacteristic
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—22no—
182C4×C47—22no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—42no—
186C4×C47—42no—
187C4×C47—41no—
188C4×C47—42no—
189C4×C47#41312yes—
190C2×#556D4711yescharacteristic
191C4×#101—471no—
192#1527—472no—
193#1491—472no—
194#1467—471no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—82no—
198C2×C47#178511yescharacteristic
199C2×C47#178911yescharacteristic
200C2×C47#179611yescharacteristic
201#556—24no—
202#556—24no—
203C2×C4⋊Q8Dic4711yescharacteristic
204C2×#126—21no—
205C2×#126—21no—
206C2×#124Dic4711yescharacteristic
207C2×#124C2×D4711yescharacteristic
208C2×C22⋊Q8—21no—
209C2×C4×Q8Dic4711yescharacteristic
210C2×C4×Q8C2×D4711yescharacteristic
211C2×C4×Q8C2×D4711yescharacteristic
212C2×C4×D4Dic4711yescharacteristic
213C2×#117—21no—
214#432—472no—
215C2×#101—941no—
216C4×M16—472no—
217#343—941no—
218#329—944no—
219Q8⋊C8—944no—
220#324—941no—
221#324—941no—
222C47#836811yescharacteristic, complemented, Sylow
223C22×Q8C2×Dic4711yescharacteristic
224C22×Q8C2×Dic4711yescharacteristic
225C22×Q8C22⋊D4711yescharacteristic
226C22×D4C2×Dic4711yescharacteristic
227C23×C4—21no—
228C2×M16—942no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231#126—28no—
232#126—28no—
233#124—22no—
234#124—22no—
235#124—22no—
236#124C22⋊D4714yes—
237C22⋊Q8—28no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8C22⋊D4714yes—
244C4×D4—22no—
245C4×D4—22no—
246#117—28no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic4711yescharacteristic
252C2×C4⋊C4C22⋊D4711yescharacteristic
253C2×C4⋊C4C22⋊D4711yescharacteristic
254C2×C4⋊C4C22⋊D4711yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C2×Dic4711yescharacteristic
263C2×C42C2×Dic4711yescharacteristic
264C2×C42C22×D4711yescharacteristic
265C4⋊C8—942no—
266#101—944no—
267C22⋊C8—942no—
268#97—942no—
269#97—942no—
270C4×C8—942no—
271C24—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8C22⋊Dic4712yes—
280C2×Q8C22⋊Dic4712yes—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4C22⋊Dic4712yes—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊Dic4711yescharacteristic
291C22×C4C22⋊Dic4711yescharacteristic
292C22×C4C22⋊Dic4711yescharacteristic
293C22×C4C22⋊Dic4711yescharacteristic
294C22×C4C22×Dic4711yescharacteristic
295C22×C4C2×C22⋊D4711yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297M16—944no—
298M16—1882no—
299C2×C8—1882no—
300C2×C8—1882no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—21no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4C22⋊Dic4712yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C42—24no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—24no—
321C42C2×C22⋊D4712yes—
322C42C2×C22⋊D4712yes—
323C23—22no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23C2×C22⋊Dic4711yescharacteristic
329Q8—24no—
330Q8—44no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334D4—42no—
335D4—42no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4#1309395112yes—
358C2×C4C2×C22⋊Dic4711yescharacteristic
359C2×C4C2×C22⋊Dic4711yescharacteristic
360C2×C4#1309399312yes—
361C2×C4#1309399312yes—
362C8—3762no—
363C8—1884no—
364C22—21no—
365C22—21no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—82no—
370C22—82no—
371C22#1568897911yescharacteristic
372C22C2×#1309395111yescharacteristic
373C22#1568914011yescharacteristic
374C4—21no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4#1568910712yes—
385C2—21no—
386C2—21no—
387C2—82no—
388C2#2024331611yescharacteristic
389C2#2024333211yescharacteristic
390C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#556C2
socleC22×C47#379
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC2×#556D47
hypocenterC47#8368
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C4×#13093943C2semidirect
3. #20241835C2semidirect
4. #20241732C2semidirect
222. C47#8368semidirect

Maximal subgroups

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

subgroupindex
C4×#130939432
#202419862
#202418352
#202417322
C2×C47×#5562
#836847