Finite Group Databasebeta

← #26107866

Subgroups — #26107866

Order 12032.

846 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243487C212yesmaximal, complemented
2C4×#13093939C211yesmaximal, characteristic, complemented
3#20241904C212yesmaximal, complemented
4#20241726C211yesmaximal, characteristic, complemented
5C2×C47×#555C211yesmaximal, characteristic
6C2×#13093939—21no—
7#15689087C2212yes—
8C4×C22⋊Dic47C2212yes—
9#15688890—21no—
10#15688882—24no—
11#15688857—24no—
12#15688849—21no—
13#15688847—21no—
14C47×#555—24no—
15C47×#555—24no—
16C2×C4⋊Q8×C47C411yescharacteristic, complemented
17C2×#124×C47—21no—
18C2×#124×C47—21no—
19C2×#121×C47—21no—
20C2×#121×C47C411yescharacteristic, complemented
21C2×#121×C47C2211yescharacteristic
22C2×C4×D4×C47C411yescharacteristic, complemented
23C2×C4×D4×C47C2211yescharacteristic
24C2×#117×C47—21no—
25C2×#117×C47C411yescharacteristic, complemented
26C2×#117×C47C2211yescharacteristic
27C2×#10625329—22no—
28C2×C22⋊Dic47—22no—
29C2×C4×Dic47—22no—
30#13093939—24no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—22no—
34C22×Q8×C47—21no—
35C22×D4×C47—21no—
36C22×D4×C47—21no—
37C22×D4×C47C2×C411yescharacteristic
38C22×D4×C47D411yescharacteristic
39C23×C4×C47—21no—
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C4⋊Q8×C47—22no—
43C4⋊Q8×C47—22no—
44#124×C47—28no—
45#124×C47—28no—
46#121×C47—28no—
47#121×C47—22no—
48#121×C47—22no—
49#121×C47—22no—
50#121×C47D414yes—
51C4×D4×C47—22no—
52C4×D4×C47—22no—
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55#117×C47—28no—
56#117×C47—22no—
57#117×C47—22no—
58#117×C47—22no—
59#117×C47D414yes—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47C2×C411yescharacteristic
62C2×C4⋊C4×C47C2×C411yescharacteristic
63C2×C4⋊C4×C47C2×C411yescharacteristic
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×C22⋊C4×C47D411yescharacteristic
71C2×C22⋊C4×C47D411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47—21no—
74C2×C42×C47C2×C411yescharacteristic
75C24×C47—21no—
76C24×C47C22⋊C411yescharacteristic
77C24×C47C2×D411yescharacteristic
78C2×Q8×C47—24no—
79C2×Q8×C47—82no—
80C2×D4×C47—22no—
81C2×D4×C47—24no—
82C2×D4×C47—24no—
83C2×D4×C47—22no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C2×D4×C47—42no—
87C2×D4×C47—82no—
88C2×D4×C47—82no—
89C2×D4×C47C22⋊C412yes—
90C22×C4×C47—24no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—42no—
97C22×C4×C47—42no—
98C22×C4×C47—42no—
99C22×C4×C47—42no—
100C22×C4×C47C22⋊C411yescharacteristic
101C22×C4×C47C22⋊C411yescharacteristic
102C22×C4×C47C22⋊C411yescharacteristic
103C22×C4×C47C22×C411yescharacteristic
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2×D411yescharacteristic
107C22×Dic47—42no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47C22⋊C412yes—
114C4⋊C4×C47C22⋊C412yes—
115C4⋊C4×C47C22⋊C412yes—
116C22⋊C4×C47—24no—
117C22⋊C4×C47—24no—
118C22⋊C4×C47—24no—
119C22⋊C4×C47—24no—
120C22⋊C4×C47—22no—
121C22⋊C4×C47—24no—
122C22⋊C4×C47—22no—
123C42×C47—24no—
124C42×C47—21no—
125C42×C47—21no—
126C42×C47—24no—
127C22⋊Dic47—24no—
128C22⋊Dic47—42no—
129C4×Dic47—44no—
130#10625329—44no—
131C2×C47⋊C8—42no—
132C23×C47—21no—
133C23×C47—21no—
134C23×C47—21no—
135C23×C47—22no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—42no—
141C23×C47—42no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—41no—
146C23×C47C2×C22⋊C411yescharacteristic
147C23×C47#12012yes—
148Q8×C47—84no—
149D4×C47—44no—
150D4×C47—42no—
151D4×C47—42no—
152D4×C47—84no—
153D4×C47—84no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—24no—
157C2×C4×C47—22no—
158C2×C4×C47—21no—
159C2×C4×C47—22no—
160C2×C4×C47—21no—
161C2×C4×C47—22no—
162C2×C4×C47—22no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—48no—
167C2×C4×C47—42no—
168C2×C4×C47—42no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—42no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47#10412yes—
175C2×C4×C47C2×C22⋊C411yescharacteristic
176C2×C4×C47C2×C22⋊C411yescharacteristic
177C2×C4×C47#12012yes—
178C2×Dic47—44no—
179C2×Dic47—82no—
180C2×Dic47—82no—
181C47⋊C8—82no—
182#8360—471nomaximal, Sylow
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—22no—
186C22×C47—22no—
187C22×C47—22no—
188C22×C47—41no—
189C22×C47—42no—
190C22×C47—41no—
191C22×C47—42no—
192C22×C47—42no—
193C22×C47—41no—
194C22×C47—81no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47—81no—
198C22×C47—82no—
199C22×C47—82no—
200C22×C47#37911yescharacteristic
201C22×C47C2×#10411yescharacteristic
202C22×C47#42111yescharacteristic
203C4×C47—22no—
204C4×C47—21no—
205C4×C47—22no—
206C4×C47—42no—
207C4×C47—42no—
208C4×C47—41no—
209C4×C47—41no—
210C4×C47—42no—
211C4×C47#41012yes—
212Dic47—84no—
213C2×#555D4711yescharacteristic
214#1826—472no—
215C4×#99—471no—
216#1507—472no—
217#1465—471no—
218C2×C47—21no—
219C2×C47—21no—
220C2×C47—42no—
221C2×C47—81no—
222C2×C47—82no—
223C2×C47#178211yescharacteristic
224C2×C47#178911yescharacteristic
225C2×C47#179511yescharacteristic
226#555—24no—
227#555—24no—
228C2×C4⋊Q8Dic4711yescharacteristic, complemented
229C2×#124—21no—
230C2×#124—21no—
231C2×#121—21no—
232C2×#121Dic4711yescharacteristic, complemented
233C2×#121C2×D4711yescharacteristic
234C2×C4×D4Dic4711yescharacteristic, complemented
235C2×C4×D4C2×D4711yescharacteristic
236C2×#117—21no—
237C2×#117Dic4711yescharacteristic, complemented
238C2×#117C2×D4711yescharacteristic
239C2×#99—941no—
240#407—472no—
241C4×C22⋊C4—472no—
242#342—941no—
243#339—944no—
244#327—944no—
245#324—941no—
246#323—941no—
247C47#836011yescharacteristic, complemented, Sylow
248C22×Q8—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×Dic4711yescharacteristic
252C22×D4C22⋊D4711yescharacteristic
253C23×C4—21no—
254C23×C4C2×Dic4711yescharacteristic
255C23×C4C22×D4711yescharacteristic
256C2×M16—942no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259#124—28no—
260#124—28no—
261#121—28no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121C22⋊D4714yes—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270#117—28no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117C22⋊D4714yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×Dic4711yescharacteristic
277C2×C4⋊C4C2×Dic4711yescharacteristic
278C2×C4⋊C4C2×Dic4711yescharacteristic
279C2×C4⋊C4C22⋊D4711yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—942no—
286C2×C22⋊C4C22⋊D4711yescharacteristic
287C2×C22⋊C4C22⋊D4711yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—942no—
291C2×C42C2×Dic4711yescharacteristic
292#99—944no—
293C22⋊C8—942no—
294C22⋊C8—942no—
295#95—942no—
296C24—21no—
297C24C22⋊Dic4711yescharacteristic
298C24C2×C22⋊D4711yescharacteristic
299C2×Q8—24no—
300C2×Q8—82no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4C22⋊Dic4712yes—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—1882no—
322C22×C4C22⋊Dic4711yescharacteristic
323C22×C4C22⋊Dic4711yescharacteristic
324C22×C4C22⋊Dic4711yescharacteristic
325C22×C4C22×Dic4711yescharacteristic
326C22×C4C2×C22⋊D4712yes—
327C22×C4C2×C22⋊D4712yes—
328C22×C4C2×C22⋊D4711yescharacteristic
329M16—1884no—
330C2×C8—1882no—
331C4⋊C4—24no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4C22⋊Dic4712yes—
337C4⋊C4C22⋊Dic4712yes—
338C4⋊C4C22⋊Dic4712yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—24no—
345C22⋊C4—22no—
346C22⋊C4—944no—
347C22⋊C4—1882no—
348C42—24no—
349C42—21no—
350C42—21no—
351C42—24no—
352C42—1884no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C2×C22⋊Dic4711yescharacteristic
368C23#1309399312yes—
369Q8—84no—
370D4—44no—
371D4—42no—
372D4—42no—
373D4—84no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—3762no—
396C2×C4—3762no—
397C2×C4—1884no—
398C2×C4#1309395112yes—
399C2×C4C2×C22⋊Dic4711yescharacteristic
400C2×C4C2×C22⋊Dic4711yescharacteristic
401C2×C4#1309399312yes—
402C8—3762no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—82no—
419C22—82no—
420C22#1568897911yescharacteristic
421C22C2×#1309395111yescharacteristic
422C22#1568914011yescharacteristic
423C4—22no—
424C4—21no—
425C4—22no—
426C4—41no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—3764no—
432C4#1568909712yes—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—82no—
437C2—81no—
438C2#2024330611yescharacteristic
439C2#2024333211yescharacteristic
440C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#555C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#555D47
hypocenterC47#8360
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20243487C2semidirect
2. C4×#13093939C2semidirect
3. #20241904C2semidirect
4. #20241726C2semidirect
16. C2×C4⋊Q8×C47C4semidirect
20. C2×#121×C47C4semidirect
22. C2×C4×D4×C47C4semidirect
25. C2×#117×C47C4semidirect
228. C2×C4⋊Q8Dic47semidirect
232. C2×#121Dic47semidirect
234. C2×C4×D4Dic47semidirect
237. C2×#117Dic47semidirect
247. C47#8360semidirect

Maximal subgroups

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

subgroupindex
#202434872
C4×#130939392
#202419042
#202417262
C2×C47×#5552
#836047