Finite Group Databasebeta

← #26134571

Subgroups — #26134571

Order 12032.

624 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688913C212yesmaximal, complemented
2C2×#15688910C212yesmaximal
3#20243376C211yesmaximal, characteristic, complemented
4C4×#13093941C211yesmaximal, characteristic, complemented
5C47×#2534C211yesmaximal, characteristic
6C2×#13093941C2212yes—
7C2×#13093939C2212yes—
8C4×#10625329—21no—
9#15688983—21no—
10#15688913—24no—
11#15688910—24no—
12#15688892—21no—
13#15688890—21no—
14C2×#123×C47C2211yescharacteristic
15C2×#120×C47C2211yescharacteristic
16C2×C4×D4×C47C2211yescharacteristic
17C22×C22⋊C4×C47C411yescharacteristic, complemented
18C47×#396—21no—
19C47×#394—21no—
20C47×#388—21no—
21C47×#386—21no—
22C47×#381C411yescharacteristic, complemented
23C47×#380C411yescharacteristic, complemented
24C47×#379C411yescharacteristic, complemented
25C2×#10625329—22no—
26C2×C4⋊Dic47—41no—
27C2×C22⋊Dic47—22no—
28#13093941—22no—
29#13093941D414yes—
30#13093939—22no—
31#13093939—22no—
32#13093935—41no—
33C4×C47⋊C8—41no—
34C47⋊#95—41no—
35C25×C47C2×C411yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47C2311yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47C2×C411yescharacteristic
40C23×C4×C47D411yescharacteristic
41#123×C47—22no—
42#123×C47—22no—
43#120×C47—22no—
44#120×C47—22no—
45C4×D4×C47—41no—
46C4×D4×C47—41no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47—41no—
57C2×C22⋊C4×C47—41no—
58C2×C22⋊C4×C47C2×C411yescharacteristic
59C2×C22⋊C4×C47C2×C411yescharacteristic
60C2×C22⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47D411yescharacteristic
62C2×C42×C47D411yescharacteristic
63#95×C47—21no—
64#95×C47—21no—
65#95×C47—21no—
66C24×C47—21no—
67C24×C47—22no—
68C24×C47—41no—
69C24×C47—42no—
70C24×C47—41no—
71C24×C47—81no—
72C24×C47C22×C411yescharacteristic
73C24×C47C2×D411yescharacteristic
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—42no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—81no—
80C2×D4×C47C2×D412yes—
81C2×D4×C47#4112yes—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—81no—
93C22×C4×C47—81no—
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C22⋊C411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×Dic47—42no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—81no—
103C22⋊C4×C47—22no—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—42no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—41no—
113C22⋊C4×C47—42no—
114C22⋊C4×C47—81no—
115C42×C47—21no—
116C42×C47—21no—
117C22⋊Dic47—42no—
118C22⋊Dic47—42no—
119C4⋊Dic47—82no—
120#10625329—24no—
121#10625329—42no—
122C2×C47⋊C8—42no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—22no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—41no—
135C23×C47—42no—
136C23×C47—42no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—82no—
141C23×C47—84no—
142C23×C47—84no—
143C23×C47—82no—
144C23×C47C2×C22⋊C411yescharacteristic
145C23×C47C4×D412yes—
146C23×C47#12411yescharacteristic
147C23×C47#12711yescharacteristic
148D4×C47—42no—
149D4×C47—42no—
150D4×C47—82no—
151D4×C47—82no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—81no—
172C2×C4×C47—81no—
173C2×C4×C47—81no—
174C2×C4×C47—82no—
175C2×C4×C47#9911yescharacteristic
176C2×C4×C47#9911yescharacteristic
177C2×C4×C47#12112yes—
178C2×Dic47—82no—
179C2×Dic47—82no—
180C2×Dic47—82no—
181C47⋊C8—44no—
182#13219—471nomaximal, Sylow
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—22no—
187C22×C47—21no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—42no—
191C22×C47—42no—
192C22×C47—42no—
193C22×C47—42no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47—41no—
197C22×C47—41no—
198C22×C47—42no—
199C22×C47—81no—
200C22×C47—81no—
201C22×C47—81no—
202C22×C47—84no—
203C22×C47—84no—
204C22×C47—81no—
205C22×C47—81no—
206C22×C47#39011yescharacteristic
207C22×C47C2×#9911yescharacteristic
208C22×C47#41011yescharacteristic
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—22no—
212C4×C47—41no—
213C4×C47—41no—
214C4×C47—41no—
215C4×C47—81no—
216C4×C47—81no—
217C4×C47—81no—
218Dic47—162no—
219#2534D4711yescharacteristic
220C2×#352—472no—
221C2×AΣL(1,𝔽16)—472no—
222#1799—471no—
223C4×#100—471no—
224C2×C47—21no—
225C2×C47—21no—
226C2×C47—41no—
227C2×C47—42no—
228C2×C47—41no—
229C2×C47—82no—
230C2×C47#196411yescharacteristic
231C2×C47#212011yescharacteristic
232C2×C47#212111yescharacteristic
233C2×#123C2×D4711yescharacteristic
234C2×#120C2×D4711yescharacteristic
235C2×C4×D4C2×D4711yescharacteristic
236C22×C22⋊C4Dic4711yescharacteristic, complemented
237C2×#100—472no—
238C2×#99—472no—
239C4×M16—941no—
240#396—21no—
241#394—21no—
242#388—21no—
243#386—21no—
244#381Dic4711yescharacteristic, complemented
245#380—941no—
246#380Dic4711yescharacteristic, complemented
247#379Dic4711yescharacteristic, complemented
248#352—944no—
249AΣL(1,𝔽16)—944no—
250#343—941no—
251#342—941no—
252C47#1321911yescharacteristic, complemented, Sylow
253C25C2×Dic4711yescharacteristic
254C22×D4—21no—
255C22×D4C22×D4711yescharacteristic
256C23×C4C2×Dic4711yescharacteristic
257C23×C4C2×Dic4711yescharacteristic
258C23×C4C22⋊D4711yescharacteristic
259C2×M16—942no—
260#123—22no—
261#123—22no—
262#120—22no—
263#120—22no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1881no—
269C2×C4⋊C4C22⋊D4711yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—942no—
279C2×C22⋊C4C2×Dic4711yescharacteristic
280C2×C22⋊C4C2×Dic4711yescharacteristic
281C2×C22⋊C4C2×Dic4711yescharacteristic
282C2×C22⋊C4C22⋊D4711yescharacteristic
283C2×C42C22⋊D4711yescharacteristic
284#100—474no—
285#100—942no—
286#99—942no—
287#99—942no—
288#97—1881no—
289C4×C8—1881no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—1881no—
294C24—21no—
295C24—22no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—81no—
300C24C22×Dic4711yescharacteristic
301C24C2×C22⋊D4711yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4D4×D4712yes—
309C2×D4#1062535212yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—1882no—
323C22×C4C22⋊Dic4711yescharacteristic
324C22×C4C22⋊Dic4711yescharacteristic
325C22×C4C22⋊Dic4711yescharacteristic
326C22×C4C22⋊Dic4711yescharacteristic
327C22×C4C2×C22⋊D4711yescharacteristic
328M16—944no—
329M16—1882no—
330C2×C8—1882no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—3762no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—1882no—
348C22⋊C4—1882no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23C2×C22⋊Dic4711yescharacteristic
373C23D4×Dic4712yes—
374C23#1309401111yescharacteristic
375C23#1309402011yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—3762no—
404C2×C4—3762no—
405C2×C4—3762no—
406C2×C4#1309393911yescharacteristic
407C2×C4#1309393911yescharacteristic
408C2×C4#1309399512yes—
409C8—1884no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22#1568902611yescharacteristic
434C22C2×#1309393911yescharacteristic
435C22#1568909711yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—7522no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2#2024422511yescharacteristic
453C2#2024502911yescharacteristic
454C2#2024503411yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#2534C2
socleC22×C47C2×#99
prime radicalC2×#120×C47C22
cosocleC22×D4×C47C23
hypercenter#2534D47
hypocenterC47#13219
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#15688913C2semidirect
3. #20243376C2semidirect
4. C4×#13093941C2semidirect
17. C22×C22⋊C4×C47C4semidirect
22. C47×#381C4semidirect
23. C47×#380C4semidirect
24. C47×#379C4semidirect
236. C22×C22⋊C4Dic47semidirect
244. #381Dic47semidirect
246. #380Dic47semidirect
247. #379Dic47semidirect
252. C47#13219semidirect

Maximal subgroups

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

subgroupindex
C2×#156889132
C2×#156889102
#202433762
C4×#130939412
C47×#25342
#1321947