Finite Group Databasebeta

← #26125455

Subgroups — #26125455

Order 12032.

562 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#20246000C211yesmaximal, characteristic
2C2×#15688847C211yesmaximal, characteristic
3#20241337C212yesmaximal, complemented
4#20241335C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C22×C22⋊Dic47C411yescharacteristic
7C22×C22⋊Dic47C411yescharacteristic
8C2×C22⋊(C47⋊C8)—21no—
9#15688996—21no—
10#15688996—21no—
11#15688995C411yescharacteristic
12#15688995C411yescharacteristic
13#15688995C2211yescharacteristic
14#15688995C2211yescharacteristic
15#15688981—21no—
16#15688981—21no—
17#15688847—21no—
18#15688847C2212yes—
19C22×C22⋊C4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#323—21no—
22C47×#323C2212yes—
23C23×Dic47—21no—
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C22×C47⋊C8—41no—
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—41no—
36C2×C22⋊Dic47—41no—
37C2×C22⋊Dic47C814yescomplemented
38C2×C22⋊Dic47C2×C411yescharacteristic
39C2×C22⋊Dic47C2×C411yescharacteristic
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—41no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C25×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C22×C8×C47—41no—
50C2×C22⋊C4×C47—41no—
51C2×C22⋊C4×C47—41no—
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C22⋊C4×C47C2311yescharacteristic
56C22⋊C8×C47—22no—
57C22⋊C8×C47—41no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—22no—
61C24×C47—81no—
62C24×C47—81no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C22⋊C411yescharacteristic
65C24×C47C2×C812yes—
66C24×C47M1612yes—
67C24×C47C22×C411yescharacteristic
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—81no—
73C22×C4×C47—81no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—24no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—41no—
84C22×Dic47—41no—
85C22×Dic47—81no—
86C22×Dic47—81no—
87C22×Dic47—81no—
88C22×Dic47—81no—
89C2×C8×C47—42no—
90C2×C8×C47—82no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C22⋊C4×C47—41no—
95C22⋊Dic47—24no—
96C22⋊Dic47—21no—
97C22⋊Dic47—21no—
98C22⋊Dic47—42no—
99C22⋊Dic47—41no—
100C22⋊Dic47—41no—
101C22⋊Dic47C2×C812yescomplemented
102C22⋊Dic47C2×C812yescomplemented
103Dic47⋊C4—81no—
104Dic47⋊C4—81no—
105Dic47⋊C4—81no—
106Dic47⋊C4—81no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—82no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—81no—
124C23×C47—81no—
125C23×C47—81no—
126C23×C47—81no—
127C23×C47—82no—
128C23×C47—82no—
129C23×C47—84no—
130C23×C47C22⋊C812yes—
131C23×C47C22⋊C812yes—
132C23×C47#9911yescharacteristic
133C23×C47#9911yescharacteristic
134C23×C47#10112yes—
135C23×C47C2×C22⋊C411yescharacteristic
136C23×C47C22×C811yescharacteristic
137C23×C47C2×M1611yescharacteristic
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—81no—
149C2×C4×C47—81no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162C2×Dic47—84no—
163C2×Dic47—82no—
164C2×Dic47—82no—
165C8×C47—82no—
166C47⋊C8—82no—
167#11453—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—22no—
175C22×C47—21no—
176C22×C47—21no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—41no—
181C22×C47—41no—
182C22×C47—42no—
183C22×C47—42no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—41no—
188C22×C47—82no—
189C22×C47—82no—
190C22×C47—82no—
191C22×C47—82no—
192C22×C47#32412yes—
193C22×C47#32412yes—
194C22×C47C2×C22⋊C811yescharacteristic
195C22×C47C2×#9911yescharacteristic
196C22×C47C2×#10111yescharacteristic
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201Dic47—42no—
202Dic47—42no—
203Dic47—81no—
204Dic47—81no—
205#2315—471no—
206C2×#323—471no—
207C2×#323D4711yescharacteristic
208#1326—472no—
209#1326—472no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—22no—
213C2×C47—22no—
214C2×C47—42no—
215C2×C47—41no—
216C2×C47—41no—
217C2×C47C2×#32411yescharacteristic
218C2×C47#212111yescharacteristic
219C2×C47#212111yescharacteristic
220C22×C22⋊C4—471no—
221C22×C22⋊C4—471no—
222C22×C22⋊C4C2×D4711yescharacteristic
223C2×C22⋊C8—21no—
224C2×C22⋊C8—941no—
225#385—471no—
226#385—471no—
227#385—471no—
228#385—471no—
229#385—941no—
230#385—941no—
231#380—941no—
232#380—941no—
233#323—21no—
234#323—472no—
235#323—941no—
236#323C2×D4712yes—
237C47#1145311yescharacteristic, complemented, Sylow
238C25C4×D4711yescharacteristic
239C23×C4—471no—
240C23×C4—471no—
241C23×C4—941no—
242C23×C4D18811yescharacteristic
243C23×C4C22⋊D4711yescharacteristic
244C22×C8—41no—
245C22×C8—1881no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4—941no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—474no—
253C2×C22⋊C4—471no—
254C2×C22⋊C4—471no—
255C2×C22⋊C4—942no—
256C2×C22⋊C4—941no—
257C2×C22⋊C4—941no—
258C2×C22⋊C4—941no—
259C2×C22⋊C4—1881no—
260C2×C22⋊C4—1881no—
261C2×C22⋊C4C4×D4711yescharacteristic
262C2×C22⋊C4D18811yescharacteristic
263C2×C22⋊C4C22⋊D4711yescharacteristic
264C2×C22⋊C4C22×D4711yescharacteristic
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—942no—
268C22⋊C8—1881no—
269#95—941no—
270#95—941no—
271#95—941no—
272#95—941no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—81no—
277C24—81no—
278C24C8×D4712yes—
279C24#1062532412yes—
280C24(C2×D47)⋊C411yescharacteristic
281C24(C2×D47)⋊C411yescharacteristic
282C24C2×C4×D4711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—944no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—941no—
294C22×C4—3761no—
295C22×C4—3761no—
296C22×C4—3761no—
297C22×C4—3761no—
298C22×C4—1881no—
299C22×C4—1881no—
300C22×C4(C2×D47)⋊C411yescharacteristic
301C22×C4(C2×D47)⋊C411yescharacteristic
302C22×C4C2×D18811yescharacteristic
303C22×C4C2×C22⋊D4711yescharacteristic
304C2×C8—42no—
305C2×C8—82no—
306C2×C8—3762no—
307C2×C8—1882no—
308C4⋊C4—3761no—
309C4⋊C4—3761no—
310C4⋊C4—3761no—
311C4⋊C4—3761no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—472no—
317C22⋊C4—472no—
318C22⋊C4—944no—
319C22⋊C4—941no—
320C22⋊C4—941no—
321C22⋊C4—1882no—
322C22⋊C4—1881no—
323C22⋊C4—1881no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—82no—
343C23—82no—
344C23—84no—
345C23(C2×D47)⋊C812yes—
346C23(C2×D47)⋊C812yes—
347C23#1309393811yescharacteristic
348C23#1309393811yescharacteristic
349C23#1309394212yes—
350C23C2×(C2×D47)⋊C411yescharacteristic
351C23C2×C8×D4711yescharacteristic
352C23C2×#1062532411yescharacteristic
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—942no—
366C2×C4—942no—
367C2×C4—3762no—
368C2×C4—3762no—
369C2×C4—3764no—
370C2×C4—1881no—
371C2×C4—1881no—
372C2×C4—1881no—
373C2×C4—1881no—
374C2×C4—1881no—
375C2×C4—1881no—
376C2×C4—1881no—
377C2×C4—1881no—
378C2×C4—1881no—
379C2×C4—1881no—
380C8—82no—
381C8—3762no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22#1568884812yes—
407C22#1568884812yes—
408C22C2×(C2×D47)⋊C811yescharacteristic
409C22C2×#1309393811yescharacteristic
410C22C2×#1309394211yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—3761no—
416C4—3761no—
417C4—1882no—
418C4—1882no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—22no—
423C2—42no—
424C2—41no—
425C2—41no—
426C2C2×#1568884811yescharacteristic
427C2#2024503211yescharacteristic
428C2#2024503511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC23×C47C22×C8
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#323C2
socleC22×C47C2×#99
prime radicalC25×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#323D47
hypocenterC47#11453
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20241337C2semidirect
4. #20241335C2semidirect
37. C2×C22⋊Dic47C8semidirect
101. C22⋊Dic47C2×C8semidirect
102. C22⋊Dic47C2×C8semidirect
237. C47#11453semidirect

Maximal subgroups

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

subgroupindex
#202460002
C2×#156888472
#202413372
#202413352
C2×C47×#3232
#1145347