Finite Group Databasebeta

← #26115803

Subgroups — #26115803

Order 12032.

510 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 111 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×#15689362C211yesmaximal, characteristic
2#20244373C211yesmaximal, characteristic
3#20243411C211yesmaximal, characteristic
4#20243166C211yesmaximal, characteristic
5#20241960C211yesmaximal, characteristic
6#20241959C211yesmaximal, characteristic
7C47×#1996C211yesmaximal, characteristic
8C22×#10625337C411yescharacteristic, complemented
9C23×C47⋊C8C2211yescharacteristic
10C2×#13094003C2211yescharacteristic
11#15689362—21no—
12#15689362—21no—
13#15689362—21no—
14#15689362—21no—
15C2×#13093957C411yescharacteristic, complemented
16C2×#13093955—21no—
17C2×#13093948C411yescharacteristic, complemented
18C2×#13093947C411yescharacteristic, complemented
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093935—21no—
21#15688983C2211yescharacteristic
22#15688889—22no—
23#15688885—21no—
24#15688878—21no—
25#15688860—22no—
26#15688859—22no—
27C2×C22⋊Q8×C47C2211yescharacteristic
28C47×#380C2211yescharacteristic
29C4×C22⋊C4×C47C2211yescharacteristic
30C47×#342—22no—
31C22×Dic94C2×C411yescharacteristic
32C2×#10625337—21no—
33C2×#10625337—21no—
34C2×#10625337—42no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8—42no—
37C22×C47⋊C8C2×C411yescharacteristic
38C22×C47⋊C8C2×C411yescharacteristic
39C22×C47⋊C8D411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41C22×C47⋊C8D411yescharacteristic
42C22×C47⋊C8D411yescharacteristic
43#13094003—21no—
44#13094003—21no—
45#13094003—21no—
46#13094003—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47C2×C411yescharacteristic
49C2×Dic47⋊C4—21no—
50C2×C22⋊Dic47—21no—
51C2×C22⋊Dic47—21no—
52#13093957—21no—
53#13093957—21no—
54#13093955—42no—
55#13093948—21no—
56#13093948—21no—
57#13093947—21no—
58#13093947—21no—
59C22⋊(C47⋊C8)—21no—
60C22⋊(C47⋊C8)D412yes—
61#13093935—42no—
62C47⋊#95—21no—
63C22×Q8×C47C2×C411yescharacteristic
64C23×C4×C47C2311yescharacteristic
65C22⋊Q8×C47—21no—
66C22⋊Q8×C47—21no—
67C22⋊Q8×C47—21no—
68C22⋊Q8×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47C2×C411yescharacteristic
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47D412yes—
75C2×C42×C47—21no—
76#95×C47—21no—
77#95×C47—21no—
78C24×C47C2×D411yescharacteristic
79C2×Q8×C47—21no—
80C2×Q8×C47—21no—
81C2×Q8×C47—41no—
82C2×Q8×C47—41no—
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47C22×C411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×C4×C47#4111yescharacteristic
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—41no—
98C2×Dic94—21no—
99C2×Dic94—21no—
100C2×Dic94—41no—
101C2×Dic94—41no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—22no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—42no—
113C42×C47—41no—
114C42×C47—41no—
115C22⋊Dic47—41no—
116C22⋊Dic47—41no—
117C22⋊Dic47—42no—
118#10625337—42no—
119#10625337—42no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122C4⋊Dic47—41no—
123C4⋊Dic47—41no—
124Dic47⋊C4—41no—
125Dic47⋊C4—41no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8C22⋊C411yescharacteristic
136C2×C47⋊C8C22⋊C411yescharacteristic
137C2×C47⋊C8C22⋊C411yescharacteristic
138C2×C47⋊C8C22⋊C411yescharacteristic
139C23×C47—41no—
140C23×C47—41no—
141C23×C47C4×D411yescharacteristic
142C23×C47#12111yescharacteristic
143C23×C47#12711yescharacteristic
144C23×C47#13712yes—
145C23×C47#13712yes—
146Q8×C47—41no—
147Q8×C47—41no—
148Q8×C47—81no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—81no—
164C2×C4×C47—81no—
165C2×C4×C47—81no—
166C2×C4×C47—81no—
167C2×C4×C47—81no—
168C2×C4×C47C2×C22⋊C411yescharacteristic
169C2×C4×C47C4×D411yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47#12411yescharacteristic
172C2×Dic47—41no—
173C2×Dic47—41no—
174C2×Dic47—81no—
175C2×Dic47—81no—
176C2×Dic47—81no—
177C2×Dic47—81no—
178C2×Dic47—81no—
179Dic94—41no—
180Dic94—41no—
181Dic94—81no—
182C47⋊C8—21no—
183C47⋊C8—21no—
184C47⋊C8—21no—
185C47⋊C8—42no—
186C47⋊C8#9912yes—
187#9784—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—42no—
195C22×C47—42no—
196C22×C47#39011yescharacteristic
197C22×C47#44112yes—
198C22×C47#47012yes—
199C22×C47#49711yescharacteristic
200C22×C47#49711yescharacteristic
201C4×C47—21no—
202C4×C47—21no—
203C4×C47—21no—
204C4×C47—82no—
205C4×C47—81no—
206C4×C47—81no—
207C4×C47—161no—
208C4×C47C2×#9911yescharacteristic
209C4×C47#41011yescharacteristic
210Dic47—81no—
211Dic47—81no—
212Dic47—161no—
213C2×#467—471no—
214#1996—471no—
215#1996D4711yescharacteristic
216#1807—471no—
217#1752—471no—
218#1520—471no—
219#1520—471no—
220C2×C47—21no—
221C2×C47—22no—
222C2×C47—21no—
223C2×C47—22no—
224C2×C47#196411yescharacteristic
225C2×C47#197111yescharacteristic
226C2×C47#197611yescharacteristic
227C22×Q16—471no—
228C23×C8—471no—
229C2×C22⋊Q8—471no—
230C2×C22⋊Q8C2×D4711yescharacteristic
231#467—941no—
232#467—941no—
233#467—941no—
234#467—941no—
235C2×#107—471no—
236C2×#106—941no—
237C2×#103—471no—
238C2×#103—471no—
239C2×C22⋊C8—471no—
240C2×#97—941no—
241#380—471no—
242#380C2×D4711yescharacteristic
243C4×C22⋊C4C2×D4711yescharacteristic
244#342—22no—
245#342—942no—
246#340—941no—
247#336—941no—
248#328—942no—
249#328—942no—
250C47#978411yescharacteristic, complemented, Sylow
251C22×Q8—471no—
252C22×Q8C4×D4711yescharacteristic
253C23×C4C22×D4711yescharacteristic
254C2×Q16—941no—
255C2×Q16—941no—
256C2×Q16—1882no—
257C22×C8—471no—
258C22×C8—471no—
259C22×C8—471no—
260C22×C8—471no—
261C22×C8—471no—
262C22×C8—471no—
263C22×C8—941no—
264C22×C8—1882no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—941no—
270C22⋊Q8—941no—
271C22⋊Q8—941no—
272C22⋊Q8—941no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—471no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4—941no—
278C2×C4⋊C4C4×D4711yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—941no—
282C2×C22⋊C4—941no—
283C2×C22⋊C4C22⋊D4712yes—
284C2×C42—21no—
285#107—941no—
286#107—941no—
287#106—1882no—
288#103—941no—
289#103—941no—
290#103—941no—
291#103—941no—
292C22⋊C8—472no—
293C22⋊C8—941no—
294#97—1882no—
295#95—21no—
296#95—21no—
297#95—941no—
298C24C2×C22⋊D4711yescharacteristic
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—941no—
304C2×Q8—941no—
305C2×Q8—1881no—
306C2×Q8—1881no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—941no—
314C22×C4—941no—
315C22×C4—1881no—
316C22×C4C2×C4×D4711yescharacteristic
317C22×C4#1062535011yescharacteristic
318C22×C4D4×D4711yescharacteristic
319C22×C4D4×D4711yescharacteristic
320C22×C4D4×D4711yescharacteristic
321C22×C4#1062535211yescharacteristic
322Q16—1882no—
323Q16—1882no—
324C2×C8—471no—
325C2×C8—471no—
326C2×C8—471no—
327C2×C8—471no—
328C2×C8—942no—
329C2×C8—942no—
330C2×C8—942no—
331C2×C8—942no—
332C2×C8—1881no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1881no—
336C2×C8—1881no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—941no—
344C4⋊C4—941no—
345C4⋊C4—1881no—
346C4⋊C4—1881no—
347C4⋊C4—1881no—
348C4⋊C4—1881no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—1882no—
355C22⋊C4—1881no—
356C22⋊C4—1881no—
357C42—41no—
358C42—41no—
359C23—41no—
360C23—41no—
361C23C4×C22⋊D4711yescharacteristic
362C23#1309399611yescharacteristic
363C23#1309402011yescharacteristic
364C23#1309405412yes—
365C23#1309405512yes—
366Q8—41no—
367Q8—41no—
368Q8—81no—
369Q8—3761no—
370Q8—1881no—
371Q8—1881no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—3761no—
392C2×C4—3761no—
393C2×C4—3761no—
394C2×C4—3761no—
395C2×C4—3761no—
396C2×C4—1881no—
397C2×C4—1881no—
398C2×C4C22⋊C4×D4711yescharacteristic
399C2×C4(C22⋊D47)⋊C411yescharacteristic
400C2×C4#1309399711yescharacteristic
401C2×C4#1309401011yescharacteristic
402C8—472no—
403C8—941no—
404C8—941no—
405C8—941no—
406C8—1882no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—42no—
414C22—42no—
415C22#1568902311yescharacteristic
416C22#1568922812yes—
417C22#1568937912yes—
418C22#1568951511yescharacteristic
419C22#1568951811yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—161no—
427C4—3761no—
428C4—3761no—
429C4—7521no—
430C4#99×D4711yescharacteristic
431C4#1568909511yescharacteristic
432C2—22no—
433C2—21no—
434C2—21no—
435C2—22no—
436C2#2024422211yescharacteristic
437C2#2024425811yescharacteristic
438C2#2024428311yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1996C2
socleC22×C47#390
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1996D47
hypocenterC47#9784
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
8. C22×#10625337C4semidirect
15. C2×#13093957C4semidirect
17. C2×#13093948C4semidirect
18. C2×#13093947C4semidirect
250. C47#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#156893622
#202443732
#202434112
#202431662
#202419602
#202419592
C47×#19962
#978447