Finite Group Databasebeta

← #26133833

Subgroups — #26133833

Order 12032.

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

normal subgroups only

This group has 694 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×#15689107C211yesmaximal, characteristic
2C2×#15688922C212yesmaximal, complemented
3#20243316C211yesmaximal, characteristic
4#20241515C212yesmaximal, complemented
5C47×#2432C211yesmaximal, characteristic
6C22×#10625329—21no—
7#15689107—24no—
8C2×#13093943C2212yes—
9C2×#13093943C2212yes—
10C2×#13093941—21no—
11C2×#13093941—21no—
12#15689087—21no—
13#15688922—24no—
14C22×#41×C47C2211yescharacteristic
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#124×C47C411yescharacteristic
17C2×C22⋊Q8×C47C411yescharacteristic
18C47×#394—21no—
19C47×#394—21no—
20C47×#391C2211yescharacteristic
21C47×#390C411yescharacteristic
22C47×#385—21no—
23C47×#385—21no—
24C4×C22⋊C4×C47C411yescharacteristic
25C2×#10625329—24no—
26C2×#10625329—22no—
27C2×#10625329—22no—
28C2×#10625329—42no—
29C2×#10625329—41no—
30C22×C47⋊C8—41no—
31#13093943—22no—
32#13093943—22no—
33#13093943—22no—
34#13093943D414yes—
35#13093941—24no—
36#13093941—42no—
37C22⋊(C47⋊C8)—41no—
38C22⋊(C47⋊C8)—41no—
39C2×#41×C47—22no—
40C2×#41×C47—22no—
41C2×#41×C47—41no—
42C2×#41×C47—41no—
43C2×#41×C47—42no—
44C22×Q8×C47C2×C411yescharacteristic
45C22×Q8×C47C2311yescharacteristic
46C22×D4×C47C2×C411yescharacteristic
47C22×D4×C47D411yescharacteristic
48C22×D4×C47D411yescharacteristic
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C4⋊Q8×C47—22no—
53C4⋊Q8×C47—22no—
54#124×C47—42no—
55C22⋊Q8×C47—41no—
56C22⋊Q8×C47—41no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—21no—
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C42×C47C2×C411yescharacteristic
68C2×C42×C47C2×C411yescharacteristic
69#95×C47—21no—
70#95×C47—21no—
71#95×C47—21no—
72C24×C47C22⋊C411yescharacteristic
73C24×C47C22⋊C411yescharacteristic
74C24×C47C22⋊C411yescharacteristic
75#41×C47—44no—
76#41×C47—44no—
77#41×C47—44no—
78#41×C47—44no—
79C2×Q8×C47—22no—
80C2×Q8×C47—41no—
81C2×Q8×C47—41no—
82C2×Q8×C47—41no—
83C2×Q8×C47—41no—
84C2×Q8×C47C2×D412yes—
85C2×Q8×C47#4112yes—
86C2×D4×C47—22no—
87C2×D4×C47—22no—
88C2×D4×C47—22no—
89C2×D4×C47—21no—
90C2×D4×C47—22no—
91C2×D4×C47—21no—
92C2×D4×C47—41no—
93C2×D4×C47—41no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—41no—
112C22×C4×C47—41no—
113C22×C4×C47C22⋊C411yescharacteristic
114C22×C4×C47C22×C411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C4⋊C4×C47—41no—
118C4⋊C4×C47—41no—
119C4⋊C4×C47—41no—
120C4⋊C4×C47—41no—
121C4⋊C4×C47—42no—
122C4⋊C4×C47—41no—
123C22⋊C4×C47—42no—
124C22⋊C4×C47—42no—
125C22⋊C4×C47—42no—
126C22⋊C4×C47—42no—
127C22⋊C4×C47—42no—
128C22⋊C4×C47—42no—
129C42×C47—22no—
130C42×C47—41no—
131#10625329—24no—
132#10625329—44no—
133#10625329—42no—
134#10625329—42no—
135#10625329—42no—
136C2×C47⋊C8—44no—
137C2×C47⋊C8—42no—
138C2×C47⋊C8—42no—
139C23×C47—21no—
140C23×C47—21no—
141C23×C47—21no—
142C23×C47—21no—
143C23×C47—21no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—41no—
148C23×C47—41no—
149C23×C47—41no—
150C23×C47—41no—
151C23×C47—41no—
152C23×C47—41no—
153C23×C47#9912yes—
154C23×C47C2×C22⋊C411yescharacteristic
155Q8×C47—42no—
156Q8×C47—42no—
157Q8×C47—82no—
158D4×C47—44no—
159D4×C47—44no—
160D4×C47—42no—
161D4×C47—42no—
162C2×C4×C47—22no—
163C2×C4×C47—22no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—21no—
169C2×C4×C47—21no—
170C2×C4×C47—21no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47—22no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
179C2×C4×C47—41no—
180C2×C4×C47—41no—
181C2×C4×C47—41no—
182C2×C4×C47—41no—
183C2×C4×C47—41no—
184C2×C4×C47—41no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47#9911yescharacteristic
191C2×C4×C47#9911yescharacteristic
192C2×C4×C47C4×D412yes—
193C2×C4×C47#12011yescharacteristic
194C2×C4×C47#12112yes—
195C2×C4×C47#12311yescharacteristic
196C47⋊C8—44no—
197C47⋊C8—82no—
198#13095—471nomaximal, Sylow
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—22no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47—41no—
205C22×C47—41no—
206C22×C47—41no—
207C22×C47—41no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47—41no—
211C22×C47—41no—
212C22×C47—81no—
213C22×C47—81no—
214C22×C47—81no—
215C22×C47—81no—
216C22×C47#38611yescharacteristic
217C22×C47C2×#9911yescharacteristic
218C22×C47C2×#9911yescharacteristic
219C4×C47—21no—
220C4×C47—22no—
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—22no—
225C4×C47—42no—
226C4×C47—81no—
227C4×C47—81no—
228C4×C47—81no—
229C2×#413—471no—
230#2432D4711yescharacteristic
231C2×#356—472no—
232#1785—471no—
233#1394—472no—
234C2×C47—21no—
235C2×C47—21no—
236C2×C47—42no—
237C2×C47—81no—
238C2×C47—81no—
239C2×C47#189811yescharacteristic
240C2×C47#213211yescharacteristic
241C2×C47#213511yescharacteristic
242C22×#41C2×D4711yescharacteristic
243C22×M16—941no—
244C2×C4⋊Q8C2×D4711yescharacteristic
245C2×#124Dic4711yescharacteristic
246C2×C22⋊Q8Dic4711yescharacteristic
247#413—944no—
248C2×#101—472no—
249C2×#101—472no—
250C2×#100—941no—
251C2×#100—941no—
252#407—941no—
253#394—21no—
254#394—21no—
255#391C2×D4711yescharacteristic
256#390Dic4711yescharacteristic
257#385—21no—
258#385—21no—
259C4×C22⋊C4Dic4711yescharacteristic
260#356—944no—
261C47#1309511yescharacteristic, complemented, Sylow
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—41no—
265C2×#41—41no—
266C2×#41—42no—
267C22×Q8C2×Dic4711yescharacteristic
268C22×Q8C22×D4711yescharacteristic
269C22×D4C2×Dic4711yescharacteristic
270C22×D4C22⋊D4711yescharacteristic
271C22×D4C22⋊D4711yescharacteristic
272C23×C4C2×Dic4711yescharacteristic
273C23×C4C22⋊D4711yescharacteristic
274C23×C4C22⋊D4711yescharacteristic
275C2×M16—944no—
276C2×M16—942no—
277C2×M16—942no—
278C2×M16—1882no—
279C2×M16—1881no—
280C22×C8—1881no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283#124—42no—
284C22⋊Q8—41no—
285C22⋊Q8—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C2×Dic4711yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42C2×Dic4711yescharacteristic
297C2×C42C2×Dic4711yescharacteristic
298#101—474no—
299#101—942no—
300#101—942no—
301#101—942no—
302#100—944no—
303#100—1882no—
304C22⋊C8—1881no—
305C22⋊C8—1881no—
306#95—21no—
307#95—21no—
308#95—21no—
309C24C22⋊Dic4711yescharacteristic
310C24C22⋊Dic4711yescharacteristic
311C24C22⋊Dic4711yescharacteristic
312#41—44no—
313#41—44no—
314#41—44no—
315#41—44no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8D4×D4712yes—
322C2×Q8#1062535212yes—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4C22⋊Dic4711yescharacteristic
351C22×C4C22×Dic4711yescharacteristic
352C22×C4C2×C22⋊D4711yescharacteristic
353C22×C4C2×C22⋊D4711yescharacteristic
354M16—944no—
355M16—1882no—
356M16—1882no—
357M16—1884no—
358M16—1882no—
359C2×C8—1882no—
360C2×C8—1884no—
361C2×C8—1882no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C42—22no—
375C42—41no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23#1309393912yes—
391C23C2×C22⋊Dic4711yescharacteristic
392Q8—42no—
393Q8—42no—
394Q8—82no—
395D4—44no—
396D4—44no—
397D4—42no—
398D4—42no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4#1309393911yescharacteristic
428C2×C4#1309393911yescharacteristic
429C2×C4D4×Dic4712yes—
430C2×C4#1309399211yescharacteristic
431C2×C4#1309399612yes—
432C2×C4#1309400511yescharacteristic
433C8—3762no—
434C8—1884no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#1568900311yescharacteristic
453C22C2×#1309393911yescharacteristic
454C22C2×#1309393911yescharacteristic
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—81no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—81no—
469C2—81no—
470C2#2024384211yescharacteristic
471C2#2024507311yescharacteristic
472C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#2432C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenter#2432D47
hypocenterC47#13095
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2432

Complements

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

NH ≅ G/Ntype
2. C2×#15688922C2semidirect
4. #20241515C2semidirect
261. C47#13095semidirect

Maximal subgroups

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

subgroupindex
C2×#156891072
C2×#156889222
#202433162
#202415152
C47×#24322
#1309547