Finite Group Databasebeta

← #12016850

Subgroups — #12016850

Order 1280.

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

normal subgroups only

This group has 676 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
1C5×#2434C211yesmaximal, characteristic
2C22×#3968C211yesmaximal, characteristic
3#10602926C211yesmaximal, characteristic
4C2×#64768C212yesmaximal, complemented
5#10598135C212yesmaximal, complemented
6C23×C5×Q8C2211yescharacteristic
7C2×C5×C4⋊Q8C411yescharacteristic
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×C5×C22⋊Q8C411yescharacteristic
10C22×#681—21no—
11C5×#395—21no—
12C5×#395—21no—
13C5×#391C411yescharacteristic
14C5×#391C2211yescharacteristic
15C5×#385—21no—
16C5×#385—21no—
17C4×C5×C22⋊C4C411yescharacteristic
18C2×#3968—24no—
19C2×#3968—21no—
20C2×#3968—21no—
21C2×#3968C2212yes—
22C2×#3968C2212yes—
23#65496—21no—
24#64768—24no—
25#13094—51nomaximal, Sylow
26C22×C5×Q8—22no—
27C22×C5×Q8—22no—
28C22×C5×Q8—41no—
29C22×C5×Q8—41no—
30C22×C5×Q8—42no—
31C22×C5×Q8C2×C411yescharacteristic
32C22×C5×Q8C2×C411yescharacteristic
33C22×C5×Q8D411yescharacteristic
34C22×C5×Q8D411yescharacteristic
35C22×C5×Q8C2311yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5D411yescharacteristic
38C23×C4×C5D411yescharacteristic
39C5×C4⋊Q8—22no—
40C5×C4⋊Q8—22no—
41C5×C4⋊Q8—42no—
42C5×C22⋊Q8—41no—
43C5×C22⋊Q8—41no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4—21no—
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4C2×C411yescharacteristic
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—21no—
52C2×C42×C5C2×C411yescharacteristic
53C2×C42×C5C2×C411yescharacteristic
54C2×#681—24no—
55C2×#681—22no—
56C2×#681—22no—
57C2×#681—41no—
58C2×#681—42no—
59C22×#168—41no—
60C5×#95—21no—
61C5×#95—21no—
62C5×#95—21no—
63C5×#95—21no—
64C5×#95—21no—
65#3968—24no—
66#3968—22no—
67#3968—22no—
68#3968—22no—
69#3968—42no—
70#3968D414yes—
71C22⋊#168—41no—
72C22⋊#168—41no—
73C22×#101—51no—
74#2434D511yescharacteristic
75C2×#356—52no—
76#1784—51no—
77#1394—52no—
78C24×C5C22⋊C411yescharacteristic
79C2×C5×Q8—22no—
80C2×C5×Q8—22no—
81C2×C5×Q8—22no—
82C2×C5×Q8—21no—
83C2×C5×Q8—22no—
84C2×C5×Q8—21no—
85C2×C5×Q8—22no—
86C2×C5×Q8—41no—
87C2×C5×Q8—41no—
88C2×C5×Q8—41no—
89C2×C5×Q8—41no—
90C2×C5×Q8—41no—
91C2×C5×Q8—41no—
92C2×C5×Q8—44no—
93C2×C5×Q8—44no—
94C2×C5×Q8—44no—
95C2×C5×Q8—44no—
96C2×C5×Q8—41no—
97C2×C5×Q8—41no—
98C2×C5×Q8C2×D412yes—
99C2×C5×Q8#4112yes—
100C22×C4×C5—21no—
101C22×C4×C5—21no—
102C22×C4×C5—22no—
103C22×C4×C5—21no—
104C22×C4×C5—21no—
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—41no—
116C22×C4×C5—41no—
117C22×C4×C5C22⋊C411yescharacteristic
118C22×C4×C5C22⋊C411yescharacteristic
119C22×C4×C5C22⋊C411yescharacteristic
120C22×C4×C5C22×C411yescharacteristic
121C22×C4×C5C2×D411yescharacteristic
122C22×C4×C5C2×D411yescharacteristic
123C5×C4⋊C4—41no—
124C5×C4⋊C4—42no—
125C5×C4⋊C4—42no—
126C5×C4⋊C4—42no—
127C5×C4⋊C4—41no—
128C5×C4⋊C4—42no—
129C5×C4⋊C4—41no—
130C5×C22⋊C4—24no—
131C5×C22⋊C4—41no—
132C5×C22⋊C4—41no—
133C42×C5—22no—
134C42×C5—41no—
135#681—24no—
136#681—44no—
137#681—42no—
138#681—42no—
139#681—42no—
140C2×#168—44no—
141C2×#168—42no—
142C2×#168—42no—
143C23×Q8C2×D511yescharacteristic
144C22×M16—101no—
145C2×C4⋊Q8Dic511yescharacteristic
146C2×C4⋊Q8C2×D511yescharacteristic
147C2×C22⋊Q8Dic511yescharacteristic
148C2×#101—52no—
149C2×#101—52no—
150C2×#101—104no—
151C2×#101—101no—
152C2×#101—101no—
153#407—101no—
154#395—21no—
155#395—21no—
156#391Dic511yescharacteristic
157#391C2×D511yescharacteristic
158#385—21no—
159#385—21no—
160C4×C22⋊C4Dic511yescharacteristic
161#356—104no—
162C23×C5—22no—
163C23×C5—21no—
164C23×C5—21no—
165C23×C5—41no—
166C23×C5#9911yescharacteristic
167C23×C5#9911yescharacteristic
168C23×C5C2×C22⋊C411yescharacteristic
169C5×Q8—42no—
170C5×Q8—42no—
171C5×Q8—44no—
172C5×Q8—44no—
173C5×Q8—42no—
174C5×Q8—42no—
175C5×Q8—82no—
176C2×C4×C5—22no—
177C2×C4×C5—22no—
178C2×C4×C5—22no—
179C2×C4×C5—22no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—21no—
186C2×C4×C5—21no—
187C2×C4×C5—21no—
188C2×C4×C5—22no—
189C2×C4×C5—22no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—21no—
196C2×C4×C5—42no—
197C2×C4×C5—42no—
198C2×C4×C5—42no—
199C2×C4×C5—42no—
200C2×C4×C5—41no—
201C2×C4×C5—41no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5—42no—
205C2×C4×C5—41no—
206C2×C4×C5—41no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5#9912yes—
210C2×C4×C5C4×D412yes—
211C2×C4×C5#12011yescharacteristic
212C2×C4×C5#12112yes—
213C2×C4×C5#12311yescharacteristic
214#168—44no—
215#168—82no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—41no—
219C22×Q8—41no—
220C22×Q8—42no—
221C22×Q8C2×Dic511yescharacteristic
222C22×Q8C2×Dic511yescharacteristic
223C22×Q8C22⋊D511yescharacteristic
224C22×Q8C22⋊D511yescharacteristic
225C22×Q8C22×D511yescharacteristic
226C23×C4C2×Dic511yescharacteristic
227C23×C4C22⋊D511yescharacteristic
228C23×C4C22⋊D511yescharacteristic
229C2×M16—104no—
230C2×M16—102no—
231C2×M16—102no—
232C2×M16—201no—
233C2×M16—202no—
234C22×C8—201no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—42no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×Dic511yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C42C2×Dic511yescharacteristic
249C2×C42C2×Dic511yescharacteristic
250#101—54no—
251#101—104no—
252#101—102no—
253#101—102no—
254#101—102no—
255#101—202no—
256C22⋊C8—201no—
257C22⋊C8—201no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263C22×C5—21no—
264C22×C5—21no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—21no—
268C22×C5—21no—
269C22×C5—21no—
270C22×C5—42no—
271C22×C5—42no—
272C22×C5#35612yescomplemented
273C22×C5#38611yescharacteristic
274C22×C5C2×#9911yescharacteristic
275C22×C5C2×#9911yescharacteristic
276C4×C5—21no—
277C4×C5—22no—
278C4×C5—22no—
279C4×C5—21no—
280C4×C5—22no—
281C4×C5—42no—
282C4×C5—42no—
283C4×C5—41no—
284C4×C5—41no—
285C4×C5—81no—
286C4×C5—81no—
287C24C22⋊Dic511yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8D4×D512yes—
308C2×Q8#71112yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4C22⋊Dic511yescharacteristic
327C22×C4C22⋊Dic511yescharacteristic
328C22×C4C22⋊Dic511yescharacteristic
329C22×C4C22×Dic511yescharacteristic
330C22×C4C2×C22⋊D511yescharacteristic
331C22×C4C2×C22⋊D511yescharacteristic
332M16—104no—
333M16—204no—
334M16—202no—
335M16—202no—
336M16—202no—
337C2×C8—204no—
338C2×C8—202no—
339C2×C8—202no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C22⋊C4—24no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—22no—
351C42—41no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—22no—
355C2×C5—41no—
356C2×C5#189811yescharacteristic
357C2×C5C2×#35611yescharacteristic
358C2×C5#213311yescharacteristic
359C23—22no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23#396611yescharacteristic
364C23#396611yescharacteristic
365C23C2×C22⋊Dic511yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—44no—
369Q8—44no—
370Q8—42no—
371Q8—42no—
372Q8—82no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4#396612yes—
407C2×C4D4×Dic512yes—
408C2×C4#408111yescharacteristic
409C2×C4#408311yescharacteristic
410C2×C4#408512yes—
411C8—204no—
412C8—402no—
413C5#1309411yescharacteristic, complemented, Sylow
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#6476812yescomplemented
424C22C2×#396611yescharacteristic
425C22C2×#396611yescharacteristic
426C22#6550911yescharacteristic
427C4—21no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#1060246311yescharacteristic
443C2C2×#6476811yescharacteristic
444C2#1060266511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#2434C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC22×C5×Q8C23
hypercenter#2434D5
hypocenterC5#13094
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2434

Complements

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

NH ≅ G/Ntype
4. C2×#64768C2semidirect
5. #10598135C2semidirect
272. C22×C5#356semidirect
413. C5#13094semidirect
423. C22#64768semidirect

Maximal subgroups

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

subgroupindex
C5×#24342
C22×#39682
#106029262
C2×#647682
#105981352
#130945