Finite Group Databasebeta

← #30601814

Subgroups — #30601814

Order 15616.

742 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461676C211yesmaximal, characteristic, complemented
2C2×#16840615C212yesmaximal, complemented
3C2×#16840614C212yesmaximal
4C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
5#21454121C211yesmaximal, characteristic
6C22×C4×#6923—21no—
7C2×#14519844—21no—
8C2×#14519834C2212yes—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840615—24no—
13#16840614—24no—
14C2×#14519765C411yescharacteristic, complemented
15C2×#14519757C2211yescharacteristic
16C2×#14519752C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839667C411yescharacteristic, complemented
19#16839663C411yescharacteristic, complemented
20#16839654—21no—
21#16839654—21no—
22#16839640—21no—
23#16839640—21no—
24C61×#382C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4×#6923—24no—
28C2×C4×#6923—42no—
29C2×C4×#6923—42no—
30#14519844—42no—
31#14519834—22no—
32#14519834—22no—
33(C2×#6923)⋊C4—24no—
34(C2×#6923)⋊C4—22no—
35(C2×#6923)⋊C4D414yes—
36C22×Dic122C2×C411yescharacteristic
37C22×D244—21no—
38C22×D244C2×C411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42#14519765—42no—
43#14519757—22no—
44#14519757—22no—
45#14519752—22no—
46#14519752—22no—
47C2×C4⋊Dic61—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×C4×Dic61C2×C411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56C2×C4×Dic61D411yescharacteristic
57C42×D61—42no—
58#14519684—21no—
59#14519684—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C42×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61—21no—
65C23×D61—21no—
66C23×D61C2×D411yescharacteristic
67C22×#6923—24no—
68C22×#6923—24no—
69C22×#6923—24no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×D244—22no—
84C2×D244—41no—
85C2×D244—41no—
86C2×D244—42no—
87C2×D244—42no—
88C2×D244—41no—
89C2×D244—41no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61C2×D412yes—
100C2×C4×D61#4112yes—
101C2×Dic122—22no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104C2×#92420—42no—
105C2×#92420—42no—
106C2×#92420—42no—
107C4×#6923—48no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C42×C61—22no—
111#11758198—42no—
112#11758198—42no—
113#11758198—42no—
114#11758198—42no—
115#11758198—42no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120C4×Dic61—22no—
121C4×Dic61—41no—
122C4×Dic61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61#12112yes—
136C22×D61#13512yes—
137C22×D61AGL(1,ℤ/8)11yescharacteristic
138C22×D61#13711yescharacteristic
139C2×#6923—28no—
140C2×#6923—44no—
141C2×#6923—44no—
142C2×#6923—44no—
143C2×#6923—44no—
144C2×#6923—44no—
145C2×#6923—44no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61#9912yes—
160C2×C4×C61C4×D412yes—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61#10412yes—
171C2×Dic61#10412yes—
172C2×Dic61#12411yescharacteristic
173C2×Dic61#12711yescharacteristic
174D244—82no—
175D244—82no—
176D244—82no—
177C4×D61—24no—
178C4×D61—24no—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61—44no—
182Dic122—82no—
183#92420—82no—
184#9713—611nomaximal, Sylow
185C22×C61#39011yescharacteristic
186C22×C61C2×#9911yescharacteristic
187C22×C61#41011yescharacteristic
188C22×C61#41611yescharacteristic
189C22×C61#41711yescharacteristic
190C22×C61C2×#10411yescharacteristic
191C22×C61C2×#10411yescharacteristic
192C2×D61—22no—
193C2×D61—22no—
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—21no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61#46312yes—
208C2×D61#46412yes—
209#6923—48no—
210C4×C61—22no—
211C4×C61—22no—
212C4×C61—41no—
213C4×C61—81no—
214C4×C61—81no—
215Dic61—21no—
216Dic61—21no—
217Dic61—22no—
218Dic61—22no—
219Dic61—81no—
220#2850—611no—
221#1797—611no—
222C2×#328—612no—
223C2×#327—612no—
224C4×#95—611no—
225C2×C61#150011yescharacteristic
226C2×C61#150111yescharacteristic
227C2×C61#150411yescharacteristic
228C2×C61#150911yescharacteristic
229C2×C61#196211yescharacteristic
230C2×C61#196411yescharacteristic
231C2×C61#198311yescharacteristic
232D61—22no—
233D61—22no—
234D61—161no—
235C2×#124—611no—
236C2×C22⋊Q8—611no—
237C2×#121—611no—
238C22×C42—611no—
239C22×C42—1221no—
240C2×C4⋊C8—1221no—
241C2×C22⋊C8—612no—
242#391—611no—
243#390—611no—
244#388—1221no—
245#388—1221no—
246#386—1221no—
247#386—1221no—
248#382#692311yescharacteristic, complemented
249C2×#95—612no—
250C2×#95—1221no—
251#336—1221no—
252#328—1224no—
253#327—1224no—
254C61#971311yescharacteristic, complemented, Sylow
255C22×Q8—611no—
256C22×D4—611no—
257C22×D4—1221no—
258C23×C4—611no—
259C23×C4—611no—
260C23×C4—611no—
261C23×C4—1222no—
262C22×C8—1222no—
263#124—2442no—
264C22⋊Q8—1222no—
265C22⋊Q8—1222no—
266#121—1222no—
267#121—1222no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4C2×#692311yescharacteristic
270C2×C4⋊C4C2×#692311yescharacteristic
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C42—611no—
278C2×C42—611no—
279C2×C42—611no—
280C2×C42—1224no—
281C2×C42—2442no—
282C2×C42—2442no—
283C2×C42—2442no—
284C2×C42C2×#692311yescharacteristic
285C4⋊C8—2442no—
286C22⋊C8—1222no—
287C22⋊C8—1222no—
288#95—21no—
289#95—21no—
290#95—614no—
291#95—1221no—
292#95—1221no—
293#95—1222no—
294#95—1224no—
295C24—611no—
296C24—1221no—
297C2×Q8—1222no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×D4—1222no—
301C2×D4—2441no—
302C2×D4—2441no—
303C2×D4—2442no—
304C2×D4—2442no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—612no—
313C22×C4—612no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—1224no—
321C22×C4—1224no—
322C22×C4—1224no—
323C22×C4—1221no—
324C22×C4—2441no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2442no—
331C22×C4C22⋊#692311yescharacteristic
332C22×C4C22⋊#692311yescharacteristic
333C22×C4C22×#692311yescharacteristic
334C2×C8—2442no—
335C2×C8—2442no—
336C2×C8—2442no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2442no—
347C22⋊C4—2442no—
348C42—22no—
349C42—1222no—
350C42—2441no—
351C42—2441no—
352C42—2448no—
353C23—612no—
354C23—612no—
355C23—611no—
356C23—611no—
357C23—1221no—
358C23—1221no—
359C23—1221no—
360C23—1221no—
361C23—2441no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23—2441no—
368C23C2×C22⋊#692311yescharacteristic
369Q8—4882no—
370D4—4882no—
371D4—4882no—
372D4—4882no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—612no—
389C2×C4—612no—
390C2×C4—1224no—
391C2×C4—1224no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1228no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2444no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2444no—
406C2×C4—2444no—
407C2×C4—2444no—
408C2×C4—2444no—
409C2×C4—2444no—
410C2×C4—2444no—
411C2×C4#1451983612yes—
412C2×C4D4×#692312yes—
413C8—4882no—
414C22—612no—
415C22—612no—
416C22—1222no—
417C22—1222no—
418C22—1222no—
419C22—1222no—
420C22—1221no—
421C22—1221no—
422C22—1221no—
423C22—1221no—
424C22—4881no—
425C22—4881no—
426C22—4881no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22#1684071411yescharacteristic
432C22C2×#1451983611yescharacteristic
433C22#1684074311yescharacteristic
434C22#1684075211yescharacteristic
435C22#1684075311yescharacteristic
436C22D61⋊#10411yescharacteristic
437C22D61⋊#10411yescharacteristic
438C4—22no—
439C4—22no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—1221no—
444C4—1221no—
445C4—1222no—
446C4—1222no—
447C4—4881no—
448C4—2448no—
449C2—1222no—
450C2—1222no—
451C2—9761no—
452C2#2146119311yescharacteristic
453C2#2146119611yescharacteristic
454C2#2146120711yescharacteristic
455C2#2146122211yescharacteristic
456C2#2146195411yescharacteristic
457C2#2146195911yescharacteristic
458C2#2146198711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#382C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C22
cosocleC22×C4×D61C23
hypercenter#382#6923
hypocenterC61#9713
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #21461676C2semidirect
2. C2×#16840615C2semidirect
4. C4×(C2×#6923)⋊C4C2semidirect
14. C2×#14519765C4semidirect
18. #16839667C4semidirect
19. #16839663C4semidirect
24. C61×#382C4semidirect
248. #382#6923semidirect
254. C61#9713semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406152
C2×#168406142
C4×(C2×#6923)⋊C42
#214541212
#971361