Finite Group Databasebeta

← #30601756

Subgroups — #30601756

Order 15616.

700 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461546C211yesmaximal, characteristic, complemented
2C2×#16840616C212yesmaximal, complemented
3C2×#16840613C212yesmaximal
4#21460997C211yesmaximal, characteristic, complemented
5#21454115C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×#14519834C2212yes—
8C2×#14519832—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840616—24no—
13#16840613—24no—
14C2×#14519753C2211yescharacteristic
15C2×#14519748C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839669C411yescharacteristic, complemented
18#16839665C411yescharacteristic, complemented
19#16839650—21no—
20#16839650—21no—
21#16839638—21no—
22#16839638—21no—
23#16839628C411yescharacteristic, complemented
24C2×#124×C61C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4⋊#6923—24no—
28C2×C4⋊#6923—42no—
29C2×C4⋊#6923—42no—
30#14519834—22no—
31#14519834D414yes—
32#14519832—42no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—22no—
35(C2×#6923)⋊C4—42no—
36C22×C22⋊D61—21no—
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40#14519753—22no—
41#14519753—22no—
42#14519748—22no—
43#14519748—22no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C42×D61—42no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59C22×Q8×C61C2×C411yescharacteristic
60C22×D4×C61C2×C411yescharacteristic
61#124×C61—42no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61C2×C411yescharacteristic
65C24×C61—21no—
66C23×D61C2×D411yescharacteristic
67C22×#6923—24no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C2×Q8×C61—22no—
75C2×Q8×C61—41no—
76C2×Q8×C61—41no—
77C2×D4×C61—22no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C22×C4×C61—21no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—21no—
100C2×C4×D61—21no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C4×D61C2×D412yes—
106C2×C4×D61#4112yes—
107C2×#92420—24no—
108C2×#92420—42no—
109C2×#92420—42no—
110C4⋊#6923—48no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C42×C61—22no—
114C22⋊Dic61—42no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—22no—
120C4⋊Dic61—22no—
121#11758196—42no—
122C4×Dic61—22no—
123C4×Dic61—41no—
124C4×Dic61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61C2×C22⋊C411yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—21no—
137C22×D61#12411yescharacteristic
138C22×D61#12711yescharacteristic
139C22×D61AGL(1,ℤ/8)11yescharacteristic
140C22×D61AGL(1,ℤ/8)11yescharacteristic
141C22×D61#13711yescharacteristic
142C22×D61#13711yescharacteristic
143C2×#6923—44no—
144C2×#6923—44no—
145C2×#6923—44no—
146C2×#6923—82no—
147C2×#6923—82no—
148C2×#6923—82no—
149C2×#6923—82no—
150Q8×C61—82no—
151D4×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61#9912yes—
163C2×C4×C61C4×D412yes—
164C22⋊D61—81no—
165C22⋊D61—81no—
166C22⋊D61—81no—
167C22⋊D61—81no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61#10412yes—
182C2×Dic61#12112yes—
183C4×D61—24no—
184C4×D61—24no—
185C4×D61—42no—
186C4×D61—42no—
187C4×D61—42no—
188C4×D61—42no—
189#92420—44no—
190#9682—611nomaximal, Sylow
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61#39011yescharacteristic
199C22×C61C2×#9911yescharacteristic
200C22×C61#41011yescharacteristic
201C22×C61#41711yescharacteristic
202C22×C61#41711yescharacteristic
203C22×C61C2×#10411yescharacteristic
204C22×C61#42111yescharacteristic
205C2×D61—21no—
206C2×D61—21no—
207C2×D61—21no—
208C2×D61—21no—
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—21no—
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—21no—
217C2×D61#48911yescharacteristic
218C2×D61#48911yescharacteristic
219C2×D61#49611yescharacteristic
220C2×D61#49711yescharacteristic
221#6923—84no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—41no—
225C4×C61—81no—
226Dic61—22no—
227Dic61—22no—
228Dic61—41no—
229Dic61—81no—
230Dic61—81no—
231#2850—611no—
232#1749—611no—
233C2×#328—612no—
234C2×#327—612no—
235C4⋊#95—611no—
236C2×C61—161no—
237C2×C61#150411yescharacteristic
238C2×C61#150911yescharacteristic
239C2×C61#151211yescharacteristic
240C2×C61#151311yescharacteristic
241C2×C61#195911yescharacteristic
242C2×C61#196411yescharacteristic
243C2×C61#198011yescharacteristic
244D61—21no—
245D61—21no—
246D61—21no—
247D61—21no—
248C2×#124#692311yescharacteristic, complemented
249C2×C22⋊Q8—611no—
250C2×#121—611no—
251C22×C4⋊C4—1221no—
252C22×C42—611no—
253C2×C22⋊C8—612no—
254C2×#97—1221no—
255#391—611no—
256#390—611no—
257#388—1221no—
258#388—1221no—
259#386—1221no—
260#386—1221no—
261#382—611no—
262C2×#95—612no—
263C2×#95—1221no—
264#336—1221no—
265#328—1224no—
266#327—1224no—
267C61#968211yescharacteristic, complemented, Sylow
268C22×Q8C2×#692311yescharacteristic
269C22×D4—1221no—
270C22×D4C2×#692311yescharacteristic
271C23×C4—611no—
272C23×C4—611no—
273C23×C4—611no—
274C23×C4—1222no—
275C22×C8—1222no—
276#124—42no—
277C22⋊Q8—1222no—
278C22⋊Q8—1222no—
279#121—1222no—
280#121—1222no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1224no—
285C2×C4⋊C4—2442no—
286C2×C4⋊C4—2442no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C42—611no—
294C2×C42—611no—
295C2×C42—611no—
296C2×C42—2442no—
297C2×C42C2×#692311yescharacteristic
298C22⋊C8—614no—
299C22⋊C8—1222no—
300#97—2442no—
301#95—1221no—
302#95—1221no—
303#95—1222no—
304#95—1222no—
305#95—1221no—
306#95—1221no—
307#95—2442no—
308C24—21no—
309C24—611no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C22×C4—21no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—612no—
326C22×C4—612no—
327C22×C4—611no—
328C22×C4—1222no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1222no—
334C22×C4—1224no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—2441no—
338C22×C4—2441no—
339C22×C4—2441no—
340C22×C4—2441no—
341C22×C4—2442no—
342C22×C4—2442no—
343C22×C4—2442no—
344C22×C4—2442no—
345C22×C4—2442no—
346C22×C4—2442no—
347C22×C4C22⋊#692311yescharacteristic
348C22×C4C22⋊#692311yescharacteristic
349C22×C4C22×#692311yescharacteristic
350C2×C8—1224no—
351C2×C8—2442no—
352C2×C8—2442no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—2442no—
356C4⋊C4—2448no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2442no—
362C22⋊C4—2442no—
363C22⋊C4—2442no—
364C42—22no—
365C42—1222no—
366C42—2441no—
367C42—2441no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—611no—
376C23—611no—
377C23—611no—
378C23—611no—
379C23—611no—
380C23—611no—
381C23—1221no—
382C23—1221no—
383C23—1221no—
384C23—1221no—
385C23C2×C22⋊#692311yescharacteristic
386Q8—82no—
387D4—82no—
388D4—4881no—
389D4—4881no—
390D4—4881no—
391D4—4881no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—612no—
403C2×C4—612no—
404C2×C4—1224no—
405C2×C4—1224no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—4882no—
414C2×C4—4882no—
415C2×C4—4882no—
416C2×C4—4882no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2444no—
422C2×C4—2444no—
423C2×C4—2444no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4#1451983512yes—
431C2×C4D4×#692312yes—
432C8—2444no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—611no—
441C22—611no—
442C22—611no—
443C22—611no—
444C22—1221no—
445C22—1221no—
446C22—1221no—
447C22—1221no—
448C22—1221no—
449C22—1221no—
450C22—1221no—
451C22—1221no—
452C22—1221no—
453C22—1221no—
454C22—1221no—
455C22—1221no—
456C22#1684071411yescharacteristic
457C22C2×#1451983511yescharacteristic
458C22#1684074211yescharacteristic
459C22#1684075311yescharacteristic
460C22#1684075311yescharacteristic
461C22D61⋊#10411yescharacteristic
462C22#1684075911yescharacteristic
463C4—22no—
464C4—22no—
465C4—41no—
466C4—81no—
467C4—1222no—
468C4—1222no—
469C4—4884no—
470C4—4881no—
471C4—4881no—
472C4—2441no—
473C2—161no—
474C2—1221no—
475C2—1221no—
476C2—1221no—
477C2—1221no—
478C2#2146120511yescharacteristic
479C2#2146122011yescharacteristic
480C2#2146123111yescharacteristic
481C2#2146123411yescharacteristic
482C2#2146194911yescharacteristic
483C2#2146195811yescharacteristic
484C2#2146198411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenterC2×#124#6923
hypocenterC61#9682
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461546C2semidirect
2. C2×#16840616C2semidirect
4. #21460997C2semidirect
17. #16839669C4semidirect
18. #16839665C4semidirect
23. #16839628C4semidirect
24. C2×#124×C61C4semidirect
248. C2×#124#6923semidirect
267. C61#9682semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406162
C2×#168406132
#214609972
#214541152
#968261