Finite Group Databasebeta

← #29652389

Subgroups — #29652389

Order 15616.

1310 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16840388C211yesmaximal, characteristic, complemented
2#21455079C212yesmaximal, complemented
3#21444775C214yesmaximal, complemented
4#21444741C214yesmaximal, complemented
5#21444351C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C61×#2404C211yesmaximal, characteristic
8C2×#14519807—22no—
9#16840388—24no—
10#16840388—22no—
11#16840388C2214yes—
12C2×#14519756—21no—
13C2×#14519756C2212yes—
14C2×#14519754—22no—
15C2×#14519754—22no—
16C22×C22⋊Dic61—21no—
17#16839924—24no—
18#16839918C2214yescomplemented
19#16839913C2214yescomplemented
20#16839763—22no—
21#16839726C2212yes—
22#16839498C2218yescomplemented
23#16839486C2214yes—
24C22×#41×C61C2211yescharacteristic
25C2×#120×C61—21no—
26C22×C22⋊C4×C61C2211yescharacteristic
27C61×#393C2212yes—
28C61×#392C2212yes—
29C61×#386—21no—
30C61×#386—22no—
31C61×#381C2211yescharacteristic
32C22×Dic122—21no—
33C22×Dic122D412yescomplemented
34C23×Dic61—21no—
35C23×Dic61—21no—
36#14519807—48no—
37C2×#11758202—24no—
38C2×#11758200—24no—
39C2×#11758194—22no—
40#14519756—24no—
41#14519756—24no—
42#14519756—22no—
43#14519756—22no—
44#14519756D414yescomplemented
45#14519756C2314yes—
46#14519754—28no—
47#14519754—28no—
48#14519754—28no—
49#14519754—28no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61D412yescomplemented
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—24no—
58#14519701—24no—
59#14519698—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yes—
62C25×C61D411yescharacteristic
63C2×#41×C61—22no—
64C2×#41×C61—22no—
65C2×#41×C61—44no—
66C22×Q8×C61C2311yescharacteristic
67C22×D4×C61—21no—
68C22×D4×C61—22no—
69C22×D4×C61C2311yescharacteristic
70C23×C4×C61—21no—
71C23×C4×C61D411yescharacteristic
72C23×C4×C61C2311yescharacteristic
73#120×C61—44no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yes—
80C2×C22⋊C4×C61C2312yes—
81#95×C61—21no—
82#95×C61—22no—
83#95×C61C2312yes—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—24no—
87C24×C61—21no—
88C24×C61—42no—
89C24×C61—81no—
90C24×C61C2×D411yescharacteristic
91C24×C61C2×D412yes—
92#41×C61—28no—
93#41×C61—44no—
94#41×C61—44no—
95#41×C61—44no—
96C2×Q8×C61—22no—
97C2×Q8×C61—42no—
98C2×D4×C61—24no—
99C2×D4×C61—22no—
100C2×D4×C61—42no—
101C2×D4×C61—42no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—44no—
105C2×D4×C61—44no—
106C2×D4×C61—82no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61—81no—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—24no—
124C22×Dic61—24no—
125C22×Dic61—82no—
126C22×Dic61—82no—
127C2×Dic122—24no—
128C2×Dic122—22no—
129C2×Dic122—44no—
130C2×Dic122—82no—
131C2×Dic122C2×D414yes—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—44no—
135C22⋊C4×C61—42no—
136C22⋊C4×C61—82no—
137C22⋊Dic61—28no—
138C22⋊Dic61—24no—
139C22⋊Dic61—24no—
140#11758202—48no—
141#11758200—48no—
142C4⋊Dic61—24no—
143C4⋊Dic61—22no—
144C4⋊Dic61C2×D414yes—
145#11758196—28no—
146#11758196—24no—
147#11758196—24no—
148#11758194—44no—
149C2×#92418—24no—
150C23×C61—24no—
151C23×C61—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—41no—
163C23×C61—42no—
164C23×C61—41no—
165C23×C61—42no—
166C23×C61—81no—
167C23×C61—81no—
168C23×C61—88no—
169C23×C61#12011yescharacteristic
170C23×C61#12011yescharacteristic
171C23×C61#13714yescomplemented
172C23×C61C22×D411yescharacteristic
173Q8×C61—24no—
174Q8×C61—42no—
175D4×C61—24no—
176D4×C61—42no—
177D4×C61—44no—
178D4×C61—44no—
179D4×C61—88no—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—48no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—81no—
195C2×C4×C61#12012yes—
196C2×C4×C61C22×D412yes—
197C2×Dic61—24no—
198C2×Dic61—24no—
199C2×Dic61—44no—
200C2×Dic61—44no—
201C2×Dic61—84no—
202C2×Dic61—84no—
203Dic122—44no—
204Dic122—82no—
205Dic122—84no—
206#92418—44no—
207#18133—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—24no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—44no—
217C22×C61—44no—
218C22×C61—48no—
219C22×C61—44no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61—84no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61B(4,𝔽2)12yes—
229C22×C61C2×#12011yescharacteristic
230C22×C61C2×#13712yes—
231C4×C61—22no—
232C4×C61—22no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—82no—
236Dic61—44no—
237Dic61—44no—
238C2×#543—611no—
239#3007—612no—
240#2404D6111yescharacteristic
241#1602—614no—
242#1596—614no—
243#1511—612no—
244#1462—611no—
245C2×C61—22no—
246C2×C61—22no—
247C2×C61—41no—
248C2×C61—44no—
249C2×C61—82no—
250C2×C61#300411yescharacteristic
251C2×C61#301911yescharacteristic
252C2×C61C2×B(4,𝔽2)11yescharacteristic
253C22×#41C2×D6111yescharacteristic
254C2×#137—1222no—
255#543—614no—
256#543—1224no—
257#543—1222no—
258C2×C22⋊Q8—612no—
259C2×C22⋊Q8—1222no—
260C2×C22⋊Q8—1222no—
261C2×C22⋊Q8—1221no—
262C2×#120—21no—
263C22×C22⋊C4—1221no—
264C22×C22⋊C4C2×D6111yescharacteristic
265#452—1224no—
266#451—614no—
267#450—614no—
268#417—1222no—
269#407—612no—
270#393C2×D6112yes—
271#392C2×D6112yes—
272#386—21no—
273#386—22no—
274#381C2×D6111yescharacteristic
275#328—618no—
276#323—614no—
277C61#1813311yescharacteristic, complemented, Sylow
278C25C22⋊D6111yescharacteristic
279C2×#41—22no—
280C2×#41—22no—
281C2×#41—44no—
282C22×Q8—612no—
283C22×Q8—1221no—
284C22×Q8C22×D6111yescharacteristic
285C22×D4—21no—
286C22×D4—22no—
287C22×D4C22×D6111yescharacteristic
288C23×C4—21no—
289C23×C4—1221no—
290C23×C4—1221no—
291C23×C4C22⋊D6111yescharacteristic
292C23×C4C22×D6111yescharacteristic
293#137—2448no—
294C2×Q16—1224no—
295C2×SD16—1224no—
296C2×M16—1222no—
297C22⋊Q8—614no—
298C22⋊Q8—614no—
299C22⋊Q8—1228no—
300C22⋊Q8—1228no—
301C22⋊Q8—1228no—
302C22⋊Q8—1228no—
303C22⋊Q8—1224no—
304C22⋊Q8—1222no—
305C22⋊Q8—1222no—
306C22⋊Q8—1224no—
307#120—44no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1224no—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4—1224no—
321C2×C22⋊C4C22⋊D6112yes—
322C2×C22⋊C4C22×D6112yes—
323#103—1224no—
324#102—1224no—
325C22⋊C8—614no—
326C22⋊C8—1222no—
327#95—21no—
328#95—22no—
329#95C22×D6112yes—
330C24—21no—
331C24—21no—
332C24—24no—
333C24—21no—
334C24—42no—
335C24—81no—
336C24C2×C22⋊D6111yescharacteristic
337C24C2×C22⋊D6112yes—
338#41—28no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—614no—
345C2×Q8—1224no—
346C2×Q8—1222no—
347C2×Q8—4882no—
348C2×Q8—2444no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—82no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—81no—
367C22×C4—1224no—
368C22×C4—1224no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—4882no—
372C22×C4—4882no—
373C22×C4D4×D6112yes—
374C22×C4D4×D6112yes—
375C22×C4C2×C22⋊D6112yes—
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378Q16—2448no—
379SD16—2448no—
380M16—2444no—
381C2×C8—1224no—
382C4⋊C4—614no—
383C4⋊C4—1228no—
384C4⋊C4—1224no—
385C4⋊C4—1224no—
386C4⋊C4—1222no—
387C4⋊C4—1224no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—82no—
393C22⋊C4—1228no—
394C22⋊C4—1224no—
395C22⋊C4—1224no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—81no—
413C23—81no—
414C23—88no—
415C23#1451974511yescharacteristic
416C23#1451974511yescharacteristic
417C23#1451980714yescomplemented
418C23C22×C22⋊D6111yescharacteristic
419Q8—24no—
420Q8—42no—
421Q8—4882no—
422Q8—4884no—
423Q8—2444no—
424D4—24no—
425D4—42no—
426D4—44no—
427D4—44no—
428D4—88no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—1224no—
445C2×C4—1224no—
446C2×C4—4884no—
447C2×C4—4884no—
448C2×C4—2444no—
449C2×C4—2444no—
450C2×C4#1451974512yes—
451C2×C4C2×D4×D6112yes—
452C8—2444no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—24no—
461C22—44no—
462C22—44no—
463C22—48no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—82no—
470C22—84no—
471C22—81no—
472C22—81no—
473C22#1683994812yes—
474C22C2×#1451974511yescharacteristic
475C22C2×#1451980712yes—
476C4—22no—
477C4—22no—
478C4—42no—
479C4—42no—
480C4—82no—
481C4—2444no—
482C4—2444no—
483C2—22no—
484C2—22no—
485C2—41no—
486C2—44no—
487C2—82no—
488C2#2145505211yescharacteristic
489C2#2145517511yescharacteristic
490C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2404C2
socleC22×C61C2×#120
hypercenter#2404D61
hypocenterC61#18133
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#16840388C2semidirect
2. #21455079C2semidirect
3. #21444775C2semidirect
4. #21444741C2semidirect
5. #21444351C2semidirect
6. #21444146C2semidirect
18. #16839918C22semidirect
19. #16839913C22semidirect
22. #16839498C22semidirect
33. C22×Dic122D4semidirect
44. #14519756D4semidirect
51. C2×C4⋊Dic61D4semidirect
171. C23×C61#137semidirect
277. C61#18133semidirect
417. C23#14519807semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214550792
#214447752
#214447412
#214443512
#214441462
C61×#24042
#1813361