Finite Group Databasebeta

← #29575983

Subgroups — #29575983

Order 15616.

582 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 91 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#21454846C211yesmaximal, characteristic, complemented
2#21451704C211yesmaximal, characteristic
3C4×#14519691C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic, complemented
5#21443673C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D61C2211yescharacteristic
9#16839836—22no—
10#16839831—22no—
11#16839816—21no—
12C2×#14519691—21no—
13#16839726C2211yescharacteristic
14C22⋊C8×D61—21no—
15#16839700C411yescharacteristic, complemented
16#16839697C411yescharacteristic, complemented
17#16839676—21no—
18#16839676C411yescharacteristic, complemented
19#16839670—21no—
20#16839670C411yescharacteristic, complemented
21#16839650—21no—
22#16839638—21no—
23#16839625C2211yescharacteristic
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—21no—
26#16839496—22no—
27C2×#121×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C4×C22⋊C4×C61C2211yescharacteristic
30C61×#327—22no—
31C22×C22⋊D61—21no—
32C22×C22⋊D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C2×#11758194—21no—
35C2×C8×D61—22no—
36C4×C22⋊D61—21no—
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39C4×C22⋊D61—21no—
40C2×#11758196—21no—
41C2×#11758196C2×C411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61D411yescharacteristic
48C2×#11758198—21no—
49C2×#11758198C2×C411yescharacteristic
50C2×C4×Dic61—21no—
51C2×C4×Dic61D411yescharacteristic
52#14519706—22no—
53#14519691—24no—
54C22⋊#92418—21no—
55C22⋊#92418D412yescomplemented
56#14519689—22no—
57#14519687—22no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684C2×C411yescharacteristic
63#14519684C2×C411yescharacteristic
64C22×D4×C61—21no—
65C22×D4×C61C2×C411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C2×M16×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61D412yescomplemented
78#95×C61—21no—
79C24×C61—21no—
80C24×C61C2×D411yescharacteristic
81C23×D61C22⋊C411yescharacteristic
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—81no—
87C2×D4×C61—81no—
88C2×D4×C61—81no—
89C22×C4×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61—81no—
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—21no—
99C2×C22⋊D61—21no—
100C2×C22⋊D61—22no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—42no—
103C2×C22⋊D61—81no—
104C2×C22⋊D61—81no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—41no—
108C22×Dic61C22⋊C411yescharacteristic
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C411yescharacteristic
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—81no—
114M16×C61—81no—
115C2×C8×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61—81no—
123C22⋊Dic61—21no—
124C22⋊Dic61—21no—
125C22⋊Dic61—22no—
126C22⋊Dic61—42no—
127C22⋊Dic61—41no—
128C22⋊Dic61—81no—
129C22⋊Dic61—81no—
130C22⋊Dic61—81no—
131#11758198—21no—
132#11758198—21no—
133#11758198—81no—
134#11758196—21no—
135#11758196—21no—
136#11758196—81no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—81no—
140#11758194—81no—
141C2×#92418—22no—
142C8×D61—82no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61C2×C22⋊C411yescharacteristic
155C23×C61#12111yescharacteristic
156C23×C61#12111yescharacteristic
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61#10412yes—
162D4×C61—81no—
163D4×C61—81no—
164D4×C61—82no—
165D4×C61—82no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—81no—
177C2×C4×C61—81no—
178C2×C4×C61C4×D411yescharacteristic
179C2×C4×C61C4×D411yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12311yescharacteristic
182C22⋊D61—42no—
183C22⋊D61—42no—
184C22⋊D61—84no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—81no—
200C2×Dic61—81no—
201C4×D61—42no—
202C4×D61—81no—
203C8×C61—42no—
204#92418—42no—
205#9185—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61#38611yescharacteristic
224C22×C61C2×#10411yescharacteristic
225C22×C61#42111yescharacteristic
226C2×D61—22no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—42no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—82no—
236C4×C61—81no—
237Dic61—41no—
238Dic61—41no—
239Dic61—41no—
240Dic61—81no—
241Dic61—81no—
242Dic61—82no—
243#2973—611no—
244#2635—611no—
245C4×#99—611no—
246#1507—611no—
247#1507D6111yescharacteristic, complemented
248#1284—611no—
249#1284—611no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—41no—
253C2×C61—41no—
254C2×C61—161no—
255C2×C61#187111yescharacteristic
256C2×C61#191011yescharacteristic
257C2×C61#191311yescharacteristic
258D61—42no—
259C2×#121C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#435—1222no—
262#435—1222no—
263#432—1221no—
264C2×#99—1221no—
265#407—611no—
266#407C2×D6111yescharacteristic
267C2×C22⋊C8—1221no—
268#400—611no—
269#399—611no—
270#394—611no—
271#394—1221no—
272#392—611no—
273#392—1221no—
274#388—1221no—
275#386—1221no—
276#381—611no—
277C4×C22⋊C4—611no—
278C4×C22⋊C4C2×D6111yescharacteristic
279#342—1221no—
280#327—22no—
281#327—1222no—
282C61#918511yescharacteristic, complemented, Sylow
283C22×D4—21no—
284C22×D4—611no—
285C22×D4—1221no—
286C22×D4C4×D6111yescharacteristic
287C23×C4—611no—
288C23×C4C22×D6111yescharacteristic
289C2×M16—21no—
290C2×M16—1221no—
291C22×C8—1222no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296C4×D4—1221no—
297C4×D4—1221no—
298C4×D4—1221no—
299C4×D4—1221no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4C4×D6111yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—611no—
306C2×C22⋊C4—611no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C42—21no—
314C2×C42—611no—
315C2×C42—1221no—
316C4⋊C8—1222no—
317#99—1224no—
318C22⋊C8—21no—
319C22⋊C8—612no—
320C22⋊C8—1222no—
321C22⋊C8—1221no—
322C22⋊C8C22⋊D6112yescomplemented
323#97—1222no—
324#95—21no—
325#95—611no—
326#95—611no—
327#95—1221no—
328#95—1221no—
329#95—1221no—
330#95—1221no—
331C24—21no—
332C24—611no—
333C24D4×D6111yescharacteristic
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4—81no—
341C2×D4—1222no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—4881no—
345C2×D4—4881no—
346C2×D4—2441no—
347C2×D4—2442no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—81no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—4881no—
359C22×C4—2441no—
360C22×C4C2×C4×D6111yescharacteristic
361C22×C4#1175822211yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4D4×D6111yescharacteristic
364C22×C4#1175822411yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366M16—81no—
367M16—4881no—
368C2×C8—22no—
369C2×C8—1222no—
370C2×C8—4882no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—4881no—
376C22⋊C4—22no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C22⋊C4—1221no—
384C22⋊C4—1222no—
385C22⋊C4—4881no—
386C22⋊C4—4881no—
387C22⋊C4—4881no—
388C22⋊C4—4881no—
389C22⋊C4—2442no—
390C22⋊C4—2441no—
391C42—81no—
392C42—1221no—
393C42—1221no—
394C42—4881no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—612no—
407C23—1221no—
408C23—1221no—
409C23—2441no—
410C23—2441no—
411C23C22⋊C4×D6111yescharacteristic
412C23#1451974811yescharacteristic
413C23#1451975011yescharacteristic
414D4—81no—
415D4—81no—
416D4—82no—
417D4—82no—
418D4—4884no—
419D4—2442no—
420D4—2442no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—4881no—
439C2×C4—4881no—
440C2×C4—4881no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451973711yescharacteristic
452C2×C4C4×C22⋊D6111yescharacteristic
453C2×C4#1451974511yescharacteristic
454C2×C4#1451975911yescharacteristic
455C8—42no—
456C8—2442no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—1222no—
475C22—4881no—
476C22—2442no—
477C22—2441no—
478C22—2441no—
479C22#1683963811yescharacteristic
480C22#104×D6111yescharacteristic
481C22#1683977711yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—82no—
487C4—81no—
488C4—4881no—
489C4—4881no—
490C4—4882no—
491C4—2441no—
492C4—2441no—
493C4—2441no—
494C2—21no—
495C2—21no—
496C2—41no—
497C2—41no—
498C2—161no—
499C2—2442no—
500C2#2144611811yescharacteristic
501C2#2144634311yescharacteristic
502C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#386
prime radical#21451704C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#9185
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
1. #21454846C2semidirect
3. C4×#14519691C2semidirect
4. #21444335C2semidirect
5. #21443673C2semidirect
6. #21443672C2semidirect
7. C61×#1507C2semidirect
15. #16839700C4semidirect
16. #16839697C4semidirect
18. #16839676C4semidirect
20. #16839670C4semidirect
55. C22⋊#92418D4semidirect
77. C22⋊C8×C61D4semidirect
247. #1507D61semidirect
282. C61#9185semidirect
322. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214548462
#214517042
C4×#145196912
#214443352
#214436732
#214436722
C61×#15072
#918561