Finite Group Databasebeta

← #30600205

Subgroups — #30600205

Order 15616.

928 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 126 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×#16840732C211yesmaximal, characteristic
2#21461641C211yesmaximal, characteristic
3C2×#16840607C212yesmaximal, complemented
4#21460657C212yesmaximal, complemented
5C2×#16839637C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840732—24no—
8C2×C22⋊#92420—21no—
9C2×C22⋊#92420—21no—
10C2×C22⋊#92420—21no—
11C2×C22⋊#92420C2212yes—
12C2×C22⋊#92420C2212yes—
13#16840607—24no—
14C24×Dic61C2211yescharacteristic
15C22×C4⋊Dic61C411yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C411yescharacteristic
18#16839637—21no—
19#16839637—21no—
20#16839637—21no—
21#16839637—21no—
22C2×#14519684C411yescharacteristic
23C2×#14519684C2211yescharacteristic
24C22×C22⋊C4×C61C411yescharacteristic
25C2×#11758217—24no—
26C2×#11758217—42no—
27C2×#11758217—42no—
28C22×#92420—22no—
29C22×#92420—22no—
30C22⋊#92420—24no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34C22⋊#92420—42no—
35C22⋊#92420—42no—
36C22⋊#92420D414yes—
37C23×Dic61—22no—
38C23×Dic61—22no—
39C23×Dic61—42no—
40C23×Dic61—42no—
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C23×Dic61D411yescharacteristic
45C23×Dic61D411yescharacteristic
46C23×Dic61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48C23×Dic61C2311yescharacteristic
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—42no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—42no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63C25×C61C2×C411yescharacteristic
64C23×C4×C61C2×C411yescharacteristic
65C23×C4×C61C2×C411yescharacteristic
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—22no—
70C24×C61—22no—
71C24×C61—22no—
72C24×C61—21no—
73C24×C61—21no—
74C24×C61—22no—
75C24×C61—22no—
76C24×C61—41no—
77C24×C61—41no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C22⋊C411yescharacteristic
80C24×C61C22×C411yescharacteristic
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61M1612yescomplemented
90C22×C4×C61M1612yescomplemented
91C22×Dic61—22no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—22no—
106C22×Dic61—22no—
107C22×Dic61—41no—
108C22×Dic61—41no—
109C22×Dic61—41no—
110C22×Dic61—41no—
111C22×Dic61—41no—
112C22×Dic61—41no—
113C22×Dic61—41no—
114C22×Dic61—48no—
115C22×Dic61—48no—
116C22×Dic61—41no—
117C22×Dic61—41no—
118C22×Dic61—41no—
119C22×Dic61—41no—
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C22⋊C411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D412yes—
125C22×Dic61#4112yes—
126#11758217—48no—
127C2×#92420—24no—
128C2×#92420—42no—
129C2×#92420—42no—
130C2×#92420—42no—
131C2×#92420—42no—
132C2×#92420—42no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—24no—
135C22⋊Dic61—42no—
136C22⋊Dic61—42no—
137C4⋊Dic61—42no—
138C4⋊Dic61—42no—
139C4⋊Dic61—41no—
140C4⋊Dic61—42no—
141C4⋊Dic61—41no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—24no—
147C23×C61—24no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61—44no—
161C23×C61—44no—
162C23×C61—44no—
163C23×C61—44no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61#9912yes—
169C23×C61#10112yes—
170C23×C61C2×C22⋊C411yescharacteristic
171C23×C61C4×D412yes—
172C23×C61C2×M1611yescharacteristic
173C23×C61C2×M1611yescharacteristic
174C23×C61#13112yes—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—24no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×Dic61—24no—
186C2×Dic61—24no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—44no—
199C2×Dic61—44no—
200C2×Dic61—44no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—82no—
210C2×Dic61—82no—
211C2×Dic61#9912yes—
212C2×Dic61#12011yescharacteristic
213C2×Dic61#12112yes—
214C2×Dic61#12311yescharacteristic
215#92420—44no—
216#92420—82no—
217#9085—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—48no—
233C22×C61—48no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61#38611yescharacteristic
243C22×C61#40711yescharacteristic
244C22×C61#40811yescharacteristic
245C22×C61C2×#9911yescharacteristic
246C22×C61C2×#9911yescharacteristic
247C22×C61C2×#10111yescharacteristic
248C22×C61#41311yescharacteristic
249C22×C61#43512yes—
250C22×C61#43612yes—
251C4×C61—42no—
252C4×C61—42no—
253Dic61—21no—
254Dic61—22no—
255Dic61—22no—
256Dic61—21no—
257Dic61—41no—
258Dic61—81no—
259Dic61—81no—
260C2×#407—611no—
261C2×#385—611no—
262#1779—611no—
263C2×#324—612no—
264#1280—612no—
265C2×C61—22no—
266C2×C61—22no—
267C2×C61—42no—
268C2×C61—42no—
269C2×C61#146311yescharacteristic
270C2×C61#146511yescharacteristic
271C2×C61#146811yescharacteristic
272C2×C61#147211yescharacteristic
273C2×C61#187011yescharacteristic
274C2×C61#189811yescharacteristic
275C2×C61#190111yescharacteristic
276C24×C4—611no—
277C22×M16—1221no—
278C22×C4⋊C4—611no—
279C22×C4⋊C4—611no—
280C22×C22⋊C4—611no—
281C22×C22⋊C4#692311yescharacteristic
282#407—1224no—
283C2×C22⋊C8—612no—
284C2×C22⋊C8—612no—
285C2×C22⋊C8—1221no—
286C2×C22⋊C8—1221no—
287C2×C22⋊C8—1221no—
288#385—1221no—
289#385—1221no—
290#385—1221no—
291#385—1221no—
292C2×#95—611no—
293C2×#95—611no—
294#324—1224no—
295C61#908511yescharacteristic, complemented, Sylow
296C25C2×#692311yescharacteristic
297C23×C4—611no—
298C23×C4—611no—
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—611no—
305C23×C4—1222no—
306C23×C4—1222no—
307C23×C4—2442no—
308C23×C4—2442no—
309C23×C4C2×#692311yescharacteristic
310C23×C4C2×#692311yescharacteristic
311C2×M16—1224no—
312C2×M16—2442no—
313C2×M16—2442no—
314C22×C8—1222no—
315C22×C8—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—2442no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—2442no—
330C22⋊C8—614no—
331C22⋊C8—1222no—
332C22⋊C8—1224no—
333C22⋊C8—1222no—
334C22⋊C8—1222no—
335C22⋊C8—2442no—
336C22⋊C8—2442no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—41no—
348C24—41no—
349C24C22⋊#692311yescharacteristic
350C24C22⋊#692311yescharacteristic
351C24C22×#692311yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—612no—
364C22×C4—612no—
365C22×C4—611no—
366C22×C4—1221no—
367C22×C4—1222no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1221no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2448no—
390C22×C4—2448no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2441no—
395C22×C4#1175821712yescomplemented
396C22×C4#1175821712yescomplemented
397M16—2448no—
398C2×C8—1224no—
399C2×C8—2442no—
400C2×C8—2442no—
401C2×C8—2442no—
402C2×C8—2442no—
403C2×C8—2442no—
404C4⋊C4—2441no—
405C4⋊C4—2442no—
406C4⋊C4—2441no—
407C4⋊C4—2442no—
408C4⋊C4—2442no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—2442no—
412C22⋊C4—2442no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—24no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#1451983612yes—
440C23#1451983812yes—
441C23C2×C22⋊#692311yescharacteristic
442C23D4×#692312yes—
443C23C2×#1175821711yescharacteristic
444C23C2×#1175821711yescharacteristic
445C23#1451986412yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—611no—
457C2×C4—612no—
458C2×C4—611no—
459C2×C4—612no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—4882no—
472C2×C4—4882no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2444no—
476C2×C4—2444no—
477C2×C4—2444no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C8—4882no—
487C8—2444no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—48no—
503C22—48no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22#1684071011yescharacteristic
513C22#1684073211yescharacteristic
514C22#1684073511yescharacteristic
515C22D61⋊#9911yescharacteristic
516C22C2×#1451983611yescharacteristic
517C22C2×#1451983811yescharacteristic
518C22#1684074911yescharacteristic
519C22#1684079312yes—
520C22#1684079712yes—
521C4—42no—
522C4—42no—
523C4—1222no—
524C4—1221no—
525C4—1222no—
526C4—1221no—
527C4—4881no—
528C4—4881no—
529C4—2441no—
530C2—22no—
531C2—22no—
532C2—42no—
533C2—42no—
534C2#2146106711yescharacteristic
535C2#2146107511yescharacteristic
536C2#2146108611yescharacteristic
537C2#2146109811yescharacteristic
538C2#2146179611yescharacteristic
539C2#2146184111yescharacteristic
540C2#2146184611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#9085
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16840607C2semidirect
4. #21460657C2semidirect
89. C22×C4×C61M16semidirect
90. C22×C4×C61M16semidirect
295. C61#9085semidirect
395. C22×C4#11758217semidirect
396. C22×C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407322
#214616412
C2×#168406072
#214606572
C2×#168396372
#908561