Finite Group Databasebeta

← #29577059

Subgroups — #29577059

Order 15616.

678 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455077C211yesmaximal, characteristic
2#21454265C211yesmaximal, characteristic
3#21445757C211yesmaximal, characteristic
4#21444183C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C61×#1520C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519808C411yescharacteristic, complemented
10C2×#14519808C411yescharacteristic, complemented
11C2×#14519783—21no—
12#16839922—21no—
13#16839922—21no—
14#16839917—21no—
15#16839912—21no—
16#16839761C411yescharacteristic, complemented
17#16839761C411yescharacteristic, complemented
18C2×#14519700—21no—
19C2×#14519697—21no—
20#16839728—21no—
21#16839727—22no—
22#16839726C2211yescharacteristic
23#16839726C2211yescharacteristic
24#16839724—21no—
25C2×C22⋊#92418C2211yescharacteristic
26C22⋊C8×D61—21no—
27#16839488—22no—
28C2×C22⋊Q8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C61×#328—21no—
32C61×#328—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—42no—
36C2×#11758222—42no—
37C2×#11758222C2×C411yescharacteristic
38C2×#11758222C2×C411yescharacteristic
39C2×#11758222D411yescharacteristic
40C2×#11758222D411yescharacteristic
41C22×Dic122D411yescharacteristic
42C22×D244D411yescharacteristic
43C22×C22⋊D61—21no—
44C22×C4×D61—21no—
45#14519808—42no—
46#14519808—42no—
47C2×#11758202—21no—
48C2×#11758202—21no—
49C2×#11758201—21no—
50C2×#11758201—21no—
51#14519783—42no—
52#14519783—42no—
53C2×#11758194—21no—
54C2×#11758194C2×C411yescharacteristic
55C2×#11758194C2×C411yescharacteristic
56C2×#11758189—22no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59#14519700—21no—
60#14519700—21no—
61#14519700—42no—
62#14519697—21no—
63#14519697—21no—
64#14519697—42no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70C22⋊#92418D412yes—
71#14519689—22no—
72#14519689—22no—
73C22×Q8×C61C2×C411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C22×C8×C61—21no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2×C411yescharacteristic
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—21no—
85C22⋊C8×C61—21no—
86C22⋊C8×C61D411yescharacteristic
87C22⋊C8×C61D411yescharacteristic
88#95×C61—21no—
89C24×C61#4111yescharacteristic
90C23×D61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C22×C411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109C2×C22⋊D61—21no—
110C2×C22⋊D61—44no—
111C22×Dic61—21no—
112#11758222—22no—
113#11758222—22no—
114#11758222—22no—
115#11758222—22no—
116#11758222—41no—
117#11758222—41no—
118#11758222—41no—
119#11758222—41no—
120#11758222—44no—
121#11758222—44no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—41no—
125C2×D244C22⋊C411yescharacteristic
126C2×D244C22⋊C411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—44no—
132C2×Dic122—22no—
133C2×Dic122—21no—
134C2×Dic122—41no—
135C2×Dic122C22⋊C411yescharacteristic
136C2×Dic122C22⋊C411yescharacteristic
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—41no—
140C2×C8×C61—41no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—81no—
150#11758202—42no—
151#11758202—42no—
152#11758201—42no—
153#11758201—42no—
154#11758194—41no—
155#11758194—41no—
156#11758194—41no—
157#11758194—41no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—41no—
162C2×#92418—41no—
163C2×#92418—41no—
164#11758189—44no—
165C8×D61—44no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61C4×D411yescharacteristic
171C23×C61#12111yescharacteristic
172C23×C61#12311yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—42no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—81no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—81no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—81no—
195C2×C4×C61—81no—
196C2×C4×C61—81no—
197C2×C4×C61—81no—
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C4×D411yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C22⋊D61—24no—
203C22⋊D61—42no—
204C22⋊D61—42no—
205C22⋊D61—42no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—42no—
209D244—21no—
210D244—21no—
211D244—21no—
212D244—42no—
213D244#9912yes—
214C4×D61—24no—
215C4×D61—42no—
216C4×D61—42no—
217C4×D61—42no—
218Dic122—21no—
219Dic122—21no—
220Dic122—21no—
221Dic122—42no—
222Dic122#9912yes—
223C8×C61—41no—
224C8×C61—41no—
225#92418—42no—
226#92418—41no—
227#92418—41no—
228#92418—81no—
229#9345—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61#38611yescharacteristic
241C22×C61#47111yescharacteristic
242C22×C61#47311yescharacteristic
243C2×D61—22no—
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61—41no—
249C2×D61—82no—
250C2×D61—82no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61—161no—
257C4×C61C2×#9911yescharacteristic
258C4×C61C2×#9911yescharacteristic
259Dic61—22no—
260Dic61—41no—
261#3007—611no—
262#2880—611no—
263#1788—611no—
264#1520D6111yescharacteristic, complemented
265#1472—611no—
266#1305—611no—
267#1298—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—41no—
271C2×C61—41no—
272C2×C61#188111yescharacteristic
273C2×C61#189811yescharacteristic
274C2×C61#190811yescharacteristic
275D61—42no—
276D61—81no—
277C22×#41—611no—
278C2×#137—611no—
279C2×#137—611no—
280C2×#131—1221no—
281C2×C22⋊Q8C2×D6111yescharacteristic
282#452—1221no—
283#452—1221no—
284#451—1221no—
285#450—1221no—
286#417—611no—
287#417—611no—
288C2×#103—1221no—
289C2×#102—1221no—
290#408—1222no—
291#408—1221no—
292#407—611no—
293#407—611no—
294#407—1221no—
295C2×C22⋊C8—611no—
296C2×C22⋊C8—1221no—
297C2×C22⋊C8C2×D6111yescharacteristic
298#380C2×D6111yescharacteristic
299#328—21no—
300#328—21no—
301#324—1222no—
302C61#934511yescharacteristic, complemented, Sylow
303C2×#41—611no—
304C2×#41—611no—
305C2×#41—611no—
306C2×#41—611no—
307C2×#41—1221no—
308C2×#41—1221no—
309C2×#41—2442no—
310C2×#41—2442no—
311C22×Q8—611no—
312C22×Q8C4×D6111yescharacteristic
313C22×D4—611no—
314C22×D4—1221no—
315C23×C4—1221no—
316C23×C4C22×D6111yescharacteristic
317#137—2442no—
318#137—2442no—
319C2×Q16—1221no—
320C2×Q16—1221no—
321C2×SD16—1221no—
322C2×SD16—1221no—
323#131—2442no—
324#131—2442no—
325C2×M16—611no—
326C2×M16—611no—
327C2×M16—1222no—
328C2×M16—1221no—
329C22×C8—21no—
330C22×C8—1222no—
331C22×C8—1221no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C4×D6111yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341#103—1221no—
342#103—1221no—
343#103—2442no—
344#102—1221no—
345#102—1221no—
346#102—2442no—
347C22⋊C8—21no—
348C22⋊C8—612no—
349C22⋊C8—1222no—
350C22⋊C8—1222no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C22⋊C8—1221no—
354C22⋊C8—1221no—
355C22⋊C8—1221no—
356C22⋊C8C22⋊D6111yescharacteristic
357C22⋊C8C22⋊D6111yescharacteristic
358#95—21no—
359C24—1221no—
360C24#1175822411yescharacteristic
361#41—1222no—
362#41—1222no—
363#41—1222no—
364#41—1222no—
365#41—2441no—
366#41—2441no—
367#41—2441no—
368#41—2441no—
369#41—2444no—
370#41—2444no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—611no—
376C2×Q8—611no—
377C2×Q8—1222no—
378C2×Q8—1221no—
379C2×Q8—2441no—
380C2×D4—611no—
381C2×D4—611no—
382C2×D4—1222no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2441no—
389C2×D4—2444no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—2444no—
401C22×C4C2×C4×D6111yescharacteristic
402C22×C4#1175822211yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407Q16—2442no—
408Q16—2442no—
409SD16—2442no—
410SD16—2442no—
411M16—2444no—
412M16—2441no—
413M16—2441no—
414M16—2441no—
415M16—2441no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—1221no—
421C2×C8—1221no—
422C2×C8—1222no—
423C2×C8—2444no—
424C2×C8—2441no—
425C2×C8—2441no—
426C2×C8—2441no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—81no—
436C23—21no—
437C23—21no—
438C23—41no—
439C23—41no—
440C23—1221no—
441C23—1221no—
442C23—4881no—
443C23—2442no—
444C23—2441no—
445C23—2441no—
446C23#1451973711yescharacteristic
447C23#1451974811yescharacteristic
448C23#1451975911yescharacteristic
449Q8—41no—
450Q8—41no—
451Q8—81no—
452Q8—612no—
453Q8—1221no—
454Q8—1221no—
455Q8—1221no—
456Q8—2442no—
457D4—612no—
458D4—1224no—
459D4—1221no—
460D4—1221no—
461D4—1221no—
462D4—2442no—
463D4—2442no—
464D4—2442no—
465D4—2442no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—1224no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4C22⋊C4×D6111yescharacteristic
490C2×C4C4×C22⋊D6111yescharacteristic
491C2×C4#1451974511yescharacteristic
492C2×C4#1451975011yescharacteristic
493C8—41no—
494C8—41no—
495C8—4881no—
496C8—2442no—
497C8—2441no—
498C8—2441no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—81no—
508C22—81no—
509C22—1222no—
510C22—4882no—
511C22—4882no—
512C22—2441no—
513C22—2441no—
514C22—2441no—
515C22—2441no—
516C22—2441no—
517C22#1683963811yescharacteristic
518C22#1684002211yescharacteristic
519C22#1684003411yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4—1222no—
527C4—2441no—
528C4#99×D6111yescharacteristic
529C4#99×D6111yescharacteristic
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—4881no—
535C2—2442no—
536C2#2144617811yescharacteristic
537C2#2144626911yescharacteristic
538C2#2144633711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1520C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1520D61
hypocenterC61#9345
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
4. #21444183C2semidirect
5. #21443725C2semidirect
6. #21443707C2semidirect
7. C61×#1520C2semidirect
9. C2×#14519808C4semidirect
10. C2×#14519808C4semidirect
16. #16839761C4semidirect
17. #16839761C4semidirect
264. #1520D61semidirect
302. C61#9345semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214542652
#214457572
#214441832
#214437252
#214437072
C61×#15202
#934561