Finite Group Databasebeta

← #29577794

Subgroups — #29577794

Order 15616.

912 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 109 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#21454845C211yesmaximal, characteristic, complemented
2#21449169C211yesmaximal, characteristic
3#21445804C211yesmaximal, characteristic, complemented
4#21445750C211yesmaximal, characteristic, complemented
5#21443687C211yesmaximal, characteristic, complemented
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1786C211yesmaximal, characteristic, complemented
8C2×#14519748—21no—
9C2×#14519746—21no—
10C2×C4×C22⋊D61C2211yescharacteristic
11#16839833—22no—
12#16839830—22no—
13#16839816—21no—
14C2×#14519698—22no—
15C2×#14519691—21no—
16#16839724—21no—
17C2×C22⋊#92418C2211yescharacteristic
18#16839676C412yescomplemented
19#16839670C412yescomplemented
20#16839665—21no—
21#16839622C2211yescharacteristic
22#16839622C2211yescharacteristic
23#16839618—21no—
24#16839528—21no—
25C23×D4×C61C2211yescharacteristic
26C2×#102×C61—22no—
27C2×C22⋊C8×C61C2211yescharacteristic
28C61×#380C2211yescharacteristic
29C22×C22⋊D61—22no—
30C22×C22⋊D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×#11758189—22no—
33C22×#92418D412yescomplemented
34#14519748—44no—
35#14519746—42no—
36#14519746—42no—
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39C4×C22⋊D61—21no—
40C4×C22⋊D61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×#11758196C2×C411yescharacteristic
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—22no—
49C2×C22⋊Dic61D411yescharacteristic
50C2×#11758198C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52#14519706—22no—
53#14519698—44no—
54#14519691—42no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519689—22no—
58#14519687—22no—
59#14519684—21no—
60#14519684C2×C412yes—
61C25×C61—21no—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61—21no—
67C22×D4×C61—44no—
68C22×D4×C61C2×C412yes—
69C23×C4×C61C2311yescharacteristic
70C22×C8×C61D412yescomplemented
71C2×C4⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73#102×C61—44no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76#95×C61—21no—
77C24×C61—22no—
78C24×C61—44no—
79C24×C61—41no—
80C24×C61—41no—
81C24×C61—41no—
82C24×C61—41no—
83C24×C61—41no—
84C24×C61—41no—
85C24×C61—82no—
86C24×C61C2×D411yescharacteristic
87C23×D61C22⋊C411yescharacteristic
88C2×D4×C61—24no—
89C2×D4×C61—24no—
90C2×D4×C61—24no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—42no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—44no—
97C2×D4×C61—44no—
98C2×D4×C61—44no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C22×C4×C61—42no—
102C22×C4×C61—41no—
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×C22⋊D61—21no—
110C2×C22⋊D61—21no—
111C2×C22⋊D61—21no—
112C2×C22⋊D61—21no—
113C2×C22⋊D61—21no—
114C2×C22⋊D61—21no—
115C2×C22⋊D61—44no—
116C2×C22⋊D61—44no—
117C2×C22⋊D61—82no—
118C22×Dic61—22no—
119C22×Dic61—41no—
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C22⋊C411yescharacteristic
122C22×Dic61C22⋊C411yescharacteristic
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—42no—
126C2×C8×C61—22no—
127C2×C8×C61—42no—
128C4⋊C4×C61—42no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊Dic61—21no—
132C22⋊Dic61—21no—
133C22⋊Dic61—44no—
134C22⋊Dic61—44no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—41no—
138C22⋊Dic61—41no—
139C22⋊Dic61—82no—
140#11758198—21no—
141#11758198—21no—
142C4⋊Dic61—42no—
143C4⋊Dic61—42no—
144#11758196—21no—
145#11758196—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C2×#92418—22no—
149C2×#92418—42no—
150#11758189—82no—
151C23×C61—24no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—44no—
157C23×C61—44no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61—84no—
165C23×C61—84no—
166C23×C61—84no—
167C23×C61—84no—
168C23×C61—81no—
169C23×C61—81no—
170C23×C61—81no—
171C23×C61—81no—
172C23×C61C2×C22⋊C411yescharacteristic
173C23×C61#12111yescharacteristic
174C23×C61#12111yescharacteristic
175C23×C61AGL(1,ℤ/8)11yescharacteristic
176C23×C61AGL(1,ℤ/8)11yescharacteristic
177C23×C61AGL(1,ℤ/8)11yescharacteristic
178C23×C61#13711yescharacteristic
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—41no—
183C22×D61—41no—
184D4×C61—24no—
185D4×C61—24no—
186D4×C61—44no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—44no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—81no—
197C2×C4×C61—81no—
198C2×C4×C61—81no—
199C2×C4×C61C4×D411yescharacteristic
200C2×C4×C61C4×D411yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12311yescharacteristic
203C22⋊D61—41no—
204C22⋊D61—41no—
205C22⋊D61—41no—
206C22⋊D61—41no—
207C22⋊D61—88no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—81no—
219C2×Dic61—81no—
220C2×Dic61—81no—
221C4×D61—42no—
222C4×D61—81no—
223C8×C61—42no—
224#92418—42no—
225#9472—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—44no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—82no—
240C22×C61—82no—
241C22×C61—84no—
242C22×C61—84no—
243C22×C61—84no—
244C22×C61—84no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61#38611yescharacteristic
248C22×C61#41711yescharacteristic
249C22×C61#41811yescharacteristic
250C22×C61#46811yescharacteristic
251C22×C61#46811yescharacteristic
252C22×C61#46911yescharacteristic
253C22×C61#46911yescharacteristic
254C2×D61—42no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—81no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—161no—
263C4×C61B(4,𝔽2)12yes—
264Dic61—41no—
265Dic61—41no—
266Dic61—41no—
267Dic61—82no—
268Dic61—161no—
269#2973—611no—
270#2408—611no—
271#1799—611no—
272#1786—611no—
273#1786D6111yescharacteristic, complemented
274#1289—611no—
275#1289—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—84no—
283C2×C61#188911yescharacteristic
284C2×C61#189511yescharacteristic
285C2×C61#191011yescharacteristic
286D61—81no—
287C23×D4C2×D6111yescharacteristic
288C2×#121—1221no—
289C2×#120—1221no—
290C2×C4×D4—611no—
291#435—1222no—
292#435—1222no—
293#432—1221no—
294C2×#102—22no—
295C2×#102—1222no—
296C2×#99—1221no—
297#407—1221no—
298C2×C22⋊C8—611no—
299C2×C22⋊C8C2×D6111yescharacteristic
300#394—612no—
301#392—612no—
302#390—1221no—
303#380—611no—
304#380—611no—
305#380C2×D6111yescharacteristic
306#379—1221no—
307#342—1221no—
308C61#947211yescharacteristic, complemented, Sylow
309C25—21no—
310C22×D4—21no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—44no—
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4C4×D6112yes—
319C23×C4—611no—
320C23×C4C22×D6111yescharacteristic
321C2×M16—1222no—
322C22×C8—612no—
323C22×C8C22⋊D6112yescomplemented
324#121—2444no—
325#120—2442no—
326#120—2442no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—611no—
337C2×C22⋊C4—611no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C42—611no—
344C4⋊C8—1222no—
345#102—44no—
346#102—2444no—
347#99—2442no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—1222no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353#97—1222no—
354#95—21no—
355#95—612no—
356#95—1221no—
357C24—22no—
358C24—44no—
359C24—41no—
360C24—41no—
361C24—41no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—82no—
366C24—611no—
367C24D4×D6111yescharacteristic
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—4882no—
388C2×D4—2444no—
389C2×D4—2444no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1222no—
398C22×C4—2442no—
399C22×C4—2441no—
400C22×C4C2×C4×D6111yescharacteristic
401C22×C4#1175822211yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406M16—4882no—
407C2×C8—22no—
408C2×C8—42no—
409C2×C8—1222no—
410C2×C8—2442no—
411C4⋊C4—42no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—4882no—
423C22⋊C4—2444no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2444no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42—1221no—
430C42—1221no—
431C23—24no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—82no—
443C23—82no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—84no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—1221no—
453C23—1221no—
454C23—1221no—
455C23—2441no—
456C23—2441no—
457C23C22⋊C4×D6111yescharacteristic
458C23#1451974811yescharacteristic
459C23#1451975011yescharacteristic
460C23#1451980111yescharacteristic
461C23#1451980111yescharacteristic
462C23#1451980411yescharacteristic
463C23#1451980611yescharacteristic
464D4—24no—
465D4—24no—
466D4—44no—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—4888no—
471D4—2441no—
472D4—2441no—
473D4—2441no—
474D4—2441no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—4881no—
490C2×C4—4881no—
491C2×C4—4881no—
492C2×C4—4881no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4#1451973711yescharacteristic
500C2×C4C4×C22⋊D6111yescharacteristic
501C2×C4#1451974511yescharacteristic
502C2×C4#1451975911yescharacteristic
503C8—42no—
504C8—2442no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—82no—
519C22—82no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—4881no—
527C22—2442no—
528C22—2441no—
529C22—2441no—
530C22—2441no—
531C22#1683963811yescharacteristic
532C22#1683976011yescharacteristic
533C22#1683976511yescharacteristic
534C22#1684000511yescharacteristic
535C22#1684000711yescharacteristic
536C22#1684001211yescharacteristic
537C22#1684001411yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—161no—
542C4—9761no—
543C4—4882no—
544C4—2441no—
545C4—2441no—
546C4—2441no—
547C4#1683994812yes—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—84no—
555C2—4881no—
556C2#2144622411yescharacteristic
557C2#2144625411yescharacteristic
558C2#2144634311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1786C2
socleC22×C61#386
prime radical#21449169C2
cosocleC23×C4×C61C23
hypercenter#1786D61
hypocenterC61#9472
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. #21454845C2semidirect
3. #21445804C2semidirect
4. #21445750C2semidirect
5. #21443687C2semidirect
6. #21443686C2semidirect
7. C61×#1786C2semidirect
18. #16839676C4semidirect
19. #16839670C4semidirect
33. C22×#92418D4semidirect
70. C22×C8×C61D4semidirect
273. #1786D61semidirect
308. C61#9472semidirect
323. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214548452
#214491692
#214458042
#214457502
#214436872
#214436862
C61×#17862
#947261