Finite Group Databasebeta

← #29597492

Subgroups — #29597492

Order 15616.

864 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 111 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#21449142C211yesmaximal, characteristic
2C2×#16839553C212yesmaximal, complemented
3#21445734C211yesmaximal, characteristic
4#21443938C212yesmaximal, complemented
5C22×#100×C61C211yesmaximal, characteristic
6C2×#14519758C411yescharacteristic, complemented
7C2×#14519758C2211yescharacteristic
8C22×C22⋊Dic61C411yescharacteristic, complemented
9C2×#14519692—21no—
10C2×#14519692C2212yes—
11#16839676—21no—
12#16839676—21no—
13#16839642C411yescharacteristic, complemented
14#16839642C2211yescharacteristic
15#16839635—21no—
16#16839618—21no—
17#16839618—21no—
18#16839618C411yescharacteristic, complemented
19#16839553—24no—
20C23×D4×C61C2211yescharacteristic
21C22×M16×C61—21no—
22C2×#100×C61—21no—
23C2×#100×C61—24no—
24C2×#100×C61C2212yes—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27#14519758—22no—
28#14519758—22no—
29#14519758—42no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—41no—
42C2×C22⋊Dic61—41no—
43C2×C22⋊Dic61—41no—
44C2×C22⋊Dic61—41no—
45C2×C22⋊Dic61C2×C411yescharacteristic
46C2×C22⋊Dic61C2×C411yescharacteristic
47#14519692—22no—
48#14519692—22no—
49#14519692—42no—
50#14519684—21no—
51#14519684—21no—
52C25×C61C2×C411yescharacteristic
53C25×C61D411yescharacteristic
54C22×D4×C61—22no—
55C22×D4×C61—22no—
56C22×D4×C61—42no—
57C22×D4×C61—41no—
58C22×D4×C61—41no—
59C22×D4×C61C2×C411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C22×D4×C61D411yescharacteristic
62C22×D4×C61C2311yescharacteristic
63C23×C4×C61D411yescharacteristic
64C2×M16×C61—24no—
65C2×M16×C61—22no—
66C2×M16×C61—41no—
67C2×M16×C61—42no—
68C22×C8×C61—41no—
69#100×C61—24no—
70#100×C61—22no—
71#100×C61D414yes—
72C24×C61—21no—
73C24×C61—21no—
74C24×C61—21no—
75C24×C61—21no—
76C24×C61—22no—
77C24×C61—22no—
78C24×C61—41no—
79C24×C61—41no—
80C24×C61—42no—
81C24×C61—42no—
82C24×C61—81no—
83C24×C61—81no—
84C24×C61C22⋊C411yescharacteristic
85C24×C61C22⋊C411yescharacteristic
86C24×C61C22⋊C411yescharacteristic
87C24×C61C22×C411yescharacteristic
88C24×C61C2×D411yescharacteristic
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—44no—
102C2×D4×C61—44no—
103C2×D4×C61—44no—
104C2×D4×C61—44no—
105C2×D4×C61C2×D412yes—
106C2×D4×C61#4112yes—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—41no—
120C22×Dic61—41no—
121C22×Dic61—41no—
122C22×Dic61—42no—
123C22×Dic61—81no—
124M16×C61—24no—
125M16×C61—44no—
126M16×C61—42no—
127C2×C8×C61—44no—
128C2×C8×C61—42no—
129C22⋊Dic61—22no—
130C22⋊Dic61—42no—
131C22⋊Dic61—42no—
132C22⋊Dic61—42no—
133C22⋊Dic61—41no—
134C22⋊Dic61—41no—
135C22⋊Dic61—42no—
136C22⋊Dic61—41no—
137C22⋊Dic61—42no—
138C22⋊Dic61—42no—
139C22⋊Dic61—81no—
140C22⋊Dic61—81no—
141C22⋊Dic61—81no—
142C22⋊Dic61—82no—
143C22⋊Dic61—82no—
144#11758196—42no—
145#11758196—41no—
146#11758196—41no—
147C23×C61—22no—
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—41no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—81no—
178C23×C61—81no—
179C23×C61—81no—
180C23×C61—81no—
181C23×C61—84no—
182C23×C61—84no—
183C23×C61—84no—
184C23×C61—84no—
185C23×C61#9911yescharacteristic
186C23×C61#9911yescharacteristic
187C23×C61C2×C22⋊C411yescharacteristic
188C23×C61C4×D412yes—
189C23×C61#12112yes—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—42no—
195D4×C61—42no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61#9912yes—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12311yescharacteristic
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—82no—
220C2×Dic61—82no—
221C2×Dic61—82no—
222C2×Dic61—82no—
223C2×Dic61—82no—
224C8×C61—44no—
225#13080—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—21no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61—84no—
253C22×C61—84no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—84no—
258C22×C61—84no—
259C22×C61#35212yescomplemented
260C22×C61#38611yescharacteristic
261C22×C61C2×#9911yescharacteristic
262C22×C61C2×#9911yescharacteristic
263C4×C61—21no—
264C4×C61—22no—
265C4×C61—21no—
266Dic61—81no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270Dic61—162no—
271C22×#100D6111yescharacteristic
272#2405—611no—
273C2×#352—612no—
274#1781—611no—
275#1390—612no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—22no—
279C2×C61—42no—
280C2×C61—41no—
281C2×C61—42no—
282C2×C61—81no—
283C2×C61—81no—
284C2×C61#189811yescharacteristic
285C2×C61C2×#35211yescharacteristic
286C2×C61#212111yescharacteristic
287C23×D4C2×D6111yescharacteristic
288C22×M16—21no—
289C2×#123—611no—
290C2×#123—611no—
291C22×C22⋊C4—611no—
292C2×#100—21no—
293C2×#100—24no—
294C2×#100C2×D6112yes—
295C2×#99—612no—
296C2×#99—1221no—
297#394—1221no—
298#394—1221no—
299#386—611no—
300#386—611no—
301#385—1221no—
302#379—611no—
303#379—1221no—
304#379—1221no—
305#352—1224no—
306C61#1308011yescharacteristic, complemented, Sylow
307C25C4×D6111yescharacteristic
308C25D24411yescharacteristic
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—42no—
312C22×D4—41no—
313C22×D4—41no—
314C22×D4C4×D6111yescharacteristic
315C22×D4D24411yescharacteristic
316C22×D4C22⋊D6111yescharacteristic
317C22×D4C22×D6111yescharacteristic
318C23×C4—611no—
319C23×C4—611no—
320C23×C4C22⋊D6111yescharacteristic
321C2×M16—24no—
322C2×M16—22no—
323C2×M16—41no—
324C2×M16—42no—
325C22×C8—41no—
326#123—1222no—
327#123—1222no—
328#123—2442no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C22⋊C4—611no—
332C2×C22⋊C4—611no—
333C2×C22⋊C4—1221no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—2441no—
343C2×C22⋊C4—2441no—
344C2×C22⋊C4—2441no—
345C2×C22⋊C4—2441no—
346#100—24no—
347#100—22no—
348#100C22⋊D6114yes—
349#99—1222no—
350#99—1222no—
351#99—2442no—
352#95—1221no—
353#95—1221no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—42no—
363C24—42no—
364C24—81no—
365C24—81no—
366C24#1175819811yescharacteristic
367C24#1175819811yescharacteristic
368C24#1175819811yescharacteristic
369C24C2×C4×D6111yescharacteristic
370C24C2×D24411yescharacteristic
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4#1175822212yes—
388C2×D4C2×C22⋊D6112yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—4881no—
400C22×C4—2442no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4—2441no—
404C22×C4#1175819811yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406M16—24no—
407M16—44no—
408M16—42no—
409C2×C8—44no—
410C2×C8—42no—
411C4⋊C4—2442no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—1222no—
415C22⋊C4—4882no—
416C22⋊C4—4882no—
417C22⋊C4—4881no—
418C22⋊C4—4881no—
419C22⋊C4—4881no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2442no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2442no—
428C22⋊C4—2441no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23#1451969111yescharacteristic
468C23#1451969111yescharacteristic
469C23C2×#1175819811yescharacteristic
470C23C4×C22⋊D6112yes—
471C23#1451975112yes—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—42no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—4882no—
487C2×C4—4882no—
488C2×C4—4882no—
489C2×C4—4882no—
490C2×C4—4882no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4#1451969112yes—
504C2×C4#1451974611yescharacteristic
505C2×C4#1451976011yescharacteristic
506C8—44no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—84no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—84no—
539C22—84no—
540C22#1683955312yescomplemented
541C22#1683964111yescharacteristic
542C22C2×#1451969111yescharacteristic
543C22C2×#1451969111yescharacteristic
544C4—21no—
545C4—22no—
546C4—21no—
547C4—9762no—
548C4—4881no—
549C4—4881no—
550C4—4881no—
551C4—4881no—
552C2—21no—
553C2—21no—
554C2—22no—
555C2—42no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—81no—
560C2#2144627211yescharacteristic
561C2C2×#1683955311yescharacteristic
562C2#2144746611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC22×#100×C61C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenterC22×#100D61
hypocenterC61#13080
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16839553C2semidirect
4. #21443938C2semidirect
6. C2×#14519758C4semidirect
8. C22×C22⋊Dic61C4semidirect
13. #16839642C4semidirect
18. #16839618C4semidirect
259. C22×C61#352semidirect
306. C61#13080semidirect
540. C22#16839553semidirect

Maximal subgroups

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

subgroupindex
#214491422
C2×#168395532
#214457342
#214439382
C22×#100×C612
#1308061