Finite Group Databasebeta

← #29594961

Subgroups — #29594961

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519692C211yesmaximal, characteristic
2#21448397C211yesmaximal, characteristic
3#21443938C214yesmaximal, complemented
4C22×#100×C61C211yesmaximal, characteristic
5C22×C22⋊Dic61—21no—
6C22×C22⋊Dic61C411yescharacteristic, complemented
7C22×C22⋊Dic61C411yescharacteristic, complemented
8C2×#14519692—21no—
9C2×#14519692—22no—
10C2×#14519692C414yescomplemented
11C2×#14519692C2212yes—
12#16839642—21no—
13#16839642—21no—
14#16839642C412yescomplemented
15#16839642C2212yes—
16#16839625—21no—
17#16839618—21no—
18C23×D4×C61C2211yescharacteristic
19C22×M16×C61—21no—
20C2×#100×C61—21no—
21C2×#100×C61—22no—
22C2×#100×C61C414yescomplemented
23C2×#100×C61C2212yes—
24C23×Dic61—21no—
25C23×Dic61—41no—
26C23×Dic61C2×C412yescomplemented
27C2×C22⋊Dic61—24no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61—41no—
35C2×C22⋊Dic61—42no—
36C2×C22⋊Dic61—41no—
37C2×C22⋊Dic61—41no—
38C2×C22⋊Dic61—42no—
39C2×C22⋊Dic61C2×C412yescomplemented
40#14519692—24no—
41#14519692—22no—
42#14519692C2×C414yescomplemented
43#14519684—22no—
44#14519684—22no—
45C25×C61C2×C411yescharacteristic
46C25×C61D411yescharacteristic
47C22×D4×C61—41no—
48C22×D4×C61—41no—
49C22×D4×C61—42no—
50C22×D4×C61C2×C411yescharacteristic
51C22×D4×C61C2×C412yes—
52C22×D4×C61C2×C412yes—
53C22×D4×C61D411yescharacteristic
54C22×D4×C61D412yes—
55C22×D4×C61D411yescharacteristic
56C22×D4×C61Q812yes—
57C22×D4×C61C2311yescharacteristic
58C23×C4×C61D411yescharacteristic
59C2×M16×C61—24no—
60C2×M16×C61—22no—
61C2×M16×C61—42no—
62C2×M16×C61—41no—
63C22×C8×C61—41no—
64#100×C61—24no—
65#100×C61—22no—
66#100×C61C2×C414yescomplemented
67C24×C61—21no—
68C24×C61—21no—
69C24×C61—21no—
70C24×C61—22no—
71C24×C61—22no—
72C24×C61—21no—
73C24×C61—82no—
74C24×C61—82no—
75C24×C61C4212yes—
76C24×C61C22⋊C411yescharacteristic
77C24×C61C22⋊C411yescharacteristic
78C24×C61C22⋊C412yes—
79C24×C61C22⋊C411yescharacteristic
80C24×C61C4⋊C412yes—
81C24×C61C4⋊C412yes—
82C24×C61C22×C411yescharacteristic
83C24×C61C2×D411yescharacteristic
84C2×D4×C61—42no—
85C2×D4×C61—42no—
86C2×D4×C61—42no—
87C2×D4×C61—42no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61—42no—
91C2×D4×C61—42no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—42no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61C4212yes—
101C2×D4×C61C22⋊C412yes—
102C2×D4×C61C22⋊C412yes—
103C2×D4×C61C4⋊C412yes—
104C2×D4×C61C4⋊C412yes—
105C2×D4×C61C4⋊C412yes—
106C2×D4×C61C22×C411yescharacteristic
107C2×D4×C61C22×C411yescharacteristic
108C2×D4×C61C2×D411yescharacteristic
109C2×D4×C61C2×Q811yescharacteristic
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C412yes—
115C22×C4×C61C4⋊C412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—42no—
120C22×Dic61—44no—
121C22×Dic61—42no—
122C22×Dic61—42no—
123C22×Dic61—82no—
124C22×Dic61—82no—
125C22×Dic61—84no—
126M16×C61—24no—
127M16×C61—44no—
128M16×C61—42no—
129C2×C8×C61—44no—
130C2×C8×C61—42no—
131C22⋊Dic61—24no—
132C22⋊Dic61—44no—
133C22⋊Dic61—42no—
134C22⋊Dic61—42no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—42no—
138C22⋊Dic61—42no—
139C22⋊Dic61—82no—
140C22⋊Dic61—82no—
141C23×C61—22no—
142C23×C61—22no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—42no—
158C23×C61—44no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—82no—
164C23×C61—82no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61—82no—
168C23×C61—84no—
169C23×C61—84no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61#9512yes—
173C23×C61#9512yes—
174C23×C61#9512yes—
175C23×C61#9911yescharacteristic
176C23×C61#9911yescharacteristic
177C23×C61#9912yes—
178C23×C61C2×C4211yescharacteristic
179C23×C61C2×C22⋊C411yescharacteristic
180C23×C61C2×C22⋊C411yescharacteristic
181C23×C61C2×C4⋊C411yescharacteristic
182C23×C61C2×C4⋊C411yescharacteristic
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—41no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61#9512yes—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C4⋊C411yescharacteristic
204C2×Dic61—44no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—84no—
212C2×Dic61—88no—
213C2×Dic61—84no—
214C2×Dic61—84no—
215C8×C61—44no—
216#12574—611nomaximal, Sylow
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—42no—
235C22×C61—44no—
236C22×C61—42no—
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61—82no—
250C22×C61#34212yes—
251C22×C61#34212yes—
252C22×C61C2×#9511yescharacteristic
253C22×C61C2×#9911yescharacteristic
254C22×C61C2×#9911yescharacteristic
255C4×C61—21no—
256C4×C61—22no—
257C4×C61—21no—
258Dic61—82no—
259Dic61—84no—
260Dic61—82no—
261C22×#100D6111yescharacteristic
262C22×#99—611no—
263#2308—611no—
264#1390—614no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—22no—
268C2×C61—22no—
269C2×C61—41no—
270C2×C61—42no—
271C2×C61—41no—
272C2×C61—42no—
273C2×C61—41no—
274C2×C61—41no—
275C2×C61C2×#34211yescharacteristic
276C2×C61#212112yes—
277C23×D4C2×D6111yescharacteristic
278C22×M16—21no—
279C22×C22⋊C4—611no—
280C22×C22⋊C4—611no—
281C22×C22⋊C4—1221no—
282C2×#100—21no—
283C2×#100—22no—
284C2×#100Dic6114yescomplemented
285C2×#100C2×D6112yes—
286C2×#99—614no—
287C2×#99—612no—
288C2×#99—1221no—
289C2×#99—1222no—
290#386—612no—
291#386—612no—
292#386—1221no—
293#386—1221no—
294#381—1221no—
295#379—1221no—
296C61#1257411yescharacteristic, complemented, Sylow
297C25C4×D6111yescharacteristic
298C25D24411yescharacteristic
299C22×D4—41no—
300C22×D4—41no—
301C22×D4—42no—
302C22×D4Dic12212yes—
303C22×D4C4×D6112yes—
304C22×D4C4×D6111yescharacteristic
305C22×D4D24411yescharacteristic
306C22×D4C2×Dic6112yes—
307C22×D4C22⋊D6112yes—
308C22×D4C22⋊D6111yescharacteristic
309C22×D4C22×D6111yescharacteristic
310C23×C4—612no—
311C23×C4—1221no—
312C23×C4—2441no—
313C23×C4C22⋊D6111yescharacteristic
314C2×M16—24no—
315C2×M16—22no—
316C2×M16—41no—
317C2×M16—42no—
318C22×C8—41no—
319C2×C22⋊C4—612no—
320C2×C22⋊C4—1224no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—2442no—
328C2×C22⋊C4—2441no—
329C2×C22⋊C4—2441no—
330C2×C22⋊C4—2441no—
331C2×C22⋊C4—2442no—
332#100—24no—
333#100—22no—
334#100C2×Dic6114yescomplemented
335#99—614no—
336#99—1224no—
337#99—1222no—
338#95—1222no—
339#95—1222no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—82no—
347C24—82no—
348C24C4×Dic6112yes—
349C24#1175819612yes—
350C24C4⋊Dic6112yes—
351C24#1175819811yescharacteristic
352C24#1175819811yescharacteristic
353C24#1175819812yes—
354C24#1175819811yescharacteristic
355C24C2×C4×D6111yescharacteristic
356C24C2×D24411yescharacteristic
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4C4×Dic6112yes—
374C2×D4#1175819612yes—
375C2×D4#1175819612yes—
376C2×D4C4⋊Dic6112yes—
377C2×D4#1175819812yes—
378C2×D4C22⋊Dic6112yes—
379C2×D4C2×Dic12211yescharacteristic
380C2×D4C2×C4×D6111yescharacteristic
381C2×D4C22×Dic6111yescharacteristic
382C2×D4C2×C22⋊D6111yescharacteristic
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—4884no—
389C22×C4—4882no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4—2444no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4#1175819612yes—
396C22×C4#1175819811yescharacteristic
397C22×C4C22⋊Dic6112yes—
398C22×C4C2×C22⋊D6111yescharacteristic
399M16—24no—
400M16—42no—
401M16—44no—
402C2×C8—42no—
403C2×C8—44no—
404C22⋊C4—1224no—
405C22⋊C4—4882no—
406C22⋊C4—4882no—
407C22⋊C4—2442no—
408C22⋊C4—2444no—
409C22⋊C4—2442no—
410C22⋊C4—2442no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2442no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—84no—
444C23—84no—
445C23#1451968412yes—
446C23#1451968412yes—
447C23#1451968412yes—
448C23#1451969111yescharacteristic
449C23#1451969111yescharacteristic
450C23#1451969112yes—
451C23C2×C4×Dic6111yescharacteristic
452C23C2×#1175819811yescharacteristic
453C23C2×#1175819811yescharacteristic
454C23C2×#1175819611yescharacteristic
455C23C2×C4⋊Dic6111yescharacteristic
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—4884no—
475C2×C4—4888no—
476C2×C4—4884no—
477C2×C4—4884no—
478C2×C4—2444no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4#1451968412yes—
486C2×C4C2×C22⋊Dic6111yescharacteristic
487C2×C4C2×#1175819611yescharacteristic
488C8—44no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—44no—
508C22—42no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22#1683952812yes—
523C22#1683952812yes—
524C22C2×#1451968411yescharacteristic
525C22C2×#1451969111yescharacteristic
526C22C2×#1451969111yescharacteristic
527C4—21no—
528C4—22no—
529C4—21no—
530C4—4882no—
531C4—4884no—
532C4—4882no—
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—41no—
538C2—42no—
539C2—41no—
540C2—42no—
541C2—41no—
542C2—41no—
543C2C2×#1683952811yescharacteristic
544C2#2144746612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC23×C61C2×C42
FrattiniC22×D4C22×D61
FittingC22×#100×C61C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenterC22×#100D61
hypocenterC61#12574
LCSG ▷ C23×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
3. #21443938C2semidirect
6. C22×C22⋊Dic61C4semidirect
7. C22×C22⋊Dic61C4semidirect
10. C2×#14519692C4semidirect
14. #16839642C4semidirect
22. C2×#100×C61C4semidirect
26. C23×Dic61C2×C4semidirect
39. C2×C22⋊Dic61C2×C4semidirect
42. #14519692C2×C4semidirect
66. #100×C61C2×C4semidirect
284. C2×#100Dic61semidirect
296. C61#12574semidirect
334. #100C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C22×#145196922
#214483972
#214439382
C22×#100×C612
#1257461