Finite Group Databasebeta

← #29689104

Subgroups — #29689104

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C2×#16839622C211yesmaximal, characteristic, complemented
3#21446257C212yesmaximal, complemented
4#21446252C212yesmaximal, complemented
5#21446134C212yesmaximal, complemented
6#21446124C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445760C212yesmaximal, complemented
9C2×C61×#407C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840273—24no—
12#16840273C418yescomplemented
13C2×#14519760—21no—
14C2×#14519756C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519746—21no—
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61—21no—
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2212yes—
23C22×C22⋊Dic61—21no—
24C2×#14519702C2214yescomplemented
25C2×#14519699C2214yescomplemented
26#16839622—21no—
27#16839622—21no—
28#16839622C2212yes—
29#16839622C2212yes—
30C2×#14519684—21no—
31#16839525C2214yescomplemented
32#16839525C2214yescomplemented
33C24×C4×C61C2211yescharacteristic
34C22×M16×C61C2212yes—
35C61×#407—24no—
36C2×C22⋊C8×C61C2212yes—
37C2×C22⋊C8×C61C2212yes—
38C2×#11758222—22no—
39C2×#11758222—44no—
40C2×#11758222C2×C414yescomplemented
41C22×Dic122C2311yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C22⋊D61—21no—
44C22×C22⋊D61—21no—
45C22×C22⋊D61—21no—
46C23×Dic61—22no—
47C22×C4×D61—21no—
48#14519760—28no—
49#14519756—22no—
50#14519756—22no—
51#14519756C2×C414yes—
52#14519756C2×C414yescomplemented
53#14519751—22no—
54#14519751—22no—
55#14519751C2×C414yes—
56#14519751C2×C414yescomplemented
57#14519746—28no—
58C4×C22⋊D61—28no—
59C4×C22⋊D61—28no—
60#14519736—22no—
61#14519736C2×C414yescomplemented
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—44no—
64C2×C4⋊Dic61C2314yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—44no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—21no—
78#14519702—44no—
79#14519699—44no—
80#14519684—22no—
81#14519684—22no—
82C25×C61D411yescharacteristic
83C23×C4×C61—44no—
84C23×C4×C61C2×C414yes—
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D414yes—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—24no—
93C2×M16×C61—44no—
94C22×C8×C61C2314yes—
95C22⋊C8×C61—24no—
96C22⋊C8×C61—24no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61C22⋊C414yes—
102C24×C61C2×D411yescharacteristic
103C24×C61C2×D411yescharacteristic
104C24×C61C2×D411yescharacteristic
105C23×D61—21no—
106C22×C4×C61—24no—
107C22×C4×C61—24no—
108C22×C4×C61—24no—
109C22×C4×C61—48no—
110C22×C4×C61—48no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61C22⋊C414yes—
115C22×C4×C61C22⋊C414yes—
116C22×C4×C61C22⋊C414yes—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—24no—
138C22×Dic61—44no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141#11758222—48no—
142#11758222—48no—
143C2×D244—24no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—24no—
147C2×C4×D61—44no—
148C2×Dic122—24no—
149C2×Dic122—21no—
150C2×Dic122C22×C412yes—
151M16×C61—48no—
152C2×C8×C61—24no—
153C2×C8×C61—24no—
154C2×C8×C61—24no—
155C22⋊Dic61—24no—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158C22⋊Dic61—44no—
159C22⋊Dic61—44no—
160#11758198—24no—
161#11758198—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—44no—
167C4⋊Dic61C22×C412yes—
168C4⋊Dic61C22×C412yes—
169#11758196—24no—
170#11758196—24no—
171C4×Dic61—24no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—28no—
183C23×C61—44no—
184C23×C61—44no—
185C23×C61—48no—
186C23×C61—48no—
187C23×C61C2×C22⋊C412yes—
188C23×C61C2×C22⋊C412yes—
189C23×C61C2×C22⋊C412yes—
190C23×C61AGL(1,ℤ/8)12yes—
191C23×C61#13712yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—42no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—42no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—44no—
210C2×C4×C61C2×C22⋊C412yes—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61H(5,𝔽2)14yes—
216C22⋊D61—48no—
217C22⋊D61—48no—
218C22⋊D61—48no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—44no—
227C2×Dic61—44no—
228C2×Dic61—44no—
229C2×Dic61—88no—
230D244—24no—
231D244—42no—
232C4×D61—48no—
233Dic122—24no—
234Dic122—42no—
235C8×C61—44no—
236#21769—611nomaximal, Sylow
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—416no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61#41714yes—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61#53412yes—
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C22×C61#57512yes—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61#51812yes—
267C4×C61#53412yes—
268Dic61—42no—
269Dic61—42no—
270Dic61—42no—
271Dic61—84no—
272C2×#525—611no—
273C2×#407D6111yescharacteristic, complemented
274C2×#380—611no—
275#1895—612no—
276#1894—612no—
277#1874—612no—
278#1872—612no—
279#1825—612no—
280#1788—612no—
281C2×C61—24no—
282C2×C61—42no—
283C2×C61—42no—
284C2×C61#230811yescharacteristic
285C2×C61C2×#41711yescharacteristic
286C2×C61#289611yescharacteristic
287C2×C61#319412yes—
288C2×C61#319612yes—
289D61—82no—
290C22×#41—611no—
291C24×C4C2×D6111yescharacteristic
292C22×M16C2×D6112yes—
293#525—618no—
294#525—1224no—
295C2×#123—1221no—
296C2×C22⋊Q8—611no—
297C2×C22⋊Q8—611no—
298C2×#121—611no—
299C2×#121—611no—
300C2×#120—1221no—
301C2×C4×D4—1221no—
302C2×C4×D4—1221no—
303C2×#117—611no—
304C22×C4⋊C4—612no—
305C22×C22⋊C4—1221no—
306C2×#103—614no—
307C2×#102—614no—
308#407—24no—
309C2×C22⋊C8C2×D6112yes—
310C2×C22⋊C8C2×D6112yes—
311#380—612no—
312#380—612no—
313#380—1221no—
314#380—1221no—
315C2×#95—1221no—
316#340—614no—
317#340—614no—
318C61#2176911yescharacteristic, complemented, Sylow
319C25D24411yescharacteristic
320C2×#41—614no—
321C2×#41—1222no—
322C2×#41—2444no—
323C22×Q8—611no—
324C22×D4—611no—
325C22×D4—1221no—
326C22×D4—1221no—
327C22×D4—1221no—
328C23×C4—44no—
329C23×C4—1221no—
330C23×C4—1222no—
331C23×C4C4×D6114yes—
332C23×C4D24411yescharacteristic
333C23×C4D24411yescharacteristic
334C23×C4D24411yescharacteristic
335C23×C4C22⋊D6114yes—
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C23×C4C22×D6111yescharacteristic
339C2×M16—24no—
340C2×M16—44no—
341C22×C8C22×D6114yes—
342#123—1228no—
343C22⋊Q8—614no—
344C22⋊Q8—614no—
345C22⋊Q8—1222no—
346C22⋊Q8—1222no—
347#121—614no—
348#121—614no—
349#121—1222no—
350#121—1222no—
351#120—1228no—
352C4×D4—1228no—
353C4×D4—1228no—
354#117—614no—
355#117—1222no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—614no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—2444no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—2444no—
371C2×C42—1221no—
372#103—2444no—
373#102—2444no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#95—1222no—
377#95—1222no—
378C24—22no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—1221no—
383C24#1175819814yes—
384C24C2×D24411yescharacteristic
385C24C2×D24411yescharacteristic
386C24C2×D24411yescharacteristic
387#41—2448no—
388#41—2448no—
389C2×Q8—612no—
390C2×Q8—1221no—
391C2×Q8—1224no—
392C2×D4—612no—
393C2×D4—1224no—
394C2×D4—1224no—
395C2×D4—1224no—
396C2×D4—1221no—
397C2×D4—1224no—
398C2×D4—2444no—
399C2×D4—2444no—
400C2×D4—2442no—
401C2×D4—2442no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1224no—
414C22×C4—1224no—
415C22×C4—2444no—
416C22×C4—2444no—
417C22×C4—2444no—
418C22×C4—2444no—
419C22×C4#1175819814yes—
420C22×C4#1175819814yes—
421C22×C4#1175819814yes—
422C22×C4C2×C4×D6112yes—
423C22×C4C2×C4×D6112yes—
424C22×C4C2×C4×D6112yes—
425C22×C4C2×D24411yescharacteristic
426C22×C4C2×D24411yescharacteristic
427C22×C4C2×D24411yescharacteristic
428C22×C4C2×C22⋊D6112yes—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C23×D6111yescharacteristic
432M16—48no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—612no—
437C4⋊C4—612no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1224no—
443C4⋊C4—1224no—
444C4⋊C4—2444no—
445C22⋊C4—1224no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—2444no—
451C22⋊C4—2444no—
452C42—1224no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—44no—
465C23—44no—
466C23—48no—
467C23—48no—
468C23—1222no—
469C23—2441no—
470C23—2441no—
471C23—2442no—
472C23—2442no—
473C23C2×#1175819812yes—
474C23C2×#1175819812yes—
475C23C2×#1175819812yes—
476C23#1451980512yes—
477C23#1451981012yes—
478C23C22×D24411yescharacteristic
479Q8—1224no—
480Q8—2442no—
481D4—1224no—
482D4—2448no—
483D4—2448no—
484D4—2448no—
485D4—2442no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—4888no—
502C2×C4—2448no—
503C2×C4—2444no—
504C2×C4—2444no—
505C2×C4—2444no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4C2×#1175819812yes—
511C2×C4C2×#1175819812yes—
512C2×C4C2×#1175819812yes—
513C2×C4C22×C4×D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4#1451982214yes—
516C8—44no—
517C22—22no—
518C22—24no—
519C22—22no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—416no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—4882no—
533C22—4882no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22#1683976414yes—
538C22C22×#1175819811yescharacteristic
539C22#1684032412yes—
540C22C2×#1451980511yescharacteristic
541C22C2×#1451981011yescharacteristic
542C22#1684056212yes—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—4884no—
547C4—2442no—
548C4—2442no—
549C4—2442no—
550C4#1684023512yes—
551C4#1684032112yes—
552C2—24no—
553C2—42no—
554C2—42no—
555C2—4882no—
556C2#2144839411yescharacteristic
557C2C2×#1683976411yescharacteristic
558C2#2145433911yescharacteristic
559C2#2145663512yes—
560C2#2145665112yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#407C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC2×#407D61
hypocenterC61#21769
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#16839622C2semidirect
3. #21446257C2semidirect
4. #21446252C2semidirect
5. #21446134C2semidirect
6. #21446124C2semidirect
7. #21445915C2semidirect
8. #21445760C2semidirect
9. C2×C61×#407C2semidirect
12. #16840273C4semidirect
24. C2×#14519702C22semidirect
25. C2×#14519699C22semidirect
31. #16839525C22semidirect
32. #16839525C22semidirect
40. C2×#11758222C2×C4semidirect
52. #14519756C2×C4semidirect
56. #14519751C2×C4semidirect
61. #14519736C2×C4semidirect
273. C2×#407D61semidirect
318. C61#21769semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
C2×#168396222
#214462572
#214462522
#214461342
#214461242
#214459152
#214457602
C2×C61×#4072
#2176961