Finite Group Databasebeta

← #29696182

Subgroups — #29696182

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840451C211yesmaximal, characteristic
2M16×C22⋊D61C212yesmaximal
3#21446810C212yesmaximal, complemented
4#21446287C212yesmaximal, complemented
5#21446286C212yesmaximal, complemented
6#21445902C211yesmaximal, characteristic, complemented
7#21445898C211yesmaximal, characteristic, complemented
8#21445814C211yesmaximal, characteristic, complemented
9#21445614C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C61×#1742C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840451—22no—
14#16840451—22no—
15#16840451—22no—
16#16840451—22no—
17C2×#14519770C412yes—
18C2×#14519769C412yes—
19C2×#14519768C2211yescharacteristic
20C2×#14519759C412yes—
21C2×#14519757C2211yescharacteristic
22C2×#14519756C2211yescharacteristic
23C2×#14519742C2211yescharacteristic
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519734C412yes—
26C2×C4⋊C4×D61C2211yescharacteristic
27#16839838—22no—
28#16839833—22no—
29#16839831—22no—
30C8×C22⋊D61—22no—
31#16839785C2212yes—
32C2×#14519695C2212yes—
33C2×#14519695C2212yes—
34#16839726C2212yes—
35#16839725C2212yes—
36M16×Dic61C2212yes—
37#16839531C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2211yescharacteristic
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2212yes—
43C2×#117×C61C2211yescharacteristic
44C61×#343C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×C22⋊D61C2×C411yescharacteristic
49C22×C4×D61C2×C412yes—
50C22×C4×D61C2311yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54M16×D61—24no—
55M16×D61—42no—
56C2×C8×D61—22no—
57#14519770—24no—
58#14519770—24no—
59#14519769—24no—
60#14519769—24no—
61#14519768—22no—
62#14519768—22no—
63#14519759—24no—
64#14519759—24no—
65#14519757—22no—
66#14519757—22no—
67#14519756—22no—
68#14519756—22no—
69#14519742—22no—
70#14519742—22no—
71C4×C22⋊D61D412yes—
72C4×C22⋊D61D412yes—
73C4×C22⋊D61D412yes—
74C4×C22⋊D61D412yes—
75#14519734—22no—
76#14519734—22no—
77#14519734—22no—
78#14519734—22no—
79C2×C4⋊Dic61C2×C412yes—
80C2×C4⋊Dic61C2×C412yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196C2×C412yes—
83C2×#11758196C2×C412yes—
84C2×#11758196C2×C412yes—
85C2×#11758196C2×C412yes—
86C2×#11758196C2×C411yescharacteristic
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C2×C22⋊Dic61C2×C412yes—
90C2×C22⋊Dic61C2×C411yescharacteristic
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C411yescharacteristic
93C2×C4×Dic61C2×C412yes—
94C2×C4×Dic61C2311yescharacteristic
95#14519706—22no—
96#14519706—22no—
97#14519695—24no—
98#14519695—42no—
99C22⋊#92418—22no—
100C22⋊#92418—22no—
101#14519689—22no—
102#14519689—22no—
103#14519687—22no—
104C8×Dic61—22no—
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—22no—
107C2×M16×C61—24no—
108C2×M16×C61—42no—
109C2×M16×C61C2312yes—
110C2×M16×C61C2312yes—
111C22×C8×C61—22no—
112#117×C61—22no—
113#117×C61—22no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61C2×C412yes—
117C2×C42×C61C2×C412yes—
118C24×C61C22×C411yescharacteristic
119C23×D61C22×C411yescharacteristic
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—42no—
130C2×C22⋊D61C22⋊C412yes—
131C2×C22⋊D61C22⋊C412yes—
132C22×Dic61C22×C412yes—
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C412yes—
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—44no—
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D412yes—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61—22no—
152M16×C61—44no—
153M16×C61#4114yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—44no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C22⋊C4×C61—24no—
160C42×C61—22no—
161C42×C61—22no—
162C22⋊Dic61—24no—
163C22⋊Dic61C22⋊C412yes—
164C22⋊Dic61C22⋊C412yes—
165#11758198—22no—
166#11758198—22no—
167#11758198C22⋊C411yescharacteristic
168#11758198C22⋊C411yescharacteristic
169#11758198C22⋊C411yescharacteristic
170#11758198C22⋊C411yescharacteristic
171C4⋊Dic61—22no—
172C4⋊Dic61—24no—
173C4⋊Dic61—24no—
174#11758196—22no—
175#11758196—22no—
176#11758196—24no—
177#11758196—22no—
178#11758196—22no—
179#11758196—24no—
180#11758196C22⋊C412yes—
181#11758196C22⋊C412yes—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61C2×D412yes—
185C4×Dic61C2×D412yes—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192#11758189—44no—
193C8×D61—44no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C2×C22⋊C412yes—
199C23×C61C23×C411yescharacteristic
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—22no—
203C22×D61—41no—
204C22×D61C2×C22⋊C411yescharacteristic
205C22×D61C2×C22⋊C411yescharacteristic
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C22×D412yes—
219C2×C4×C61C2×#4112yes—
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—24no—
223C22⋊D61—42no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61C2×C22⋊C411yescharacteristic
234C2×Dic61C2×C22⋊C411yescharacteristic
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—44no—
242C4×D61—44no—
243Dic122—42no—
244Dic122—42no—
245C8×C61—24no—
246C8×C61—42no—
247#92418—42no—
248#92418—42no—
249#22480—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C22×C22⋊C411yescharacteristic
258C22×C61C4×#4111yescharacteristic
259C22×C61#52011yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61#41312yes—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61#54612yes—
273C4×C61#54712yes—
274Dic61—22no—
275Dic61—22no—
276Dic61—21no—
277Dic61—21no—
278Dic61—42no—
279Dic61—42no—
280C2×#552—611no—
281D4×M16—612no—
282#1998—612no—
283#1901—612no—
284#1901—612no—
285#1822—611no—
286#1821—611no—
287#1801—611no—
288C4×#101—611no—
289#1742D6111yescharacteristic, complemented
290C2×#343—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—42no—
294C2×C61#233911yescharacteristic
295C2×C61C2×#41311yescharacteristic
296C2×C61#288911yescharacteristic
297D61—22no—
298D61—41no—
299C22×M16—612no—
300C22×M16C2×D6112yes—
301#552—1222no—
302#552—1222no—
303#552—1222no—
304#552—1222no—
305C2×#126—612no—
306C2×#125—611no—
307C2×#125—612no—
308C2×#123—612no—
309C2×C22⋊Q8—611no—
310C2×C22⋊Q8—611no—
311C2×C4×Q8—611no—
312C2×C4×D4—611no—
313C2×#117—612no—
314C2×#117C2×D6111yescharacteristic
315C22×C4⋊C4—611no—
316#436—1222no—
317#435—1222no—
318#435—1222no—
319C8×D4—1222no—
320#423—612no—
321C2×#101—612no—
322C2×#101—612no—
323#407—612no—
324#407—612no—
325C4×M16—612no—
326#343—612no—
327#343—612no—
328#343—611no—
329#343—611no—
330#343—612no—
331#343C2×D6111yescharacteristic
332#343C2×D6112yes—
333#343C2×D6111yescharacteristic
334C61#2248011yescharacteristic, complemented, Sylow
335C22×Q8—611no—
336C22×D4—611no—
337C23×C4—611no—
338C23×C4—612no—
339C23×C4C22×D6111yescharacteristic
340C2×M16—22no—
341C2×M16—24no—
342C2×M16—42no—
343C2×M16—612no—
344C2×M16—612no—
345C2×M16—1222no—
346C2×M16—1224no—
347C2×M16—2442no—
348C2×M16C22×D6112yes—
349C2×M16C22×D6112yes—
350C22×C8—22no—
351C22×C8—1222no—
352#126—1224no—
353#126—1224no—
354#125—1224no—
355#125—1224no—
356#125—1222no—
357#125—1222no—
358#123—1224no—
359#123—1224no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C4×Q8—1222no—
365C4×Q8—1222no—
366C4×D4—612no—
367C4×D4—612no—
368C4×D4—612no—
369C4×D4—612no—
370#117—22no—
371#117—22no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375#117—1222no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—612no—
391C2×C22⋊C4—612no—
392C2×C22⋊C4C4×D6112yes—
393C2×C42—611no—
394C2×C42—612no—
395C2×C42C4×D6112yes—
396C4⋊C8—1222no—
397C4⋊C8—1222no—
398#101—1224no—
399#101—2442no—
400C22⋊C8—1222no—
401C22⋊C8—1222no—
402C22⋊C8—1222no—
403C22⋊C8—1222no—
404#97—1222no—
405C4×C8—1222no—
406C24—611no—
407C24C2×C4×D6111yescharacteristic
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—2442no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—2444no—
431C22×C4C2×C4×D6111yescharacteristic
432C22×C4C2×C4×D6112yes—
433C22×C4C2×C4×D6112yes—
434C22×C4D4×D6112yes—
435C22×C4D4×D6112yes—
436C22×C4C23×D6111yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—44no—
441M16—1222no—
442M16—1222no—
443M16—1222no—
444M16—1222no—
445M16—2444no—
446M16#1175822214yes—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—44no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—2444no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C22⋊C4—24no—
467C22⋊C4—612no—
468C22⋊C4—612no—
469C22⋊C4—611no—
470C22⋊C4—611no—
471C22⋊C4—611no—
472C22⋊C4—611no—
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C22⋊C4—1224no—
476C42—22no—
477C42—22no—
478C42—612no—
479C42—612no—
480C42—1222no—
481C42—1222no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—611no—
487C23—611no—
488C23—1222no—
489C23—1221no—
490C23—1221no—
491C23—2441no—
492C23C22⋊C4×D6112yes—
493C23C22×C4×D6111yescharacteristic
494Q8—2442no—
495Q8—2442no—
496D4—1224no—
497D4—2442no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4C22⋊C4×D6111yescharacteristic
528C2×C4C22⋊C4×D6111yescharacteristic
529C2×C4C2×D4×D6112yes—
530C2×C4C2×#1175822212yes—
531C2×C4#1451982211yescharacteristic
532C2×C4#1451982511yescharacteristic
533C8—24no—
534C8—42no—
535C8—2442no—
536C8—2442no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—81no—
543C22—81no—
544C22—612no—
545C22—1221no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1221no—
550C22—2442no—
551C22—2442no—
552C22C2×C22⋊C4×D6111yescharacteristic
553C22C4×#1175822211yescharacteristic
554C22#1684024311yescharacteristic
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—1222no—
560C4—1222no—
561C4—1221no—
562C4—1221no—
563C4—2442no—
564C4—2442no—
565C4#1684040212yes—
566C4#1684041312yes—
567C2—21no—
568C2—21no—
569C2—42no—
570C2—1222no—
571C2—2441no—
572C2#2144864611yescharacteristic
573C2#413×D6111yescharacteristic
574C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#22480
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #21446810C2semidirect
4. #21446287C2semidirect
5. #21446286C2semidirect
6. #21445902C2semidirect
7. #21445898C2semidirect
8. #21445814C2semidirect
9. #21445614C2semidirect
10. #343×D61C2semidirect
11. C61×#1742C2semidirect
289. #1742D61semidirect
334. C61#22480semidirect

Maximal subgroups

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

subgroupindex
C2×#168404512
M16×C22⋊D612
#214468102
#214462872
#214462862
#214459022
#214458982
#214458142
#214456142
#343×D612
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