Finite Group Databasebeta

← #29586405

Subgroups — #29586405

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839768C211yesmaximal, characteristic
2C2×#16839766C211yesmaximal, characteristic
3#21454237C211yesmaximal, characteristic, complemented
4#21448399C211yesmaximal, characteristic
5#21445551C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12#16839768—22no—
13#16839768C414yescomplemented
14#16839766—22no—
15#16839766C414yescomplemented
16C2×#14519699—22no—
17C2×#14519697—22no—
18#16839720—22no—
19#16839717—22no—
20M16×Dic61—22no—
21#16839664C414yescomplemented
22#16839663—22no—
23#16839617—22no—
24#16839528—22no—
25#16839518—22no—
26C22×M16×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C2×#117×C61C2211yescharacteristic
29C61×H(3,ℤ/4)—22no—
30C24×D61—21no—
31C22×D244—21no—
32C22×D244—21no—
33C22×D244—21no—
34C22×D244—21no—
35C22×D244—21no—
36C22×D244—21no—
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41C22×D244—21no—
42C22×D244C2×C412yes—
43C22×D244C2×C412yes—
44C22×D244C2×C412yes—
45C2×#11758194—21no—
46C2×#11758194—21no—
47C2×#11758194—21no—
48C2×#11758194—21no—
49C2×#11758194C2×C412yes—
50C2×#11758194D412yes—
51C22×#92418D412yes—
52#14519736—21no—
53#14519736—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736C2×C412yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2×C412yes—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61D412yes—
64C2×#11758198—28no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61D412yes—
67#14519699—28no—
68#14519697—28no—
69#14519687—24no—
70C8×Dic61—24no—
71C23×C4×C61C2311yescharacteristic
72C2×M16×C61—21no—
73C2×M16×C61—21no—
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C2×M16×C61C2×C412yes—
77C2×M16×C61D412yes—
78C22×C8×C61D412yes—
79#117×C61—21no—
80#117×C61—21no—
81#117×C61—21no—
82#117×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61C2×C412yes—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61C2×C412yes—
93C24×C61C2×D411yescharacteristic
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—41no—
98C23×D61—41no—
99C23×D61—41no—
100C23×D61—41no—
101C23×D61—84no—
102C22×C4×C61—22no—
103C22×C4×C61—41no—
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C22×C411yescharacteristic
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×Dic61—41no—
119C22×Dic61C22⋊C414yes—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—41no—
136C2×D244—41no—
137C2×D244—41no—
138C2×D244—41no—
139C2×D244—41no—
140C2×D244—41no—
141C2×D244—41no—
142C2×D244—41no—
143C2×D244—41no—
144C2×D244—41no—
145C2×D244—41no—
146C2×D244—41no—
147C2×D244C4214yes—
148M16×C61—22no—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61C22⋊C414yescomplemented
155C4⋊C4×C61—22no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61—41no—
163C42×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—41no—
167C22⋊Dic61—22no—
168C22⋊Dic61—22no—
169C22⋊Dic61—22no—
170C22⋊Dic61—41no—
171#11758198—88no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—41no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C4×Dic61—41no—
180#11758194—22no—
181#11758194—22no—
182#11758194—22no—
183#11758194—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C22⋊C414yescomplemented
187C23×C61—42no—
188C23×C61C2×C22⋊C411yescharacteristic
189C23×C61C4×D411yescharacteristic
190C23×C61C4×D411yescharacteristic
191C23×C61#12111yescharacteristic
192C23×C61#12111yescharacteristic
193C23×C61#12411yescharacteristic
194C23×C61#12711yescharacteristic
195C22×D61—24no—
196C22×D61—48no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—84no—
204C22×D61—84no—
205C22×D61—84no—
206C22×D61—84no—
207C22×D61—84no—
208C22×D61—84no—
209C2×C4×C61—44no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—82no—
213C2×C4×C61C2×C4211yescharacteristic
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C2×C4×C61C2×C22⋊C411yescharacteristic
216C2×C4×C61#11711yescharacteristic
217C2×C4×C61#11711yescharacteristic
218C2×C4×C61#11711yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12411yescharacteristic
239C2×C4×C61#12611yescharacteristic
240C2×C4×C61#12611yescharacteristic
241C2×Dic61—24no—
242C2×Dic61—44no—
243C2×Dic61—42no—
244C2×Dic61—82no—
245D244—24no—
246D244—24no—
247D244—42no—
248D244—42no—
249D244—42no—
250D244—42no—
251D244—42no—
252D244—42no—
253C8×C61—24no—
254#92418—24no—
255#10948—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61C4×C22⋊C411yescharacteristic
264C22×C61#38611yescharacteristic
265C22×C61#38811yescharacteristic
266C22×C61#38811yescharacteristic
267C22×C61#39011yescharacteristic
268C22×C61#39011yescharacteristic
269C22×C61#39011yescharacteristic
270C2×D61—48no—
271C2×D61—44no—
272C2×D61—84no—
273C2×D61—84no—
274C2×D61—84no—
275C2×D61—84no—
276C2×D61—84no—
277C2×D61—84no—
278C4×C61—82no—
279C4×C61—82no—
280C4×C61C4×C22⋊C411yescharacteristic
281C4×C61C4×C22⋊C411yescharacteristic
282C4×C61#38611yescharacteristic
283C4×C61#38611yescharacteristic
284C4×C61#38811yescharacteristic
285C4×C61#38811yescharacteristic
286C4×C61#38811yescharacteristic
287C4×C61#38811yescharacteristic
288Dic61—44no—
289Dic61—82no—
290C2×#418—611no—
291C2×#418—611no—
292#2873—611no—
293#2309—611no—
294#1740—611no—
295C2×H(3,ℤ/4)—611no—
296C2×H(3,ℤ/4)D6111yescharacteristic, complemented
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61#143711yescharacteristic
304C2×C61#191012yes—
305D61—84no—
306C23×D4—611no—
307C22×M16—611no—
308C22×M16C2×D6111yescharacteristic
309C2×#117—611no—
310C2×#117—611no—
311C2×#117C2×D6111yescharacteristic
312C2×#117C2×D6111yescharacteristic
313#418—614no—
314#418—614no—
315#418—1222no—
316#418—1222no—
317C2×#102—1222no—
318C2×#102—1222no—
319#405—1222no—
320#405—1222no—
321C4×M16—1222no—
322#390—614no—
323#390—1222no—
324#379—1222no—
325#342—1222no—
326H(3,ℤ/4)—22no—
327H(3,ℤ/4)—1222no—
328C61#1094811yescharacteristic, complemented, Sylow
329C25—1221no—
330C22×D4—612no—
331C22×D4—612no—
332C22×D4—612no—
333C22×D4—1221no—
334C22×D4—1221no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—1221no—
338C22×D4—1221no—
339C22×D4—1221no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4—1221no—
344C23×C4C22×D6111yescharacteristic
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—612no—
350C2×M16—612no—
351C2×M16—1221no—
352C2×M16—1221no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16C4×D6112yes—
356C2×M16C22⋊D6112yes—
357C22×C8—612no—
358C22×C8D24412yes—
359#117—21no—
360#117—21no—
361#117—21no—
362#117—21no—
363#117—21no—
364#117—21no—
365#117—21no—
366#117—612no—
367#117—1221no—
368#117—1221no—
369#117—1221no—
370#117—1221no—
371#117—1221no—
372#117—1221no—
373#117—1221no—
374#117C4×D6112yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—1221no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1228no—
384C2×C42—21no—
385C2×C42—612no—
386C2×C42—1221no—
387C2×C42C4×D6112yes—
388#102—1228no—
389#102—1228no—
390#97—1224no—
391C4×C8—1224no—
392C24—1222no—
393C24—1222no—
394C24—1222no—
395C24—4884no—
396C24—2441no—
397C24—2441no—
398C24—2441no—
399C24—2441no—
400C24D4×D6111yescharacteristic
401C2×D4—614no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—2441no—
418C2×D4—2441no—
419C2×D4—2441no—
420C2×D4—2441no—
421C2×D4—2441no—
422C2×D4—2441no—
423C2×D4—2441no—
424C2×D4—2441no—
425C2×D4—2441no—
426C2×D4—2441no—
427C2×D4—2441no—
428C2×D4—2441no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—614no—
432C22×C4—2441no—
433C22×C4C2×C4×D6111yescharacteristic
434C22×C4C2×C4×D6111yescharacteristic
435C22×C4C2×C4×D6111yescharacteristic
436C22×C4C2×D24411yescharacteristic
437C22×C4#1175822211yescharacteristic
438C22×C4#1175822211yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4D4×D6111yescharacteristic
442C22×C4#1175822411yescharacteristic
443C22×C4#1175822411yescharacteristic
444C22×C4#1175822611yescharacteristic
445C22×C4#1175822611yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—1222no—
452M16—1222no—
453M16—1222no—
454M16—1222no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—614no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8#1175819814yescomplemented
461C4⋊C4—22no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—2441no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—4888no—
477C22⋊C4—2441no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—1222no—
483C42—1222no—
484C42—1222no—
485C42—2441no—
486C23—42no—
487C23—1224no—
488C23—4884no—
489C23—4884no—
490C23—4884no—
491C23—4884no—
492C23—4884no—
493C23—4884no—
494C23—2448no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23—2442no—
500C23—2442no—
501C23C22⋊C4×D6111yescharacteristic
502C23#1451973711yescharacteristic
503C23#1451973911yescharacteristic
504C23#1451975011yescharacteristic
505C23#1451975211yescharacteristic
506C23#1451976311yescharacteristic
507C23#1451977311yescharacteristic
508D4—1224no—
509D4—1224no—
510D4—2442no—
511D4—2442no—
512D4—2442no—
513D4—2442no—
514D4—2442no—
515D4—2442no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—1224no—
521C2×C4—4882no—
522C2×C4—2442no—
523C2×C4—2444no—
524C2×C4C42×D6111yescharacteristic
525C2×C4C22⋊C4×D6111yescharacteristic
526C2×C4C2×#1175819811yescharacteristic
527C2×C4#1451973411yescharacteristic
528C2×C4#1451973511yescharacteristic
529C2×C4#1451973511yescharacteristic
530C2×C4#1451973711yescharacteristic
531C2×C4#1451973711yescharacteristic
532C2×C4#1451973711yescharacteristic
533C2×C4C4×C22⋊D6111yescharacteristic
534C2×C4C4×C22⋊D6111yescharacteristic
535C2×C4#1451973911yescharacteristic
536C2×C4#1451973911yescharacteristic
537C2×C4#1451973911yescharacteristic
538C2×C4C4×D24411yescharacteristic
539C2×C4C4×D24411yescharacteristic
540C2×C4#1451974711yescharacteristic
541C2×C4#1451974911yescharacteristic
542C2×C4#1451974911yescharacteristic
543C2×C4#1451975011yescharacteristic
544C2×C4#1451975211yescharacteristic
545C2×C4#1451976111yescharacteristic
546C2×C4#1451976211yescharacteristic
547C2×C4#1451976211yescharacteristic
548C2×C4#1451976311yescharacteristic
549C2×C4#1451976611yescharacteristic
550C2×C4#1451977211yescharacteristic
551C2×C4#1451977211yescharacteristic
552C8—24no—
553C8—1224no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—4884no—
562C22—4884no—
563C22—4884no—
564C22—4884no—
565C22—4884no—
566C22—4884no—
567C22—2448no—
568C22—2444no—
569C22#1683960911yescharacteristic
570C22#1683964011yescharacteristic
571C22#1683965111yescharacteristic
572C22#1683965411yescharacteristic
573C22#1683966211yescharacteristic
574C22#1683966211yescharacteristic
575C22#1683966311yescharacteristic
576C4—82no—
577C4—82no—
578C4—4882no—
579C4—2444no—
580C4#1683960911yescharacteristic
581C4C4×#1175819811yescharacteristic
582C4#1683963911yescharacteristic
583C4#1683964011yescharacteristic
584C4#1683965111yescharacteristic
585C4#1683965211yescharacteristic
586C4#1683965211yescharacteristic
587C4#1683965411yescharacteristic
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—4884no—
595C2#2144405511yescharacteristic
596C2#2144634512yes—
subgroupG/H
centerC22#16839640
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC22×C61#386
prime radicalC23×D244C22
cosocleC23×C4×C61C23
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#10948
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #21454237C2semidirect
5. #21445551C2semidirect
6. C2×#16839518C2semidirect
7. C2×C61×H(3,ℤ/4)C2semidirect
13. #16839768C4semidirect
15. #16839766C4semidirect
21. #16839664C4semidirect
154. C2×C8×C61C22⋊C4semidirect
186. C2×#92418C22⋊C4semidirect
296. C2×H(3,ℤ/4)D61semidirect
328. C61#10948semidirect
460. C2×C8#11758198semidirect

Maximal subgroups

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

subgroupindex
C2×#168397682
C2×#168397662
#214542372
#214483992
#214455512
C2×#168395182
C2×C61×H(3,ℤ/4)2
#1094861