Finite Group Databasebeta

← #29587585

Subgroups — #29587585

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839744C211yesmaximal, characteristic
2C2×#16839735C211yesmaximal, characteristic
3#21448309C211yesmaximal, characteristic
4C2×#16839522C211yesmaximal, characteristic, complemented
5C2×#16839519C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12#16839744—22no—
13#16839744—22no—
14C2×#14519695—21no—
15C2×#14519693—21no—
16#16839735—22no—
17#16839735—22no—
18C2×#14519691—22no—
19#16839668—21no—
20#16839664—21no—
21#16839641—22no—
22C4×#11758198C414yescomplemented
23#16839522—22no—
24#16839519—22no—
25#16839518—22no—
26C22×M16×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C22×C42×C61C2211yescharacteristic
29C61×#339—22no—
30C2×#11758222—21no—
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C22×Dic122D411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C22⋊D61D412yes—
45C22×C4×D61C2×C412yes—
46C2×#11758194—21no—
47C2×#11758194—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194D412yes—
52C22×#92418—21no—
53#14519736—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736D412yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×C22⋊Dic61—21no—
65C2×#11758198—24no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2×C412yes—
68#14519695—24no—
69#14519693—24no—
70#14519691—28no—
71#14519684—24no—
72C23×C4×C61—21no—
73C23×C4×C61C2311yescharacteristic
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61D412yes—
80C22×C8×C61—21no—
81#117×C61—21no—
82#117×C61—21no—
83#117×C61—21no—
84#117×C61D412yes—
85C2×C4⋊C4×C61—21no—
86C2×C22⋊C4×C61—21no—
87C2×C42×C61—21no—
88C2×C42×C61—21no—
89C2×C42×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61C2×C412yes—
94C24×C61C2×D411yescharacteristic
95C23×D61C22⋊C412yes—
96C22×C4×C61—22no—
97C22×C4×C61—44no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—22no—
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C22×Dic61—22no—
122C22×Dic61—41no—
123C22×Dic61C22⋊C412yes—
124#11758222—41no—
125#11758222—41no—
126#11758222—41no—
127#11758222—41no—
128#11758222—41no—
129#11758222—41no—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—41no—
161M16×C61—41no—
162C2×C8×C61—22no—
163C2×C8×C61—41no—
164C2×C8×C61—41no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—41no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—41no—
173C42×C61—41no—
174C42×C61—41no—
175C22⋊Dic61—22no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179#11758198—48no—
180C4⋊Dic61—22no—
181C4⋊Dic61—41no—
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—41no—
188#11758194—22no—
189#11758194—22no—
190#11758194—41no—
191#11758194—41no—
192C2×#92418—22no—
193C2×#92418—41no—
194C2×#92418—41no—
195C23×C61—42no—
196C23×C61C2×C22⋊C411yescharacteristic
197C23×C61#12011yescharacteristic
198C23×C61#12111yescharacteristic
199C23×C61#12111yescharacteristic
200C23×C61#12311yescharacteristic
201C23×C61#12311yescharacteristic
202C23×C61#12311yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—44no—
207C2×C4×C61—24no—
208C2×C4×C61—42no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—82no—
214C2×C4×C61#11711yescharacteristic
215C2×C4×C61#11711yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61#12311yescharacteristic
228C2×C4×C61#12311yescharacteristic
229C2×C4×C61#12311yescharacteristic
230C2×C4×C61#12311yescharacteristic
231C2×C4×C61#12311yescharacteristic
232C2×C4×C61#12311yescharacteristic
233C2×C4×C61#12311yescharacteristic
234C2×C4×C61#12411yescharacteristic
235C2×C4×C61#12411yescharacteristic
236C2×C4×C61#12411yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12611yescharacteristic
239C2×C4×C61#12611yescharacteristic
240C2×C4×C61#12611yescharacteristic
241C2×C4×C61#12611yescharacteristic
242C22⋊D61—42no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—44no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—82no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—42no—
261Dic122—41no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265C8×C61—42no—
266#92418—42no—
267#11148—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61#38111yescharacteristic
276C22×C61#38611yescharacteristic
277C22×C61#38611yescharacteristic
278C22×C61#39411yescharacteristic
279C22×C61#39411yescharacteristic
280C22×C61#39711yescharacteristic
281C22×C61#39711yescharacteristic
282C2×D61—44no—
283C2×D61—42no—
284C2×D61—82no—
285C2×D61—82no—
286C4×C61—44no—
287C4×C61—82no—
288C4×C61#38811yescharacteristic
289C4×C61#38811yescharacteristic
290C4×C61#38811yescharacteristic
291C4×C61#38811yescharacteristic
292C4×C61#39411yescharacteristic
293C4×C61#39411yescharacteristic
294C4×C61#39711yescharacteristic
295C4×C61#39711yescharacteristic
296Dic61—42no—
297Dic61—82no—
298Dic61—82no—
299C2×#413—611no—
300C2×#410—611no—
301#2297—611no—
302C2×#339—611no—
303C2×#339D6111yescharacteristic, complemented
304C2×H(3,ℤ/4)—611no—
305C2×H(3,ℤ/4)—611no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61#144011yescharacteristic
313C2×C61#182712yes—
314D61—82no—
315C22×#41—611no—
316C22×M16—611no—
317C22×M16C2×D6111yescharacteristic
318C2×#117—611no—
319C2×#117—611no—
320C2×#117C2×D6111yescharacteristic
321C22×C42C2×D6111yescharacteristic
322#413—1222no—
323#413—1222no—
324C2×#101—1221no—
325C2×#100—1221no—
326#410—1222no—
327#410—1222no—
328C2×#99—1222no—
329#391—1221no—
330#390—1221no—
331#386—1222no—
332C4×C22⋊C4—614no—
333#339—22no—
334#339—1222no—
335H(3,ℤ/4)—1222no—
336H(3,ℤ/4)—1222no—
337C61#1114811yescharacteristic, complemented, Sylow
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C2×#41—1221no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C22×Q8—611no—
351C22×D4—612no—
352C22×D4—611no—
353C23×C4—21no—
354C23×C4—612no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—612no—
362C2×M16—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16C22⋊D6112yes—
368C22×C8—21no—
369C22×C8—1221no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—612no—
374#117—1221no—
375#117—1221no—
376#117—1221no—
377#117—1221no—
378#117—1221no—
379#117—1221no—
380#117—1221no—
381#117D24412yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1221no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—612no—
396C2×C42—1221no—
397C2×C42C4×D6112yes—
398#101—1224no—
399#100—1224no—
400#99—1228no—
401#95—1224no—
402C24—612no—
403C24D4×D6111yescharacteristic
404#41—2441no—
405#41—2441no—
406#41—2441no—
407#41—2441no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—612no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—2441no—
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4C2×D24411yescharacteristic
454C22×C4#1175822211yescharacteristic
455C22×C4#1175822211yescharacteristic
456C22×C4#1175822211yescharacteristic
457C22×C4#1175822211yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4#1175822411yescharacteristic
462C22×C4#1175822411yescharacteristic
463C22×C4#1175822611yescharacteristic
464C22×C4#1175822611yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—1222no—
471M16—1222no—
472M16—2441no—
473M16—2441no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—1222no—
478C2×C8—2441no—
479C2×C8—2441no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—1222no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—1222no—
489C22⋊C4—2448no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—1222no—
500C42—1222no—
501C42—1222no—
502C42—2441no—
503C23—42no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—2444no—
508C23C22⋊C4×D6111yescharacteristic
509C23#1451974711yescharacteristic
510C23#1451974811yescharacteristic
511C23#1451975011yescharacteristic
512C23#1451975911yescharacteristic
513C23#1451976211yescharacteristic
514C23#1451976211yescharacteristic
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518Q8—2441no—
519D4—2442no—
520D4—2442no—
521D4—2442no—
522D4—2442no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—4882no—
538C2×C4—2444no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4#1451973411yescharacteristic
546C2×C4#1451973511yescharacteristic
547C2×C4#1451973711yescharacteristic
548C2×C4C4×C22⋊D6111yescharacteristic
549C2×C4#1451973911yescharacteristic
550C2×C4C4×D24411yescharacteristic
551C2×C4#1451974511yescharacteristic
552C2×C4#1451974911yescharacteristic
553C2×C4#1451975011yescharacteristic
554C2×C4#1451975011yescharacteristic
555C2×C4#1451975111yescharacteristic
556C2×C4#1451975211yescharacteristic
557C2×C4#1451975211yescharacteristic
558C2×C4#1451975911yescharacteristic
559C2×C4#1451975911yescharacteristic
560C2×C4#1451976011yescharacteristic
561C2×C4#1451976111yescharacteristic
562C2×C4#1451976211yescharacteristic
563C2×C4#1451976211yescharacteristic
564C2×C4#1451976211yescharacteristic
565C2×C4#1451976311yescharacteristic
566C2×C4#1451976311yescharacteristic
567C2×C4#1451976511yescharacteristic
568C2×C4#1451976611yescharacteristic
569C2×C4#1451977111yescharacteristic
570C2×C4#1451977211yescharacteristic
571C2×C4#1451977211yescharacteristic
572C2×C4#1451977211yescharacteristic
573C8—42no—
574C8—2442no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—4882no—
583C22—4882no—
584C22—2444no—
585C22—2442no—
586C22#1683962411yescharacteristic
587C22#1683963811yescharacteristic
588C22#1683964011yescharacteristic
589C22#1683967811yescharacteristic
590C22#1683968011yescharacteristic
591C22#1683968811yescharacteristic
592C22#1683969111yescharacteristic
593C4—44no—
594C4—82no—
595C4—4882no—
596C4—4882no—
597C4—2442no—
598C4#1683965111yescharacteristic
599C4#1683965211yescharacteristic
600C4#1683965311yescharacteristic
601C4#1683965411yescharacteristic
602C4#1683967711yescharacteristic
603C4#1683967911yescharacteristic
604C4#1683968911yescharacteristic
605C4#1683969111yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—4882no—
613C2#2144406711yescharacteristic
614C2#2144592212yes—
subgroupG/H
centerC2×C4#14519735
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC22×C61#381
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#11148
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C2×#16839522C2semidirect
5. C2×#16839519C2semidirect
6. C2×#16839518C2semidirect
7. C2×C61×#339C2semidirect
22. C4×#11758198C4semidirect
303. C2×#339D61semidirect
337. C61#11148semidirect

Maximal subgroups

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

subgroupindex
C2×#168397442
C2×#168397352
#214483092
C2×#168395222
C2×#168395192
C2×#168395182
C2×C61×#3392
#1114861