Finite Group Databasebeta

← #29587365

Subgroups — #29587365

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839810C211yesmaximal, characteristic, complemented
2C2×#16839780C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C2×#16839744C211yesmaximal, characteristic
5C2×#16839532C211yesmaximal, characteristic, complemented
6#21445558C211yesmaximal, characteristic, complemented
7C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16839810—22no—
13#16839810—22no—
14C2×#14519712—22no—
15#16839780—22no—
16#16839780—22no—
17C2×#14519705C414yescomplemented
18#16839770—21no—
19#16839766—21no—
20#16839761—22no—
21#16839756—22no—
22#16839744—22no—
23#16839744—22no—
24C2×#14519695—21no—
25C2×#14519693—21no—
26#16839532—22no—
27#16839531—22no—
28C22×M16×C61C2211yescharacteristic
29C2×#117×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C61×H(3,ℤ/4)—22no—
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—
42C2×#11758222C2×C412yes—
43C2×#11758222Q812yes—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61—21no—
47C22×C4×D61—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194C2×C412yes—
62C2×#11758194D412yes—
63C2×#11758194D412yes—
64C2×#11758194D412yes—
65C2×#11758194D412yes—
66C22×#92418—21no—
67C22×#92418—21no—
68C22×#92418—21no—
69#14519712—28no—
70#14519705—28no—
71#14519700—24no—
72#14519697—24no—
73#14519695—24no—
74#14519693—24no—
75C23×C4×C61C2311yescharacteristic
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61D412yes—
82C22×C8×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90#117×C61D412yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C42×C61—21no—
96C2×C42×C61C2×C412yes—
97C24×C61#4111yescharacteristic
98C23×D61—21no—
99C22×C4×C61—22no—
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×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×Q811yescharacteristic
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—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C22×Dic61—21no—
122#11758222—22no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C4⋊C412yes—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C4⋊C412yes—
156M16×C61—22no—
157M16×C61—22no—
158M16×C61—41no—
159M16×C61—41no—
160C2×C8×C61—22no—
161C2×C8×C61—41no—
162C2×C8×C61—41no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—41no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61—41no—
175#11758194—22no—
176#11758194—22no—
177#11758194—22no—
178#11758194—22no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194—41no—
183#11758194—41no—
184#11758194—41no—
185#11758194—41no—
186#11758194C22⋊C414yes—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—41no—
193C2×#92418—41no—
194C2×#92418—41no—
195C2×#92418—41no—
196C23×C61—81no—
197C23×C61C4×D411yescharacteristic
198C23×C61#12111yescharacteristic
199C23×C61C22⋊Q811yescharacteristic
200C23×C61#12311yescharacteristic
201C23×C61#12411yescharacteristic
202C23×C61#12411yescharacteristic
203C23×C61#12611yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—82no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—82no—
211C2×C4×C61—82no—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C4⋊C411yescharacteristic
214C2×C4×C61#11711yescharacteristic
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
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#12111yescharacteristic
228C2×C4×C61C22⋊Q811yescharacteristic
229C2×C4×C61C22⋊Q811yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61C22⋊Q811yescharacteristic
233C2×C4×C61#12311yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12611yescharacteristic
239C2×C4×C61#12711yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—42no—
244C2×Dic61—22no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255Dic122—22no—
256Dic122—22no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—42no—
260#92418—24no—
261#92418—42no—
262#92418—42no—
263#11116—611nomaximal, Sylow
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#38511yescharacteristic
272C22×C61#38811yescharacteristic
273C22×C61#39011yescharacteristic
274C22×C61#39311yescharacteristic
275C22×C61#39411yescharacteristic
276C22×C61#39611yescharacteristic
277C22×C61#39711yescharacteristic
278C2×D61—42no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61—82no—
282C4×C61—82no—
283C4×C61—82no—
284C4×C61#38011yescharacteristic
285C4×C61#38511yescharacteristic
286C4×C61#38611yescharacteristic
287C4×C61#38811yescharacteristic
288C4×C61#39211yescharacteristic
289C4×C61#39411yescharacteristic
290C4×C61#39611yescharacteristic
291C4×C61#39911yescharacteristic
292Dic61—42no—
293C2×#430—611no—
294C2×#421—611no—
295#2896—611no—
296C2×#413—611no—
297C2×#344—611no—
298#1742—611no—
299C2×H(3,ℤ/4)D6111yescharacteristic, complemented
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61#143911yescharacteristic
308D61—82no—
309C22×#41—611no—
310C22×M16—611no—
311C22×M16—611no—
312C22×M16—611no—
313C22×M16C2×D6111yescharacteristic
314C2×#117C2×D6111yescharacteristic
315C2×#117C2×D6111yescharacteristic
316#430—1222no—
317#430—1222no—
318C2×#108—1222no—
319#421—1222no—
320#421—1222no—
321C2×#104—614no—
322#419—1221no—
323#418—1221no—
324#417—1222no—
325#416—1222no—
326#413—1222no—
327#413—1222no—
328C2×#101—1221no—
329C2×#100—1221no—
330#344—1222no—
331#343—1222no—
332H(3,ℤ/4)—22no—
333C61#1111611yescharacteristic, complemented, Sylow
334C2×#41—612no—
335C2×#41—612no—
336C2×#41—1221no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C22×Q8—611no—
347C22×D4—611no—
348C22×D4—1221no—
349C23×C4—1221no—
350C23×C4C22×D6111yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—612no—
357C2×M16—612no—
358C2×M16—612no—
359C2×M16—612no—
360C2×M16—612no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16—1221no—
374C2×M16D24412yes—
375C22×C8—21no—
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117C22⋊D6112yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42C4×D6112yes—
393#108—1228no—
394#104—1228no—
395#103—1224no—
396#102—1224no—
397#101—1224no—
398#100—1224no—
399C24—1221no—
400C24#1175822411yescharacteristic
401#41—1222no—
402#41—1222no—
403#41—1222no—
404#41—1222no—
405#41—1222no—
406#41—1222no—
407#41—1222no—
408#41—1222no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417C2×Q8—612no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×D4—612no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2441no—
434C2×D4—2441no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—1221no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4C2×C4×D6111yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4#1175822211yescharacteristic
447C22×C4#1175822211yescharacteristic
448C22×C4#1175822211yescharacteristic
449C22×C4#1175822211yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4Q8×D6111yescharacteristic
456C22×C4#1175822611yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—614no—
463M16—1222no—
464M16—1222no—
465M16—1222no—
466M16—1222no—
467M16—1222no—
468M16—1222no—
469M16—1222no—
470M16—2441no—
471M16—2441no—
472M16—2441no—
473M16—2441no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—2441no—
483C2×C8—2441no—
484C2×C8—2441no—
485C2×C8—2441no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—41no—
498C23—81no—
499C23—1222no—
500C23—4882no—
501C23—2441no—
502C23—2441no—
503C23#1451973711yescharacteristic
504C23#1451974911yescharacteristic
505C23#1451975711yescharacteristic
506C23#1451975911yescharacteristic
507C23#1451976311yescharacteristic
508C23#1451976311yescharacteristic
509C23#1451977211yescharacteristic
510Q8—1222no—
511Q8—1222no—
512Q8—2441no—
513Q8—2441no—
514D4—1222no—
515D4—1222no—
516D4—2442no—
517D4—2442no—
518D4—2442no—
519D4—2442no—
520D4—2441no—
521D4—2441no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—1222no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4C22⋊C4×D6111yescharacteristic
534C2×C4C4⋊C4×D6111yescharacteristic
535C2×C4#1451973411yescharacteristic
536C2×C4C4×C22⋊D6111yescharacteristic
537C2×C4#1451973911yescharacteristic
538C2×C4C4×D24411yescharacteristic
539C2×C4#1451974511yescharacteristic
540C2×C4#1451974711yescharacteristic
541C2×C4#1451974711yescharacteristic
542C2×C4#1451975011yescharacteristic
543C2×C4#1451975011yescharacteristic
544C2×C4#1451975011yescharacteristic
545C2×C4#1451975011yescharacteristic
546C2×C4#1451975011yescharacteristic
547C2×C4#1451975111yescharacteristic
548C2×C4#1451975211yescharacteristic
549C2×C4#1451975311yescharacteristic
550C2×C4#1451975511yescharacteristic
551C2×C4#1451975511yescharacteristic
552C2×C4#1451975511yescharacteristic
553C2×C4#1451975511yescharacteristic
554C2×C4#1451976011yescharacteristic
555C2×C4#1451976211yescharacteristic
556C2×C4#1451976211yescharacteristic
557C2×C4#1451976211yescharacteristic
558C2×C4#1451976511yescharacteristic
559C2×C4#1451977111yescharacteristic
560C2×C4#1451977311yescharacteristic
561C8—42no—
562C8—1224no—
563C8—2442no—
564C8—2442no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—4882no—
573C22—4882no—
574C22—4882no—
575C22—2442no—
576C22#1683963611yescharacteristic
577C22#1683965111yescharacteristic
578C22#1683966211yescharacteristic
579C22#1683967511yescharacteristic
580C22#1683967711yescharacteristic
581C22#1683968411yescharacteristic
582C22#1683969111yescharacteristic
583C4—82no—
584C4—82no—
585C4—2442no—
586C4#1683962111yescharacteristic
587C4#1683963311yescharacteristic
588C4#1683964011yescharacteristic
589C4#1683965311yescharacteristic
590C4#1683967111yescharacteristic
591C4#1683968011yescharacteristic
592C4#1683968611yescharacteristic
593C4#1683969711yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—4882no—
602C2#2144406311yescharacteristic
subgroupG/H
centerC4#16839653
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC2×C61#1439
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#11116
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#16839810C2semidirect
5. C2×#16839532C2semidirect
6. #21445558C2semidirect
7. C2×C61×H(3,ℤ/4)C2semidirect
17. C2×#14519705C4semidirect
299. C2×H(3,ℤ/4)D61semidirect
333. C61#11116semidirect

Maximal subgroups

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

subgroupindex
C2×#168398102
C2×#168397802
#214543362
C2×#168397442
C2×#168395322
#214455582
C2×C61×H(3,ℤ/4)2
#1111661