Finite Group Databasebeta

← #29587594

Subgroups — #29587594

Order 15616.

818 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×#16839780C211yesmaximal, characteristic
2C2×#16839756C211yesmaximal, characteristic
3C2×#16839744C211yesmaximal, characteristic
4C2×#16839532C211yesmaximal, characteristic, complemented
5C2×#16839526C211yesmaximal, characteristic, complemented
6#21445516C211yesmaximal, characteristic, complemented
7C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12#16839780—22no—
13#16839780—22no—
14C2×#14519705—22no—
15#16839756—22no—
16#16839756—22no—
17C2×#14519700—21no—
18C2×#14519697—21no—
19#16839744—22no—
20#16839744C414yescomplemented
21C2×#14519695—21no—
22C2×#14519693—21no—
23#16839532—22no—
24#16839526—22no—
25#16839517—22no—
26C22×M16×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C2×#117×C61C2211yescharacteristic
29C61×H(3,ℤ/4)—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×#11758222C2×C412yes—
41C2×#11758222D412yes—
42C22×Dic122D411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C22⋊D61—21no—
45C22×C4×D61—21no—
46C2×#11758194—21no—
47C2×#11758194—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×#11758194C2×C412yes—
57C2×#11758194D412yes—
58C22×#92418—21no—
59C22×#92418—21no—
60C22×#92418—21no—
61C22×#92418—21no—
62C22×#92418—21no—
63C22×#92418—21no—
64C22×#92418—21no—
65C22×#92418—21no—
66C22×#92418D412yes—
67#14519705—28no—
68#14519700—24no—
69#14519697—24no—
70#14519695—24no—
71#14519693—24no—
72C23×C4×C61C2311yescharacteristic
73C2×M16×C61—21no—
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61C2×C412yes—
79C22×C8×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×C61—21no—
87#117×C61D412yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61D412yes—
94C24×C61#4111yescharacteristic
95C23×D61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—41no—
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—41no—
117C2×C22⋊D61—41no—
118C22×Dic61—21no—
119#11758222—22no—
120#11758222—22no—
121#11758222—22no—
122#11758222—22no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—41no—
128#11758222—41no—
129#11758222—41no—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C22⋊C412yes—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122C22⋊C412yes—
153M16×C61—22no—
154M16×C61—22no—
155M16×C61—41no—
156M16×C61—41no—
157C2×C8×C61—22no—
158C2×C8×C61—41no—
159C2×C8×C61—41no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—41no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—41no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—41no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61—41no—
172#11758194—22no—
173#11758194—22no—
174#11758194—22no—
175#11758194—22no—
176#11758194—41no—
177#11758194—41no—
178#11758194—41no—
179#11758194—41no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—41no—
186C2×#92418—41no—
187C2×#92418—41no—
188C2×#92418—41no—
189C2×#92418—41no—
190C2×#92418—41no—
191C2×#92418—41no—
192C2×#92418—41no—
193C23×C61—42no—
194C23×C61#11711yescharacteristic
195C23×C61#12111yescharacteristic
196C23×C61#12311yescharacteristic
197C23×C61#12311yescharacteristic
198C23×C61#12311yescharacteristic
199C23×C61#12411yescharacteristic
200C23×C61#12611yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—82no—
205C2×C4×C61—44no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—82no—
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61#11711yescharacteristic
211C2×C4×C61C4×D411yescharacteristic
212C2×C4×C61C4×D411yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61C4×D411yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12311yescharacteristic
225C2×C4×C61#12311yescharacteristic
226C2×C4×C61#12311yescharacteristic
227C2×C4×C61#12311yescharacteristic
228C2×C4×C61#12311yescharacteristic
229C2×C4×C61#12311yescharacteristic
230C2×C4×C61#12311yescharacteristic
231C2×C4×C61#12411yescharacteristic
232C2×C4×C61#12411yescharacteristic
233C2×C4×C61#12411yescharacteristic
234C2×C4×C61#12611yescharacteristic
235C2×C4×C61#12611yescharacteristic
236C2×C4×C61#12611yescharacteristic
237C22⋊D61—42no—
238C22⋊D61—42no—
239C22⋊D61—42no—
240C22⋊D61—42no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—22no—
245D244—22no—
246D244—41no—
247D244—41no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—42no—
252Dic122—22no—
253Dic122—22no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—42no—
257#92418—42no—
258#92418—42no—
259#92418—42no—
260#11150—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61#38111yescharacteristic
269C22×C61#38811yescharacteristic
270C22×C61#38811yescharacteristic
271C22×C61#39411yescharacteristic
272C22×C61#39411yescharacteristic
273C22×C61#39711yescharacteristic
274C22×C61#39711yescharacteristic
275C2×D61—42no—
276C2×D61—82no—
277C2×D61—82no—
278C2×D61—82no—
279C4×C61—82no—
280C4×C61—82no—
281C4×C61#38611yescharacteristic
282C4×C61#38611yescharacteristic
283C4×C61#38811yescharacteristic
284C4×C61#38811yescharacteristic
285C4×C61#39411yescharacteristic
286C4×C61#39411yescharacteristic
287C4×C61#39711yescharacteristic
288C4×C61#39711yescharacteristic
289Dic61—42no—
290C2×#421—611no—
291C2×#416—611no—
292C2×#413—611no—
293C2×#344—611no—
294C2×#341—611no—
295C2×H(3,ℤ/4)D6111yescharacteristic, complemented
296#1729—611no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61#144011yescharacteristic
304C2×C61#208012yes—
305D61—82no—
306C22×#41—611no—
307C22×M16—611no—
308C22×M16—611no—
309C22×M16C2×D6111yescharacteristic
310C23×C8—611no—
311C2×#117C2×D6111yescharacteristic
312C2×#117C2×D6111yescharacteristic
313#421—1222no—
314#421—1222no—
315C2×#104—1222no—
316#416—1222no—
317#416—1222no—
318C2×#103—1221no—
319C2×#102—1221no—
320#413—614no—
321#413—1222no—
322C2×#101—1221no—
323C2×#100—1221no—
324#344—1222no—
325#341—1222no—
326H(3,ℤ/4)—22no—
327#336—1222no—
328C61#1115011yescharacteristic, complemented, Sylow
329C2×#41—612no—
330C2×#41—612no—
331C2×#41—1221no—
332C2×#41—1221no—
333C2×#41—1221no—
334C2×#41—1221no—
335C2×#41—1221no—
336C2×#41—1221no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C22×Q8—611no—
342C22×D4—611no—
343C22×D4—1221no—
344C23×C4—1221no—
345C23×C4C22×D6111yescharacteristic
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—612no—
352C2×M16—612no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16—1221no—
356C2×M16—1221no—
357C2×M16—1221no—
358C2×M16—1221no—
359C2×M16—1221no—
360C2×M16—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16C4×D6112yes—
364C22×C8—21no—
365C22×C8—612no—
366C22×C8—1221no—
367C22×C8—1221no—
368C22×C8—1221no—
369C22×C8—1221no—
370C22×C8—1221no—
371C22×C8—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117C22⋊D6112yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42D24412yes—
388#104—1228no—
389#103—1224no—
390#102—1224no—
391#101—1224no—
392#100—1224no—
393C24—1221no—
394C24#1175822411yescharacteristic
395#41—1222no—
396#41—1222no—
397#41—1222no—
398#41—1222no—
399#41—1222no—
400#41—1222no—
401#41—1222no—
402#41—1222no—
403#41—2441no—
404#41—2441no—
405#41—2441no—
406#41—2441no—
407#41—2441no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411C2×Q8—612no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×D4—612no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—2441no—
426C2×D4—2441no—
427C2×D4—2441no—
428C2×D4—2441no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2441no—
438C22×C4C2×C4×D6111yescharacteristic
439C22×C4C2×D24411yescharacteristic
440C22×C4#1175822211yescharacteristic
441C22×C4#1175822211yescharacteristic
442C22×C4#1175822211yescharacteristic
443C22×C4#1175822211yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4#1175822611yescharacteristic
450C22×C4#1175822611yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—1222no—
457M16—1222no—
458M16—1222no—
459M16—1222no—
460M16—2441no—
461M16—2441no—
462M16—2441no—
463M16—2441no—
464C2×C8—22no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—2441no—
473C2×C8—2441no—
474C2×C8—2441no—
475C2×C8—2441no—
476C2×C8—2441no—
477C2×C8—2441no—
478C2×C8—2441no—
479C2×C8—2441no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C22⋊C4—22no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C23—42no—
493C23—1222no—
494C23—4882no—
495C23—2441no—
496C23—2441no—
497C23#1451973511yescharacteristic
498C23#1451974911yescharacteristic
499C23#1451975911yescharacteristic
500C23#1451975911yescharacteristic
501C23#1451976111yescharacteristic
502C23#1451976311yescharacteristic
503C23#1451977211yescharacteristic
504Q8—1222no—
505Q8—1222no—
506Q8—2441no—
507Q8—2441no—
508D4—1222no—
509D4—1222no—
510D4—2442no—
511D4—2442no—
512D4—2442no—
513D4—2442no—
514D4—2441no—
515D4—2441no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—1222no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C22⋊C4×D6111yescharacteristic
528C2×C4#1451973411yescharacteristic
529C2×C4#1451973711yescharacteristic
530C2×C4C4×C22⋊D6111yescharacteristic
531C2×C4#1451973911yescharacteristic
532C2×C4C4×D24411yescharacteristic
533C2×C4#1451974511yescharacteristic
534C2×C4#1451974711yescharacteristic
535C2×C4#1451974811yescharacteristic
536C2×C4#1451975011yescharacteristic
537C2×C4#1451975011yescharacteristic
538C2×C4#1451975011yescharacteristic
539C2×C4#1451975111yescharacteristic
540C2×C4#1451975211yescharacteristic
541C2×C4#1451975211yescharacteristic
542C2×C4#1451975911yescharacteristic
543C2×C4#1451976011yescharacteristic
544C2×C4#1451976211yescharacteristic
545C2×C4#1451976211yescharacteristic
546C2×C4#1451976211yescharacteristic
547C2×C4#1451976211yescharacteristic
548C2×C4#1451976211yescharacteristic
549C2×C4#1451976311yescharacteristic
550C2×C4#1451976511yescharacteristic
551C2×C4#1451976611yescharacteristic
552C2×C4#1451977111yescharacteristic
553C2×C4#1451977211yescharacteristic
554C2×C4#1451977211yescharacteristic
555C8—42no—
556C8—2442no—
557C8—2442no—
558C8—2442no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—4882no—
567C22—4882no—
568C22—4882no—
569C22—2442no—
570C22#1683962411yescharacteristic
571C22#1683965211yescharacteristic
572C22#1683965411yescharacteristic
573C22#1683967711yescharacteristic
574C22#1683967811yescharacteristic
575C22#1683968911yescharacteristic
576C22#1683969111yescharacteristic
577C4—82no—
578C4—82no—
579C4—2442no—
580C4#1683963811yescharacteristic
581C4#1683964011yescharacteristic
582C4#1683965111yescharacteristic
583C4#1683965311yescharacteristic
584C4#1683967911yescharacteristic
585C4#1683968011yescharacteristic
586C4#1683968811yescharacteristic
587C4#1683969111yescharacteristic
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—4882no—
595C2#2144406711yescharacteristic
596C2#2144727112yes—
subgroupG/H
centerC2×C4#14519765
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC22×C61#397
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#11150
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. C2×#16839532C2semidirect
5. C2×#16839526C2semidirect
6. #21445516C2semidirect
7. C2×C61×H(3,ℤ/4)C2semidirect
20. #16839744C4semidirect
295. C2×H(3,ℤ/4)D61semidirect
328. C61#11150semidirect

Maximal subgroups

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

subgroupindex
C2×#168397802
C2×#168397562
C2×#168397442
C2×#168395322
C2×#168395262
#214455162
C2×C61×H(3,ℤ/4)2
#1115061