Finite Group Databasebeta

← #29572054

Subgroups — #29572054

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758222C211yesmaximal, characteristic
2#21445840C211yesmaximal, characteristic
3#21444317C211yesmaximal, characteristic, complemented
4#21444275C211yesmaximal, characteristic, complemented
5#21444248C211yesmaximal, characteristic, complemented
6#21444202C211yesmaximal, characteristic
7C4×#103×C61C211yesmaximal, characteristic, complemented
8Q8×Dic122C411yescharacteristic, complemented
9#16840490—21no—
10#16840490C411yescharacteristic, complemented
11#16840472—21no—
12#16840472C411yescharacteristic, complemented
13#16840467—22no—
14#16840467—21no—
15Q8×D244C411yescharacteristic, complemented
16Q8×C22⋊D61—21no—
17#16840319—21no—
18#16840319C2211yescharacteristic
19#16840316—22no—
20#16840316—21no—
21C4×#11758222—21no—
22C4×#11758222C2211yescharacteristic
23C4×Q8×D61—21no—
24#16839815C2211yescharacteristic
25#16839786C2211yescharacteristic
26C2×#14519705—21no—
27#16839522—21no—
28#16839498—22no—
29#16839494—21no—
30#16839494—22no—
31Q8⋊#92418—21no—
32Q8⋊#92418—21no—
33#16839491—21no—
34C2×C4×Q8×C61C2211yescharacteristic
35C2×#103×C61—21no—
36C2×C4×C8×C61C2211yescharacteristic
37C4×C4⋊C4×C61C2211yescharacteristic
38C61×#340—21no—
39C2×#11758222D411yescharacteristic
40C2×Q8×D61—21no—
41C2×#11758194—21no—
42#14519777—21no—
43#14519777—21no—
44#14519777C2×C411yescharacteristic
45#14519777C2×C411yescharacteristic
46#14519776—22no—
47#14519776—21no—
48#14519775—22no—
49#14519775—21no—
50#14519769—21no—
51#14519769—21no—
52#14519769—21no—
53#14519767—22no—
54#14519767—22no—
55#14519767—22no—
56#14519757—21no—
57#14519757—22no—
58#14519756—22no—
59#14519756—22no—
60#14519756—21no—
61#14519756D412yes—
62#14519755—22no—
63#14519755—22no—
64#14519755—22no—
65#14519753—22no—
66#14519753—21no—
67Q8×Dic61—21no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×Dic122—21no—
73C4×Dic122C2×C411yescharacteristic
74C4×Dic122D411yescharacteristic
75C4×Dic122D412yes—
76#14519742—22no—
77#14519742—21no—
78C4×D244—21no—
79C4×D244—21no—
80C4×D244C2×C411yescharacteristic
81C4×D244D411yescharacteristic
82C4×C22⋊D61—22no—
83C4×C22⋊D61—22no—
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86#14519736—21no—
87#14519736D411yescharacteristic
88#14519734—22no—
89#14519734—21no—
90C4⋊C4×D61—22no—
91C4⋊C4×D61—21no—
92C42×D61—22no—
93C42×D61—21no—
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97#14519705—42no—
98C22⋊#92418—21no—
99C4×#92418—21no—
100C22×Q8×C61D411yescharacteristic
101C22×C8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104C4×Q8×C61C2×C411yescharacteristic
105C4×Q8×C61C2×C411yescharacteristic
106C4×Q8×C61D411yescharacteristic
107C4×Q8×C61D411yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#103×C61—24no—
115C4×C8×C61—21no—
116C4×C8×C61—21no—
117#95×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—41no—
121C2×Q8×C61C22⋊C411yescharacteristic
122C2×Q8×C61C22⋊C411yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C2×C22⋊D61—21no—
130Q8×D61—42no—
131Q8×D61—42no—
132#11758222—21no—
133#11758222—21no—
134#11758222SD1612yes—
135#11758222Q1612yes—
136C2×D244C22⋊C411yescharacteristic
137C2×C4×D61—21no—
138C2×C4×D61—22no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—21no—
142C2×Dic122—22no—
143C2×Dic122—21no—
144C2×Dic122—22no—
145C2×Dic122C22⋊C411yescharacteristic
146C2×Dic122C22⋊C412yes—
147C2×C8×C61—22no—
148C2×C8×C61—41no—
149C2×C8×C61—41no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61C22⋊C411yescharacteristic
158C4⋊C4×C61C22⋊C411yescharacteristic
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—41no—
164C42×C61—41no—
165C42×C61C22×C411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C22⋊Dic61—22no—
170C22⋊Dic61—21no—
171#11758198—22no—
172#11758198—22no—
173#11758198—21no—
174#11758198—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61C22⋊C411yescharacteristic
181C4⋊Dic61C22⋊C411yescharacteristic
182C4⋊Dic61C22⋊C412yes—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—21no—
190#11758196—22no—
191#11758196—21no—
192#11758196—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—21no—
196C4×Dic61—22no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199#11758194—41no—
200#11758194—41no—
201C2×#92418—41no—
202C2×#92418—41no—
203C23×C61#12011yescharacteristic
204C22×D61—21no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—42no—
209Q8×C61#10412yescomplemented
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—81no—
220C2×C4×C61—81no—
221C2×C4×C61—81no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61C2×SD1611yescharacteristic
229C2×C4×C61C2×Q1611yescharacteristic
230C2×C4×C61#13711yescharacteristic
231C2×C4×C61#13711yescharacteristic
232C22⋊D61—22no—
233C22⋊D61—22no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239D244—21no—
240D244#10312yescomplemented
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—44no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—81no—
247C4×D61—82no—
248Dic122—21no—
249Dic122—44no—
250Dic122—41no—
251Dic122—41no—
252Dic122—42no—
253Dic122—82no—
254Dic122#10312yescomplemented
255C8×C61—42no—
256#92418—81no—
257#92418—81no—
258#8439—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61#37911yescharacteristic
262C22×C61#44811yescharacteristic
263C22×C61#45111yescharacteristic
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—41no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—82no—
275C4×C61C2×#10311yescharacteristic
276C4×C61#41911yescharacteristic
277C4×C61C2×#10411yescharacteristic
278C4×C61#42111yescharacteristic
279C4×C61#44811yescharacteristic
280C4×C61#45111yescharacteristic
281Dic61—22no—
282Dic61—42no—
283Dic61—41no—
284Dic61—41no—
285Dic61—42no—
286Q8×#41—611no—
287#1806—611no—
288C4×#103D6111yescharacteristic, complemented
289#1503—611no—
290#1493—611no—
291#1487—611no—
292#1476—611no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61#178711yescharacteristic
296C2×C61#178911yescharacteristic
297C2×C61#179411yescharacteristic
298D61—42no—
299Q82—611no—
300#557—611no—
301#557—1221no—
302#554—611no—
303#554—1222no—
304#554—1221no—
305#554—1221no—
306D4×Q8—611no—
307D4×Q8—1221no—
308#533—611no—
309#533—1221no—
310#533—1222no—
311#533—1221no—
312C4×#41—611no—
313C4×#41—1221no—
314C2×C4×Q8—1221no—
315C2×C4×Q8C2×D6111yescharacteristic
316#431—611no—
317#423—611no—
318C2×#104—1221no—
319C2×#103—21no—
320C2×C4×C8C2×D6111yescharacteristic
321C4×C4⋊C4C2×D6111yescharacteristic
322#340—21no—
323#339—1221no—
324#328—1222no—
325Q8⋊C8—1221no—
326Q8⋊C8—1222no—
327Q8⋊C8—1221no—
328Q8⋊C8—1221no—
329D4⋊C8—1221no—
330C61#843911yescharacteristic, complemented, Sylow
331C2×#41—611no—
332C22×Q8—1221no—
333C22×Q8C22⋊D6111yescharacteristic
334C2×M16—1221no—
335C22×C8—21no—
336C4⋊Q8—611no—
337C4⋊Q8—611no—
338C4⋊Q8—1221no—
339C4⋊Q8—1221no—
340C4⋊Q8—1222no—
341C4⋊Q8—1222no—
342C4⋊Q8—1221no—
343C4⋊Q8—1221no—
344#125—1221no—
345#125—1221no—
346#125—1222no—
347#125—1222no—
348#125—1222no—
349#125—1221no—
350C22⋊Q8—612no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C22⋊Q8—1222no—
354C22⋊Q8—1222no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358C22⋊Q8—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—612no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8—1222no—
369C4×Q8—1222no—
370C4×Q8—1222no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8—1221no—
374C4×Q8C4×D6111yescharacteristic
375C4×Q8C4×D6111yescharacteristic
376C4×Q8D24411yescharacteristic
377C4×Q8D24411yescharacteristic
378C4×D4—611no—
379C4×D4—611no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1221no—
385C4×D4—1221no—
386#117—611no—
387#117—1222no—
388#117—1221no—
389#117—1221no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1222no—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403#104—2442no—
404#103—24no—
405C22⋊C8—1221no—
406C4×C8—21no—
407C4×C8—21no—
408C4×C8—1221no—
409#95—21no—
410#41—612no—
411#41—612no—
412#41—1221no—
413#41—1221no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—41no—
417C2×Q8—611no—
418C2×Q8—612no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1221no—
422C2×Q8—2442no—
423C2×Q8—2442no—
424C2×Q8#1175819811yescharacteristic
425C2×Q8#1175819811yescharacteristic
426C2×D4—611no—
427C2×D4—1221no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439M16—2441no—
440M16—2441no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—2441no—
445C2×C8—2441no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—41no—
453C4⋊C4—611no—
454C4⋊C4—611no—
455C4⋊C4—612no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4#1175819811yescharacteristic
472C4⋊C4#1175819811yescharacteristic
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C22⋊C4—1221no—
476C22⋊C4—1222no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—41no—
484C42—41no—
485C42—1222no—
486C42—1222no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—1222no—
491C42C2×C4×D6111yescharacteristic
492C42C2×D24411yescharacteristic
493C42D4×D6111yescharacteristic
494C42D4×D6111yescharacteristic
495C23—1221no—
496C23#1451974511yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—42no—
501Q8—612no—
502Q8—1221no—
503Q8—4882no—
504Q8—2442no—
505Q8—2444no—
506Q8—2441no—
507Q8—2441no—
508Q8#1451970512yescomplemented
509D4—612no—
510D4—1222no—
511D4—1222no—
512D4—1221no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1222no—
532C2×C4—4881no—
533C2×C4—4882no—
534C2×C4—2444no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4C22⋊C4×D6111yescharacteristic
538C2×C4C22⋊C4×D6111yescharacteristic
539C2×C4C2×#1175819811yescharacteristic
540C2×C4#1451974511yescharacteristic
541C2×C4#1451974711yescharacteristic
542C2×C4#1451974711yescharacteristic
543C2×C4SD16×D6111yescharacteristic
544C2×C4Q16×D6111yescharacteristic
545C2×C4#1451980611yescharacteristic
546C2×C4#1451980911yescharacteristic
547C8—42no—
548C8—4881no—
549C8—4881no—
550C22—21no—
551C22—21no—
552C22—1222no—
553C22—2441no—
554C22—2441no—
555C22#1683961711yescharacteristic
556C22#1683989911yescharacteristic
557C22#1683991611yescharacteristic
558C4—21no—
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—1222no—
567C4—2442no—
568C4—2441no—
569C4—2441no—
570C4—2442no—
571C4#103×D6111yescharacteristic
572C4#1683976911yescharacteristic
573C4C2×#1451970511yescharacteristic
574C4#1683978011yescharacteristic
575C4#1683990211yescharacteristic
576C4#1683991911yescharacteristic
577C2—21no—
578C2—21no—
579C2—2442no—
580C2#2144575211yescharacteristic
581C2#2144576411yescharacteristic
582C2#2144578411yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×#103×C61C2
socleC22×C61#379
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenterC4×#103D61
hypocenterC61#8439
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #21444317C2semidirect
4. #21444275C2semidirect
5. #21444248C2semidirect
7. C4×#103×C61C2semidirect
8. Q8×Dic122C4semidirect
10. #16840490C4semidirect
12. #16840472C4semidirect
15. Q8×D244C4semidirect
209. Q8×C61#104semidirect
240. D244#103semidirect
254. Dic122#103semidirect
288. C4×#103D61semidirect
330. C61#8439semidirect
508. Q8#14519705semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582222
#214458402
#214443172
#214442752
#214442482
#214442022
C4×#103×C612
#843961