Finite Group Databasebeta

← #29582316

Subgroups — #29582316

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3#21446624C211yesmaximal, characteristic, complemented
4C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
5C2×D4⋊#92418C211yesmaximal, characteristic, complemented
6#21443702C211yesmaximal, characteristic, complemented
7C61×#2538C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C22×#11758199—21no—
10C2×#14519774C2211yescharacteristic
11#16839962—22no—
12#16839962—22no—
13#16839957—22no—
14#16839957—22no—
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519697—21no—
18C2×#14519697—21no—
19C2×#14519697—22no—
20C2×C4×#92418C2211yescharacteristic
21Q8⋊#92418—22no—
22D4⋊#92418—22no—
23C2×C22⋊Q8×C61—21no—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×D4×C61C2211yescharacteristic
26C61×#396—22no—
27C61×#390—21no—
28C61×#389—21no—
29C61×#384C2211yescharacteristic
30C61×#380—21no—
31C22×D244—21no—
32C22×D244—21no—
33C22×D244—21no—
34C2×#11758201—24no—
35C2×#11758201—42no—
36C2×#11758201—42no—
37C2×#11758199—24no—
38C2×#11758199—42no—
39C2×#11758199—42no—
40C22×#92418—21no—
41C22×#92418—21no—
42C22×#92418—21no—
43#14519774—21no—
44#14519774—21no—
45#14519774—21no—
46#14519774D412yes—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418D412yes—
50C4⋊#92418D412yes—
51#14519697—24no—
52#14519697—24no—
53#14519697—44no—
54C4×#92418—21no—
55C4×#92418—21no—
56C22×Q8×C61D411yescharacteristic
57C22×D4×C61D411yescharacteristic
58C23×C4×C61—21no—
59C22⋊Q8×C61—42no—
60C22⋊Q8×C61—42no—
61C4×Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C4⋊C4×C61D412yes—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—22no—
79C2×C42×C61—21no—
80C2×C42×C61C2311yescharacteristic
81#95×C61—22no—
82C24×C61—21no—
83C23×D61—41no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61D812yescomplemented
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61SD1612yescomplemented
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—42no—
113C2×D244—42no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244—41no—
119C2×D244—41no—
120C2×D244—41no—
121C2×D244—41no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—44no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—44no—
138C22⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—82no—
141C42×C61—41no—
142C42×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147#11758201—48no—
148#11758199—48no—
149C2×#92418—22no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—41no—
153C2×#92418—41no—
154C2×#92418—41no—
155C2×#92418—41no—
156C2×#92418—41no—
157C2×#92418—41no—
158C23×C61—22no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61#12011yescharacteristic
166C22×D61—42no—
167C22×D61—81no—
168C22×D61—81no—
169C22×D61—81no—
170C22×D61—81no—
171C22×D61—81no—
172C22×D61—81no—
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—41no—
176Q8×C61—41no—
177D4×C61—22no—
178D4×C61—22no—
179D4×C61—41no—
180D4×C61—41no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—81no—
196C2×C4×C61—81no—
197C2×C4×C61—81no—
198C2×C4×C61—81no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12711yescharacteristic
205C2×C4×C61C2×D811yescharacteristic
206C2×C4×C61C2×SD1611yescharacteristic
207C2×C4×C61#13511yescharacteristic
208C2×C4×C61#13511yescharacteristic
209C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
210C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C61#13711yescharacteristic
213D244—44no—
214D244—42no—
215D244—42no—
216D244—42no—
217D244—42no—
218D244—82no—
219D244—81no—
220D244—81no—
221D244—81no—
222D244—81no—
223#92418—42no—
224#92418—42no—
225#92418—42no—
226#10229—611nomaximal, Sylow
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#39211yescharacteristic
235C22×C61#44711yescharacteristic
236C22×C61#44811yescharacteristic
237C22×C61#44911yescharacteristic
238C22×C61#44911yescharacteristic
239C22×C61#45311yescharacteristic
240C22×C61#45411yescharacteristic
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—161no—
245C2×D61—161no—
246C2×D61—161no—
247C2×D61—161no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—42no—
251C4×C61—82no—
252C4×C61—81no—
253C4×C61—81no—
254C4×C61#44711yescharacteristic
255C4×C61#44911yescharacteristic
256C4×C61#45011yescharacteristic
257C4×C61#45211yescharacteristic
258C4×C61#46511yescharacteristic
259C4×C61#46611yescharacteristic
260C4×C61#46811yescharacteristic
261C4×C61#46911yescharacteristic
262C2×#460—611no—
263C2×#459—611no—
264#2538D6111yescharacteristic, complemented
265#1957—611no—
266C2×Q8⋊C8—611no—
267C2×D4⋊C8—611no—
268#1296—611no—
269C2×C61—82no—
270C2×C61#162111yescharacteristic
271C2×C61#162211yescharacteristic
272C2×C61#162711yescharacteristic
273C2×C61#162811yescharacteristic
274C2×C61#199911yescharacteristic
275C2×C61#200011yescharacteristic
276C2×C61#200611yescharacteristic
277D61—162no—
278C22×SD16—1221no—
279C22×D8—1221no—
280C2×#127—611no—
281C2×C22⋊Q8—21no—
282C2×C4×Q8C2×D6111yescharacteristic
283C2×C4×D4C2×D6111yescharacteristic
284#460—1222no—
285#460—1222no—
286#459—1222no—
287#459—1222no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—611no—
290C2×#102—1221no—
291C2×#102—1221no—
292C2×#102—1222no—
293C2×C4×C8—611no—
294#396—22no—
295#390—21no—
296#389—21no—
297#384C2×D6111yescharacteristic
298#380—21no—
299Q8⋊C8—1222no—
300D4⋊C8—1222no—
301C61#1022911yescharacteristic, complemented, Sylow
302C22×Q8C22⋊D6111yescharacteristic
303C22×D4—1221no—
304C22×D4—1221no—
305C22×D4—1221no—
306C22×D4C22⋊D6111yescharacteristic
307C23×C4—21no—
308C2×SD16—1224no—
309C2×SD16—2442no—
310C2×SD16—2442no—
311C2×D8—1224no—
312C2×D8—2442no—
313C2×D8—2442no—
314C22×C8—1221no—
315C22×C8—1221no—
316C22×C8—1221no—
317#127—612no—
318#127—1221no—
319#127—1221no—
320#127—1221no—
321C22⋊Q8—42no—
322C22⋊Q8—42no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C4⋊C4C22⋊D6111yescharacteristic
337C2×C4⋊C4C22⋊D6112yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C42—21no—
342C2×C42C22×D6111yescharacteristic
343C4⋊C8—612no—
344C4⋊C8—612no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347#102—1224no—
348#102—1224no—
349#102—2444no—
350C4×C8—1221no—
351C4×C8—1221no—
352#95—22no—
353C24—21no—
354C24—2441no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#1175819912yescomplemented
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—2442no—
370C2×D4—2442no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4—2441no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C2×D4—2441no—
380C2×D4—2441no—
381C2×D4—2441no—
382C2×D4#1175820112yescomplemented
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396SD16—2448no—
397D8—2448no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—2441no—
402C2×C8—2441no—
403C2×C8—2441no—
404C2×C8—2441no—
405C2×C8—2441no—
406C2×C8—2441no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C22⋊C4—44no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—82no—
423C42—41no—
424C42—41no—
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42D4×D6111yescharacteristic
428C42#1175822411yescharacteristic
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—4881no—
437C23—4881no—
438C23—4881no—
439C23—4881no—
440C23—4881no—
441C23—4881no—
442C23—2442no—
443C23#1451974611yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—4882no—
453D4—4881no—
454D4—4881no—
455D4—4881no—
456D4—4881no—
457D4—2444no—
458D4—2442no—
459D4—2442no—
460D4—2442no—
461D4—2442no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974511yescharacteristic
482C2×C4#1451974811yescharacteristic
483C2×C4#1451974911yescharacteristic
484C2×C4#1451974911yescharacteristic
485C2×C4#1451977311yescharacteristic
486C2×C4C2×#1175819911yescharacteristic
487C2×C4C2×#1175820111yescharacteristic
488C2×C4#1451979711yescharacteristic
489C2×C4#1451979711yescharacteristic
490C2×C4#1451980211yescharacteristic
491C2×C4#1451980211yescharacteristic
492C2×C4#1451980311yescharacteristic
493C2×C4#1451980811yescharacteristic
494C8—2442no—
495C8—2442no—
496C8—2442no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—9761no—
505C22—9761no—
506C22—9761no—
507C22—9761no—
508C22—4882no—
509C22—4882no—
510C22—4882no—
511C22#1683967011yescharacteristic
512C22#1683989611yescharacteristic
513C22#1683990111yescharacteristic
514C22#1683990711yescharacteristic
515C22#1683990711yescharacteristic
516C22#1683993011yescharacteristic
517C22#1683993511yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4#1683989511yescharacteristic
525C4#1683990511yescharacteristic
526C4#1683991211yescharacteristic
527C4#1683992211yescharacteristic
528C4#1683999311yescharacteristic
529C4#1683999611yescharacteristic
530C4#1684001011yescharacteristic
531C4#1684001311yescharacteristic
532C2—82no—
533C2—9762no—
534C2#2144488411yescharacteristic
535C2#2144489111yescharacteristic
536C2#2144492211yescharacteristic
537C2#2144492911yescharacteristic
538C2#2144681511yescharacteristic
539C2#2144682011yescharacteristic
540C2#2144686111yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2538C2
socleC23×C61#120
prime radicalC2×#16839957C2
hypercenter#2538D61
hypocenterC61#10229
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16839962C2semidirect
3. #21446624C2semidirect
4. C2×Q8⋊#92418C2semidirect
5. C2×D4⋊#92418C2semidirect
6. #21443702C2semidirect
7. C61×#2538C2semidirect
89. C2×Q8×C61D8semidirect
95. C2×D4×C61SD16semidirect
264. #2538D61semidirect
301. C61#10229semidirect
360. C2×Q8#11758199semidirect
382. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
C2×#168399572
#214466242
C2×Q8⋊#924182
C2×D4⋊#924182
#214437022
C61×#25382
#1022961