Finite Group Databasebeta

← #29582318

Subgroups — #29582318

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×#16839963C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic, complemented
4#21446625C211yesmaximal, characteristic, complemented
5C2×#16839494C211yesmaximal, characteristic, complemented
6#21443703C211yesmaximal, characteristic, complemented
7C2×C61×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×#11758190—21no—
9C22×#11758199—21no—
10C2×#14519774C2211yescharacteristic
11C2×#14519756—21no—
12C2×C4×Dic122C2211yescharacteristic
13#16839963—22no—
14#16839963—22no—
15#16839957—22no—
16#16839957—22no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519699—21no—
19C2×#14519699—22no—
20C2×#14519697—21no—
21#16839685—22no—
22#16839665—21no—
23#16839658—21no—
24#16839632C2211yescharacteristic
25#16839622—21no—
26#16839494—22no—
27C2×C4×D4×C61C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C2×C4×C8×C61C2211yescharacteristic
30C61×D4⋊C8—22no—
31C22×Dic122D411yescharacteristic
32C22×D244—21no—
33C22×D244—21no—
34C22×D244—21no—
35C2×#11758190—24no—
36C2×#11758190—42no—
37C2×#11758190—42no—
38C2×#11758199—24no—
39C2×#11758199—42no—
40C2×#11758199—42no—
41C22×#92418—21no—
42#14519774—21no—
43#14519774—21no—
44#14519774—21no—
45#14519774D412yes—
46#14519756—42no—
47#14519756—42no—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×C4⋊Dic61D412yes—
56C2×#11758196—22no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C4×Dic61—21no—
61C4⋊#92418—21no—
62C4⋊#92418D412yes—
63#14519699—24no—
64#14519699—44no—
65#14519697—24no—
66#14519684—22no—
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—21no—
69C22×C8×C61—21no—
70C22×C8×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80C4×C8×C61—21no—
81C4×C8×C61—21no—
82C24×C61—21no—
83C23×D61—41no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61SD1612yescomplemented
90C22×C4×C61—21no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—42no—
107C2×D244—42no—
108C2×D244—41no—
109C2×D244—41no—
110C2×D244—41no—
111C2×D244—41no—
112C2×D244—41no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244—41no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122D812yescomplemented
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61—41no—
128C2×C8×C61—41no—
129C2×C8×C61—41no—
130C2×C8×C61—41no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C22⋊Dic61—44no—
140C22⋊Dic61—82no—
141#11758199—48no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148C4⋊Dic61—42no—
149#11758196—44no—
150#11758196—41no—
151#11758196—41no—
152C4×Dic61—41no—
153C4×Dic61—41no—
154C2×#92418—22no—
155C2×#92418—41no—
156C2×#92418—41no—
157#11758190—48no—
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—
173D4×C61—22no—
174D4×C61—22no—
175D4×C61—41no—
176D4×C61—41no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—81no—
183C2×C4×C61—81no—
184C2×C4×C61—81no—
185C2×C4×C61—81no—
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12711yescharacteristic
192C2×C4×C61C2×D811yescharacteristic
193C2×C4×C61C2×SD1611yescharacteristic
194C2×C4×C61#13511yescharacteristic
195C2×C4×C61#13511yescharacteristic
196C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C61#13711yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209D244—44no—
210D244—42no—
211D244—42no—
212D244—42no—
213D244—42no—
214D244—82no—
215D244—81no—
216D244—81no—
217D244—81no—
218D244—81no—
219Dic122—22no—
220Dic122—22no—
221Dic122—41no—
222Dic122—41no—
223C8×C61—42no—
224C8×C61—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—81no—
251C4×C61#44711yescharacteristic
252C4×C61#44911yescharacteristic
253C4×C61#45011yescharacteristic
254C4×C61#45211yescharacteristic
255C4×C61#46511yescharacteristic
256C4×C61#46611yescharacteristic
257C4×C61#46811yescharacteristic
258C4×C61#46911yescharacteristic
259Dic61—42no—
260Dic61—82no—
261Dic61—81no—
262C2×#460—611no—
263C2×#459—611no—
264#2538—611no—
265#1957—611no—
266C2×Q8⋊C8—611no—
267C2×D4⋊C8D6111yescharacteristic, complemented
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—1221no—
282C2×C4×Q8—611no—
283C2×C4×D4C2×D6111yescharacteristic
284#460—1222no—
285#460—1222no—
286#459—1222no—
287#459—1222no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8C2×D6111yescharacteristic
290C2×#102—1221no—
291C2×#102—1221no—
292C2×#102—1222no—
293C2×C4×C8C2×D6111yescharacteristic
294#396—1222no—
295#390—1221no—
296#389—1221no—
297#384—611no—
298#380—1221no—
299Q8⋊C8—1222no—
300D4⋊C8—22no—
301C61#1022911yescharacteristic, complemented, Sylow
302C22×Q8—611no—
303C22×D4—1221no—
304C22×D4—1221no—
305C22×D4—1221no—
306C22×D4D24411yescharacteristic
307C23×C4—21no—
308C2×SD16—1224no—
309C2×SD16—2442no—
310C2×SD16—2442no—
311C2×D8—1224no—
312C2×D8—2442no—
313C2×D8—2442no—
314C22×C8—21no—
315C22×C8—21no—
316C22×C8—1221no—
317#127—612no—
318#127—1221no—
319#127—1221no—
320#127—1221no—
321C22⋊Q8—2442no—
322C22⋊Q8—2442no—
323C4×Q8—1221no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4D24411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C42—1221no—
342C2×C42C22×D6111yescharacteristic
343C4⋊C8—21no—
344C4⋊C8—612no—
345C4⋊C8—1221no—
346C4⋊C8C22⋊D6112yes—
347#102—1224no—
348#102—1224no—
349#102—2444no—
350C4×C8—21no—
351C4×C8—21no—
352#95—1222no—
353C24—21no—
354C24—2441no—
355C2×Q8—612no—
356C2×Q8—1221no—
357C2×Q8—1221no—
358C2×Q8—1221no—
359C2×Q8—1221no—
360C2×Q8—1221no—
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#1175819012yescomplemented
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—1222no—
393C22×C4C2×D24411yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4D4×D6111yescharacteristic
396SD16—2448no—
397D8—2448no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—1222no—
405C2×C8—2441no—
406C2×C8—2441no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—2442no—
412C4⋊C4—2444no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—4882no—
422C22⋊C4—2444no—
423C42—2441no—
424C42—2441no—
425C42#1175822211yescharacteristic
426C42D4×D6111yescharacteristic
427C42D4×D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
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#1451974711yescharacteristic
444Q8—1222no—
445Q8—1222no—
446Q8—2441no—
447Q8—2441no—
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—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—1222no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974711yescharacteristic
482C2×C4#1451975011yescharacteristic
483C2×C4#1451975011yescharacteristic
484C2×C4#1451975111yescharacteristic
485C2×C4#1451977311yescharacteristic
486C2×C4D8×D6111yescharacteristic
487C2×C4C2×#1175819011yescharacteristic
488C2×C4#1451979811yescharacteristic
489C2×C4#1451979911yescharacteristic
490C2×C4#1451980111yescharacteristic
491C2×C4#1451980411yescharacteristic
492C2×C4#1451980511yescharacteristic
493C2×C4#1451981011yescharacteristic
494C8—42no—
495C8—42no—
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#1683967111yescharacteristic
512C22#1683989711yescharacteristic
513C22#1683990311yescharacteristic
514C22#1683990811yescharacteristic
515C22#1683990911yescharacteristic
516C22#1683993111yescharacteristic
517C22#1683993611yescharacteristic
518C4—21no—
519C4—21no—
520C4—81no—
521C4—4882no—
522C4—4881no—
523C4—2442no—
524C4#1683989411yescharacteristic
525C4#1683990611yescharacteristic
526C4#1683991511yescharacteristic
527C4#1683992611yescharacteristic
528C4#1683999111yescharacteristic
529C4#1683999711yescharacteristic
530C4#1684000811yescharacteristic
531C4#1684001411yescharacteristic
532C2—82no—
533C2—9762no—
534C2#2144488711yescharacteristic
535C2#2144489311yescharacteristic
536C2#2144492511yescharacteristic
537C2#2144493111yescharacteristic
538C2#2144681611yescharacteristic
539C2#2144682411yescharacteristic
540C2#2144686211yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×D4⋊C8C2
socleC23×C61#120
prime radicalC2×#16839957C2
hypercenterC2×D4⋊C8D61
hypocenterC61#10229
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16839963C2semidirect
3. #21450741C2semidirect
4. #21446625C2semidirect
5. C2×#16839494C2semidirect
6. #21443703C2semidirect
7. C2×C61×D4⋊C8C2semidirect
89. C2×D4×C61SD16semidirect
124. C2×Dic122D8semidirect
267. C2×D4⋊C8D61semidirect
301. C61#10229semidirect
382. C2×D4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399632
C2×#168399572
#214507412
#214466252
C2×#168394942
#214437032
C2×C61×D4⋊C82
#1022961