Finite Group Databasebeta

← #29658602

Subgroups — #29658602

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic, complemented
2#21455850C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21445082C214yesmaximal, complemented
5#21444922C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840158C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—22no—
15#16840080—21no—
16#16840049—22no—
17#16840047C2212yescomplemented
18#16839977—22no—
19#16839967C2212yes—
20#16839962C2212yescomplemented
21#16839957C2212yes—
22#16839901—21no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839508C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C23×Q8×C61—21no—
32D4×Q8×C61—22no—
33D4×Q8×C61—22no—
34D4×Q8×C61—22no—
35D4×Q8×C61C2214yes—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C22⋊Q8×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519803—44no—
45C2×#11758200—22no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519708—22no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—22no—
64#14519697—22no—
65#14519697—22no—
66#14519697—24no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61—22no—
72C22×Q8×C61—24no—
73C22×Q8×C61—24no—
74C22×Q8×C61—21no—
75C22×Q8×C61—48no—
76C22×Q8×C61D411yescharacteristic
77C22×D4×C61D411yescharacteristic
78C23×C4×C61—22no—
79C23×C4×C61—21no—
80C4⋊Q8×C61—24no—
81C4⋊Q8×C61—24no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61D412yes—
85C4⋊Q8×C61C2312yes—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—28no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—28no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61D412yes—
93C4×Q8×C61C2312yes—
94C4×D4×C61—24no—
95C4×D4×C61—24no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C24×C61—21no—
111C2×Q8×C61—24no—
112C2×Q8×C61—28no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—48no—
117C2×Q8×C61—44no—
118C2×Q8×C61—416no—
119C2×Q8×C61—48no—
120C2×Q8×C61—48no—
121C2×Q8×C61C2×D412yes—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61SD1614yescomplemented
125C2×D4×C61C2×D412yes—
126C22×C4×C61—21no—
127C22×C4×C61—28no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—48no—
134C22×C4×C61—44no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C2×C22⋊D61—41no—
139#11758222—42no—
140#11758222—42no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×C4×D61—41no—
145C2×Dic122—21no—
146C4⋊C4×C61—28no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61C2×D414yes—
157C22⋊C4×C61—28no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C42×C61—22no—
161C42×C61—22no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2411yescharacteristic
166#11758201—44no—
167#11758200—44no—
168#11758199—44no—
169#11758198—41no—
170#11758198—41no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173#11758194—42no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61#12011yescharacteristic
182C22×D61—41no—
183Q8×C61—24no—
184Q8×C61—22no—
185Q8×C61—416no—
186Q8×C61—44no—
187Q8×C61—44no—
188Q8×C61AGL(1,ℤ/8)14yescomplemented
189D4×C61—22no—
190D4×C61—24no—
191D4×C61C2×SD1614yes—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—416no—
200C2×C4×C61—44no—
201C2×C4×C61—42no—
202C2×C4×C61—44no—
203C2×C4×C61—42no—
204C2×C4×C61—44no—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61C2×SD1612yes—
209C2×C4×C61#13712yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C22⋊D61—82no—
214C2×Dic61—41no—
215D244—42no—
216D244—41no—
217D244—42no—
218D244—41no—
219C4×D61—82no—
220Dic122—41no—
221Dic122—41no—
222#92418—42no—
223#92418—42no—
224#92418—42no—
225#18758—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—24no—
229C22×C61—48no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61#44812yes—
233C22×C61C2×#12011yescharacteristic
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C4×C61—24no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—44no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61#44812yes—
245C4×C61C22×SD1611yescharacteristic
246C4×C61C2×AGL(1,ℤ/8)12yes—
247C4×C61C2×#13711yescharacteristic
248Dic61—81no—
249C2×D4×Q8D6111yescharacteristic, complemented
250#3227—611no—
251#3103—611no—
252#3036—611no—
253#1651—614no—
254#1627—614no—
255#1549—611no—
256#1487—611no—
257#1478—611no—
258C2×C61—22no—
259C2×C61—44no—
260C2×C61C2×#44811yescharacteristic
261C2×C61#300311yescharacteristic
262C2×C61#302011yescharacteristic
263D61—161no—
264C23×Q8—21no—
265C2×AGL(1,ℤ/8)—1221no—
266D4×Q8—22no—
267D4×Q8—22no—
268D4×Q8—22no—
269D4×Q8C2×D6114yes—
270#532—611no—
271C2×C4⋊Q8—21no—
272C2×C4⋊Q8C2×D6111yescharacteristic
273C2×C22⋊Q8—21no—
274C2×C22⋊Q8—22no—
275C2×C4×Q8C2×D6111yescharacteristic
276C2×C4×D4—21no—
277C2×C4×D4C2×D6111yescharacteristic
278#498—612no—
279#486—612no—
280#481—1221no—
281#481—1222no—
282#476—1222no—
283#475—612no—
284#463—1222no—
285#461—612no—
286#460—612no—
287#459—612no—
288#448—1221no—
289C4×SD16—1222no—
290#431—611no—
291#431—611no—
292#423—611no—
293#417—1221no—
294#332—614no—
295Q8⋊C8—614no—
296D4⋊C8—614no—
297C61#1875811yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×Q8—22no—
300C22×Q8—24no—
301C22×Q8—24no—
302C22×Q8—21no—
303C22×Q8—48no—
304C22×Q8C22⋊D6111yescharacteristic
305C22×D4C22⋊D6111yescharacteristic
306C23×C4—22no—
307C23×C4—21no—
308AGL(1,ℤ/8)—2444no—
309C2×SD16—1222no—
310C2×SD16—1222no—
311C2×D8—1222no—
312C2×M16—1221no—
313C4⋊Q8—24no—
314C4⋊Q8—24no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—611no—
318C4⋊Q8C22⋊D6112yes—
319C4⋊Q8C22×D6112yes—
320#127—611no—
321#124—1221no—
322C22⋊Q8—24no—
323C22⋊Q8—28no—
324C22⋊Q8—24no—
325C22⋊Q8—28no—
326#121—1221no—
327#121—1221no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8C22⋊D6112yes—
331C4×Q8C22×D6112yes—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4C22⋊D6112yes—
339C4×D4C22×D6112yes—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22⋊D6112yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C42—21no—
349C2×C42C22×D6111yescharacteristic
350#106—1222no—
351#106—1224no—
352C4⋊C8—612no—
353C4⋊C8—612no—
354C4⋊C8—1221no—
355C4⋊C8—1221no—
356#103—1222no—
357#103—1222no—
358#102—1222no—
359#102—1222no—
360#102—1224no—
361C22⋊C8—1221no—
362C22⋊C8—1221no—
363C4×C8—612no—
364C4×C8—1221no—
365C24—21no—
366#41—2442no—
367#41—2442no—
368C2×Q8—24no—
369C2×Q8—28no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—1221no—
379C2×Q8C2×C22⋊D6112yes—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—2441no—
386C2×D4#1175820114yescomplemented
387C2×D4C2×C22⋊D6112yes—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—2441no—
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401SD16—2444no—
402SD16—2444no—
403D8—2444no—
404M16—2442no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4C2×C22⋊D6112yes—
419C4⋊C4C2×C22⋊D6112yes—
420C4⋊C4C2×C22⋊D6114yes—
421C22⋊C4—28no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C42—22no—
427C42—22no—
428C42C2×C22⋊D6111yescharacteristic
429C42C2×C22⋊D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C42C23×D6111yescharacteristic
432C23—22no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—2441no—
437C23#1451974611yescharacteristic
438Q8—24no—
439Q8—22no—
440Q8—416no—
441Q8—44no—
442Q8—44no—
443Q8—2441no—
444Q8—2441no—
445Q8#1451980314yescomplemented
446D4—22no—
447D4—24no—
448D4—4882no—
449D4—2442no—
450D4—2441no—
451D4—2442no—
452D4—2441no—
453D4C2×#1175820114yes—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—416no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—4882no—
468C2×C4—2441no—
469C2×C4#1451974611yescharacteristic
470C2×C4#1451974611yescharacteristic
471C2×C4#1451974611yescharacteristic
472C2×C4C2×#1175820112yes—
473C2×C4#1451980812yes—
474C2×C4C22×C22⋊D6111yescharacteristic
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C8—2442no—
478C8—2442no—
479C8—2442no—
480C22—21no—
481C22—21no—
482C22—24no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22#1683990112yes—
490C22C2×#1451974611yescharacteristic
491C4—21no—
492C4—21no—
493C4—24no—
494C4—21no—
495C4—42no—
496C4—44no—
497C4—42no—
498C4—4881no—
499C4#1683990112yes—
500C4C22×#1175820111yescharacteristic
501C4C2×#1451980312yes—
502C4C2×#1451980811yescharacteristic
503C2—22no—
504C2—44no—
505C2—9761no—
506C2C2×#1683990111yescharacteristic
507C2#2145504611yescharacteristic
508C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×D4×Q8×C61C2
socleC22×C61C2×#120
prime radical#21455283C2
hypercenterC2×D4×Q8D61
hypocenterC61#18758
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
2. #21455850C2semidirect
4. #21445082C2semidirect
5. #21444922C2semidirect
6. #21444504C2semidirect
7. #21444248C2semidirect
8. #21444210C2semidirect
9. C2×D4×Q8×C61C2semidirect
12. #16840158C22semidirect
13. #16840110C22semidirect
17. #16840047C22semidirect
20. #16839962C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
124. C2×D4×C61SD16semidirect
188. Q8×C61AGL(1,ℤ/8)semidirect
249. C2×D4×Q8D61semidirect
297. C61#18758semidirect
386. C2×D4#11758201semidirect
445. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214558502
#214552832
#214450822
#214449222
#214445042
#214442482
#214442102
C2×D4×Q8×C612
#1875861