Finite Group Databasebeta

← #29698290

Subgroups — #29698290

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839872C212yesmaximal
2C2×C4×#11758199C211yesmaximal, characteristic
3C4⋊C4×D244C211yesmaximal, characteristic, complemented
4#21446554C212yesmaximal, complemented
5#21446080C212yesmaximal, complemented
6#21446079C212yesmaximal, complemented
7#21445975C211yesmaximal, characteristic, complemented
8#21445819C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic, complemented
10#21445688C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×C4×D244C2212yes—
14C2×C4×D244C2211yescharacteristic
15C2×C4⋊C4×D61—21no—
16#16839872—28no—
17C4×#11758199C418yescomplemented
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2212yes—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519698C2211yescharacteristic
24C2×#14519697C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519688C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839660C2212yes—
29#16839636—22no—
30#16839632C2211yescharacteristic
31#16839619—21no—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4×D4×C61C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C22×C4⋊C4×C61—21no—
38C61×#389C2212yes—
39C61×#385—22no—
40C61×#384C2211yescharacteristic
41C61×#380—21no—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—22no—
46C2×#11758199C2×C414yes—
47C2×#11758199D414yes—
48C2×#11758199Q814yes—
49C22×#92418C2312yes—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52C4×D244—22no—
53C4×D244—24no—
54C4×D244—42no—
55C4×D244C2×C414yescomplemented
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C4⋊C4×D61—44no—
62C2×#11758198—21no—
63C2×#11758198—22no—
64#14519710C2×C414yescomplemented
65#14519708—24no—
66#14519708—42no—
67C4⋊#92418—22no—
68#14519698—24no—
69#14519698C2×C414yescomplemented
70#14519697—24no—
71#14519697C2×C414yescomplemented
72#14519688—24no—
73C4×#92418C2×C414yescomplemented
74#14519684—22no—
75C22×D4×C61C2311yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61—22no—
78C4×D4×C61—24no—
79C4×D4×C61—22no—
80C4×D4×C61—42no—
81C4×D4×C61C2×C414yescomplemented
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—44no—
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—22no—
92C2×C42×C61C2312yes—
93C2×C42×C61C2311yescharacteristic
94#95×C61—22no—
95#95×C61D412yes—
96C24×C61—21no—
97C23×D61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61C22×C412yes—
102C2×D4×C61C2×D412yes—
103C2×D4×C61C2×Q812yes—
104C22×C4×C61—21no—
105C22×C4×C61—24no—
106C22×C4×C61—22no—
107C22×C4×C61—44no—
108C22×C4×C61—44no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—82no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C22×C412yes—
123C2×D244C2×D412yes—
124C2×D244C2×Q812yes—
125C2×C4×D61—24no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61—82no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—44no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61—81no—
139C4⋊C4×C61C22×C412yes—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—44no—
142C42×C61—22no—
143C42×C61—22no—
144C42×C61C22×C412yes—
145C42×C61#4112yes—
146#11758199C4⋊C4116yescomplemented
147#11758198—24no—
148#11758198—44no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—42no—
153C4⋊Dic61—81no—
154C4⋊Dic61C22×C412yes—
155#11758196—44no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418C22×C412yes—
162C2×#92418C22×C412yes—
163C2×#92418C2×D412yes—
164C2×#92418C2×Q812yes—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61C22×D411yescharacteristic
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180D4×C61—22no—
181D4×C61—22no—
182D4×C61—22no—
183D4×C61C2×C4⋊C414yescomplemented
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—48no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—88no—
200C2×C4×C61C4×D412yes—
201C2×C4×C61C4×D412yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×Q811yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—22no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216D244—22no—
217D244—22no—
218D244—22no—
219D244C2×C4⋊C414yescomplemented
220C4×D61—48no—
221C4×D61—88no—
222#92418—22no—
223#92418—42no—
224#92418C2×C4⋊C414yescomplemented
225#22670—611nomaximal, Sylow
226C22×C61—24no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61C2×C4×D411yescharacteristic
234C22×C61D4211yescharacteristic
235C22×C61D4×Q811yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C61#57711yescharacteristic
239C22×C61#57711yescharacteristic
240C2×D61—24no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C4×C61—22no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—82no—
252C4×C61#44214yescomplemented
253C4×C61C22×C4⋊C411yescharacteristic
254C4×C61#52011yescharacteristic
255C4×C61#54711yescharacteristic
256C4×C61#55411yescharacteristic
257Dic61—42no—
258Dic61—82no—
259C2×#442—612no—
260C2×C4×D8—611no—
261D4×C4⋊C4—611no—
262D4×C4⋊C4D6111yescharacteristic, complemented
263#1943—612no—
264#1864—612no—
265#1864—612no—
266#1844—611no—
267#1802—611no—
268#1802—611no—
269#1770—611no—
270C2×C61—44no—
271C2×C61D4×C4⋊C411yescharacteristic
272C2×C61C2×#44211yescharacteristic
273C2×C61#295211yescharacteristic
274C2×C61#327211yescharacteristic
275C2×C61#327211yescharacteristic
276C2×C61#338411yescharacteristic
277C2×C61#338411yescharacteristic
278D61—44no—
279C22×D8—611no—
280C2×C4×D4—611no—
281C2×C4×D4—612no—
282C2×C4×D4C2×D6112yes—
283C2×C4×D4C2×D6111yescharacteristic
284C22×C4⋊C4—21no—
285C22×C4⋊C4—1221no—
286#442—1228no—
287C4×D8—618no—
288C2×#107—611no—
289C2×#106—612no—
290C2×#106—612no—
291C2×C4⋊C8—611no—
292C2×#102—611no—
293C2×#102—611no—
294C2×#102—612no—
295C2×#102—612no—
296C2×#97—612no—
297C2×C4×C8—611no—
298#389—612no—
299#389C2×D6112yes—
300#385—22no—
301#385—1222no—
302#384—611no—
303#384C2×D6111yescharacteristic
304#380—21no—
305#380—1221no—
306C4×C4⋊C4C2×D6111yescharacteristic
307#340—612no—
308#340—612no—
309#336—612no—
310C61#2267011yescharacteristic, complemented, Sylow
311C22×D4—611no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—1221no—
316C23×C4—1222no—
317C2×D8—614no—
318C2×D8—614no—
319C2×D8—614no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—612no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4—614no—
327C4×D4—1224no—
328C4×D4—1222no—
329C4×D4—2442no—
330C4×D4C4×D6114yescomplemented
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—2444no—
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1222no—
349C2×C42C22×D6112yes—
350C2×C42C22×D6111yescharacteristic
351#107—614no—
352#106—1224no—
353#106—2442no—
354C4⋊C8—1222no—
355#102—614no—
356#102—614no—
357#102—1224no—
358#102—1224no—
359#97—1224no—
360C4×C8—614no—
361#95—22no—
362#95—1222no—
363#95C22⋊D6112yes—
364C24—21no—
365C24—1221no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—612no—
370C2×D4—612no—
371C2×D4—612no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4C2×C4×D6112yes—
376C2×D4D4×D6112yes—
377C2×D4Q8×D6112yes—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—1221no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—4882no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4C2×C22⋊D6112yes—
395C22×C4C2×C22⋊D6112yes—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C23×D6111yescharacteristic
399D8—6116no—
400C2×C8—612no—
401C2×C8—612no—
402C2×C8—612no—
403C2×C8—612no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—612no—
418C4⋊C4—1221no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C4⋊C4—4881no—
422C4⋊C4—2444no—
423C4⋊C4—2442no—
424C4⋊C4C2×C4×D6112yes—
425C22⋊C4—24no—
426C22⋊C4—44no—
427C22⋊C4—1224no—
428C22⋊C4—2444no—
429C42—22no—
430C42—22no—
431C42C2×C4×D6112yes—
432C42#1175822212yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—1222no—
441C23—1222no—
442C23—1222no—
443C23—2441no—
444C23—2441no—
445C23—2441no—
446C23—2441no—
447C23C22×C22⋊D6111yescharacteristic
448D4—22no—
449D4—22no—
450D4—22no—
451D4—614no—
452D4—1222no—
453D4—1222no—
454D4—1222no—
455D4C4⋊C4×D6114yescomplemented
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—88no—
472C2×C4—1222no—
473C2×C4—4888no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2448no—
480C2×C4C4×C22⋊D6112yes—
481C2×C4C4×C22⋊D6112yes—
482C2×C4#1451980212yes—
483C2×C4#1451980312yes—
484C2×C4C22×C4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×Q8×D6111yescharacteristic
487C2×C4C2×#1175822211yescharacteristic
488C2×C4#1451982211yescharacteristic
489C2×C4#1451982511yescharacteristic
490C8—614no—
491C8—1222no—
492C8—2442no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—1224no—
501C22—2442no—
502C22—2442no—
503C22—2442no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22C2×C4×C22⋊D6111yescharacteristic
508C22D4×C22⋊D6111yescharacteristic
509C22Q8×C22⋊D6111yescharacteristic
510C22C2×#1451980211yescharacteristic
511C22C2×#1451980311yescharacteristic
512C22#1684057011yescharacteristic
513C22#1684057311yescharacteristic
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—4882no—
520C4—2442no—
521C4#1683987214yescomplemented
522C4C2×C4⋊C4×D6111yescharacteristic
523C4#1684024311yescharacteristic
524C4#1684041311yescharacteristic
525C4#1684046711yescharacteristic
526C2—44no—
527C2—2444no—
528C2C4⋊C4×C22⋊D6111yescharacteristic
529C2C2×#1683987211yescharacteristic
530C2#2145470011yescharacteristic
531C2#2145741511yescharacteristic
532C2#2145741911yescharacteristic
533C2#2145871411yescharacteristic
534C2#2145871811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenterD4×C4⋊C4D61
hypocenterC61#22670
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×D244C2semidirect
4. #21446554C2semidirect
5. #21446080C2semidirect
6. #21446079C2semidirect
7. #21445975C2semidirect
8. #21445819C2semidirect
9. #21445816C2semidirect
10. #21445688C2semidirect
11. D4×C4⋊C4×C61C2semidirect
17. C4×#11758199C4semidirect
55. C4×D244C2×C4semidirect
64. #14519710C2×C4semidirect
69. #14519698C2×C4semidirect
71. #14519697C2×C4semidirect
73. C4×#92418C2×C4semidirect
81. C4×D4×C61C2×C4semidirect
146. #11758199C4⋊C4semidirect
183. D4×C61C2×C4⋊C4semidirect
219. D244C2×C4⋊C4semidirect
224. #92418C2×C4⋊C4semidirect
252. C4×C61#442semidirect
262. D4×C4⋊C4D61semidirect
310. C61#22670semidirect
330. C4×D4C4×D61semidirect
455. D4C4⋊C4×D61semidirect
521. C4#16839872semidirect

Maximal subgroups

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

subgroupindex
C2×#168398722
C2×C4×#117581992
C4⋊C4×D2442
#214465542
#214460802
#214460792
#214459752
#214458192
#214458162
#214456882
D4×C4⋊C4×C612
#2267061