Finite Group Databasebeta

← #29729612

Subgroups — #29729612

Order 15616.

934 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839958C211yesmaximal, characteristic
2#21450818C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic, complemented
4#21447192C212yesmaximal, complemented
5#21447130C212yesmaximal, complemented
6#21446695C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446583C211yesmaximal, characteristic, complemented
9#21446516C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C61×#446C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15C2×#14519753—21no—
16C2×C4×D244C2211yescharacteristic
17#16839958—24no—
18C2×#14519709C2212yes—
19C2×#14519699C2211yescharacteristic
20C2×#14519699C2211yescharacteristic
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24#16839684—22no—
25#16839679—22no—
26#16839664C2212yes—
27#16839664C2212yes—
28#16839664C2212yes—
29#16839663—21no—
30#16839660C2212yes—
31#16839632C2211yescharacteristic
32#16839619—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C61×#446—24no—
39C2×C4⋊C8×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#97×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×D488—24no—
49C2×D488—24no—
50C2×D488—42no—
51#14519774—22no—
52#14519774—22no—
53#14519766—44no—
54#14519753—44no—
55C4×D244—22no—
56C4×D244—22no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—22no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—21no—
72#14519709—42no—
73#14519699—22no—
74#14519699—24no—
75#14519699—22no—
76#14519699—24no—
77#14519699—42no—
78#14519684—22no—
79#14519684—21no—
80C22×Q8×C61D411yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C2×C4⋊C4×C61D412yes—
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C42×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—24no—
91C4⋊C8×C61D414yes—
92#97×C61—24no—
93C4×C8×C61—22no—
94C23×D61—21no—
95C23×D61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2411yescharacteristic
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—22no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244—81no—
124C2×D244—81no—
125C2×D244#4112yes—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C8×C61—21no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61C2×D414yes—
137C2×C8×C61#4112yes—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—24no—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61C2×D412yes—
143C42×C61#4112yes—
144#11758198—44no—
145#11758198—44no—
146#11758198—44no—
147#11758198—42no—
148#11758198—82no—
149#11758198—82no—
150#11758198—82no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—42no—
160C4⋊Dic61—81no—
161#11758196—44no—
162C4×Dic61—42no—
163D488—48no—
164C23×C61C22×D411yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—44no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61#12112yes—
189C2×C4×C61#12112yes—
190C2×C4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210D244—22no—
211D244—22no—
212D244—42no—
213D244—41no—
214D244—42no—
215D244—42no—
216D244—41no—
217D244—42no—
218D244—84no—
219D244—84no—
220C4×D61—84no—
221C8×C61—24no—
222C8×C61—22no—
223C8×C61—41no—
224#25300—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61#54811yescharacteristic
227C22×C61#55011yescharacteristic
228C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C61#57611yescharacteristic
231C22×C61#57711yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61#46914yes—
250C4×C61#52911yescharacteristic
251C4×C61#54211yescharacteristic
252C4×C61#54611yescharacteristic
253C4×C61#54811yescharacteristic
254Dic61—81no—
255Dic61—81no—
256Dic61—82no—
257Dic61—81no—
258C2×#459—611no—
259C2×#446D6111yescharacteristic, complemented
260#2548—611no—
261#2538—611no—
262#2065—612no—
263#2054—612no—
264#1972—612no—
265#1957—611no—
266#1948—611no—
267#1936—612no—
268C4×#102—611no—
269C2×C61#242611yescharacteristic
270C2×C61C2×#46911yescharacteristic
271C2×C61#306211yescharacteristic
272C2×C61#336611yescharacteristic
273C2×C61#337311yescharacteristic
274C2×C61#339711yescharacteristic
275C2×C61#339911yescharacteristic
276D61—82no—
277D61—162no—
278C22×D8—612no—
279C2×#127—611no—
280C2×#124—1221no—
281C2×C22⋊Q8—1221no—
282C2×C4×Q8C2×D6111yescharacteristic
283C2×C4×D4—611no—
284#459—1224no—
285#446—24no—
286C2×#106—612no—
287C2×C4⋊C8C2×D6112yes—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×#102—611no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#102—611no—
293C2×#102—612no—
294C2×#97C2×D6112yes—
295C2×C4×C8C2×D6111yescharacteristic
296#396—1222no—
297#394—1222no—
298#390—612no—
299#390—612no—
300#390—612no—
301#390—1221no—
302#389—612no—
303#384—611no—
304#380—1221no—
305C4×C4⋊C4—611no—
306#340—612no—
307#340—611no—
308#340—611no—
309C61#2530011yescharacteristic, complemented, Sylow
310C22×Q8D24411yescharacteristic
311C22×D4—611no—
312C22×D4—612no—
313C22×D4—1221no—
314C22×D4—1221no—
315C23×C4—1221no—
316C2×D8—1224no—
317C2×D8—1224no—
318C2×D8—2442no—
319C22×C8C22×D6111yescharacteristic
320C22×C8C22×D6111yescharacteristic
321C22×C8C22×D6112yes—
322#127—1222no—
323#127—1222no—
324#124—2444no—
325C22⋊Q8—2444no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—1222no—
329C4×D4—1222no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4D24411yescharacteristic
338C2×C4⋊C4D24412yes—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1222no—
346C2×C42—1221no—
347C2×C42C22×D6112yes—
348C2×C42C22×D6111yescharacteristic
349#106—2442no—
350C4⋊C8—24no—
351C4⋊C8C22⋊D6114yes—
352#102—1222no—
353#102—1222no—
354#102—1224no—
355#102—1224no—
356#102—2442no—
357#97—24no—
358C4×C8—22no—
359#95—1221no—
360#95—1222no—
361C24—1221no—
362C24—1222no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×D4—612no—
367C2×D4—1221no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—1221no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—1221no—
375C2×D4—1222no—
376C2×D4—1221no—
377C2×D4—1222no—
378C2×D4—4881no—
379C2×D4—4881no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4—2442no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4C2×D24412yes—
392C22×C4C2×D24412yes—
393C22×C4C2×D24411yescharacteristic
394C22×C4C2×D24411yescharacteristic
395C22×C4C23×D6111yescharacteristic
396D8—2448no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8#1175822612yes—
405C2×C8C2×C22⋊D6114yes—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—4881no—
417C4⋊C4—2444no—
418C4⋊C4—2442no—
419C22⋊C4—4882no—
420C22⋊C4—4882no—
421C22⋊C4—4882no—
422C22⋊C4—2444no—
423C22⋊C4—2444no—
424C22⋊C4—2444no—
425C22⋊C4—2442no—
426C42—22no—
427C42—22no—
428C42—2442no—
429C42#1175822212yes—
430C42C2×C22⋊D6112yes—
431C23—1222no—
432C23—2442no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23—2441no—
440C23—2441no—
441C23—2441no—
442C23—2441no—
443C23—2441no—
444C23—2441no—
445C23C22×D24411yescharacteristic
446Q8—44no—
447D4—1222no—
448D4—1222no—
449D4—4884no—
450D4—4884no—
451D4—2442no—
452D4—2442no—
453D4—2441no—
454D4—2442no—
455D4—2442no—
456D4—2441no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—4884no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4#1451975112yes—
479C2×C4#1451975112yes—
480C2×C4#1451980512yes—
481C2×C4#1451980512yes—
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982311yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—24no—
489C8—22no—
490C8—41no—
491C22—4882no—
492C22—4882no—
493C22—4882no—
494C22—4882no—
495C22—4882no—
496C22—4882no—
497C22—4882no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22—2442no—
503C22—2442no—
504C22—2442no—
505C22C2×#1451975111yescharacteristic
506C22#1684042411yescharacteristic
507C22#1684043411yescharacteristic
508C22C2×#1451980511yescharacteristic
509C22C2×#1451980511yescharacteristic
510C22#1684056711yescharacteristic
511C22#1684057411yescharacteristic
512C4—22no—
513C4—42no—
514C4—42no—
515C4—4881no—
516C4—4881no—
517C4—4881no—
518C4—4882no—
519C4#1684001514yes—
520C4#1684029911yescharacteristic
521C4#1684038411yescharacteristic
522C4#1684040011yescharacteristic
523C4#1684042311yescharacteristic
524C2—9762no—
525C2—4882no—
526C2#2144937911yescharacteristic
527C2C2×#1684001511yescharacteristic
528C2#2145549911yescharacteristic
529C2#2145852811yescharacteristic
530C2#2145861311yescharacteristic
531C2#2145884111yescharacteristic
532C2#2145886311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#446C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
hypercenterC2×#446D61
hypocenterC61#25300
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21450818C2semidirect
3. #21450737C2semidirect
4. #21447192C2semidirect
5. #21447130C2semidirect
6. #21446695C2semidirect
7. #21446625C2semidirect
8. #21446583C2semidirect
9. #21446516C2semidirect
10. C4×#14519699C2semidirect
11. C2×C61×#446C2semidirect
259. C2×#446D61semidirect
309. C61#25300semidirect

Maximal subgroups

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

subgroupindex
C2×#168399582
#214508182
#214507372
#214471922
#214471302
#214466952
#214466252
#214465832
#214465162
C4×#145196992
C2×C61×#4462
#2530061