Finite Group Databasebeta

← #29737026

Subgroups — #29737026

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16840131C211yesmaximal, characteristic, complemented
3C2×#16839968C211yesmaximal, characteristic
4#21451031C211yesmaximal, characteristic, complemented
5#21446971C212yesmaximal, complemented
6#21446961C212yesmaximal, complemented
7#21446774C211yesmaximal, characteristic, complemented
8#21446152C212yesmaximal
9#21446152C212yesmaximal
10#21445645C211yesmaximal, characteristic
11C61×#1985C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519755—21no—
21C2×#14519753—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26#16840131—24no—
27#16839968—24no—
28C2×#14519702C2212yes—
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519699C2212yes—
32#16839701—21no—
33#16839701—21no—
34#16839669—21no—
35#16839668C2212yes—
36#16839668C2212yes—
37#16839627—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839525C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C61×#336C2212yes—
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2312yes—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C2×#11758190—24no—
54C2×#11758190—28no—
55#14519777—22no—
56#14519777—22no—
57#14519777—22no—
58#14519777—22no—
59#14519769—22no—
60#14519769—22no—
61#14519769—22no—
62#14519769D414yes—
63#14519755—28no—
64#14519755—28no—
65#14519753—28no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244D414yes—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—28no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88#14519702—24no—
89#14519702—24no—
90#14519702—24no—
91#14519699—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61D411yescharacteristic
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C4×Q8×C61—22no—
101C4×Q8×C61—22no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#97×C61—22no—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C2×D244—21no—
127C2×D244—22no—
128C2×D244C2×D412yes—
129C2×D244#4114yes—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—24no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61—82no—
136C2×C4×D61—82no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—24no—
140C2×Dic122—24no—
141C2×Dic122—24no—
142C2×Dic122—82no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—24no—
147C2×C8×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61—21no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61—42no—
168C4⋊Dic61C2×D412yes—
169C4⋊Dic61C2×D412yes—
170C4⋊Dic61C2×D412yes—
171#11758196—24no—
172#11758196—24no—
173#11758196—24no—
174#11758196—24no—
175C4×Dic61—42no—
176#11758190—48no—
177C23×C61C22×D411yescharacteristic
178C22×D61—22no—
179C22×D61—24no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—42no—
183Q8×C61—44no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61#12012yes—
197C2×C4×C61#12012yes—
198C2×C4×C61#13712yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4112yes—
202C2×C4×C612−1+412yes—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—44no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215D244—22no—
216D244—24no—
217D244#12414yes—
218C4×D61—48no—
219C4×D61—84no—
220C4×D61—84no—
221Dic122—42no—
222Dic122—42no—
223Dic122—44no—
224Dic122—84no—
225C8×C61—42no—
226C8×C61—42no—
227#25945—611nomaximal, Sylow
228C22×C61C2×#12011yescharacteristic
229C22×C61#54812yes—
230C22×C61C2×#13711yescharacteristic
231C22×C61#57511yescharacteristic
232C22×C61#57712yes—
233C2×D61—24no—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—42no—
237C2×D61—42no—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—41no—
243C4×C61—41no—
244C4×C61C2×#12411yescharacteristic
245C4×C61#53611yescharacteristic
246C4×C61#54812yes—
247Dic61—42no—
248Dic61—42no—
249Dic61—42no—
250Dic61—82no—
251Dic61—82no—
252C2×#554—611no—
253C2×#491—611no—
254C2×#461—611no—
255#2568—611no—
256#2026—612no—
257#2024—612no—
258#1989—611no—
259#1985D6111yescharacteristic, complemented
260#1877—612no—
261#1877—612no—
262#1762—611no—
263C2×C61#243611yescharacteristic
264C2×C61#300711yescharacteristic
265C2×C61#300911yescharacteristic
266C2×C61#337212yes—
267C2×C61#337612yes—
268D61—44no—
269C22×SD16—612no—
270#554—1224no—
271#554—1224no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—611no—
274C2×#125—611no—
275C2×#125—611no—
276C2×C22⋊Q8—1221no—
277C2×C22⋊Q8—1221no—
278C2×C22⋊Q8—1221no—
279C2×C4×Q8C2×D6111yescharacteristic
280C2×C4×D4—611no—
281C2×C4×D4—611no—
282C2×C4×D4—611no—
283C22×C4⋊C4—1221no—
284#491—1224no—
285#461—1224no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#103—612no—
288C2×#103—612no—
289C2×#103—612no—
290C2×#102—612no—
291C2×#97C2×D6111yescharacteristic
292#400—1221no—
293#400—1221no—
294#391—612no—
295#391—612no—
296#391—1221no—
297#382—1221no—
298C4×C4⋊C4—611no—
299#340—612no—
300#340—612no—
301#340—612no—
302#336C2×D6112yes—
303#336C2×D6112yes—
304C61#2594511yescharacteristic, complemented, Sylow
305C22×Q8—612no—
306C22×Q8—1221no—
307C22×Q8D24411yescharacteristic
308C22×D4—611no—
309C23×C4—1221no—
310C23×C4—1221no—
311C23×C4—1221no—
312C2×SD16—1224no—
313C2×SD16—1228no—
314C22×C8C22×D6112yes—
315C22×C8C22×D6112yes—
316C4⋊Q8—1222no—
317C4⋊Q8—1222no—
318C4⋊Q8—1222no—
319C4⋊Q8—1222no—
320#125—614no—
321#125—1222no—
322#125—1222no—
323#125—1222no—
324C22⋊Q8—1228no—
325C22⋊Q8—1228no—
326C22⋊Q8—1228no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×D4—614no—
330C4×D4—1222no—
331C4×D4—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C4×D4—1222no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1228no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4D24411yescharacteristic
348C2×C4⋊C4D24411yescharacteristic
349C2×C4⋊C4D24411yescharacteristic
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355C2×C42C22×D6111yescharacteristic
356C2×C42C22×D6111yescharacteristic
357C4⋊C8—22no—
358#103—1224no—
359#103—1224no—
360#103—1224no—
361#102—1224no—
362#97—22no—
363#95—1221no—
364#95—1222no—
365#95—1222no—
366#95—1222no—
367#95—1222no—
368C24—1221no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—1224no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1224no—
376C2×Q8—1224no—
377C2×Q8—4882no—
378C2×D4—612no—
379C2×D4—614no—
380C2×D4—1221no—
381C2×D4—1222no—
382C22×C4—1224no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—4882no—
391C22×C4—4882no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4C2×D24411yescharacteristic
395C22×C4C2×D24411yescharacteristic
396C22×C4C2×D24411yescharacteristic
397C22×C4C2×D24411yescharacteristic
398C22×C4C2×D24411yescharacteristic
399C22×C4C2×D24411yescharacteristic
400C22×C4C23×D6111yescharacteristic
401SD16—2448no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—612no—
411C4⋊C4—612no—
412C4⋊C4—612no—
413C4⋊C4—1224no—
414C4⋊C4—1224no—
415C4⋊C4—1224no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2442no—
424C4⋊C4—2442no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—2442no—
433C42D4×D6112yes—
434C42D4×D6112yes—
435C23—1222no—
436C23—1224no—
437C23—2441no—
438C23—2441no—
439C23—2442no—
440C23C22×D24411yescharacteristic
441Q8—44no—
442Q8—4884no—
443Q8—2442no—
444Q8—2442no—
445Q8—2444no—
446D4—614no—
447D4—1222no—
448D4—1224no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—4884no—
465C2×C4—4884no—
466C2×C4—2448no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2444no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4#1451974712yes—
477C2×C4#1451974712yes—
478C2×C4#1451981012yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#41×D6112yes—
482C2×C4#1451982512yes—
483C8—42no—
484C8—42no—
485C22—1224no—
486C22—2444no—
487C22—2444no—
488C22—2442no—
489C22—2442no—
490C22C2×#1451974711yescharacteristic
491C22#1684042412yes—
492C22C2×#1451981011yescharacteristic
493C22#1684056211yescharacteristic
494C22#1684057412yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—4882no—
502C4—4882no—
503C4—2442no—
504C4—2442no—
505C4—2442no—
506C4#124×D6111yescharacteristic
507C4#1684033711yescharacteristic
508C4#1684042212yes—
509C2—2444no—
510C2#2144947611yescharacteristic
511C2#2145508111yescharacteristic
512C2#2145509811yescharacteristic
513C2#2145860212yes—
514C2#2145864812yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1985C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1985D61
hypocenterC61#25945
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. C2×#16840131C2semidirect
4. #21451031C2semidirect
5. #21446971C2semidirect
6. #21446961C2semidirect
7. #21446774C2semidirect
11. C61×#1985C2semidirect
259. #1985D61semidirect
304. C61#25945semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168401312
C2×#168399682
#214510312
#214469712
#214469612
#214467742
#214461522
#214456452
C61×#19852
#2594561