Finite Group Databasebeta

← #29737028

Subgroups — #29737028

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×#16840130C211yesmaximal, characteristic, complemented
3C2×#16839964C211yesmaximal, characteristic
4#21446970C212yesmaximal, complemented
5#21446960C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal
9#21446156C212yesmaximal
10#21445647C211yesmaximal, characteristic
11C61×#2568C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519753—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840130—24no—
26#16839964—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519697C2212yes—
32C2×#14519688C2211yescharacteristic
33#16839524C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C61×#400—21no—
41C61×#400—21no—
42C61×#391—21no—
43C61×#391C2212yes—
44C61×#391C2212yes—
45C61×#382—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C2×#11758201—24no—
53C2×#11758201—28no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56#14519777—22no—
57#14519777—22no—
58#14519769—22no—
59#14519769—22no—
60#14519769—22no—
61#14519769D414yes—
62#14519755—28no—
63#14519755—28no—
64#14519753—28no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D414yes—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C4⋊C4×D61—28no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C4⋊#92418—22no—
84#14519701—24no—
85#14519701—24no—
86#14519701—24no—
87#14519697—24no—
88#14519688—22no—
89C22×Q8×C61D411yescharacteristic
90C22×Q8×C61C2312yes—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105#95×C61—21no—
106#95×C61—22no—
107#95×C61—22no—
108#95×C61—22no—
109#95×C61—22no—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—24no—
116C2×Q8×C61—24no—
117C2×Q8×C61—41no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C2×D244—21no—
131C2×D244—22no—
132C2×D244C2×D412yes—
133C2×D244#4114yes—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—82no—
140C2×C4×D61—82no—
141C2×Dic122—24no—
142C2×Dic122—82no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—42no—
150C42×C61—21no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61—42no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157#11758201—48no—
158#11758198—24no—
159#11758198—24no—
160#11758198—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61C2×D412yes—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—24no—
176C2×#92418—22no—
177C23×C61C22×D411yescharacteristic
178C22×D61—22no—
179C22×D61—24no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—42no—
183Q8×C61—44no—
184Q8×C61—42no—
185Q8×C61—42no—
186Q8×C61—44no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—22no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12012yes—
206C2×C4×C61#13712yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4112yes—
210C2×C4×C612−1+412yes—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218D244—22no—
219D244—24no—
220D244#12414yes—
221C4×D61—48no—
222C4×D61—84no—
223C4×D61—84no—
224Dic122—84no—
225#92418—42no—
226#92418—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×C61—82no—
245C4×C61—82no—
246C4×C61C2×#12411yescharacteristic
247C4×C61#53611yescharacteristic
248C4×C61#54812yes—
249Dic61—42no—
250Dic61—42no—
251Dic61—42no—
252C2×#554—611no—
253C2×#491—611no—
254C2×#461—611no—
255#2568D6111yescharacteristic, complemented
256#2026—612no—
257#2024—612no—
258#1989—611no—
259#1985—611no—
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⋊Q8C2×D6111yescharacteristic
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⋊C8—611no—
287C2×#103—612no—
288C2×#103—612no—
289C2×#103—612no—
290C2×#102—612no—
291C2×#97—611no—
292#400—21no—
293#400—21no—
294#391—21no—
295#391C2×D6112yes—
296#391C2×D6112yes—
297#382—21no—
298C4×C4⋊C4C2×D6111yescharacteristic
299#340—612no—
300#340—612no—
301#340—612no—
302#336—612no—
303#336—612no—
304C61#2594511yescharacteristic, complemented, Sylow
305C22×Q8—1221no—
306C22×Q8C22⋊D6111yescharacteristic
307C22×Q8C22×D6112yes—
308C22×D4—611no—
309C23×C4—1221no—
310C23×C4—1221no—
311C23×C4—1221no—
312C2×SD16—1224no—
313C2×SD16—1228no—
314C22×C8—612no—
315C22×C8—612no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
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—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—1228no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4C22⋊D6111yescharacteristic
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C4⋊C4C22×D6112yes—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—21no—
354C2×C42C22×D6111yescharacteristic
355C2×C42C22×D6111yescharacteristic
356C2×C42C22×D6111yescharacteristic
357C4⋊C8—1222no—
358#103—1224no—
359#103—1224no—
360#103—1224no—
361#102—1224no—
362#97—1222no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368C24—1221no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—41no—
376C2×Q8—1224no—
377C2×Q8—4882no—
378C2×D4—612no—
379C2×D4—614no—
380C2×D4—1221no—
381C2×D4—1222no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—1224no—
385C22×C4—1224no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1224no—
390C22×C4—4882no—
391C22×C4—4882no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C23×D6111yescharacteristic
401SD16—2448no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1224no—
406C2×C8—1222no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—612no—
415C4⋊C4—612no—
416C4⋊C4—612no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42D4×D6112yes—
434C42D4×D6112yes—
435C23—1222no—
436C23—1224no—
437C23—2441no—
438C23—2441no—
439C23—2442no—
440C23C22×C22⋊D6111yescharacteristic
441Q8—44no—
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—4884no—
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—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—4884no—
470C2×C4—4884no—
471C2×C4—2448no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4#1451974512yes—
477C2×C4#1451974512yes—
478C2×C4#1451980812yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#41×D6112yes—
482C2×C4#1451982412yes—
483C8—2442no—
484C8—2442no—
485C22—1224no—
486C22—2444no—
487C22—2444no—
488C22—2442no—
489C22—2442no—
490C22C2×#1451974511yescharacteristic
491C22#1684041912yes—
492C22C2×#1451980811yescharacteristic
493C22#1684056011yescharacteristic
494C22#1684057012yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—82no—
502C4—82no—
503C4—2442no—
504C4—2442no—
505C4—2442no—
506C4#124×D6111yescharacteristic
507C4#1684033211yescharacteristic
508C4#1684042012yes—
509C2—2444no—
510C2#2144947411yescharacteristic
511C2#2145507711yescharacteristic
512C2#2145509411yescharacteristic
513C2#2145859612yes—
514C2#2145864012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2568C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2568D61
hypocenterC61#25945
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
2. C2×#16840130C2semidirect
4. #21446970C2semidirect
5. #21446960C2semidirect
6. #21446776C2semidirect
7. #21446760C2semidirect
11. C61×#2568C2semidirect
255. #2568D61semidirect
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×#168401302
C2×#168399642
#214469702
#214469602
#214467762
#214467602
#214461562
#214456472
C61×#25682
#2594561