Finite Group Databasebeta

← #29736721

Subgroups — #29736721

Order 15616.

1110 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×#16840128C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21451569C211yesmaximal, characteristic
4#21446962C212yesmaximal, complemented
5#21446952C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446785C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446136C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C2×C61×#544C211yesmaximal, characteristic, complemented
12C2×#14519774C2211yescharacteristic
13C2×#14519751C2212yes—
14#16840128—24no—
15#16840110—24no—
16C2×#14519697C2212yes—
17C2×#14519697C2212yes—
18C2×#14519697C2212yes—
19C2×#14519697C2212yes—
20C2×C22⋊#92418C2212yes—
21C2×#14519688C2211yescharacteristic
22C2×C4×#92418C2211yescharacteristic
23#16839679—21no—
24#16839679—21no—
25#16839671—21no—
26#16839671—21no—
27#16839664C2212yes—
28#16839628C2211yescharacteristic
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#544—24no—
35C61×#544—24no—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#125×C61—21no—
40C2×#125×C61C2211yescharacteristic
41C2×#125×C61C2211yescharacteristic
42C2×C22⋊Q8×C61—21no—
43C2×C22⋊Q8×C61—21no—
44C2×#117×C61C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C22⋊D61—22no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519774—22no—
54#14519774—22no—
55#14519751—28no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60#14519697—24no—
61#14519697—24no—
62#14519697—24no—
63#14519697—24no—
64C22⋊#92418—24no—
65#14519688—22no—
66C4×#92418—22no—
67#14519684—21no—
68#14519684—21no—
69C22×Q8×C61—21no—
70C22×Q8×C61—21no—
71C23×C4×C61—21no—
72C23×C4×C61C2311yescharacteristic
73C4⋊Q8×C61—28no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D414yescomplemented
78#125×C61—28no—
79#125×C61—22no—
80#125×C61—22no—
81#125×C61—22no—
82#125×C61D414yescomplemented
83C22⋊Q8×C61—28no—
84C22⋊Q8×C61—28no—
85#117×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88#117×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C24×C61C2×D411yescharacteristic
111C23×D61—22no—
112C2×Q8×C61—24no—
113C2×Q8×C61—24no—
114C2×Q8×C61—82no—
115C2×Q8×C61—82no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—82no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4114yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—44no—
133C2×C22⋊D61—82no—
134C2×C22⋊D61—82no—
135C22×Dic61—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—24no—
139C2×D244—24no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—82no—
143C2×D244—82no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D412yes—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C42×C61—24no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61C2×D412yes—
168C42×C61C2×D412yes—
169#11758198—44no—
170#11758198—82no—
171#11758198—82no—
172C4⋊Dic61—24no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61#12412yes—
184C23×C61C22×D411yescharacteristic
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192Q8×C61—84no—
193Q8×C61—84no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—84no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12012yes—
215C2×C4×C61#12412yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C612−1+412yes—
220C22⋊D61—88no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224D244—44no—
225D244—44no—
226D244—84no—
227D244—84no—
228#92418—42no—
229#92418—42no—
230#25915—611nomaximal, Sylow
231C22×C61—24no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61C2×#12011yescharacteristic
237C22×C61C2×#12411yescharacteristic
238C22×C61#53611yescharacteristic
239C22×C61#57612yes—
240C22×C61#57712yes—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—82no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—41no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—41no—
256C4×C61—42no—
257C4×C61#54812yes—
258C4×C61#54812yes—
259Dic61—82no—
260C2×#544D6111yescharacteristic, complemented
261C2×#490—611no—
262C2×#486—611no—
263#2619—611no—
264#2025—612no—
265#2023—612no—
266#1993—611no—
267#1991—611no—
268#1875—612no—
269#1874—612no—
270#1750—611no—
271C2×C61—42no—
272C2×C61#243611yescharacteristic
273C2×C61#300811yescharacteristic
274C2×C61#301011yescharacteristic
275C2×C61#313612yes—
276C2×C61#324112yes—
277D61—162no—
278#544—24no—
279#544—24no—
280C2×C4⋊Q8—21no—
281C2×C4⋊Q8C2×D6111yescharacteristic
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#127—611no—
284C2×#125—21no—
285C2×#125C2×D6111yescharacteristic
286C2×#125C2×D6111yescharacteristic
287C2×C22⋊Q8—21no—
288C2×C22⋊Q8—21no—
289C2×#121—612no—
290C2×#117C2×D6111yescharacteristic
291C2×#117C2×D6111yescharacteristic
292C22×C4⋊C4C2×D6111yescharacteristic
293#490—1224no—
294#486—1224no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102—612no—
298C2×#102—612no—
299C2×C22⋊C8—612no—
300C2×#97—611no—
301C2×C4×C8—611no—
302#394—1221no—
303#394—1221no—
304#392—1221no—
305#392—1221no—
306#390—612no—
307#382—611no—
308#340—612no—
309#340—612no—
310#340—612no—
311#340—612no—
312#336—612no—
313C61#2591511yescharacteristic, complemented, Sylow
314C22×Q8—21no—
315C22×Q8—21no—
316C22×D4—612no—
317C22×D4—1222no—
318C22×D4—1221no—
319C22×D4—1221no—
320C23×C4—21no—
321C23×C4C22×D6111yescharacteristic
322C22×C8—612no—
323C22×C8—612no—
324C4⋊Q8—28no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8C22⋊D6114yescomplemented
329#127—1222no—
330#127—1222no—
331#125—28no—
332#125—22no—
333#125—22no—
334#125—22no—
335#125C22⋊D6114yescomplemented
336C22⋊Q8—28no—
337C22⋊Q8—28no—
338#121—1228no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4C22⋊D6111yescharacteristic
354C2×C4⋊C4C22⋊D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4C22⋊D6111yescharacteristic
365C2×C42—21no—
366C2×C42C22×D6111yescharacteristic
367C2×C42C22×D6111yescharacteristic
368#102—1224no—
369#102—1224no—
370#102—1224no—
371#102—1224no—
372C22⋊C8—1224no—
373#97—1222no—
374C4×C8—1222no—
375#95—1221no—
376#95—1221no—
377C24—1222no—
378C24C2×C22⋊D6111yescharacteristic
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1224no—
386C2×D4—1224no—
387C2×D4—4882no—
388C2×D4—4882no—
389C2×D4—4882no—
390C2×D4—4882no—
391C2×D4—2444no—
392C2×D4—2442no—
393C2×D4—2442no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—82no—
403C22×C4—1222no—
404C22×C4#1175822614yes—
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—1224no—
431C4⋊C4C2×C22⋊D6112yes—
432C4⋊C4C2×C22⋊D6112yes—
433C4⋊C4C2×C22⋊D6112yes—
434C4⋊C4C2×C22⋊D6112yes—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—4882no—
439C22⋊C4—4882no—
440C22⋊C4—2444no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42C2×C22⋊D6112yes—
445C42C2×C22⋊D6112yes—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—2442no—
451C23—2442no—
452C23—2442no—
453C23—2442no—
454C23—2442no—
455C23—2442no—
456C23—2442no—
457C23#1451976612yes—
458C23C22×C22⋊D6111yescharacteristic
459Q8—84no—
460Q8—84no—
461D4—4884no—
462D4—4884no—
463D4—4888no—
464D4—2444no—
465D4—2444no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4#1451974612yes—
489C2×C4#1451974612yes—
490C2×C4#1451976612yes—
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4C2×#1175822612yes—
494C2×C4#1451982512yes—
495C8—2442no—
496C8—2442no—
497C22—22no—
498C22—24no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—4882no—
503C22—4882no—
504C22—4882no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22—4882no—
509C22C2×#1451974611yescharacteristic
510C22C2×#1451976611yescharacteristic
511C22#1684033611yescharacteristic
512C22#1684056512yes—
513C22#1684057012yes—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—4882no—
524C4#1684042312yes—
525C4#1684042312yes—
526C2—42no—
527C2—9762no—
528C2#2144947311yescharacteristic
529C2#2145508711yescharacteristic
530C2#2145510511yescharacteristic
531C2#2145617812yes—
532C2#2145707212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC23×C61C22×D4
prime radical#21451569C2
hypercenterC2×#544D61
hypocenterC61#25915
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×#16840128C2semidirect
2. C2×#16840110C2semidirect
4. #21446962C2semidirect
5. #21446952C2semidirect
6. #21446793C2semidirect
7. #21446785C2semidirect
8. #21446141C2semidirect
9. #21446136C2semidirect
10. #21445585C2semidirect
11. C2×C61×#544C2semidirect
77. C4⋊Q8×C61D4semidirect
82. #125×C61D4semidirect
260. C2×#544D61semidirect
313. C61#25915semidirect
328. C4⋊Q8C22⋊D61semidirect
335. #125C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
C2×#168401102
#214515692
#214469622
#214469522
#214467932
#214467852
#214461412
#214461362
#214455852
C2×C61×#5442
#2591561