Finite Group Databasebeta

← #29729516

Subgroups — #29729516

Order 15616.

954 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×#16839957C211yesmaximal, characteristic
2C2×Q8×#92418C211yesmaximal, characteristic, complemented
3#21450818C211yesmaximal, characteristic, complemented
4#21447204C212yesmaximal, complemented
5#21447129C212yesmaximal, complemented
6#21446684C212yesmaximal, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446575C211yesmaximal, characteristic, complemented
9#21446501C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15#16839957—24no—
16Q8×#92418—24no—
17C2×#14519710C2212yes—
18C2×C4⋊#92418C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26C2×C4×#92418C2212yes—
27#16839679—22no—
28#16839664C2212yes—
29#16839664C2212yes—
30#16839663—21no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35C2×C22⋊Q8×C61—21no—
36C2×C4×Q8×C61C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C61×#396—22no—
39C61×#390C2212yes—
40C61×#389C2212yes—
41C61×#384C2211yescharacteristic
42C61×#380—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C2×#11758199—24no—
47C2×#11758199—24no—
48C2×#11758199—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52#14519774—22no—
53#14519774—22no—
54#14519766—44no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—21no—
62#14519710—42no—
63C4⋊#92418—24no—
64C4⋊#92418D414yes—
65#14519698—22no—
66#14519698—22no—
67#14519697—24no—
68#14519697—24no—
69#14519697—42no—
70C4×#92418—22no—
71C4×#92418—24no—
72#14519684—21no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61—21no—
76C22⋊Q8×C61—44no—
77C4×Q8×C61—22no—
78C4×Q8×C61—22no—
79C4×D4×C61—22no—
80C4×D4×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61D411yescharacteristic
85C2×C4⋊C4×C61D412yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—22no—
91C2×C42×C61C2312yes—
92C2×C42×C61C2311yescharacteristic
93#95×C61—22no—
94C24×C61—21no—
95C23×D61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61D814yescomplemented
99C2×D4×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—21no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61#4112yes—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—21no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—42no—
124C2×D244—42no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—81no—
128C2×D244—81no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—42no—
136C4⋊C4×C61—44no—
137C4⋊C4×C61—81no—
138C22⋊C4×C61—42no—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—82no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61#4112yes—
145#11758199—48no—
146#11758198—44no—
147#11758198—44no—
148#11758198—82no—
149#11758198—82no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4×Dic61—42no—
155C2×#92418—21no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418C2×D414yes—
163C2×#92418#4112yes—
164C23×C61—22no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61C22×D411yescharacteristic
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—28no—
180D4×C61—22no—
181D4×C61—22no—
182D4×C61—41no—
183D4×C61—41no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—84no—
198C2×C4×C61#12112yes—
199C2×C4×C61#12112yes—
200C2×C4×C61C2×D814yes—
201C2×C4×C61C2×D812yes—
202C2×C4×C61#13512yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)11yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215D244—42no—
216D244—42no—
217D244—42no—
218D244—42no—
219D244—84no—
220D244—84no—
221#92418—24no—
222#92418—22no—
223#92418—41no—
224#25292—611nomaximal, Sylow
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61C2×#12111yescharacteristic
233C22×C61#54811yescharacteristic
234C22×C61#55011yescharacteristic
235C22×C61C22×D811yescharacteristic
236C22×C61C2×#13511yescharacteristic
237C22×C61#57611yescharacteristic
238C22×C61#57711yescharacteristic
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C4×C61—24no—
247C4×C61—22no—
248C4×C61—42no—
249C4×C61—82no—
250C4×C61—81no—
251C4×C61#46614yes—
252C4×C61#52911yescharacteristic
253C4×C61#54211yescharacteristic
254C4×C61#54611yescharacteristic
255C4×C61#54811yescharacteristic
256Dic61—81no—
257Dic61—81no—
258C2×#459—611no—
259C2×C8×Q8—611no—
260#2548—611no—
261#2538D6111yescharacteristic, complemented
262#2067—612no—
263#2054—612no—
264#1970—612no—
265#1957—611no—
266#1947—611no—
267#1934—612no—
268C4×#102—611no—
269C2×C61—82no—
270C2×C61#242611yescharacteristic
271C2×C61C2×#46611yescharacteristic
272C2×C61#306211yescharacteristic
273C2×C61#335811yescharacteristic
274C2×C61#336111yescharacteristic
275C2×C61#339111yescharacteristic
276C2×C61#339411yescharacteristic
277D61—162no—
278C22×D8—612no—
279C2×#127—611no—
280C2×#124—1221no—
281C2×C22⋊Q8—21no—
282C2×C4×Q8C2×D6111yescharacteristic
283C2×C4×D4C2×D6111yescharacteristic
284#459—1224no—
285C8×Q8—1224no—
286C2×#107—612no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—612no—
289C2×#102—611no—
290C2×#102—611no—
291C2×#102—612no—
292C2×#102—612no—
293C2×#102—612no—
294C2×C4×C8—612no—
295C2×C4×C8—611no—
296#396—22no—
297#394—1222no—
298#390—612no—
299#390—612no—
300#390—1221no—
301#390C2×D6112yes—
302#389C2×D6112yes—
303#384C2×D6111yescharacteristic
304#380—21no—
305C4×C4⋊C4—611no—
306#340—612no—
307#340—611no—
308#340—611no—
309C61#2529211yescharacteristic, complemented, Sylow
310C22×Q8C22⋊D6111yescharacteristic
311C22×D4—612no—
312C22×D4—1221no—
313C22×D4—1221no—
314C22×D4C22×D6111yescharacteristic
315C23×C4—21no—
316C2×D8—1224no—
317C2×D8—1224no—
318C2×D8—2442no—
319C22×C8—612no—
320C22×C8—611no—
321C22×C8—611no—
322#127—1222no—
323#127—1222no—
324#124—2444no—
325C22⋊Q8—44no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—22no—
329C4×D4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C4⋊C4C22⋊D6112yes—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6112yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1222no—
346C2×C42—1221no—
347C2×C42C22×D6112yes—
348C2×C42C22×D6111yescharacteristic
349#107—2442no—
350C4⋊C8—614no—
351C4⋊C8—1224no—
352#102—1224no—
353#102—1224no—
354#102—1222no—
355#102—1222no—
356#102—2442no—
357C4×C8—1224no—
358C4×C8—1222no—
359#95—22no—
360#95—1221no—
361C24—21no—
362C24—1222no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8#1175819914yescomplemented
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—4881no—
378C2×D4—4881no—
379C2×D4—2442no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4#1175822612yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4C2×C22⋊D6112yes—
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6112yes—
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396D8—2448no—
397C2×C8—614no—
398C2×C8—612no—
399C2×C8—1221no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—81no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C22⋊C4—4882no—
423C22⋊C4—4882no—
424C22⋊C4—2444no—
425C22⋊C4—2444no—
426C42—22no—
427C42—22no—
428C42—2442no—
429C42D4×D6112yes—
430C42#1175822412yes—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23—2442no—
444C23—2442no—
445C23C22×C22⋊D6111yescharacteristic
446Q8—28no—
447D4—22no—
448D4—22no—
449D4—41no—
450D4—41no—
451D4—4884no—
452D4—4884no—
453D4—2442no—
454D4—2442no—
455D4—2442no—
456D4—2442no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4#1451974912yes—
478C2×C4#1451974912yes—
479C2×C4C2×#1175819914yes—
480C2×C4C2×#1175819912yes—
481C2×C4#1451979712yes—
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822411yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982311yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—1224no—
489C8—1222no—
490C8—2441no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22—4882no—
503C22—4882no—
504C22—4882no—
505C22C2×#1451974911yescharacteristic
506C22#1684042311yescharacteristic
507C22#1684043111yescharacteristic
508C22C22×#1175819911yescharacteristic
509C22C2×#1451979711yescharacteristic
510C22#1684056511yescharacteristic
511C22#1684057011yescharacteristic
512C4—24no—
513C4—22no—
514C4—42no—
515C4—81no—
516C4—82no—
517C4—4881no—
518C4—4881no—
519C4#1683999614yes—
520C4#1684029811yescharacteristic
521C4#1684038511yescharacteristic
522C4#1684040311yescharacteristic
523C4#1684042211yescharacteristic
524C2—82no—
525C2—9762no—
526C2#2144938711yescharacteristic
527C2C2×#1683999611yescharacteristic
528C2#2145550111yescharacteristic
529C2#2145845411yescharacteristic
530C2#2145847911yescharacteristic
531C2#2145879111yescharacteristic
532C2#2145881511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenter#2538D61
hypocenterC61#25292
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#92418C2semidirect
3. #21450818C2semidirect
4. #21447204C2semidirect
5. #21447129C2semidirect
6. #21446684C2semidirect
7. #21446624C2semidirect
8. #21446575C2semidirect
9. #21446501C2semidirect
10. C4×#14519698C2semidirect
11. C61×#2538C2semidirect
98. C2×Q8×C61D8semidirect
261. #2538D61semidirect
309. C61#25292semidirect
365. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×Q8×#924182
#214508182
#214472042
#214471292
#214466842
#214466242
#214465752
#214465012
C4×#145196982
C61×#25382
#2529261