Finite Group Databasebeta

← #29716159

Subgroups — #29716159

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840072C212yesmaximal, complemented
3#21448349C211yesmaximal, characteristic
4#21447147C211yesmaximal, characteristic, complemented
5#21446961C211yesmaximal, characteristic, complemented
6#21446957C211yesmaximal, characteristic, complemented
7#21446773C212yesmaximal, complemented
8#21446130C212yesmaximal
9#21446017C212yesmaximal
10#21445798C211yesmaximal, characteristic
11C61×#2058C211yesmaximal, characteristic, complemented
12#16840317—28no—
13C2×#14519777C2211yescharacteristic
14C2×#14519768C2212yes—
15C2×#14519757—21no—
16C2×#14519757C2211yescharacteristic
17C2×#14519753—21no—
18C2×Q8×Dic61—21no—
19C2×#14519742—21no—
20C2×#14519742C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C42×D61C2211yescharacteristic
23#16840072—28no—
24C2×#14519710C2212yes—
25C2×#14519706C2212yes—
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519689C2212yes—
29#16839668C2211yescharacteristic
30#16839645—21no—
31#16839633—21no—
32#16839619C2211yescharacteristic
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#106×C61C2212yes—
42C61×#389C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2312yes—
51#14519777—24no—
52#14519768—24no—
53#14519768D418yes—
54#14519757—28no—
55#14519757—22no—
56#14519757Q814yescomplemented
57#14519753—28no—
58Q8×Dic61—28no—
59#14519742—28no—
60#14519742—24no—
61#14519735—24no—
62#14519735—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C42×D61—24no—
80#14519710—24no—
81#14519706—24no—
82#14519702—24no—
83#14519700—24no—
84#14519689—24no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89C22×Q8×C61—21no—
90C22×C8×C61C2312yes—
91C4⋊Q8×C61—24no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61C2311yescharacteristic
98#106×C61—24no—
99#95×C61—21no—
100C23×D61C2×D411yescharacteristic
101C2×Q8×C61—24no—
102C2×Q8×C61—42no—
103C2×Q8×C61—42no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61C2×Q812yes—
125C2×C4×D61#4112yes—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—24no—
129C2×Dic122—21no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122C2×Q812yes—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61C2×D414yes—
142C4⋊C4×C61C2×Q812yes—
143C42×C61—22no—
144#11758198—28no—
145#11758198—24no—
146#11758198—42no—
147C4⋊Dic61—24no—
148C4⋊Dic61—22no—
149C4⋊Dic61—28no—
150C4⋊Dic61—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61C2×D414yes—
153#11758196—28no—
154#11758196—42no—
155C4×Dic61—28no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—42no—
159C4×Dic61—42no—
160C4×Dic61C2×D414yes—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C23×C61C22×D411yescharacteristic
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61C22⋊Q812yes—
172Q8×C61—48no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#13512yes—
187C2×C4×C61#13712yes—
188C2×C4×C61C22×D412yes—
189C2×C4×C61C22×Q811yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C612−1+412yes—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—24no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61#12012yes—
208C2×Dic61#12012yes—
209C2×Dic61C22⋊Q812yes—
210C4×D61—24no—
211C4×D61—24no—
212C4×D61—44no—
213C4×D61—44no—
214Dic122—24no—
215Dic122—48no—
216Dic122—42no—
217C8×C61—42no—
218#92418—42no—
219#24306—611nomaximal, Sylow
220C22×C61C2×#12011yescharacteristic
221C22×C61C2×C22⋊Q811yescharacteristic
222C22×C61#53611yescharacteristic
223C22×C61C2×#13511yescharacteristic
224C22×C61C2×#13711yescharacteristic
225C22×C61#57511yescharacteristic
226C22×C61#57511yescharacteristic
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C4×C61—22no—
235C4×C61—22no—
236C4×C61—42no—
237C4×C61—44no—
238C4×C61—81no—
239C4×C61#54812yes—
240C4×C61D4×Q812yes—
241Dic61—22no—
242Dic61—22no—
243Dic61—22no—
244Dic61—42no—
245Dic61—44no—
246Dic61—81no—
247C2×#533—611no—
248C2×#479—612no—
249C4×C22⋊Q8—611no—
250#2058D6111yescharacteristic, complemented
251#2057—611no—
252#2024—611no—
253#2024—611no—
254#1989—612no—
255#1873—612no—
256#1854—612no—
257#1797—611no—
258C2×C61#240911yescharacteristic
259C2×C61#300211yescharacteristic
260C2×C61#300711yescharacteristic
261C2×C61#307611yescharacteristic
262C2×C61#307811yescharacteristic
263C2×C61#323111yescharacteristic
264C2×C61#324011yescharacteristic
265D61—42no—
266#533—1228no—
267C2×C4⋊Q8—611no—
268C2×C4⋊Q8C2×D6111yescharacteristic
269C2×#125—612no—
270C2×C22⋊Q8—611no—
271C2×C22⋊Q8—1221no—
272C2×C22⋊Q8—1221no—
273C2×C4×Q8—611no—
274C2×C4×Q8—1221no—
275C2×C4×Q8—1221no—
276C2×#117—612no—
277C22×C42—611no—
278#479—1228no—
279C2×#107—612no—
280C2×#106C2×D6112yes—
281C2×C4⋊C8—612no—
282C2×#103—612no—
283C2×#103—612no—
284C2×C22⋊C8—612no—
285#391—611no—
286#389C2×D6111yescharacteristic
287#387—1221no—
288#385—1221no—
289#380—611no—
290C4×C4⋊C4—611no—
291C4×C4⋊C4—1221no—
292C4×C22⋊C4—1221no—
293#340—612no—
294#340—612no—
295#340—612no—
296#340C2×D6112yes—
297#336—611no—
298#336C2×D6111yescharacteristic
299C61#2430611yescharacteristic, complemented, Sylow
300C22×Q8—21no—
301C22×Q8—611no—
302C22×Q8—1221no—
303C23×C4—611no—
304C23×C4—611no—
305C23×C4—611no—
306C22×C8—612no—
307C22×C8C22×D6112yes—
308C4⋊Q8—24no—
309C4⋊Q8—1224no—
310#125—618no—
311#125—1224no—
312C22⋊Q8—614no—
313C22⋊Q8—1222no—
314C22⋊Q8—1228no—
315C22⋊Q8—1228no—
316C4×Q8—1224no—
317C4×Q8—1228no—
318C4×Q8—1228no—
319#117—1224no—
320#117—1224no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—612no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4C22×D6111yescharacteristic
331C2×C4⋊C4C22×D6112yes—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C42—611no—
337C2×C42—611no—
338C2×C42—611no—
339C2×C42—1224no—
340C2×C42—1222no—
341C2×C42—1221no—
342C2×C42—1221no—
343C2×C42C22×D6111yescharacteristic
344#107—1224no—
345#106—24no—
346C4⋊C8—1224no—
347#103—1224no—
348#103—1224no—
349C22⋊C8—1224no—
350#95—21no—
351#95—1221no—
352#95—1221no—
353#95—1221no—
354#95—1221no—
355C24—611no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—612no—
360C2×Q8—1224no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×Q8—2442no—
365C2×Q8—2442no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—612no—
373C22×C4—612no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1221no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1221no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4D4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4C23×D6111yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—42no—
399C4⋊C4—614no—
400C4⋊C4—1228no—
401C4⋊C4—1228no—
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C4⋊C4—1222no—
405C4⋊C4—1222no—
406C4⋊C4—2442no—
407C4⋊C4D4×D6114yes—
408C4⋊C4Q8×D6112yes—
409C22⋊C4—1224no—
410C22⋊C4—1228no—
411C22⋊C4—2442no—
412C42—22no—
413C42—614no—
414C42—1222no—
415C42—1222no—
416C42—1228no—
417C42—2442no—
418C42—2442no—
419C23—612no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23C2×D4×D6111yescharacteristic
427Q8—48no—
428Q8—1224no—
429Q8—2448no—
430Q8—2442no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—612no—
445C2×C4—612no—
446C2×C4—612no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1224no—
454C2×C4—1224no—
455C2×C4—1224no—
456C2×C4—1222no—
457C2×C4—2444no—
458C2×C4—2444no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4#1451979812yes—
467C2×C4#1451980612yes—
468C2×C4C2×D4×D6112yes—
469C2×C4C2×Q8×D6111yescharacteristic
470C2×C4#41×D6111yescharacteristic
471C2×C4#1451982512yes—
472C8—42no—
473C8—2442no—
474C22—1222no—
475C22—1222no—
476C22—1222no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22—2441no—
481C22#120×D6111yescharacteristic
482C22C22⋊Q8×D6111yescharacteristic
483C22#1684033711yescharacteristic
484C22C2×#1451979811yescharacteristic
485C22C2×#1451980611yescharacteristic
486C22#1684055811yescharacteristic
487C22#1684056111yescharacteristic
488C4—22no—
489C4—22no—
490C4—44no—
491C4—42no—
492C4—81no—
493C4—1222no—
494C4—1222no—
495C4—1222no—
496C4—4881no—
497C4—2444no—
498C4—2442no—
499C4#1684042212yes—
500C4#1684042712yes—
501C2—2442no—
502C2#2144917911yescharacteristic
503C2#2145503611yescharacteristic
504C2#2145507411yescharacteristic
505C2#2145560811yescharacteristic
506C2#2145561911yescharacteristic
507C2#2145698111yescharacteristic
508C2#2145706011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2058C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2058D61
hypocenterC61#24306
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
2. C2×#16840072C2semidirect
4. #21447147C2semidirect
5. #21446961C2semidirect
6. #21446957C2semidirect
7. #21446773C2semidirect
11. C61×#2058C2semidirect
56. #14519757Q8semidirect
250. #2058D61semidirect
299. C61#24306semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400722
#214483492
#214471472
#214469612
#214469572
#214467732
#214461302
#214460172
#214457982
C61×#20582
#2430661