Finite Group Databasebeta

← #29732134

Subgroups — #29732134

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456365C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840053C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21447331C212yesmaximal, complemented
6#21447174C212yesmaximal
7#21446807C211yesmaximal, characteristic, complemented
8#21446600C212yesmaximal, complemented
9#21446482C212yesmaximal
10C4×#14519693C211yesmaximal, characteristic
11C61×#1950C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840171—22no—
16#16840167—21no—
17#16840160—21no—
18#16840063—22no—
19#16840053—21no—
20#16840053—21no—
21#16840053—21no—
22#16840053—22no—
23#16840043—21no—
24#16839891—22no—
25#16839889—21no—
26#16839886—21no—
27#16839804C2212yes—
28#16839804C2212yes—
29#16839804C2212yes—
30C2×#14519709C2212yes—
31#16839788C2212yes—
32#16839768C2212yes—
33C2×#14519699C2212yes—
34C2×#14519693C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36#16839717C2212yes—
37C4×#11758194C2211yescharacteristic
38#16839530C2212yes—
39#16839530C2212yes—
40#16839530C2211yescharacteristic
41#16839530C2211yescharacteristic
42C2×#108×C61C2212yes—
43C61×#423C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C4×M16×C61C2211yescharacteristic
46C61×#341C2212yes—
47C22×Dic122—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C2×#11758194C2311yescharacteristic
51C2×#11758194C2311yescharacteristic
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777D411yescharacteristic
57#14519777D411yescharacteristic
58#14519769D412yes—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519736C2312yes—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758198—21no—
72#14519711—22no—
73#14519711—22no—
74#14519711—22no—
75#14519709—22no—
76#14519709—22no—
77#14519709—22no—
78#14519709—22no—
79#14519709—22no—
80#14519706—22no—
81#14519706—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85#14519699—22no—
86#14519693—22no—
87#14519693Q814yescomplemented
88#14519688—21no—
89#14519687—22no—
90C4×#92418—21no—
91C8×Dic61—22no—
92C2×M16×C61C2312yes—
93C2×M16×C61C2311yescharacteristic
94C2×M16×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97#108×C61—24no—
98C4⋊C8×C61—21no—
99C4⋊C8×C61—21no—
100C4⋊C8×C61D411yescharacteristic
101C4⋊C8×C61D411yescharacteristic
102C4⋊C8×C61D412yes—
103#97×C61—21no—
104C4×C8×C61—21no—
105C23×D61—21no—
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244C2×Q812yes—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—82no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—41no—
123C2×Dic122—41no—
124C2×Dic122—82no—
125M16×C61—21no—
126M16×C61—21no—
127M16×C61—22no—
128M16×C61—22no—
129M16×C61C2×Q812yes—
130M16×C61#4112yes—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61C2×D412yes—
137C2×C8×C61C2×D412yes—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—22no—
143#11758198—41no—
144#11758198—41no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61C2×D412yes—
154C4⋊Dic61C2×D412yes—
155#11758196—22no—
156#11758196—22no—
157C4×Dic61—22no—
158#11758194—21no—
159#11758194—21no—
160#11758194—21no—
161#11758194C2×Q812yes—
162C2×#92418—21no—
163C2×#92418—21no—
164C23×C61C2×#4111yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—81no—
169C22×D61—81no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×Q811yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C612−1+411yescharacteristic
181C2×C4×C612−1+411yescharacteristic
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—82no—
187C2×Dic61—81no—
188D244—22no—
189D244—22no—
190D244—41no—
191D244—41no—
192C4×D61—82no—
193C4×D61—82no—
194Dic122—42no—
195Dic122—81no—
196Dic122—82no—
197Dic122—82no—
198C8×C61—22no—
199C8×C61—22no—
200C8×C61—22no—
201C8×C61—22no—
202C8×C61—42no—
203#92418—22no—
204#92418—41no—
205#25490—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61C2×#12111yescharacteristic
209C22×C61#53311yescharacteristic
210C22×C61#54111yescharacteristic
211C2×D61—42no—
212C2×D61—82no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—82no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61#47112yes—
220C4×C61#54711yescharacteristic
221C4×C61#54811yescharacteristic
222C4×C61D4×Q811yescharacteristic
223C4×C61D4×Q811yescharacteristic
224Dic61—42no—
225Dic61—42no—
226Dic61—81no—
227Dic61—82no—
228#3165—611no—
229#3077—611no—
230C2×#476—611no—
231#2962—611no—
232#2090—612no—
233#2063—612no—
234#1997—611no—
235#1951—612no—
236#1950D6111yescharacteristic, complemented
237#1931—612no—
238C4×#100—611no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61#242911yescharacteristic
242C2×C61C2×#47111yescharacteristic
243C2×C61#306811yescharacteristic
244D61—82no—
245#533—611no—
246C2×C4⋊Q8—611no—
247C2×C4×D4—611no—
248#501—1222no—
249#500—1221no—
250#498—1221no—
251#478—1222no—
252#476—1221no—
253#476—1221no—
254#476—1222no—
255#476—1221no—
256#474—1221no—
257#446—1221no—
258#446—1222no—
259C8×Q8—1221no—
260C2×#108C2×D6112yes—
261#428—612no—
262#428—612no—
263#428—612no—
264C2×#106—612no—
265#424—612no—
266#423C2×D6111yescharacteristic
267C2×C4⋊C8C2×D6111yescharacteristic
268#418—612no—
269C2×#102—612no—
270C2×#100—611no—
271C2×#100—611no—
272#405—612no—
273C4×M16—611no—
274C4×M16C2×D6111yescharacteristic
275#343—612no—
276#343—611no—
277#343—612no—
278#343—611no—
279#341C2×D6112yes—
280C61#2549011yescharacteristic, complemented, Sylow
281C22×Q8—1221no—
282C22×D4—611no—
283C23×C4—1221no—
284C2×M16—611no—
285C2×M16—611no—
286C2×M16C22×D6112yes—
287C2×M16C22×D6111yescharacteristic
288C2×M16C22×D6111yescharacteristic
289C22×C8C22×D6112yes—
290C4⋊Q8—611no—
291C4⋊Q8—611no—
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8—1221no—
295C4⋊Q8—1221no—
296#125—612no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299C4×Q8—1221no—
300C4×Q8—1221no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4—1221no—
304C4×D4—1221no—
305#117—612no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—612no—
308C2×C4⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42C22×D6111yescharacteristic
311#108—24no—
312#107—1222no—
313#107—1222no—
314#107—1222no—
315#106—1222no—
316#106—1222no—
317#106—1222no—
318#106—1222no—
319#106—1222no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—1222no—
323C4⋊C8—1222no—
324C4⋊C8D24411yescharacteristic
325C4⋊C8D24411yescharacteristic
326C4⋊C8D24412yes—
327#102—1222no—
328#102—1222no—
329#102—1222no—
330#102—1222no—
331#100—614no—
332#100—1222no—
333#97—21no—
334#97—1222no—
335#97—1221no—
336C4×C8—21no—
337C4×C8—1222no—
338C4×C8—1221no—
339C24—1221no—
340C2×Q8—1221no—
341C2×Q8—1221no—
342C2×Q8—4882no—
343C2×Q8—2441no—
344C2×Q8—2441no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—1221no—
352C22×C4—1222no—
353C22×C4—4882no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4#1175822611yescharacteristic
357C22×C4#1175822611yescharacteristic
358C22×C4C23×D6111yescharacteristic
359M16—21no—
360M16—21no—
361M16—22no—
362M16—22no—
363M16—612no—
364M16—1221no—
365M16—1221no—
366M16—1221no—
367M16#1175822412yes—
368M16Q8×D6112yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—1221no—
375C2×C8—1221no—
376C2×C8C2×D24412yes—
377C2×C8C2×D24412yes—
378C4⋊C4—612no—
379C4⋊C4—612no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—2441no—
389C4⋊C4—2441no—
390C22⋊C4—1222no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C42—1222no—
394C42C2×D24411yescharacteristic
395C42C2×D24411yescharacteristic
396C42D4×D6111yescharacteristic
397C42D4×D6111yescharacteristic
398C23—1222no—
399C23—4881no—
400C23—4881no—
401C23—2441no—
402C23—2441no—
403C23C2×#1175822611yescharacteristic
404Q8—4882no—
405Q8—4882no—
406Q8—4881no—
407Q8—2442no—
408D4—1222no—
409D4—1222no—
410D4—2441no—
411D4—2441no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—4882no—
417C2×C4—4882no—
418C2×C4—4881no—
419C2×C4—4882no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4#1451975211yescharacteristic
423C2×C4#1451975211yescharacteristic
424C2×C4#1451975211yescharacteristic
425C2×C4#1451975211yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C22×D24411yescharacteristic
428C2×C4C2×Q8×D6111yescharacteristic
429C2×C4C2×#1175822411yescharacteristic
430C2×C4#1451982411yescharacteristic
431C2×C4#1451982411yescharacteristic
432C8—22no—
433C8—22no—
434C8—22no—
435C8—22no—
436C8—42no—
437C8—1222no—
438C8—2441no—
439C22—21no—
440C22—21no—
441C22—4882no—
442C22—4882no—
443C22—4881no—
444C22—4881no—
445C22—2442no—
446C22C2×#1451975211yescharacteristic
447C22#1684031711yescharacteristic
448C22#1684037511yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—4881no—
453C4—4882no—
454C4—2442no—
455C4—2442no—
456C4#1684002612yes—
457C4#1684041611yescharacteristic
458C4#1684041811yescharacteristic
459C4#1684042511yescharacteristic
460C4Q8×D24411yescharacteristic
461C2—21no—
462C2—21no—
463C2—4882no—
464C2#2144941611yescharacteristic
465C2C2×#1684002611yescharacteristic
466C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1950C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#1950D61
hypocenterC61#25490
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
1. #21456365C2semidirect
4. #21454762C2semidirect
5. #21447331C2semidirect
7. #21446807C2semidirect
8. #21446600C2semidirect
11. C61×#1950C2semidirect
87. #14519693Q8semidirect
236. #1950D61semidirect
280. C61#25490semidirect

Maximal subgroups

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

subgroupindex
#214563652
#214556182
C2×#168400532
#214547622
#214473312
#214471742
#214468072
#214466002
#214464822
C4×#145196932
C61×#19502
#2549061