Finite Group Databasebeta

← #29732126

Subgroups — #29732126

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#21456361C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840039C211yesmaximal, characteristic
4#21454766C211yesmaximal, characteristic, complemented
5#21447328C212yesmaximal, complemented
6#21447180C212yesmaximal
7#21446603C212yesmaximal, complemented
8#21446592C211yesmaximal, characteristic, complemented
9#21446479C212yesmaximal
10C4×#14519693C211yesmaximal, characteristic
11C61×#1997C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840168—22no—
15#16840165—21no—
16#16840158—21no—
17#16840063—22no—
18#16840053—21no—
19#16840043—21no—
20#16840039—21no—
21#16840039—22no—
22#16840039—21no—
23#16839892—21no—
24#16839890—22no—
25C8×Dic122—21no—
26C2×#14519712C2212yes—
27#16839804C2212yes—
28#16839802C2212yes—
29#16839802C2212yes—
30C2×#14519710C2212yes—
31#16839790C2212yes—
32C2×C4⋊#92418C2211yescharacteristic
33#16839768C2212yes—
34C2×#14519697C2212yes—
35C2×#14519693C2211yescharacteristic
36C2×#14519693C2211yescharacteristic
37#16839720C2212yes—
38C4×#11758194C2211yescharacteristic
39#16839530C2211yescharacteristic
40#16839530C2211yescharacteristic
41#16839526C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#423C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2212yes—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777D411yescharacteristic
53#14519777D411yescharacteristic
54#14519769D412yes—
55#14519756—21no—
56#14519756—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244—21no—
63#14519736C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758198—21no—
66#14519712—24no—
67#14519711—22no—
68#14519710—22no—
69#14519710—22no—
70#14519710—22no—
71#14519710—22no—
72#14519709—22no—
73#14519708—22no—
74#14519708—22no—
75C4⋊#92418D411yescharacteristic
76C4⋊#92418D412yes—
77C4⋊#92418D411yescharacteristic
78#14519706—22no—
79#14519706—22no—
80#14519699—22no—
81#14519699—22no—
82#14519697—22no—
83#14519697—22no—
84#14519693—22no—
85#14519693Q814yescomplemented
86#14519688—21no—
87#14519687—22no—
88C4×#92418—21no—
89C8×Dic61—22no—
90C22×Q8×C61—21no—
91C2×M16×C61C2311yescharacteristic
92C2×M16×C61C2312yes—
93C2×M16×C61C2311yescharacteristic
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61—21no—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C4⋊C8×C61—21no—
103#97×C61—21no—
104C4×C8×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—41no—
107C2×Q8×C61—41no—
108C2×Q8×C61—82no—
109C22×C4×C61—22no—
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244C2×Q812yes—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C2×C4×D61—82no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125M16×C61—21no—
126M16×C61—21no—
127M16×C61—22no—
128M16×C61—21no—
129M16×C61—22no—
130M16×C61C2×Q812yes—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
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—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61C2×D412yes—
150C4⋊Dic61C2×D412yes—
151#11758196—22no—
152#11758196—22no—
153C4×Dic61—22no—
154#11758194—21no—
155#11758194—21no—
156#11758194C2×Q812yes—
157#11758194#4112yes—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418C2×D412yes—
163C2×#92418C2×D412yes—
164C23×C61C2×#4111yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—81no—
169C22×D61—81no—
170Q8×C61—82no—
171Q8×C61—82no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—82no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×Q811yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—41no—
189C2×Dic61—81no—
190D244—22no—
191D244—22no—
192D244—41no—
193D244—41no—
194C4×D61—82no—
195C4×D61—82no—
196Dic122—42no—
197Dic122—81no—
198C8×C61—22no—
199C8×C61—41no—
200C8×C61—42no—
201#92418—22no—
202#92418—22no—
203#92418—22no—
204#92418—22no—
205#25489—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—82no—
220C4×C61#47212yes—
221C4×C61#54711yescharacteristic
222C4×C61#54811yescharacteristic
223C4×C61D4×Q811yescharacteristic
224C4×C61D4×Q811yescharacteristic
225Dic61—42no—
226Dic61—42no—
227Dic61—81no—
228#3165—611no—
229#3077—611no—
230C2×#474—611no—
231#2962—611no—
232#2089—612no—
233#2063—612no—
234#1997D6111yescharacteristic, complemented
235#1952—612no—
236#1950—611no—
237#1931—612no—
238C4×#100—611no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61#242911yescharacteristic
242C2×C61C2×#47211yescharacteristic
243C2×C61#306911yescharacteristic
244D61—82no—
245#533—611no—
246C2×C4⋊Q8C2×D6111yescharacteristic
247C2×C4×D4—611no—
248#501—1222no—
249#500—1221no—
250#498—1221no—
251#478—1222no—
252#476—1221no—
253#474—1221no—
254#474—1221no—
255#474—1222no—
256#474—1221no—
257#446—1221no—
258#446—1222no—
259C8×Q8—1221no—
260C2×#108—612no—
261#428—612no—
262#428—612no—
263#428—612no—
264C2×#107—612no—
265#424—612no—
266#423C2×D6111yescharacteristic
267C2×C4⋊C8—611no—
268#418—612no—
269C2×#102—612no—
270C2×#100—611no—
271C2×#100—611no—
272#405—612no—
273C4×M16—611no—
274C4×M16C2×D6111yescharacteristic
275#343—611no—
276#343—611no—
277#343C2×D6112yes—
278#343C2×D6112yes—
279#341—612no—
280C61#2548911yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×D4—611no—
283C23×C4—1221no—
284C2×M16—611no—
285C2×M16—611no—
286C2×M16C22×D6112yes—
287C2×M16C22×D6111yescharacteristic
288C2×M16C22×D6111yescharacteristic
289C22×C8—612no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—611no—
295C4⋊Q8—611no—
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—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4C22×D6112yes—
309C2×C22⋊C4—1221no—
310C2×C42C22×D6111yescharacteristic
311#108—1224no—
312#107—1222no—
313#107—1222no—
314#107—1222no—
315#107—1222no—
316#107—1222no—
317#106—1222no—
318#106—1222no—
319#106—1222no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—611no—
323C4⋊C8—611no—
324C4⋊C8—612no—
325C4⋊C8—1222no—
326C4⋊C8—1222no—
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—41no—
341C2×Q8—41no—
342C2×Q8—82no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—22no—
352C22×C4—1221no—
353C22×C4—4882no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4#1175822211yescharacteristic
357C22×C4#1175822211yescharacteristic
358C22×C4C23×D6111yescharacteristic
359M16—21no—
360M16—21no—
361M16—22no—
362M16—22no—
363M16—21no—
364M16—612no—
365M16—612no—
366M16—1221no—
367M16—1221no—
368M16Q8×D6112yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—612no—
373C2×C8—612no—
374C2×C8—1221no—
375C2×C8—1221no—
376C2×C8—1222no—
377C2×C8—1222no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—612no—
383C4⋊C4—612no—
384C4⋊C4—1222no—
385C4⋊C4—1222no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C22⋊C4—1222no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C42—1222no—
394C42D4×D6111yescharacteristic
395C42D4×D6111yescharacteristic
396C42D4×D6111yescharacteristic
397C42D4×D6111yescharacteristic
398C23—1222no—
399C23—4881no—
400C23—4881no—
401C23—2441no—
402C23—2441no—
403C23C2×#1175822211yescharacteristic
404Q8—82no—
405Q8—82no—
406Q8—4881no—
407Q8—2442no—
408D4—1222no—
409D4—1222no—
410D4—2441no—
411D4—2441no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—4882no—
419C2×C4—4882no—
420C2×C4—4881no—
421C2×C4—2441no—
422C2×C4#1451975011yescharacteristic
423C2×C4#1451975011yescharacteristic
424C2×C4#1451975011yescharacteristic
425C2×C4#1451975011yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C2×D4×D6111yescharacteristic
428C2×C4C2×Q8×D6111yescharacteristic
429C2×C4#41×D6111yescharacteristic
430C2×C4#1451982411yescharacteristic
431C2×C4#1451982511yescharacteristic
432C8—22no—
433C8—41no—
434C8—42no—
435C8—1222no—
436C8—1222no—
437C8—1222no—
438C8—1222no—
439C22—21no—
440C22—21no—
441C22—4882no—
442C22—4882no—
443C22—4881no—
444C22—4881no—
445C22—2442no—
446C22C2×#1451975011yescharacteristic
447C22#1684031611yescharacteristic
448C22#1684037011yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—82no—
453C4—4881no—
454C4—2442no—
455C4—2442no—
456C4#1684003112yes—
457C4#1684041411yescharacteristic
458C4#1684042011yescharacteristic
459C4#1684042511yescharacteristic
460C4#1684042711yescharacteristic
461C2—21no—
462C2—21no—
463C2—4882no—
464C2#2144941911yescharacteristic
465C2C2×#1684003111yescharacteristic
466C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#1997D61
hypocenterC61#25489
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
1. #21456361C2semidirect
4. #21454766C2semidirect
5. #21447328C2semidirect
7. #21446603C2semidirect
8. #21446592C2semidirect
11. C61×#1997C2semidirect
85. #14519693Q8semidirect
234. #1997D61semidirect
280. C61#25489semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214556182
C2×#168400392
#214547662
#214473282
#214471802
#214466032
#214465922
#214464792
C4×#145196932
C61×#19972
#2548961