Finite Group Databasebeta

← #29663882

Subgroups — #29663882

Order 15616.

650 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#21456394C211yesmaximal, characteristic
2#21455942C211yesmaximal, characteristic, complemented
3#21455937C211yesmaximal, characteristic, complemented
4C2×#16839886C211yesmaximal, characteristic, complemented
5#21445238C212yesmaximal, complemented
6#21445234C212yesmaximal, complemented
7#21445222C212yesmaximal, complemented
8#21445178C212yesmaximal, complemented
9#21444580C211yesmaximal, characteristic, complemented
10#21444512C211yesmaximal, characteristic, complemented
11C61×#1551C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14#16840165C2211yescharacteristic
15#16840138C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840108C2211yescharacteristic
18#16840076—21no—
19#16840073—21no—
20#16840058C2212yes—
21#16840054C2212yes—
22#16839996—21no—
23#16839996—21no—
24#16839963C2212yes—
25#16839959C2212yes—
26#16839903—21no—
27#16839900—21no—
28#16839886—21no—
29#16839886—21no—
30#16839886C2212yes—
31#16839886C2212yes—
32#16839864—22no—
33#16839862—22no—
34C4×#11758199—21no—
35C4×#11758199—21no—
36#16839819C2211yescharacteristic
37C2×#14519706—21no—
38C2×C4×#92418C2211yescharacteristic
39C2×C8×Dic61—21no—
40#16839514C2212yes—
41#16839512C2212yes—
42#16839510C2212yes—
43#16839505C2212yes—
44C61×#532C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#331C2212yes—
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51C2×#11758190—22no—
52C2×#11758200—22no—
53C2×#11758199—21no—
54C2×#11758199—21no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122D412yes—
65C4×Dic122C2312yes—
66C4×D244—21no—
67C4×D244—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×#11758196—21no—
70C2×C4×Dic61—21no—
71#14519711—22no—
72#14519710—22no—
73#14519710—21no—
74#14519710—21no—
75C4⋊#92418C2312yes—
76#14519706—22no—
77#14519706—22no—
78#14519702—22no—
79#14519700—22no—
80#14519699—22no—
81#14519698—22no—
82#14519698—21no—
83#14519698—21no—
84#14519697—21no—
85#14519697—21no—
86C22⋊#92418—21no—
87#14519688—21no—
88C4×#92418D411yescharacteristic
89C4×#92418D411yescharacteristic
90C4×#92418C2311yescharacteristic
91C4×#92418C2311yescharacteristic
92C8×Dic61—22no—
93C8×Dic61—22no—
94C2×#41×C61—21no—
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97#127×C61C2311yescharacteristic
98#124×C61—21no—
99#121×C61—21no—
100#121×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106C4⋊C8×C61C2312yes—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109#41×C61—82no—
110C2×Q8×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—41no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122#11758222—82no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×C4×D61—41no—
127C2×Dic122—21no—
128C2×Dic122—24no—
129C2×Dic122—21no—
130C2×Dic122C2×D412yes—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—42no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61#4111yescharacteristic
141C42×C61C2411yescharacteristic
142#11758200—44no—
143#11758199—42no—
144#11758199—42no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D412yes—
151#11758196—22no—
152#11758196—22no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—41no—
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162#11758190—44no—
163C23×C61—41no—
164C23×C61#12111yescharacteristic
165C22×D61—41no—
166Q8×C61—81no—
167D4×C61—42no—
168D4×C61—41no—
169D4×C61—41no—
170D4×C61—81no—
171D4×C61—82no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—41no—
177C2×C4×C61—82no—
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C22⋊D61—82no—
185C2×Dic61—24no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190D244—42no—
191D244—41no—
192D244—41no—
193D244—81no—
194C4×D61—82no—
195Dic122—24no—
196Dic122—22no—
197Dic122—81no—
198Dic122#13514yes—
199C8×C61—42no—
200C8×C61—42no—
201#92418—21no—
202#92418—21no—
203#92418—21no—
204#92418—42no—
205#92418#13712yes—
206#19264—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61#47212yes—
213C22×C61C2×#12111yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—81no—
222C4×C61C2×#13512yes—
223C4×C61C2×#13711yescharacteristic
224C4×C61#57511yescharacteristic
225Dic61—24no—
226Dic61—42no—
227Dic61—81no—
228#3169—611no—
229#3112—611no—
230#3112—611no—
231C2×C8×Q8—611no—
232#1678—612no—
233#1678—612no—
234#1676—612no—
235#1668—612no—
236#1568—611no—
237#1551—611no—
238#1551D6111yescharacteristic, complemented
239C2×C61—22no—
240C2×C61—161no—
241C2×C61#303711yescharacteristic
242C2×C61#305011yescharacteristic
243C2×C61C2×#47211yescharacteristic
244D61—161no—
245#532—611no—
246#532C2×D6111yescharacteristic
247C2×C4×Q8—611no—
248#500—611no—
249#493—611no—
250#486—611no—
251#486—611no—
252#480—1221no—
253#480—1221no—
254#477—612no—
255#477—612no—
256#466—1221no—
257#466—1221no—
258#460—612no—
259#460—612no—
260#448—1221no—
261#448—1221no—
262C8×Q8—612no—
263C8×Q8—612no—
264C8×Q8—1221no—
265C8×Q8—1221no—
266#440—1222no—
267#440—1222no—
268C4×D8—1221no—
269C4×D8—1221no—
270#432—611no—
271#432C2×D6111yescharacteristic
272C2×C4⋊C8—1221no—
273C2×C4⋊C8C2×D6111yescharacteristic
274C2×C4×C8—611no—
275C2×C4×C8—1221no—
276#335—612no—
277#334—612no—
278#333—612no—
279#333C2×D6112yes—
280#331—612no—
281#331C2×D6112yes—
282C61#1926411yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1221no—
285C22×Q8—611no—
286C2×SD16—1222no—
287C2×SD16—1222no—
288C2×D8—1221no—
289C2×D8—1221no—
290C22×C8—21no—
291C22×C8—611no—
292C22×C8—611no—
293C4⋊Q8—611no—
294C4⋊Q8C22×D6111yescharacteristic
295#127—611no—
296#127C22×D6111yescharacteristic
297#124—21no—
298#124—1221no—
299#121—21no—
300#121—21no—
301#121—1221no—
302#121—1221no—
303C4×Q8—612no—
304C4×Q8—612no—
305C4×Q8—1221no—
306C4×Q8—1221no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—1221no—
310C4×D4—1221no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—1221no—
313C2×C42—1221no—
314C2×C42C22×D6111yescharacteristic
315#107—1222no—
316#107—1221no—
317#107—1221no—
318#107—1222no—
319C4⋊C8—21no—
320C4⋊C8—612no—
321C4⋊C8—1222no—
322C4⋊C8—1222no—
323C4⋊C8C22×D6112yes—
324C4⋊C8C22×D6112yes—
325#103—1222no—
326#103—1222no—
327#102—1221no—
328#102—1221no—
329#102—1222no—
330#102—1222no—
331#102—1221no—
332#102—1221no—
333C22⋊C8—21no—
334C22⋊C8—1221no—
335#97—21no—
336#97—1221no—
337C4×C8—611no—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8—611no—
341C4×C8—1222no—
342C4×C8—1222no—
343#41—82no—
344#41—4882no—
345C2×Q8—21no—
346C2×Q8—612no—
347C2×Q8—1224no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—41no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—2441no—
362C22×C4#1175822211yescharacteristic
363C22×C4D4×D6111yescharacteristic
364C22×C4D4×D6111yescharacteristic
365SD16—2444no—
366SD16—2444no—
367D8—2442no—
368D8—2442no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—42no—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8—611no—
375C2×C8—611no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—2441no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—612no—
382C4⋊C4—1222no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C42—1222no—
392C42—1222no—
393C42#1175822211yescharacteristic
394C42D4×D6111yescharacteristic
395C42D4×D6111yescharacteristic
396C42C23×D6111yescharacteristic
397C23—41no—
398C23—2441no—
399C23#1451975011yescharacteristic
400Q8—81no—
401Q8—614no—
402Q8—1224no—
403Q8—1222no—
404Q8—4881no—
405D4—42no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—82no—
410D4—4881no—
411D4—4882no—
412D4—2442no—
413D4—2441no—
414D4—2441no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—82no—
421C2×C4—1224no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—4882no—
425C2×C4—2441no—
426C2×C4—2442no—
427C2×C4#1451975011yescharacteristic
428C2×C4#1451975011yescharacteristic
429C2×C4#1451975011yescharacteristic
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C2×#1175822211yescharacteristic
433C8—42no—
434C8—42no—
435C8—612no—
436C8—1221no—
437C8—1221no—
438C8—1221no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—4881no—
446C22—4881no—
447C22—4881no—
448C22#1684003112yes—
449C22C2×#1451975011yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—1224no—
456C4—4881no—
457C4—2442no—
458C4#135×D6112yes—
459C4#137×D6111yescharacteristic
460C4#1684055811yescharacteristic
461C2—22no—
462C2—161no—
463C2—9761no—
464C2#2145529011yescharacteristic
465C2#2145539911yescharacteristic
466C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1551C2
socleC22×C61C2×#121
prime radical#21456394C2
hypercenter#1551D61
hypocenterC61#19264
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

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

NH ≅ G/Ntype
2. #21455942C2semidirect
3. #21455937C2semidirect
4. C2×#16839886C2semidirect
5. #21445238C2semidirect
6. #21445234C2semidirect
7. #21445222C2semidirect
8. #21445178C2semidirect
9. #21444580C2semidirect
10. #21444512C2semidirect
11. C61×#1551C2semidirect
238. #1551D61semidirect
282. C61#19264semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214559422
#214559372
C2×#168398862
#214452382
#214452342
#214452222
#214451782
#214445802
#214445122
C61×#15512
#1926461