Finite Group Databasebeta

← #29730415

Subgroups — #29730415

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456215C211yesmaximal, characteristic
2C2×#16840054C211yesmaximal, characteristic, complemented
3#21455293C211yesmaximal, characteristic
4#21454565C211yesmaximal, characteristic
5#21447263C212yesmaximal, complemented
6#21447121C212yesmaximal, complemented
7#21446592C211yesmaximal, characteristic, complemented
8#21446335C212yesmaximal, complemented
9#21445882C212yesmaximal, complemented
10C4×#14519695C211yesmaximal, characteristic, complemented
11C61×#1997C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C2×#14519785C2212yes—
14#16840312C2211yescharacteristic
15C2×C4×Dic122C2211yescharacteristic
16#16840124—22no—
17#16840114—21no—
18#16840110—21no—
19#16840054—21no—
20#16840054—22no—
21#16840054—21no—
22#16839985—22no—
23#16839973—21no—
24#16839963—21no—
25#16839841—21no—
26#16839835—22no—
27C8×D244—21no—
28C2×#14519712C2212yes—
29C2×#14519710C2212yes—
30C2×C4⋊#92418C2211yescharacteristic
31#16839764C2212yes—
32#16839761C2212yes—
33#16839761C2212yes—
34C2×#14519700C2212yes—
35C2×#14519695C2211yescharacteristic
36C2×#14519695C2211yescharacteristic
37#16839730C2212yes—
38C4×#11758194C2211yescharacteristic
39#16839530C2211yescharacteristic
40#16839530C2211yescharacteristic
41#16839526C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#423C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2212yes—
47C2×#11758222C2312yes—
48C22×Dic122C2311yescharacteristic
49#14519810—24no—
50#14519810—42no—
51C2×Dic244—22no—
52C2×#11758190—22no—
53#14519785—24no—
54#14519785—42no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758189—22no—
58C22×#92418C2312yes—
59C2×C8×D61—22no—
60#14519777D411yescharacteristic
61#14519774D411yescharacteristic
62#14519765D412yes—
63#14519751—21no—
64#14519751—21no—
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×D244—21no—
70C4×D244—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×C4×Dic61—21no—
74#14519712—42no—
75#14519710—22no—
76#14519710—22no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D412yes—
79C4⋊#92418D411yescharacteristic
80#14519702—22no—
81#14519700—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519699—22no—
86#14519697—22no—
87#14519697—22no—
88#14519695—22no—
89#14519695—22no—
90#14519689—22no—
91#14519689—22no—
92#14519688—21no—
93C4×#92418—21no—
94C22×Q8×C61—21no—
95C2×M16×C61C2311yescharacteristic
96C2×M16×C61C2312yes—
97C2×M16×C61C2311yescharacteristic
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107#97×C61—21no—
108C4×C8×C61—21no—
109C2×Q8×C61—41no—
110C2×Q8×C61—41no—
111C2×Q8×C61—82no—
112C22×C4×C61—22no—
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—22no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119#11758222—22no—
120#11758222—22no—
121#11758222—22no—
122#11758222—22no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244C2×D412yes—
126C2×C4×D61—22no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122C2×D412yes—
133C2×Dic122#4112yes—
134M16×C61—21no—
135M16×C61—22no—
136M16×C61—21no—
137M16×C61—22no—
138M16×C61—21no—
139M16×C61#4112yes—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151#11758198—22no—
152#11758198—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157#11758196—41no—
158#11758196—41no—
159C4×Dic61—41no—
160C4×Dic61—41no—
161#11758194—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171Dic244—44no—
172#11758190—44no—
173#11758189—44no—
174C8×D61—44no—
175C23×C61C2×#4111yescharacteristic
176C22×D61—22no—
177Q8×C61—82no—
178Q8×C61—82no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—82no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)11yescharacteristic
192C2×C4×C612−1+411yescharacteristic
193C22⋊D61—42no—
194C22⋊D61—42no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—81no—
201D244—22no—
202D244—22no—
203D244—42no—
204D244—81no—
205C4×D61—42no—
206C4×D61—42no—
207Dic122—22no—
208Dic122—22no—
209Dic122—22no—
210Dic122—22no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—22no—
214C8×C61—42no—
215C8×C61—41no—
216#92418—22no—
217#92418—22no—
218#92418—41no—
219#92418—41no—
220#25360—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C2×#12111yescharacteristic
224C22×C61#52911yescharacteristic
225C22×C61#54211yescharacteristic
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—82no—
233C4×C61#47312yes—
234C4×C61#54611yescharacteristic
235C4×C61#54611yescharacteristic
236C4×C61#54711yescharacteristic
237C4×C61#54811yescharacteristic
238Dic61—42no—
239Dic61—42no—
240Dic61—81no—
241#3141—611no—
242C2×#477—611no—
243#3037—611no—
244#2935—611no—
245#2078—612no—
246#2052—612no—
247#1997D6111yescharacteristic, complemented
248#1950—611no—
249#1908—612no—
250#1816—612no—
251C4×#101—611no—
252C2×C61—21no—
253C2×C61—21no—
254C2×C61#242711yescharacteristic
255C2×C61C2×#47311yescharacteristic
256C2×C61#306811yescharacteristic
257D61—82no—
258C2×#137—612no—
259C2×#131—612no—
260#532—611no—
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×C4×Q8—611no—
263#489—1222no—
264#487—1221no—
265#486—1221no—
266#477—1221no—
267#477—1222no—
268#477—1221no—
269#464—1222no—
270#462—1221no—
271#460—1221no—
272#436—1221no—
273#435—1222no—
274C8×D4—1221no—
275C2×#108—612no—
276C2×#107—612no—
277#423C2×D6111yescharacteristic
278C2×C4⋊C8—611no—
279#417—612no—
280#417—612no—
281#417—612no—
282C2×#103—612no—
283C2×#101—611no—
284C2×#101—611no—
285#408—612no—
286C4×M16—611no—
287C4×M16C2×D6111yescharacteristic
288#343—611no—
289#343—611no—
290#343C2×D6112yes—
291#343C2×D6112yes—
292#341—612no—
293C61#2536011yescharacteristic, complemented, Sylow
294C2×#41—612no—
295C22×Q8—21no—
296C22×Q8—611no—
297#137—1224no—
298#137—2442no—
299C2×Q16—1222no—
300C2×SD16—1222no—
301#131—1224no—
302#131—2442no—
303C2×M16—611no—
304C2×M16—611no—
305C2×M16—1222no—
306C2×M16C22×D6112yes—
307C2×M16C22×D6111yescharacteristic
308C2×M16C22×D6111yescharacteristic
309C22×C8—612no—
310C22×C8—1222no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—611no—
316#127—611no—
317#124—612no—
318#121—1221no—
319#121—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4C22×D6112yes—
330C2×C42—1221no—
331C2×C42C22×D6111yescharacteristic
332#108—2442no—
333#107—1222no—
334#107—1222no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—611no—
338C4⋊C8—612no—
339C4⋊C8—611no—
340#103—1222no—
341#103—1222no—
342#103—1222no—
343#103—1222no—
344#103—1222no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#101—1222no—
349#101—1222no—
350C22⋊C8—1222no—
351C22⋊C8—1222no—
352#97—21no—
353#97—1221no—
354C4×C8—21no—
355C4×C8—1221no—
356#41—1222no—
357#41—1222no—
358#41—1222no—
359#41—1222no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—82no—
363C2×Q8—612no—
364C2×Q8—612no—
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1221no—
370C2×D4—612no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1221no—
374C22×C4—22no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4—1221no—
378C22×C4#1175822211yescharacteristic
379C22×C4#1175822211yescharacteristic
380C22×C4C23×D6111yescharacteristic
381Q16—2444no—
382SD16—2444no—
383M16—21no—
384M16—21no—
385M16—22no—
386M16—22no—
387M16—21no—
388M16—1221no—
389M16—1221no—
390M16—1221no—
391M16—1221no—
392M16—2444no—
393M16#1175822612yes—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—612no—
398C2×C8—612no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—2444no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—2441no—
417C42—2441no—
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42D4×D6111yescharacteristic
421C42D4×D6111yescharacteristic
422C23—1222no—
423C23C2×#1175822211yescharacteristic
424Q8—82no—
425Q8—82no—
426Q8—1222no—
427Q8—1222no—
428Q8—1222no—
429Q8—1222no—
430Q8—2441no—
431Q8—2441no—
432D4—1222no—
433D4—1222no—
434D4—4881no—
435D4—2442no—
436D4—2442no—
437D4—2442no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—4881no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4#1451975011yescharacteristic
451C2×C4#1451975011yescharacteristic
452C2×C4#1451975011yescharacteristic
453C2×C4#1451975011yescharacteristic
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C2×D4×D6111yescharacteristic
456C2×C4#41×D6111yescharacteristic
457C2×C4C2×#1175822611yescharacteristic
458C2×C4#1451982311yescharacteristic
459C2×C4#1451982511yescharacteristic
460C8—22no—
461C8—42no—
462C8—41no—
463C8—1222no—
464C8—1222no—
465C8—2441no—
466C8—2441no—
467C22—21no—
468C22—21no—
469C22—2442no—
470C22—2442no—
471C22—2442no—
472C22C2×#1451975011yescharacteristic
473C22#1684029911yescharacteristic
474C22#1684038411yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—82no—
479C4—4881no—
480C4—2442no—
481C4—2442no—
482C4#1684003612yes—
483C4#1684040311yescharacteristic
484C4#1684040511yescharacteristic
485C4#1684041511yescharacteristic
486C4#1684042211yescharacteristic
487C2—21no—
488C2—21no—
489C2—4882no—
490C2#2144939711yescharacteristic
491C2C2×#1684003611yescharacteristic
492C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1997D61
hypocenterC61#25360
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
2. C2×#16840054C2semidirect
5. #21447263C2semidirect
6. #21447121C2semidirect
7. #21446592C2semidirect
8. #21446335C2semidirect
9. #21445882C2semidirect
10. C4×#14519695C2semidirect
11. C61×#1997C2semidirect
247. #1997D61semidirect
293. C61#25360semidirect

Maximal subgroups

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

subgroupindex
#214562152
C2×#168400542
#214552932
#214545652
#214472632
#214471212
#214465922
#214463352
#214458822
C4×#145196952
C61×#19972
#2536061