Finite Group Databasebeta

← #29730414

Subgroups — #29730414

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#21456213C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic
4#21454565C211yesmaximal, characteristic
5#21447263C212yesmaximal, complemented
6#21447118C212yesmaximal, complemented
7#21446808C211yesmaximal, characteristic, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446331C212yesmaximal, complemented
10#21445882C212yesmaximal, complemented
11C4×#101×C61C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519785C2212yes—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840122—22no—
17#16840113—21no—
18#16840109—21no—
19#16840057—21no—
20#16840057—22no—
21#16840057—21no—
22#16839984—22no—
23#16839972—21no—
24#16839962—21no—
25#16839841—21no—
26#16839835—22no—
27C8×D244—21no—
28C2×#14519712C2212yes—
29C2×#14519710C2212yes—
30#16839786C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839764C2212yes—
33#16839764C2212yes—
34#16839761C2212yes—
35C2×#14519701C2212yes—
36#16839730C2212yes—
37#16839530C2212yes—
38#16839530C2212yes—
39#16839526C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×#101×C61C2211yescharacteristic
42C2×#101×C61C2211yescharacteristic
43C4×M16×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#343C2211yescharacteristic
46C61×#343C2211yescharacteristic
47C2×#11758222C2312yes—
48C22×Dic122—21no—
49#14519808—24no—
50#14519808—42no—
51C2×#11758202—22no—
52C2×#11758201—22no—
53#14519785—24no—
54#14519785—42no—
55C2×#11758194C2312yes—
56C2×#11758189—22no—
57C22×#92418C2312yes—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777D411yescharacteristic
64#14519774D411yescharacteristic
65#14519765D412yes—
66#14519751—21no—
67#14519751—21no—
68C4×D244—21no—
69C4×D244—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72#14519712—42no—
73#14519710—22no—
74#14519710—22no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D412yes—
79C4⋊#92418D411yescharacteristic
80#14519702—22no—
81#14519702—22no—
82#14519701—22no—
83#14519701—22no—
84#14519700—22no—
85#14519699—22no—
86#14519699—22no—
87#14519697—22no—
88#14519689—22no—
89#14519689—22no—
90C22×Q8×C61C2311yescharacteristic
91C2×M16×C61C2311yescharacteristic
92C2×M16×C61C2311yescharacteristic
93C2×M16×C61C2311yescharacteristic
94C2×M16×C61C2311yescharacteristic
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#101×C61—22no—
104#101×C61—22no—
105#97×C61—21no—
106#97×C61—21no—
107C4×C8×C61—21no—
108C4×C8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—22no—
121C22×Dic61—22no—
122#11758222—22no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244C2×D412yes—
129C2×C4×D61—22no—
130C2×Dic122—41no—
131C2×Dic122—41no—
132C2×Dic122—82no—
133C2×Dic122C2×D412yes—
134M16×C61—21no—
135M16×C61—21no—
136M16×C61—21no—
137M16×C61—21no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141M16×C61#4112yes—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C42×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156#11758202—44no—
157#11758201—44no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—41no—
165C4⋊Dic61—41no—
166#11758194—22no—
167#11758194—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173#11758189—44no—
174C8×D61—44no—
175C23×C61C2×#4111yescharacteristic
176C22×D61—22no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—41no—
180Q8×C61—41no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—81no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C22⋊D61—42no—
199C22⋊D61—42no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—82no—
203D244—22no—
204D244—22no—
205D244—42no—
206D244—81no—
207C4×D61—42no—
208C4×D61—42no—
209Dic122—22no—
210Dic122—22no—
211Dic122—82no—
212Dic122—82no—
213C8×C61—22no—
214C8×C61—41no—
215C8×C61—41no—
216C8×C61—41no—
217#92418—22no—
218#92418—22no—
219#92418—42no—
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—42no—
233C4×C61—81no—
234C4×C61#47312yes—
235C4×C61#54611yescharacteristic
236C4×C61#54611yescharacteristic
237C4×C61#54711yescharacteristic
238C4×C61#54811yescharacteristic
239Dic61—42no—
240Dic61—82no—
241#3141—611no—
242C2×#477—611no—
243#3037—611no—
244#2935—611no—
245#2078—612no—
246#2052—612no—
247#1997—611no—
248#1950—611no—
249#1908—612no—
250#1816—612no—
251C4×#101D6111yescharacteristic, complemented
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⋊Q8—611no—
262C2×C4×Q8C2×D6111yescharacteristic
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#423—611no—
278C2×C4⋊C8—611no—
279#417—612no—
280#417—612no—
281#417—612no—
282C2×#103—612no—
283C2×#101C2×D6111yescharacteristic
284C2×#101C2×D6111yescharacteristic
285#408—612no—
286C4×M16C2×D6111yescharacteristic
287C4×M16C2×D6111yescharacteristic
288#343—612no—
289#343—612no—
290#343C2×D6111yescharacteristic
291#343C2×D6111yescharacteristic
292#341—612no—
293C61#2536011yescharacteristic, complemented, Sylow
294C2×#41—612no—
295C22×Q8—1221no—
296C22×Q8C22×D6111yescharacteristic
297#137—1224no—
298#137—2442no—
299C2×Q16—1222no—
300C2×SD16—1222no—
301#131—1224no—
302#131—2442no—
303C2×M16—612no—
304C2×M16—1222no—
305C2×M16C22×D6111yescharacteristic
306C2×M16C22×D6111yescharacteristic
307C2×M16C22×D6111yescharacteristic
308C2×M16C22×D6111yescharacteristic
309C22×C8—612no—
310C22×C8—1222no—
311C4⋊Q8—611no—
312C4⋊Q8—1221no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8—1221no—
316#127—611no—
317#124—612no—
318#121—1221no—
319#121—1221no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4C22×D6111yescharacteristic
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#108—2442no—
333#107—1222no—
334#107—1222no—
335C4⋊C8—611no—
336C4⋊C8—612no—
337C4⋊C8—611no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
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—22no—
349#101—22no—
350C22⋊C8—1222no—
351C22⋊C8—1222no—
352#97—21no—
353#97—21no—
354C4×C8—21no—
355C4×C8—21no—
356#41—1222no—
357#41—1222no—
358#41—1222no—
359#41—1222no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—612no—
366C2×Q8—4882no—
367C2×Q8—2441no—
368C2×Q8—2441no—
369C2×Q8#1175822212yes—
370C2×D4—612no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1221no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4#1175822211yescharacteristic
379C22×C4#1175822211yescharacteristic
380C22×C4C23×D6111yescharacteristic
381Q16—2444no—
382SD16—2444no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—1222no—
391M16—1222no—
392M16—2444no—
393M16#1175822212yes—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—612no—
399C2×C8—612no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—2444no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—41no—
417C42—41no—
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42D4×D6111yescharacteristic
421C42D4×D6111yescharacteristic
422C23—1222no—
423C23C2×#1175822211yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—1222no—
429Q8—1222no—
430Q8—4882no—
431Q8—4882no—
432D4—1222no—
433D4—1222no—
434D4—4881no—
435D4—2442no—
436D4—2442no—
437D4—2442no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—1222no—
446C2×C4—4882no—
447C2×C4—2442no—
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×C4C2×#1175822211yescharacteristic
457C2×C4C2×#1175822211yescharacteristic
458C2×C4#1451982311yescharacteristic
459C2×C4#1451982411yescharacteristic
460C8—22no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—1222no—
465C8—1222no—
466C8—2442no—
467C22—21no—
468C22—21no—
469C22—2442no—
470C22—2442no—
471C22—2442no—
472C22C2×#1451975011yescharacteristic
473C22#1684029611yescharacteristic
474C22#1684038311yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—81no—
480C4—4882no—
481C4—2442no—
482C4#1684003612yes—
483C4#1684040811yescharacteristic
484C4#1684040811yescharacteristic
485C4#1684041711yescharacteristic
486C4#1684042111yescharacteristic
487C2—21no—
488C2—21no—
489C2—4882no—
490C2#2144939611yescharacteristic
491C2C2×#1684003611yescharacteristic
492C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#101×C61C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC4×#101D61
hypocenterC61#25360
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
5. #21447263C2semidirect
6. #21447118C2semidirect
7. #21446808C2semidirect
8. #21446594C2semidirect
9. #21446331C2semidirect
10. #21445882C2semidirect
11. C4×#101×C61C2semidirect
251. C4×#101D61semidirect
293. C61#25360semidirect

Maximal subgroups

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

subgroupindex
#214562132
C2×#168400572
#214552912
#214545652
#214472632
#214471182
#214468082
#214465942
#214463312
#214458822
C4×#101×C612
#2536061