Finite Group Databasebeta

← #29663231

Subgroups — #29663231

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic, complemented
2#21456245C211yesmaximal, characteristic
3#21454754C211yesmaximal, characteristic, complemented
4#21445311C212yesmaximal, complemented
5#21445301C212yesmaximal, complemented
6#21445169C212yesmaximal, complemented
7#21445165C212yesmaximal, complemented
8C2×#16839512C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C61×#3246C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13C2×#14519797—21no—
14#16840312C2211yescharacteristic
15#16840144—21no—
16#16840144—21no—
17#16840141C2211yescharacteristic
18#16840138C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840134C2211yescharacteristic
21#16840049—21no—
22#16840047C2212yes—
23#16839993—22no—
24#16839964—21no—
25#16839962C2212yes—
26#16839893—21no—
27#16839893C2212yes—
28Q8×#92418—21no—
29Q8×#92418C2212yes—
30#16839789—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839718—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839512C2212yes—
36#16839512C2212yes—
37#16839508C2212yes—
38#16839507C2212yes—
39#16839506C2212yes—
40#16839505C2212yes—
41C61×#533C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C61×#476—21no—
44C61×#475C2212yes—
45C61×#465—22no—
46C61×#461—21no—
47C61×#460C2212yes—
48C2×C4⋊C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50#14519797—44no—
51#14519797—44no—
52C2×#11758202—21no—
53C2×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758200—21no—
56C2×#11758201—22no—
57C2×#11758201—21no—
58C2×#11758201—21no—
59C2×#11758199—21no—
60C2×#11758199—21no—
61C2×#11758194—21no—
62C22×#92418—21no—
63C22×#92418D411yescharacteristic
64C22×#92418D411yescharacteristic
65#14519777C2311yescharacteristic
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×D244—21no—
71C4×D244—21no—
72#14519708—22no—
73C4⋊#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418—22no—
76C4⋊#92418—22no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2312yes—
79#14519701—22no—
80#14519697—22no—
81#14519688—22no—
82C4×#92418—22no—
83C4×#92418D411yescharacteristic
84C4×#92418D411yescharacteristic
85C4×#92418C2311yescharacteristic
86C4×#92418C2311yescharacteristic
87C2×#41×C61—21no—
88C2×SD16×C61—22no—
89C22×C8×C61—21no—
90C4⋊Q8×C61D411yescharacteristic
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61C2311yescharacteristic
94#125×C61—21no—
95#124×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61D412yes—
98#121×C61—21no—
99#121×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61C2312yes—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#106×C61—22no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#102×C61—22no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125#11758222—42no—
126#11758222—42no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132SD16×C61—82no—
133C2×C8×C61—22no—
134C2×C8×C61—81no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61—22no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61#4111yescharacteristic
150C42×C61C2411yescharacteristic
151#11758202—42no—
152#11758202—42no—
153#11758201—42no—
154#11758201—42no—
155#11758201—82no—
156#11758200—42no—
157#11758200—42no—
158#11758199—42no—
159#11758199—42no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—81no—
169C2×#92418SD1614yescomplemented
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174C23×C61—41no—
175C23×C61#12111yescharacteristic
176C22×D61—41no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—42no—
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—81no—
187D4×C61—82no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61C2×SD1612yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C22⋊D61—82no—
204C2×Dic61—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—41no—
209D244—81no—
210C4×D61—82no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—42no—
214#92418—21no—
215#92418—21no—
216#92418—42no—
217#92418—42no—
218#92418C2×SD1612yes—
219#92418C2×SD1612yes—
220#19205—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#46012yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61#46012yes—
238C4×C61C22×SD1611yescharacteristic
239C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C61#57711yescharacteristic
241C4×C61#57711yescharacteristic
242Dic61—81no—
243#3246—611no—
244#3246D6111yescharacteristic, complemented
245#3147—611no—
246#2960—611no—
247#1690—612no—
248#1688—612no—
249#1666—612no—
250#1666—612no—
251C2×#334—611no—
252#1547—611no—
253#1547—611no—
254C2×C61—22no—
255C2×C61—161no—
256C2×C61C2×#46011yescharacteristic
257C2×C61#306011yescharacteristic
258C2×C61#306711yescharacteristic
259D61—161no—
260C2×#135—1221no—
261C2×#135—1221no—
262#533C2×D6111yescharacteristic
263#532—611no—
264#532C2×D6111yescharacteristic
265#495—1221no—
266#495—1221no—
267#494—611no—
268#493—611no—
269#492—611no—
270#492—611no—
271#476—21no—
272#476—1221no—
273#475—612no—
274#475C2×D6112yes—
275#465—22no—
276#465—1222no—
277#461—21no—
278#461—1221no—
279#460—612no—
280#460C2×D6112yes—
281#446—612no—
282#446—1221no—
283C8×Q8—612no—
284C8×Q8—1221no—
285#424—1221no—
286#423—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8C2×D6111yescharacteristic
289#405—1221no—
290C2×C4×C8—611no—
291#334—612no—
292#334—612no—
293#332—612no—
294#332—612no—
295#331—612no—
296#331—612no—
297C61#1920511yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—1221no—
300#135—2444no—
301#135—2444no—
302C2×Q16—1221no—
303C2×Q16—1221no—
304C2×SD16—22no—
305C2×SD16—1221no—
306C2×SD16—1221no—
307C2×SD16—1221no—
308C2×SD16—1221no—
309C2×SD16—1222no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×M16—1221no—
313C22×C8—21no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—1221no—
317C4⋊Q8—611no—
318C4⋊Q8C22⋊D6111yescharacteristic
319C4⋊Q8C22⋊D6111yescharacteristic
320C4⋊Q8C22×D6111yescharacteristic
321#127—611no—
322#127C22×D6111yescharacteristic
323#125—21no—
324#124—21no—
325#124—1221no—
326C22⋊Q8—21no—
327C22⋊Q8C22⋊D6112yes—
328#121—21no—
329#121—21no—
330#121—1221no—
331#121—1221no—
332C4×Q8—21no—
333C4×Q8C22×D6112yes—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—1221no—
337C4×D4—1221no—
338#117—21no—
339C2×C42C22×D6111yescharacteristic
340#106—22no—
341#106—1222no—
342C4⋊C8—21no—
343C4⋊C8—612no—
344C4⋊C8—612no—
345C4⋊C8—1222no—
346C4⋊C8—1222no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349C4⋊C8C22×D6112yes—
350#103—22no—
351#103—1222no—
352#102—22no—
353#102—1222no—
354#97—1222no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—1222no—
360#41—42no—
361#41—42no—
362#41—2442no—
363#41—2442no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—1221no—
367C2×Q8C2×C22⋊D6112yes—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—2441no—
376C22×C4—41no—
377C22×C4—2441no—
378C22×C4D4×D6111yescharacteristic
379C22×C4#1175822411yescharacteristic
380C22×C4C2×C22⋊D6111yescharacteristic
381Q16—2442no—
382Q16—2442no—
383SD16—82no—
384SD16—4882no—
385SD16—2442no—
386SD16—2442no—
387SD16—2442no—
388SD16—2442no—
389D8—2442no—
390D8—2442no—
391M16—2442no—
392C2×C8—22no—
393C2×C8—81no—
394C2×C8—611no—
395C2×C8—611no—
396C2×C8—614no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—4881no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4C2×C22⋊D6112yes—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C42—22no—
419C42D4×D6111yescharacteristic
420C42#1175822411yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—41no—
424C23—2441no—
425C23#1451974911yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—42no—
429Q8—41no—
430Q8—41no—
431Q8—2441no—
432Q8—2441no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4—4881no—
440D4—4882no—
441D4—2441no—
442D4—2441no—
443D4—2441no—
444D4—2441no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—4882no—
453C2×C4—2441no—
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4#1451974911yescharacteristic
457C2×C4SD16×D6112yes—
458C2×C4#1451980412yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C2×#1175822411yescharacteristic
462C8—42no—
463C8—612no—
464C8—612no—
465C8—1221no—
466C8—1221no—
467C8—2442no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683996012yes—
478C22C2×#1451974911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—4881no—
486C4#1683996012yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451980411yescharacteristic
489C4#1684057011yescharacteristic
490C4#1684057311yescharacteristic
491C2—22no—
492C2—161no—
493C2—9761no—
494C2C2×#1683996011yescharacteristic
495C2#2145547811yescharacteristic
496C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3246C2
socleC22×C61C2×#121
prime radical#21456245C2
hypercenter#3246D61
hypocenterC61#19205
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
1. #21457131C2semidirect
3. #21454754C2semidirect
4. #21445311C2semidirect
5. #21445301C2semidirect
6. #21445169C2semidirect
7. #21445165C2semidirect
8. C2×#16839512C2semidirect
9. #21444496C2semidirect
10. #21444494C2semidirect
11. C61×#3246C2semidirect
169. C2×#92418SD16semidirect
244. #3246D61semidirect
297. C61#19205semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214562452
#214547542
#214453112
#214453012
#214451692
#214451652
C2×#168395122
#214444962
#214444942
C61×#32462
#1920561