Finite Group Databasebeta

← #29686957

Subgroups — #29686957

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455535C211yesmaximal, characteristic
2#21454720C212yesmaximal, complemented
3#21454244C211yesmaximal, characteristic, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446607C211yesmaximal, characteristic, complemented
6#21446485C212yesmaximal, complemented
7#21446323C212yesmaximal, complemented
8#21445874C211yesmaximal, characteristic, complemented
9#21445703C211yesmaximal, characteristic, complemented
10#21445619C212yesmaximal, complemented
11C2×C61×#444C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519783C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840034C412yescomplemented
17#16840029C412yescomplemented
18#16840022C414yescomplemented
19#16839882—24no—
20#16839877—24no—
21#16839810C2212yes—
22C2×#14519713C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839779C2212yes—
25#16839778C2212yes—
26#16839746C2212yes—
27#16839736C2212yes—
28#16839719—21no—
29#16839719—21no—
30#16839717C2211yescharacteristic
31#16839717C2212yes—
32#16839717C2211yescharacteristic
33C4×#11758194C2211yescharacteristic
34#16839714—21no—
35#97×D61—21no—
36#16839533C2212yes—
37#16839533C2212yes—
38#16839522C2212yes—
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2212yes—
41C61×#444—21no—
42C61×#444—21no—
43C61×#444—21no—
44C61×#444—21no—
45C2×#108×C61C2211yescharacteristic
46C2×#104×C61C2212yes—
47C2×#97×C61C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808C2×C414yes—
50#14519807—22no—
51#14519803C2×C414yes—
52#14519802—22no—
53#14519797—24no—
54#14519783—21no—
55#14519783—21no—
56#14519783—22no—
57#14519783C2×C412yes—
58#14519783C2×C412yes—
59#14519783D414yes—
60C2×#11758194C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758194C2311yescharacteristic
63C2×#11758189—21no—
64C2×#11758189—21no—
65C2×C8×D61—21no—
66C2×C8×D61—21no—
67C4×Dic122—21no—
68C4×D244—21no—
69C4×C22⋊D61—22no—
70#14519736C2312yes—
71#14519736C2311yescharacteristic
72#14519734—21no—
73C42×D61—21no—
74#14519713C2×C412yescomplemented
75#14519713C2×C412yescomplemented
76#14519712—24no—
77#14519711—21no—
78#14519709—21no—
79#14519704—24no—
80#14519703—24no—
81#14519696C2×C414yescomplemented
82#14519694C2×C414yescomplemented
83#14519692—24no—
84#14519688—21no—
85#14519687—21no—
86#14519687—21no—
87#14519687—22no—
88C4×#92418—21no—
89C8×Dic61—21no—
90C8×Dic61—21no—
91C8×Dic61—22no—
92C2×#41×C61C2312yes—
93#135×C61—21no—
94#135×C61—21no—
95#135×C61—21no—
96#135×C61—21no—
97C2×Q16×C61C2×C411yescharacteristic, complemented
98C2×SD16×C61C2×C412yescomplemented
99C2×D8×C61C2×C411yescharacteristic, complemented
100#131×C61—22no—
101#131×C61—22no—
102#131×C61—22no—
103#131×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2312yes—
106C22×C8×C61—22no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110#108×C61—21no—
111#108×C61—21no—
112#104×C61—22no—
113#104×C61—22no—
114#97×C61D411yescharacteristic
115#97×C61D411yescharacteristic
116#97×C61D411yescharacteristic
117#97×C61D411yescharacteristic
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61C22×C412yes—
123C2×D4×C61—22no—
124C2×D4×C61C22×C412yes—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—21no—
129#11758222—21no—
130#11758222—21no—
131#11758222C22×C412yes—
132#11758222C2×D412yes—
133C2×D244#4111yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×Dic122#4111yescharacteristic
137Q16×C61—21no—
138Q16×C61—21no—
139SD16×C61—22no—
140SD16×C61—22no—
141D8×C61—21no—
142D8×C61—21no—
143M16×C61—22no—
144M16×C61—22no—
145M16×C61—42no—
146M16×C61—42no—
147C2×C8×C61—22no—
148C2×C8×C61—42no—
149C2×C8×C61—42no—
150C2×C8×C61C22×C411yescharacteristic
151C2×C8×C61C22×C411yescharacteristic
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D411yescharacteristic
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61#4111yescharacteristic
158C2×C8×C61#4111yescharacteristic
159C2×C8×C61#4112yes—
160C42×C61—21no—
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C22⋊Dic61—22no—
164C22⋊Dic61#4112yes—
165#11758202—24no—
166#11758201—24no—
167#11758200—24no—
168#11758199—24no—
169#11758198—22no—
170C4⋊Dic61—21no—
171C4⋊Dic61—22no—
172#11758196—22no—
173C4×Dic61—22no—
174C4×Dic61—21no—
175C4×Dic61—22no—
176#11758194—21no—
177#11758194—21no—
178#11758194—22no—
179#11758194C22×C414yes—
180#11758194C22×C412yes—
181#11758194C2×D412yes—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—22no—
185#11758189—22no—
186#11758189—22no—
187#11758189—24no—
188C8×D61—22no—
189C8×D61—22no—
190C8×D61—24no—
191C23×C61—22no—
192C23×C61C2×#4111yescharacteristic
193C22×D61—21no—
194Q8×C61—22no—
195Q8×C61—22no—
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—42no—
199D4×C61—42no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—81no—
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C22⋊D61—22no—
213C22⋊D61—22no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—42no—
217D244—21no—
218D244C4×D412yes—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—82no—
222Dic122—21no—
223Dic122C4×D412yes—
224C8×C61—42no—
225C8×C61C4×D411yescharacteristic
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61#12112yes—
230C8×C61#12112yes—
231#92418—22no—
232#92418—24no—
233#92418—22no—
234#21559—611nomaximal, Sylow
235C22×C61—41no—
236C22×C61—42no—
237C22×C61—82no—
238C22×C61C4×#4111yescharacteristic
239C22×C61#52511yescharacteristic
240C22×C61#53211yescharacteristic
241C2×D61—22no—
242C2×D61—42no—
243C4×C61—41no—
244C4×C61—41no—
245C4×C61—42no—
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×#12111yescharacteristic
249C4×C61#52911yescharacteristic
250Dic61—22no—
251Dic61—42no—
252Dic61—82no—
253#3067—611no—
254#2955—612no—
255C2×#444D6111yescharacteristic, complemented
256#2875—611no—
257#1976—611no—
258#1953—611no—
259#1932—612no—
260#1906—612no—
261#1814—611no—
262#1773—611no—
263#1757—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—82no—
268C2×C61C4×#12111yescharacteristic
269D61—42no—
270#575—612no—
271C2×#135C2×D6111yescharacteristic
272C2×#131—611no—
273C2×#131—611no—
274C2×#131C2×D6112yes—
275C4×#41—611no—
276#473—612no—
277#472—612no—
278#471—614no—
279#444—21no—
280#444—21no—
281#444—21no—
282#444—21no—
283#444—1224no—
284#443—1224no—
285#430—612no—
286C2×#108—611no—
287C2×#108C2×D6111yescharacteristic
288#427—611no—
289#421—612no—
290#421—612no—
291C2×#104C2×D6112yes—
292#413—612no—
293#410—612no—
294#405—611no—
295#405—611no—
296#405—612no—
297#405—1221no—
298#405—1221no—
299C4×M16—611no—
300C4×M16—1221no—
301C2×#97—1221no—
302C2×#97C2×D6111yescharacteristic
303#344—612no—
304#344—612no—
305#339—612no—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41—611no—
308C2×#41C22×D6112yes—
309#137—614no—
310#137—1222no—
311AGL(1,ℤ/8)—614no—
312AGL(1,ℤ/8)—1222no—
313#135—21no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—1224no—
318C2×Q16C4×D6111yescharacteristic, complemented
319C2×SD16C4×D6112yescomplemented
320C2×D8C4×D6111yescharacteristic, complemented
321#131—22no—
322#131—22no—
323#131—22no—
324#131—22no—
325#131—612no—
326#131—612no—
327#131—614no—
328#131—1221no—
329#131—1221no—
330#131—1222no—
331C2×M16—22no—
332C2×M16—611no—
333C2×M16—611no—
334C2×M16—612no—
335C2×M16—1221no—
336C2×M16—1221no—
337C2×M16C22×D6112yes—
338C22×C8—22no—
339C22×C8—1221no—
340C22×C8—1221no—
341C22×C8C22×D6111yescharacteristic
342C22×C8C22×D6111yescharacteristic
343C4×Q8—1221no—
344C4×D4—1221no—
345C4×D4—1222no—
346#117—611no—
347#117—612no—
348#117—1221no—
349C2×C42—1221no—
350C2×C42C22×D6111yescharacteristic
351#108—21no—
352#108—21no—
353#108—612no—
354#108—612no—
355#108—1224no—
356#107—1221no—
357#106—1221no—
358#104—22no—
359#104—22no—
360#104—1224no—
361#104—1224no—
362#101—614no—
363#100—614no—
364#99—1224no—
365#97—1222no—
366#97—1221no—
367#97—1221no—
368#97—1221no—
369#97C22⋊D6111yescharacteristic
370#97C22⋊D6111yescharacteristic
371#97C22⋊D6111yescharacteristic
372#97C22⋊D6111yescharacteristic
373C4×C8—1222no—
374C4×C8—1221no—
375C4×C8—1221no—
376C4×C8—1221no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—612no—
382#41—612no—
383#41—1221no—
384#41—1221no—
385C2×Q8—611no—
386C2×Q8C2×C4×D6112yes—
387C2×D4—22no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4C2×C4×D6112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4C23×D6111yescharacteristic
396Q16—21no—
397Q16—21no—
398Q16—1224no—
399SD16—22no—
400SD16—22no—
401SD16—1224no—
402SD16—1224no—
403D8—21no—
404D8—21no—
405D8—1224no—
406M16—22no—
407M16—22no—
408M16—42no—
409M16—42no—
410M16—612no—
411M16—614no—
412M16—612no—
413M16—1222no—
414M16—1221no—
415M16—1221no—
416M16—1222no—
417M16—1222no—
418M16—1224no—
419C2×C8—22no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—1222no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1224no—
428C2×C8C2×C4×D6111yescharacteristic
429C2×C8C2×C4×D6111yescharacteristic
430C2×C8#1175822211yescharacteristic
431C2×C8#1175822211yescharacteristic
432C2×C8D4×D6112yes—
433C2×C8#1175822412yes—
434C2×C8C2×C22⋊D6111yescharacteristic
435C2×C8C2×C22⋊D6111yescharacteristic
436C2×C8C2×C22⋊D6111yescharacteristic
437C2×C8C2×C22⋊D6111yescharacteristic
438C4⋊C4—1222no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C22⋊C4—612no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C42—21no—
445C42—1222no—
446C42—1221no—
447C42—1222no—
448C42C2×C22⋊D6111yescharacteristic
449C42C2×C22⋊D6111yescharacteristic
450C23—22no—
451C23—1221no—
452C23#41×D6111yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—612no—
456Q8—1221no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—612no—
462D4—1222no—
463D4—1222no—
464D4—1221no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—1221no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—4882no—
476C2×C4—2442no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C22×C22⋊D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822411yescharacteristic
482C2×C4C2×#1175822211yescharacteristic
483C8—42no—
484C8—1224no—
485C8—1222no—
486C8—1222no—
487C8C4×C22⋊D6111yescharacteristic
488C8C4×C22⋊D6111yescharacteristic
489C8C4×C22⋊D6111yescharacteristic
490C8C4×C22⋊D6111yescharacteristic
491C8#1451974812yes—
492C8#1451974812yes—
493C22—41no—
494C22—42no—
495C22—82no—
496C22—1222no—
497C22—2442no—
498C22C4×#1175822411yescharacteristic
499C22#1684026911yescharacteristic
500C22#1684030911yescharacteristic
501C4—41no—
502C4—41no—
503C4—42no—
504C4—1222no—
505C4—4882no—
506C4—2442no—
507C4C4×D4×D6111yescharacteristic
508C4C2×C4×C22⋊D6111yescharacteristic
509C4C2×#1451974811yescharacteristic
510C4#1684029411yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—2442no—
516C2C4×#1451974811yescharacteristic
subgroupG/H
centerC4C2×#14519748
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#444C2
socleC2×C61C4×#121
prime radical#21455535C2
cosocleC22×C4×C61C24
hypercenterC2×#444D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #21454720C2semidirect
3. #21454244C2semidirect
4. #21446718C2semidirect
5. #21446607C2semidirect
6. #21446485C2semidirect
7. #21446323C2semidirect
8. #21445874C2semidirect
9. #21445703C2semidirect
10. #21445619C2semidirect
11. C2×C61×#444C2semidirect
16. #16840034C4semidirect
17. #16840029C4semidirect
18. #16840022C4semidirect
74. #14519713C2×C4semidirect
75. #14519713C2×C4semidirect
81. #14519696C2×C4semidirect
82. #14519694C2×C4semidirect
97. C2×Q16×C61C2×C4semidirect
98. C2×SD16×C61C2×C4semidirect
99. C2×D8×C61C2×C4semidirect
255. C2×#444D61semidirect
306. C61#21559semidirect
318. C2×Q16C4×D61semidirect
319. C2×SD16C4×D61semidirect
320. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555352
#214547202
#214542442
#214467182
#214466072
#214464852
#214463232
#214458742
#214457032
#214456192
C2×C61×#4442
#2155961