Finite Group Databasebeta

← #29686965

Subgroups — #29686965

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#21455540C211yesmaximal, characteristic
2#21454723C212yesmaximal, complemented
3C2×#16839885C211yesmaximal, characteristic, complemented
4#21454244C211yesmaximal, characteristic, complemented
5#21446715C211yesmaximal, characteristic, complemented
6#21446610C211yesmaximal, characteristic, complemented
7#21446489C212yesmaximal, complemented
8#21446319C212yesmaximal, complemented
9#21445870C211yesmaximal, characteristic, complemented
10#21445620C212yesmaximal, complemented
11C61×#1773C211yesmaximal, characteristic, complemented
12#16840562C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519783C2212yes—
15C2×#14519783C2211yescharacteristic
16C2×#14519783C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840038C412yescomplemented
19#16840033C412yescomplemented
20#16840026C414yescomplemented
21#16839885—24no—
22#16839885—21no—
23#16839885—21no—
24#16839885—21no—
25#16839885—21no—
26#16839880—24no—
27#16839811C2212yes—
28C2×#14519713C2211yescharacteristic
29#16839780C2212yes—
30#16839780C2212yes—
31C2×#14519705C2212yes—
32#16839744C2212yes—
33#16839735C2212yes—
34#16839719—21no—
35#16839719—21no—
36#16839717C2211yescharacteristic
37#16839717C2211yescharacteristic
38C4×#11758194C2211yescharacteristic
39#16839714—21no—
40#97×D61—21no—
41#16839534C2212yes—
42#16839532C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#427C2211yescharacteristic
45C61×#405C2212yes—
46C2×#97×C61C2211yescharacteristic
47C61×#339C2212yes—
48C2×#11758222C2312yes—
49C2×#11758222C2311yescharacteristic
50#14519810—22no—
51#14519810C2×C414yes—
52#14519805—22no—
53#14519805C2×C414yes—
54#14519799—24no—
55#14519799—21no—
56#14519799—21no—
57#14519799—21no—
58#14519799—21no—
59C2×Dic244C2×C411yescharacteristic, complemented
60C2×#11758190C2×C412yescomplemented
61C2×D488C2×C411yescharacteristic, complemented
62#14519783—21no—
63#14519783—21no—
64#14519783—22no—
65#14519783—22no—
66#14519783—22no—
67#14519783—22no—
68#14519783—22no—
69#14519783C2×C412yes—
70#14519783C2×C412yes—
71#14519783D414yes—
72C2×#11758194C2312yes—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758189—21no—
76C2×#11758189—21no—
77C2×#11758189—22no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80C2×C8×D61—22no—
81C4×Dic122—21no—
82C4×D244—21no—
83C4×C22⋊D61—22no—
84#14519736C2311yescharacteristic
85#14519734—21no—
86C42×D61—21no—
87#14519713—21no—
88#14519713—21no—
89#14519713—24no—
90#14519705—24no—
91#14519705—22no—
92#14519705—22no—
93#14519705—24no—
94#14519695C2×C414yescomplemented
95#14519693C2×C414yescomplemented
96#14519691—24no—
97#14519688—21no—
98#14519687—21no—
99#14519687—21no—
100C4×#92418—21no—
101C8×Dic61—21no—
102C8×Dic61—21no—
103C2×M16×C61C2312yes—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106#117×C61C2312yes—
107C2×C42×C61C2311yescharacteristic
108#108×C61C2×C412yescomplemented
109#108×C61C2×C412yescomplemented
110#107×C61—21no—
111#106×C61—21no—
112#97×C61—22no—
113#97×C61D411yescharacteristic
114#97×C61D411yescharacteristic
115#97×C61D411yescharacteristic
116#97×C61D411yescharacteristic
117C4×C8×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—21no—
121C2×C22⋊D61—22no—
122#11758222—21no—
123#11758222—21no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222C22×C412yes—
129#11758222C2×D412yes—
130C2×D244C22×C412yes—
131C2×D244#4111yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×Dic122C22×C412yes—
136C2×Dic122#4111yescharacteristic
137M16×C61—22no—
138M16×C61C22×C414yes—
139C2×C8×C61—22no—
140C2×C8×C61C22×C411yescharacteristic
141C2×C8×C61C22×C411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D412yes—
147C2×C8×C61#4111yescharacteristic
148C2×C8×C61#4111yescharacteristic
149C2×C8×C61#4112yes—
150C4⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C42×C61—21no—
153C42×C61—22no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C22⋊Dic61#4112yes—
157#11758198—22no—
158C4⋊Dic61—21no—
159#11758196—22no—
160C4×Dic61—22no—
161C4×Dic61—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—22no—
165#11758194—22no—
166#11758194C22×C412yes—
167#11758194C2×D412yes—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—22no—
171Dic244—24no—
172Dic244—21no—
173Dic244—21no—
174D488—24no—
175D488—21no—
176D488—21no—
177#11758190—24no—
178#11758190—24no—
179#11758190—22no—
180#11758190—22no—
181#11758189—22no—
182#11758189—22no—
183#11758189—24no—
184#11758189—42no—
185#11758189—42no—
186C8×D61—22no—
187C8×D61—22no—
188C8×D61—24no—
189C8×D61—42no—
190C8×D61—42no—
191C23×C61C2×#4111yescharacteristic
192C22×D61—21no—
193C22×D61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—42no—
197C2×C4×C61—81no—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—22no—
205C22⋊D61—22no—
206C22⋊D61—42no—
207C22⋊D61—42no—
208C2×Dic61—21no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211D244—22no—
212D244—22no—
213D244—21no—
214D244C4×D412yes—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61—82no—
220Dic122—21no—
221Dic122—22no—
222Dic122—22no—
223Dic122C4×D412yes—
224C8×C61—24no—
225C8×C61C4×D411yescharacteristic
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61#12112yes—
230C8×C61#12112yes—
231#92418—22no—
232#92418—22no—
233#92418—42no—
234#21559—611nomaximal, Sylow
235C22×C61—41no—
236C22×C61C4×#4111yescharacteristic
237C22×C61#52511yescharacteristic
238C22×C61#53211yescharacteristic
239C2×D61—22no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—82no—
243C4×C61—41no—
244C4×C61—41no—
245C4×C61—82no—
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×#12111yescharacteristic
249C4×C61#52911yescharacteristic
250Dic61—22no—
251Dic61—42no—
252Dic61—42no—
253#3067—611no—
254#2955—612no—
255C2×#444—611no—
256#2875—611no—
257#1976—611no—
258#1953—611no—
259#1932—612no—
260#1906—612no—
261#1814—611no—
262#1773D6111yescharacteristic, complemented
263#1757—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61C4×#12111yescharacteristic
268D61—42no—
269D61—82no—
270#575—612no—
271C2×#135—611no—
272C2×#131—611no—
273C2×#131—611no—
274C2×#131—612no—
275C4×#41—611no—
276#473—612no—
277#472—612no—
278#471—614no—
279#444—1224no—
280#444—1221no—
281#444—1221no—
282#444—1221no—
283#444—1221no—
284#443—1224no—
285#430—612no—
286C2×#108—611no—
287C2×#108C2×D6111yescharacteristic
288#427C2×D6111yescharacteristic
289#421—612no—
290#421—612no—
291C2×#104—612no—
292#413—612no—
293#410—612no—
294#405—611no—
295#405—611no—
296#405—1221no—
297#405—1221no—
298#405C2×D6112yes—
299C4×M16—611no—
300C4×M16—1221no—
301C2×#97—1221no—
302C2×#97C2×D6111yescharacteristic
303#344—612no—
304#344—612no—
305#339C2×D6112yes—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41—611no—
308C2×#41—612no—
309#137—614no—
310#137—1222no—
311AGL(1,ℤ/8)—614no—
312AGL(1,ℤ/8)—1222no—
313#135—1224no—
314#135—1221no—
315#135—1221no—
316#135—1221no—
317#135—1221no—
318C2×Q16—611no—
319C2×SD16—612no—
320C2×D8—611no—
321#131—612no—
322#131—612no—
323#131—614no—
324#131—1221no—
325#131—1221no—
326#131—1222no—
327#131—1222no—
328#131—1222no—
329#131—1222no—
330#131—1222no—
331C2×M16—611no—
332C2×M16—611no—
333C2×M16—612no—
334C2×M16—1221no—
335C2×M16—1221no—
336C2×M16—1222no—
337C2×M16C22×D6112yes—
338C22×C8—1221no—
339C22×C8—1221no—
340C22×C8—1222no—
341C22×C8C22×D6111yescharacteristic
342C22×C8C22×D6111yescharacteristic
343C4×Q8—1221no—
344C4×D4—1221no—
345C4×D4—1222no—
346#117—611no—
347#117—1221no—
348#117C22×D6112yes—
349C2×C42—1221no—
350C2×C42C22×D6111yescharacteristic
351#108—1224no—
352#108—1221no—
353#108—1221no—
354#108C4×D6112yescomplemented
355#108C4×D6112yescomplemented
356#107—21no—
357#106—21no—
358#104—1224no—
359#104—1224no—
360#104—1222no—
361#104—1222no—
362#101—614no—
363#100—614no—
364#99—1224no—
365#97—22no—
366#97—1221no—
367#97—1221no—
368#97—1221no—
369#97D24411yescharacteristic
370#97D24411yescharacteristic
371#97D24411yescharacteristic
372#97D24411yescharacteristic
373C4×C8—22no—
374C4×C8—1221no—
375C4×C8—1221no—
376C4×C8—1221no—
377#41—612no—
378#41—612no—
379#41—1221no—
380#41—1221no—
381#41—1222no—
382#41—1222no—
383#41—1222no—
384#41—1222no—
385C2×Q8—612no—
386C2×Q8—611no—
387C2×D4—612no—
388C2×D4—611no—
389C2×D4—1221no—
390C2×D4—1222no—
391C22×C4—21no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4C23×D6111yescharacteristic
396Q16—1224no—
397Q16—1221no—
398Q16—1221no—
399SD16—1224no—
400SD16—1224no—
401SD16—1222no—
402SD16—1222no—
403D8—1224no—
404D8—1221no—
405D8—1221no—
406M16—22no—
407M16—612no—
408M16—612no—
409M16—1221no—
410M16—1221no—
411M16—1222no—
412M16—1222no—
413M16—1224no—
414M16—1222no—
415M16—1222no—
416M16—2442no—
417M16—2442no—
418M16C2×C4×D6114yes—
419C2×C8—22no—
420C2×C8—1221no—
421C2×C8—1221no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1224no—
425C2×C8—1222no—
426C2×C8—2442no—
427C2×C8—2442no—
428C2×C8C2×C4×D6111yescharacteristic
429C2×C8C2×C4×D6111yescharacteristic
430C2×C8C2×D24411yescharacteristic
431C2×C8C2×D24411yescharacteristic
432C2×C8C2×D24411yescharacteristic
433C2×C8C2×D24411yescharacteristic
434C2×C8#1175822211yescharacteristic
435C2×C8#1175822211yescharacteristic
436C2×C8D4×D6112yes—
437C2×C8#1175822612yes—
438C4⋊C4—22no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C22⋊C4—22no—
442C22⋊C4—612no—
443C22⋊C4—1222no—
444C42—21no—
445C42—22no—
446C42—1222no—
447C42—1221no—
448C42C2×D24411yescharacteristic
449C42C2×D24411yescharacteristic
450C23—1221no—
451C23—1222no—
452C23#41×D6111yescharacteristic
453Q8—612no—
454Q8—1221no—
455Q8—1222no—
456Q8—1222no—
457D4—612no—
458D4—1222no—
459D4—1222no—
460D4—1222no—
461D4—1222no—
462D4—1221no—
463D4—2442no—
464D4—2442no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—1221no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—4882no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4C22×C4×D6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C22×D24411yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822211yescharacteristic
482C2×C4C2×#1175822611yescharacteristic
483C8—24no—
484C8—1222no—
485C8—1222no—
486C8—2442no—
487C8C4×D24411yescharacteristic
488C8C4×D24411yescharacteristic
489C8C4×D24411yescharacteristic
490C8C4×D24411yescharacteristic
491C8#1451975212yes—
492C8#1451975212yes—
493C22—41no—
494C22—1222no—
495C22—4882no—
496C22—2442no—
497C22—2442no—
498C22C4×#1175822611yescharacteristic
499C22#1684027411yescharacteristic
500C22#1684030911yescharacteristic
501C4—41no—
502C4—41no—
503C4—82no—
504C4—1222no—
505C4—2442no—
506C4—2442no—
507C4C4×D4×D6111yescharacteristic
508C4C2×C4×D24411yescharacteristic
509C4C2×#1451975211yescharacteristic
510C4#1684029911yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—4882no—
515C2—2442no—
516C2C4×#1451975211yescharacteristic
subgroupG/H
centerC4C2×#14519752
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1773C2
socleC2×C61C4×#121
prime radical#21455540C2
cosocleC22×C4×C61C24
hypercenter#1773D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #21454723C2semidirect
3. C2×#16839885C2semidirect
4. #21454244C2semidirect
5. #21446715C2semidirect
6. #21446610C2semidirect
7. #21446489C2semidirect
8. #21446319C2semidirect
9. #21445870C2semidirect
10. #21445620C2semidirect
11. C61×#1773C2semidirect
18. #16840038C4semidirect
19. #16840033C4semidirect
20. #16840026C4semidirect
59. C2×Dic244C2×C4semidirect
60. C2×#11758190C2×C4semidirect
61. C2×D488C2×C4semidirect
94. #14519695C2×C4semidirect
95. #14519693C2×C4semidirect
108. #108×C61C2×C4semidirect
109. #108×C61C2×C4semidirect
262. #1773D61semidirect
306. C61#21559semidirect
354. #108C4×D61semidirect
355. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555402
#214547232
C2×#168398852
#214542442
#214467152
#214466102
#214464892
#214463192
#214458702
#214456202
C61×#17732
#2155961