Finite Group Databasebeta

← #29686964

Subgroups — #29686964

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#21454722C212yesmaximal, complemented
2C2×#16839884C211yesmaximal, characteristic, complemented
3#21454245C211yesmaximal, characteristic, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446610C211yesmaximal, characteristic, complemented
6#21446490C212yesmaximal, complemented
7#21446322C212yesmaximal, complemented
8#21445873C211yesmaximal, characteristic, complemented
9#21445706C211yesmaximal, characteristic, complemented
10#21445622C212yesmaximal, complemented
11C61×#3067C211yesmaximal, characteristic
12C2×#14519784C2212yes—
13C2×#14519784C2211yescharacteristic
14#41×Dic61C2211yescharacteristic
15#16839884—24no—
16#16839884—21no—
17#16839884—21no—
18#16839884—21no—
19#16839884—21no—
20#16839879—24no—
21#16839811C2212yes—
22C2×#14519713C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839779C2212yes—
25#16839779C2212yes—
26C2×#14519704C2212yes—
27#16839736C2212yes—
28#16839720—21no—
29#16839720C2212yes—
30#16839720C2211yescharacteristic
31#16839717—21no—
32#16839717C2211yescharacteristic
33M16×Dic61—21no—
34M16×Dic61C2211yescharacteristic
35C2×#14519687—21no—
36C2×#14519687C2211yescharacteristic
37#16839534C2212yes—
38#16839534C2212yes—
39#16839523C2212yes—
40C61×#575C2212yes—
41C2×#135×C61C2211yescharacteristic
42C2×#131×C61C2211yescharacteristic
43C61×#473C412yescomplemented
44C61×#472C412yescomplemented
45C61×#471C414yescomplemented
46C2×#108×C61C2211yescharacteristic
47C61×#413C2212yes—
48#14519784—22no—
49#14519784—22no—
50#14519784—22no—
51#14519784—22no—
52#14519784—22no—
53#14519784D414yes—
54C2×#11758194—21no—
55C2×#11758194—22no—
56C2×#11758194C2312yes—
57C2×#11758194C2311yescharacteristic
58C22×#92418—21no—
59C22×#92418—22no—
60C22×#92418C2311yescharacteristic
61Q8×Dic61—21no—
62D4×Dic61—21no—
63D4×Dic61—22no—
64#14519736—21no—
65#14519736C2312yes—
66#14519736C2311yescharacteristic
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2311yescharacteristic
69#14519713—21no—
70#14519713—21no—
71#14519713—24no—
72#14519711—21no—
73#14519709—21no—
74#14519704—24no—
75#14519704—22no—
76#14519704—22no—
77#14519704—24no—
78#14519692—24no—
79#14519687—21no—
80#14519687—21no—
81#14519687—21no—
82#14519687—22no—
83#14519687D411yescharacteristic
84#14519687D411yescharacteristic
85#14519687D411yescharacteristic
86#14519687D411yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61—21no—
90C8×Dic61—22no—
91C2×#41×C61C2312yes—
92C2×#41×C61C2311yescharacteristic
93#137×C61—22no—
94#137×C61C2×C414yes—
95AGL(1,ℤ/8)×C61—22no—
96AGL(1,ℤ/8)×C61C2×C414yes—
97#135×C61—24no—
98#135×C61—21no—
99#135×C61—21no—
100#135×C61—21no—
101#135×C61—21no—
102C2×Q16×C61C2×C411yescharacteristic, complemented
103C2×SD16×C61C2×C412yescomplemented
104C2×D8×C61C2×C411yescharacteristic, complemented
105#131×C61—21no—
106#131×C61—21no—
107#131×C61C2×C412yes—
108#131×C61C2×C412yes—
109C2×M16×C61—21no—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2312yes—
112C22×C8×C61—21no—
113C22×C8×C61C2311yescharacteristic
114#108×C61C2×C412yescomplemented
115#108×C61C2×C412yescomplemented
116#101×C61C2×C414yescomplemented
117#100×C61C2×C414yescomplemented
118#41×C61—21no—
119#41×C61—21no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61C22×C412yes—
125#41×C61C2×D412yes—
126C2×Q8×C61C22×C412yes—
127C2×Q8×C61#4111yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—22no—
130C2×D4×C61C22×C412yes—
131C2×D4×C61#4111yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—22no—
136C22×Dic61—21no—
137Q16×C61—24no—
138Q16×C61—21no—
139Q16×C61—21no—
140SD16×C61—24no—
141SD16×C61—24no—
142SD16×C61—22no—
143SD16×C61—22no—
144D8×C61—24no—
145D8×C61—21no—
146D8×C61—21no—
147M16×C61—21no—
148M16×C61—22no—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61C22×C412yes—
152M16×C61C22×C414yes—
153C2×C8×C61—21no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61C22×C411yescharacteristic
158C2×C8×C61C22×C411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61#4111yescharacteristic
162C2×C8×C61#4111yescharacteristic
163C22⋊Dic61—22no—
164C22⋊Dic61—22no—
165C22⋊Dic61#4112yes—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4×Dic61—22no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—22no—
173C4×Dic61C2×D411yescharacteristic
174C4×Dic61C2×D411yescharacteristic
175#11758194—21no—
176#11758194—24no—
177#11758194—22no—
178#11758194—22no—
179#11758194—42no—
180#11758194—42no—
181#11758194C2×D412yes—
182C2×#92418—21no—
183C2×#92418—24no—
184C2×#92418—22no—
185C2×#92418—42no—
186C2×#92418—42no—
187C2×#92418C2×D412yes—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418#4112yes—
191C23×C61—21no—
192C23×C61—22no—
193C23×C61C2×#4111yescharacteristic
194Q8×C61—21no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61C4×D412yes—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—21no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61C4×D412yes—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61C23×C411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—21no—
221C2×Dic61—42no—
222C2×Dic61—81no—
223C2×Dic61—82no—
224C8×C61—22no—
225C8×C61—24no—
226C8×C61C4×D411yescharacteristic
227C8×C61C4×D411yescharacteristic
228C8×C61C4×D411yescharacteristic
229C8×C61C4×D411yescharacteristic
230#92418—22no—
231#92418—42no—
232#92418#12112yes—
233#92418#12112yes—
234#21559—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—82no—
240C22×C61C4×#4111yescharacteristic
241C22×C61#52511yescharacteristic
242C22×C61#53211yescharacteristic
243C4×C61—22no—
244C4×C61—42no—
245C4×C61C2×C4×D411yescharacteristic
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×#12111yescharacteristic
248C4×C61#52911yescharacteristic
249Dic61—42no—
250Dic61—41no—
251Dic61—41no—
252Dic61—82no—
253#3067D6111yescharacteristic
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#1773—611no—
263#1757—612no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—42no—
268C2×C61—82no—
269C2×C61C4×#12111yescharacteristic
270#575C2×D6112yes—
271C2×#135C2×D6111yescharacteristic
272C2×#131—611no—
273C2×#131—612no—
274C2×#131C2×D6111yescharacteristic
275C4×#41—611no—
276#473Dic6112yescomplemented
277#472Dic6112yescomplemented
278#471Dic6114yescomplemented
279#444—1221no—
280#444—1221no—
281#444—1221no—
282#444—1221no—
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×#104—612no—
292#413C2×D6112yes—
293#410—612no—
294#405—611no—
295#405—612no—
296#405—611no—
297#405—1221no—
298#405—1221no—
299C4×M16—611no—
300C4×M16—1221no—
301C2×#97—611no—
302C2×#97—1221no—
303#344—612no—
304#344—612no—
305#339—612no—
306C61#2155911yescharacteristic, complemented, Sylow
307C2×#41C22×D6111yescharacteristic
308C2×#41C22×D6112yes—
309#137—22no—
310#137C2×Dic6114yes—
311AGL(1,ℤ/8)—22no—
312AGL(1,ℤ/8)C2×Dic6114yes—
313#135—24no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—21no—
318C2×Q16C2×Dic6111yescharacteristic, complemented
319C2×SD16C2×Dic6112yescomplemented
320C2×D8C2×Dic6111yescharacteristic, complemented
321#131—21no—
322#131—21no—
323#131—614no—
324#131—1222no—
325#131—1222no—
326#131—1222no—
327#131—1222no—
328#131—1222no—
329#131C2×Dic6112yes—
330#131C2×Dic6112yes—
331C2×M16—21no—
332C2×M16—611no—
333C2×M16—612no—
334C2×M16—1221no—
335C2×M16—1222no—
336C2×M16C22×D6111yescharacteristic
337C2×M16C22×D6112yes—
338C22×C8—21no—
339C22×C8—611no—
340C22×C8—1221no—
341C22×C8—1222no—
342C22×C8C22×D6111yescharacteristic
343C4×Q8—1221no—
344C4×D4—1222no—
345C4×D4—1221no—
346#117—611no—
347#117—612no—
348#117—1221no—
349C2×C42—611no—
350C2×C42—1221no—
351#108—1224no—
352#108—1221no—
353#108—1221no—
354#108C2×Dic6112yescomplemented
355#108C2×Dic6112yescomplemented
356#107—1221no—
357#106—1221no—
358#104—1222no—
359#104—1222no—
360#104—1224no—
361#104—1224no—
362#101C2×Dic6114yescomplemented
363#100C2×Dic6114yescomplemented
364#99—1224no—
365#97—611no—
366#97—611no—
367#97—611no—
368#97—611no—
369#97—1221no—
370#97—1221no—
371#97—1222no—
372#97—1221no—
373C4×C8—1221no—
374C4×C8—1221no—
375C4×C8—1222no—
376C4×C8—1221no—
377#41—21no—
378#41—21no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—22no—
383#41D4×D6112yes—
384#41C22×Dic6112yes—
385C2×Q8#1175822411yescharacteristic
386C2×Q8C22×Dic6112yes—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4#1175822411yescharacteristic
390C2×D4C22×Dic6112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4C23×D6111yescharacteristic
396Q16—24no—
397Q16—21no—
398Q16—21no—
399SD16—24no—
400SD16—24no—
401SD16—22no—
402SD16—22no—
403D8—24no—
404D8—21no—
405D8—21no—
406M16—21no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—612no—
411M16—1224no—
412M16—1221no—
413M16—1222no—
414M16—1222no—
415M16—2442no—
416M16—2442no—
417M16C22×Dic6112yes—
418M16C22×Dic6114yes—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—611no—
424C2×C8—612no—
425C2×C8—611no—
426C2×C8—612no—
427C2×C8—1224no—
428C2×C8—1221no—
429C2×C8—1222no—
430C2×C8—2442no—
431C2×C8—2442no—
432C2×C8D4×D6111yescharacteristic
433C2×C8D4×D6111yescharacteristic
434C2×C8#1175822411yescharacteristic
435C2×C8#1175822411yescharacteristic
436C2×C8C22×Dic6111yescharacteristic
437C2×C8C22×Dic6111yescharacteristic
438C4⋊C4—1222no—
439C4⋊C4—1221no—
440C4⋊C4—1222no—
441C22⋊C4—612no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C42—611no—
445C42—611no—
446C42—1222no—
447C42—1221no—
448C42—1221no—
449C42—1222no—
450C23—21no—
451C23—22no—
452C23#41×D6111yescharacteristic
453Q8—21no—
454Q8—22no—
455Q8—22no—
456Q8D4×Dic6112yes—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—42no—
463D4—42no—
464D4D4×Dic6112yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1222no—
474C2×C4—4881no—
475C2×C4—4882no—
476C2×C4—2442no—
477C2×C4C23×Dic6111yescharacteristic
478C2×C4C2×D4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822411yescharacteristic
482C2×C4C2×#1175822411yescharacteristic
483C8—22no—
484C8—24no—
485C8—612no—
486C8—612no—
487C8—1222no—
488C8—2442no—
489C8D4×Dic6111yescharacteristic
490C8D4×Dic6111yescharacteristic
491C8D4×Dic6111yescharacteristic
492C8D4×Dic6111yescharacteristic
493C22—22no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22#41×Dic6111yescharacteristic
499C22#1684027211yescharacteristic
500C22#1684030711yescharacteristic
501C4—22no—
502C4—42no—
503C4—4882no—
504C4—2441no—
505C4—2442no—
506C4—2441no—
507C4C2×D4×Dic6111yescharacteristic
508C4C2×D4×Dic6111yescharacteristic
509C4#121×D6111yescharacteristic
510C4#1684029311yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—82no—
516C2#121×Dic6111yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61C4×#121
cosocleC22×C4×C61C24
hypercenter#3067D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #21454722C2semidirect
2. C2×#16839884C2semidirect
3. #21454245C2semidirect
4. #21446718C2semidirect
5. #21446610C2semidirect
6. #21446490C2semidirect
7. #21446322C2semidirect
8. #21445873C2semidirect
9. #21445706C2semidirect
10. #21445622C2semidirect
43. C61×#473C4semidirect
44. C61×#472C4semidirect
45. C61×#471C4semidirect
102. C2×Q16×C61C2×C4semidirect
103. C2×SD16×C61C2×C4semidirect
104. C2×D8×C61C2×C4semidirect
114. #108×C61C2×C4semidirect
115. #108×C61C2×C4semidirect
116. #101×C61C2×C4semidirect
117. #100×C61C2×C4semidirect
276. #473Dic61semidirect
277. #472Dic61semidirect
278. #471Dic61semidirect
306. C61#21559semidirect
318. C2×Q16C2×Dic61semidirect
319. C2×SD16C2×Dic61semidirect
320. C2×D8C2×Dic61semidirect
354. #108C2×Dic61semidirect
355. #108C2×Dic61semidirect
362. #101C2×Dic61semidirect
363. #100C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214547222
C2×#168398842
#214542452
#214467182
#214466102
#214464902
#214463222
#214458732
#214457062
#214456222
C61×#30672
#2155961