Finite Group Databasebeta

← #29572101

Subgroups — #29572101

Order 15616.

1016 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2#21446781C211yesmaximal, characteristic, complemented
3#21445825C211yesmaximal, characteristic, complemented
4#21445820C211yesmaximal, characteristic
5C2×#16839492C211yesmaximal, characteristic, complemented
6C2×#16839489C211yesmaximal, characteristic
7C2×C61×#332C211yesmaximal, characteristic, complemented
8C22×Q8×D61—21no—
9#16840427—22no—
10#16840427—22no—
11#16840427—22no—
12#16840427C414yescomplemented
13C2×#14519776—21no—
14C2×#14519755—22no—
15C2×#14519753—21no—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519739—21no—
19C2×#14519702—21no—
20C2×#14519699—21no—
21C2×C4×#92418C2211yescharacteristic
22#16839632C2211yescharacteristic
23#16839525—21no—
24#16839525—21no—
25#16839524—22no—
26#16839492—22no—
27#16839489—22no—
28C2×C4⋊Q8×C61C2211yescharacteristic
29C2×C4⋊C8×C61C2211yescharacteristic
30C2×C4⋊C8×C61C2211yescharacteristic
31C61×#332—22no—
32C22×Dic122—22no—
33C22×Dic122D411yescharacteristic
34C2×Q8×D61—24no—
35C2×Q8×D61—24no—
36C2×Q8×D61—42no—
37C2×Q8×D61—42no—
38C2×Q8×D61—42no—
39C2×Q8×D61—42no—
40C22×D244D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—21no—
43C22×#92418—21no—
44#14519776—24no—
45#14519776—24no—
46#14519755—28no—
47#14519755—28no—
48#14519753—24no—
49#14519753—24no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52C4×Dic122C2×C412yes—
53C4×Dic122D412yes—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244C2×C412yes—
57C4×D244D412yes—
58#14519739—24no—
59#14519739—24no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758196—22no—
66C2×#11758196—21no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×C4×Dic61—21no—
70#14519702—42no—
71#14519699—42no—
72C4×#92418—21no—
73C4×#92418—21no—
74C22×Q8×C61—21no—
75C22×C8×C61—21no—
76C22×C8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61C2×C412yes—
80C4⋊Q8×C61D412yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61D412yes—
83C2×C42×C61C2311yescharacteristic
84C4⋊C8×C61—21no—
85C4⋊C8×C61—21no—
86C4⋊C8×C61—21no—
87C4⋊C8×C61—21no—
88C23×D61—21no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—41no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—41no—
103Q8×D61—48no—
104Q8×D61—48no—
105Q8×D61—48no—
106Q8×D61—48no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244C22⋊C412yes—
111C2×D244SD1612yes—
112C2×D244Q1612yes—
113C2×C4×D61—28no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—44no—
117C2×C4×D61—44no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—28no—
126C2×Dic122—44no—
127C2×Dic122—44no—
128C2×Dic122C22⋊C412yes—
129C2×Dic122D812yes—
130C2×Dic122SD1612yes—
131C2×C8×C61—41no—
132C2×C8×C61—41no—
133C2×C8×C61—41no—
134C2×C8×C61—41no—
135C2×C8×C61—41no—
136C2×C8×C61—41no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61C22⋊C414yes—
141C42×C61C22×C411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145#11758198—28no—
146#11758198—22no—
147#11758198—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—42no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C4⋊Dic61C22⋊C412yes—
156C4⋊Dic61C22⋊C412yes—
157#11758196—28no—
158#11758196—22no—
159#11758196—22no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C2×#92418—22no—
163C2×#92418—41no—
164C2×#92418—41no—
165C23×C61#12011yescharacteristic
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173Q8×C61—42no—
174Q8×C61—42no—
175Q8×C61—42no—
176Q8×C61—42no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—24no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61C2×C22⋊C411yescharacteristic
184C2×C4×C61C2×C22⋊C411yescharacteristic
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C2×SD1611yescharacteristic
190C2×C4×C61C2×SD1611yescharacteristic
191C2×C4×C61C2×Q1611yescharacteristic
192C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C61#13711yescharacteristic
195C2×C4×C61#13711yescharacteristic
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—81no—
206C2×Dic61—81no—
207C2×Dic61—81no—
208D244—22no—
209D244—22no—
210D244—22no—
211D244#10314yescomplemented
212C4×D61—416no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—42no—
217Dic122—22no—
218Dic122—22no—
219Dic122—22no—
220Dic122—416no—
221Dic122#10214yescomplemented
222C8×C61—81no—
223C8×C61—81no—
224#92418—42no—
225#8447—611nomaximal, Sylow
226C22×C61#37911yescharacteristic
227C22×C61#44711yescharacteristic
228C22×C61#44811yescharacteristic
229C22×C61#45011yescharacteristic
230C22×C61#45111yescharacteristic
231C22×C61#45411yescharacteristic
232C22×C61#45611yescharacteristic
233C2×D61—24no—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—44no—
243C4×C61C2×#10211yescharacteristic
244C4×C61C2×#10311yescharacteristic
245C4×C61#41811yescharacteristic
246C4×C61#41911yescharacteristic
247C4×C61#44711yescharacteristic
248C4×C61#44811yescharacteristic
249C4×C61#45011yescharacteristic
250C4×C61#45111yescharacteristic
251Dic61—24no—
252Dic61—42no—
253Dic61—42no—
254Dic61—161no—
255C2×D4×Q8—611no—
256#1990—611no—
257#1803—611no—
258#1802—611no—
259C2×#332D6111yescharacteristic, complemented
260C2×Q8⋊C8—611no—
261C2×D4⋊C8—611no—
262C2×C61#162711yescharacteristic
263C2×C61#163711yescharacteristic
264C2×C61#164911yescharacteristic
265C2×C61#165111yescharacteristic
266C2×C61#178611yescharacteristic
267C2×C61#178711yescharacteristic
268C2×C61#179311yescharacteristic
269D61—44no—
270C23×Q8—1221no—
271D4×Q8—614no—
272D4×Q8—1222no—
273D4×Q8—1222no—
274D4×Q8—1222no—
275C2×C4⋊Q8—1221no—
276C2×C4⋊Q8C2×D6111yescharacteristic
277C2×C22⋊Q8—1221no—
278C2×C22⋊Q8—1222no—
279C2×C4×Q8—611no—
280C2×C4×D4—611no—
281C2×C4×D4—1221no—
282C2×C4⋊C8C2×D6111yescharacteristic
283C2×C4⋊C8C2×D6111yescharacteristic
284C2×#103—1221no—
285C2×#102—1221no—
286C2×C4×C8—611no—
287#384—611no—
288#340—1221no—
289#340—1221no—
290#340—1222no—
291#332—22no—
292Q8⋊C8—1222no—
293D4⋊C8—1222no—
294C61#844711yescharacteristic, complemented, Sylow
295C22×Q8—21no—
296C22×Q8—611no—
297C22×Q8—1222no—
298C22×Q8—1224no—
299C22×Q8—1224no—
300C22×Q8—2442no—
301C22×Q8—2442no—
302C22×Q8—2442no—
303C22×Q8—2442no—
304C22×D4—611no—
305C23×C4—1222no—
306C23×C4—1221no—
307C22×C8—21no—
308C22×C8—21no—
309C22×C8—1221no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—1224no—
313C4⋊Q8—1224no—
314C4⋊Q8C4×D6112yes—
315C4⋊Q8C22⋊D6112yes—
316C22⋊Q8—1228no—
317C22⋊Q8—1224no—
318C22⋊Q8—1224no—
319C22⋊Q8—1228no—
320C4×Q8—612no—
321C4×Q8—612no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×D4—612no—
325C4×D4—612no—
326C4×D4—1224no—
327C4×D4—1224no—
328C4×D4—1221no—
329C4×D4—1221no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4D24412yes—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C2×C42—1221no—
342C2×C42C22×D6111yescharacteristic
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347#103—2442no—
348#102—2442no—
349C4×C8—1221no—
350C4×C8—1221no—
351C24—1221no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—612no—
359C2×Q8—612no—
360C2×Q8—612no—
361C2×Q8—1228no—
362C2×Q8—1221no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—2444no—
366C2×Q8—2444no—
367C2×Q8—2448no—
368C2×Q8—2448no—
369C2×Q8—2448no—
370C2×Q8—2448no—
371C2×D4—612no—
372C2×D4—612no—
373C2×D4—612no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C22×C4—22no—
378C22×C4—1228no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—2444no—
385C22×C4—2444no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2441no—
390C22×C4—2441no—
391C22×C4C2×D24411yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—1222no—
401C2×C8—2441no—
402C2×C8—2441no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—612no—
407C4⋊C4—612no—
408C4⋊C4—1228no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—2442no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4#1175819814yes—
419C22⋊C4—1228no—
420C22⋊C4—1222no—
421C22⋊C4—1222no—
422C42—1222no—
423C42—1222no—
424C42C2×C4×D6111yescharacteristic
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C23—1222no—
429C23—1222no—
430C23—1222no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23—2441no—
435C23#1451974711yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—614no—
441Q8—1222no—
442Q8—1222no—
443Q8—1222no—
444Q8—24416no—
445D4—614no—
446D4—1222no—
447D4—1222no—
448D4—1222no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—4881no—
460C2×C4—4881no—
461C2×C4—4881no—
462C2×C4—24416no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4C22⋊C4×D6111yescharacteristic
472C2×C4C22⋊C4×D6111yescharacteristic
473C2×C4C2×#1175819811yescharacteristic
474C2×C4#1451974511yescharacteristic
475C2×C4#1451974511yescharacteristic
476C2×C4#1451974711yescharacteristic
477C2×C4D8×D6111yescharacteristic
478C2×C4SD16×D6111yescharacteristic
479C2×C4SD16×D6111yescharacteristic
480C2×C4Q16×D6111yescharacteristic
481C2×C4#1451980111yescharacteristic
482C2×C4#1451980411yescharacteristic
483C2×C4#1451980611yescharacteristic
484C2×C4#1451980911yescharacteristic
485C8—81no—
486C8—81no—
487C8—2442no—
488C22—1224no—
489C22—2442no—
490C22—2442no—
491C22—2442no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22#1683961711yescharacteristic
496C22#1683989711yescharacteristic
497C22#1683990211yescharacteristic
498C22#1683991411yescharacteristic
499C22#1683991911yescharacteristic
500C22#1683993711yescharacteristic
501C22#1683994711yescharacteristic
502C4—21no—
503C4—21no—
504C4—44no—
505C4—1224no—
506C4—9761no—
507C4—2442no—
508C4—2442no—
509C4#102×D6111yescharacteristic
510C4#103×D6111yescharacteristic
511C4#1683976511yescharacteristic
512C4#1683976911yescharacteristic
513C4#1683989411yescharacteristic
514C4#1683989911yescharacteristic
515C4#1683991111yescharacteristic
516C4#1683991611yescharacteristic
517C2—2444no—
518C2#2144492311yescharacteristic
519C2#2144498511yescharacteristic
520C2#2144506911yescharacteristic
521C2#2144508311yescharacteristic
522C2#2144574811yescharacteristic
523C2#2144575211yescharacteristic
524C2#2144578011yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#332C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenterC2×#332D61
hypocenterC61#8447
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #21446781C2semidirect
3. #21445825C2semidirect
5. C2×#16839492C2semidirect
7. C2×C61×#332C2semidirect
12. #16840427C4semidirect
211. D244#103semidirect
221. Dic122#102semidirect
259. C2×#332D61semidirect
294. C61#8447semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
#214467812
#214458252
#214458202
C2×#168394922
C2×#168394892
C2×C61×#3322
#844761