Finite Group Databasebeta

← #29736642

Subgroups — #29736642

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3C2×#16840109C211yesmaximal, characteristic, complemented
4#21454165C211yesmaximal, characteristic
5#21446964C214yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446140C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic, complemented
9C61×#1990C211yesmaximal, characteristic, complemented
10#16840317—24no—
11#16840317—24no—
12C2×#14519777C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14C2×#14519768C2212yes—
15C2×#14519757—21no—
16C2×#14519753—21no—
17C2×#14519752—21no—
18C2×#14519752—21no—
19C2×#14519752C2212yes—
20C2×Q8×Dic61—21no—
21C2×#14519742—21no—
22C2×#14519735C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840110—24no—
25#16840109—24no—
26C2×#14519699C2214yes—
27C2×#14519697C2214yes—
28C2×#14519689C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839663—21no—
31#16839663—21no—
32#16839663C2212yes—
33#16839524C2214yes—
34#16839516C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×C8×C61C2211yescharacteristic
37C61×#384C2211yescharacteristic
38C61×#340C2214yes—
39C22×Dic122—21no—
40C22×D244—22no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244C2312yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48#14519777—22no—
49#14519777D414yescomplemented
50#14519774—22no—
51#14519774—22no—
52#14519768—24no—
53#14519768—24no—
54#14519757—28no—
55#14519753—28no—
56#14519752—28no—
57#14519752—44no—
58#14519752—44no—
59Q8×Dic61—28no—
60#14519742—28no—
61#14519735—24no—
62#14519735—24no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C42×D61—22no—
76C42×D61—22no—
77#14519699—28no—
78#14519697—28no—
79#14519689—24no—
80C4×#92418—22no—
81C22×Q8×C61—21no—
82C22×C8×C61C2312yes—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D414yescomplemented
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91C4×C8×C61—22no—
92C23×D61—22no—
93C23×D61C2×D411yescharacteristic
94C2×Q8×C61—24no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—22no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—24no—
109C2×D244—24no—
110C2×D244—44no—
111C2×D244—44no—
112C2×D244—44no—
113C2×D244—42no—
114C2×D244—42no—
115C2×D244—42no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—42no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61#4114yes—
129C2×Dic122—24no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61C2×D414yes—
141C42×C61C2×D412yes—
142C42×C61C2×D412yes—
143#11758198—28no—
144#11758198—44no—
145#11758198—44no—
146#11758198—44no—
147C4⋊Dic61—28no—
148C4⋊Dic61—24no—
149C4⋊Dic61—22no—
150C4⋊Dic61C2×D414yes—
151#11758196—28no—
152C4×Dic61—28no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C23×C61C22×D411yescharacteristic
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61#12412yes—
175Q8×C61—48no—
176C2×C4×C61—24no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61#12012yes—
183C2×C4×C61#12012yes—
184C2×C4×C61#13512yes—
185C2×C4×C61#13512yes—
186C2×C4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4112yes—
191C2×C4×C612−1+412yes—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—24no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61#12412yes—
204D244—44no—
205D244—44no—
206D244—88no—
207D244—84no—
208D244—84no—
209C4×D61—24no—
210C4×D61—24no—
211C4×D61—44no—
212C4×D61—44no—
213Dic122—48no—
214C8×C61—42no—
215#92418—42no—
216#25905—611nomaximal, Sylow
217C22×C61C2×#12011yescharacteristic
218C22×C61C2×#12411yescharacteristic
219C22×C61#53611yescharacteristic
220C22×C61C2×#13511yescharacteristic
221C22×C61C2×#13511yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—82no—
232C2×D61—82no—
233C2×D61—82no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—44no—
239C4×C61—82no—
240C4×C61#44914yes—
241C4×C61#44914yes—
242C4×C61#54812yes—
243C4×C61#54812yes—
244Dic61—22no—
245Dic61—44no—
246Dic61—41no—
247Dic61—41no—
248C2×#533—611no—
249C2×#486—611no—
250C2×#486—611no—
251#2856—611no—
252#2025—614no—
253#1990—611no—
254#1990D6111yescharacteristic, complemented
255#1875—614no—
256C4×C22⋊C8—611no—
257C2×C61#243611yescharacteristic
258C2×C61C2×#44911yescharacteristic
259C2×C61C2×#44911yescharacteristic
260C2×C61#313111yescharacteristic
261C2×C61#313811yescharacteristic
262C2×C61#322811yescharacteristic
263C2×C61#322811yescharacteristic
264D61—42no—
265D61—162no—
266#533—1224no—
267#533—1224no—
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8C2×D6111yescharacteristic
270C2×#127—611no—
271C2×#125—612no—
272C2×C22⋊Q8—1221no—
273C2×C22⋊Q8—1221no—
274C2×#121—612no—
275C2×#121—1221no—
276C2×#121—1221no—
277C2×C4×Q8—1221no—
278C2×C4×Q8—1221no—
279C2×#117—612no—
280C22×C42—611no—
281#486—1224no—
282#486—1224no—
283C2×#102—614no—
284C2×#102—614no—
285C2×C22⋊C8—612no—
286C2×C4×C8—611no—
287C2×C4×C8C2×D6111yescharacteristic
288#390—612no—
289#390—1221no—
290#390—1221no—
291#384C2×D6111yescharacteristic
292#340—614no—
293#340C2×D6114yes—
294#336—612no—
295C61#2590511yescharacteristic, complemented, Sylow
296C22×Q8—21no—
297C22×Q8—1221no—
298C22×D4—612no—
299C22×D4—1222no—
300C22×D4—1221no—
301C22×D4—1221no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—611no—
305C22×C8—612no—
306C22×C8C22×D6112yes—
307C4⋊Q8—22no—
308C4⋊Q8—614no—
309C4⋊Q8—1222no—
310C4⋊Q8D24414yescomplemented
311#127—1222no—
312#127—1222no—
313#125—1224no—
314#125—1224no—
315C22⋊Q8—1228no—
316C22⋊Q8—1228no—
317#121—1228no—
318#121—2444no—
319#121—2444no—
320C4×Q8—1228no—
321C4×Q8—1228no—
322#117—1224no—
323#117—1224no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4C22×D6112yes—
332C2×C4⋊C4C22×D6112yes—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1222no—
337C2×C42—611no—
338C2×C42—611no—
339C2×C42—611no—
340C2×C42—1222no—
341C2×C42—1222no—
342C2×C42—1222no—
343C2×C42C22×D6111yescharacteristic
344#102—1228no—
345#102—1228no—
346C22⋊C8—1224no—
347C4×C8—22no—
348C4×C8—1222no—
349C24—611no—
350C24—1222no—
351C2×Q8—24no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—1224no—
355C2×Q8—2442no—
356C2×Q8—2442no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—1224no—
360C2×D4—1224no—
361C2×D4—2444no—
362C2×D4—2444no—
363C2×D4—2444no—
364C2×D4—2442no—
365C2×D4—2442no—
366C2×D4—2442no—
367C2×D4—2442no—
368C2×D4—2442no—
369C2×D4—2442no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—611no—
373C22×C4—614no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—1222no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—614no—
401C4⋊C4—1228no—
402C4⋊C4—1228no—
403C4⋊C4—1222no—
404C4⋊C4—1224no—
405C4⋊C4C2×D24414yes—
406C22⋊C4—1228no—
407C22⋊C4—2444no—
408C22⋊C4—2444no—
409C22⋊C4—2444no—
410C42—1228no—
411C42—1221no—
412C42—1221no—
413C42—1222no—
414C42—1222no—
415C42C2×D24412yes—
416C42D4×D6112yes—
417C23—612no—
418C23—1221no—
419C23—1221no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23—1221no—
424C23—2442no—
425C23—2442no—
426C23—2442no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23—2442no—
431C23C2×D4×D6111yescharacteristic
432Q8—48no—
433Q8—2448no—
434D4—4888no—
435D4—4884no—
436D4—4884no—
437D4—2444no—
438D4—2444no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—612no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1224no—
449C2×C4—1224no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1224no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2444no—
460C2×C4—2444no—
461C2×C4#1451974712yes—
462C2×C4#1451974712yes—
463C2×C4#1451979812yes—
464C2×C4#1451980012yes—
465C2×C4#1451980112yes—
466C2×C4#1451980412yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C22×D24411yescharacteristic
469C2×C4#41×D6112yes—
470C2×C4#1451982512yes—
471C8—42no—
472C8—2442no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22—4882no—
477C22—4882no—
478C22—4882no—
479C22—4882no—
480C22—4882no—
481C22—4882no—
482C22—4882no—
483C22—2441no—
484C22—2441no—
485C22—2441no—
486C22—2441no—
487C22C2×#1451974711yescharacteristic
488C22#124×D6111yescharacteristic
489C22#1684033711yescharacteristic
490C22C2×#1451979811yescharacteristic
491C22C2×#1451980011yescharacteristic
492C22C2×#1451980111yescharacteristic
493C22C2×#1451980411yescharacteristic
494C4—44no—
495C4—82no—
496C4—1222no—
497C4—2441no—
498C4—2441no—
499C4—2444no—
500C4#1683990814yes—
501C4#1683991014yes—
502C4#1684042212yes—
503C4#1684042412yes—
504C2—9762no—
505C2—2442no—
506C2#2144947611yescharacteristic
507C2C2×#1683990811yescharacteristic
508C2C2×#1683991011yescharacteristic
509C2#2145614311yescharacteristic
510C2#2145619011yescharacteristic
511C2#2145695611yescharacteristic
512C2#2145695811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radical#21454165C2
hypercenter#1990D61
hypocenterC61#25905
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
2. C2×#16840110C2semidirect
3. C2×#16840109C2semidirect
5. #21446964C2semidirect
6. #21446779C2semidirect
7. #21446140C2semidirect
8. C4×#14519689C2semidirect
9. C61×#1990C2semidirect
49. #14519777D4semidirect
84. C4⋊Q8×C61D4semidirect
254. #1990D61semidirect
295. C61#25905semidirect
310. C4⋊Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401102
C2×#168401092
#214541652
#214469642
#214467792
#214461402
C4×#145196892
C61×#19902
#2590561