Finite Group Databasebeta

← #29736549

Subgroups — #29736549

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2C2×#16840132C211yesmaximal, characteristic, complemented
3C2×#16840066C211yesmaximal, characteristic, complemented
4#21447022C212yesmaximal, complemented
5#21447015C212yesmaximal, complemented
6#21446742C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic
8#21445753C212yesmaximal, complemented
9#21445753C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2562C211yesmaximal, characteristic
12C23×Dic122C2212yes—
13D4×Dic122—24no—
14D4×Dic122—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519756C2212yes—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21#16840132—24no—
22#16840066—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2212yes—
32C2×#14519700C2212yes—
33C2×C22⋊#92418C2212yes—
34C2×C22⋊#92418C2212yes—
35C2×#14519688C2211yescharacteristic
36#16839524C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#400—21no—
40C61×#394—21no—
41C61×#388—22no—
42C61×#380C2212yes—
43C61×#380C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×Dic122—22no—
46C22×Dic122—24no—
47C22×Dic122—24no—
48C22×Dic122—24no—
49C22×Dic122—24no—
50C22×Dic122—42no—
51C22×Dic122—42no—
52C22×Dic122C2312yes—
53C22×Dic122C2311yescharacteristic
54C23×Dic61—22no—
55C22×#92418C2312yes—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519777—22no—
59#14519777D414yes—
60#14519776—28no—
61#14519756—24no—
62#14519756—24no—
63#14519754—216no—
64C4×Dic122—22no—
65C4×Dic122D414yes—
66D4×Dic61—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—21no—
74#14519710—22no—
75#14519708—22no—
76C4⋊#92418—22no—
77#14519700—22no—
78#14519700—22no—
79#14519700—24no—
80#14519700—24no—
81#14519700—24no—
82#14519700—42no—
83#14519700—42no—
84C22⋊#92418—42no—
85C22⋊#92418—42no—
86#14519688—22no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61C2312yes—
89#125×C61—22no—
90#125×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D412yes—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#95×C61—21no—
105#95×C61—22no—
106#95×C61—22no—
107C24×C61C2×D412yes—
108C2×D4×C61—24no—
109C2×D4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—28no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—44no—
128C22×Dic61—44no—
129C2×Dic122—21no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—28no—
136C2×Dic122—22no—
137C2×Dic122—48no—
138C2×Dic122—48no—
139C2×Dic122—48no—
140C2×Dic122—44no—
141C2×Dic122—44no—
142C2×Dic122—48no—
143C2×Dic122C2×D414yes—
144C2×Dic122C2×D412yes—
145C2×Dic122#4114yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—42no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—82no—
158C42×C61—81no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—28no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61C2×D412yes—
166#11758196—28no—
167#11758196—24no—
168C4×Dic61—24no—
169C2×#92418—22no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61#13714yes—
183C23×C61C22×D411yescharacteristic
184D4×C61—28no—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C61H(5,𝔽2)12yes—
209C2×Dic61—22no—
210C2×Dic61—24no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—416no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218Dic122—22no—
219Dic122—24no—
220Dic122—24no—
221Dic122—24no—
222Dic122—416no—
223Dic122—42no—
224Dic122—42no—
225Dic122#12414yes—
226#92418—41no—
227#92418—41no—
228#92418—42no—
229#25894—611nomaximal, Sylow
230C22×C61—24no—
231C22×C61—24no—
232C22×C61—24no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61C2×#12011yescharacteristic
238C22×C61#54612yes—
239C22×C61C2×#13712yes—
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—42no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61—82no—
248C4×C61C2×#12411yescharacteristic
249C4×C61#53511yescharacteristic
250C4×C61#54612yes—
251Dic61—24no—
252Dic61—44no—
253Dic61—42no—
254C2×D4×Q8—611no—
255C2×#491—611no—
256C2×#479—611no—
257#2562D6111yescharacteristic
258#2036—612no—
259#2035—612no—
260#1981—611no—
261#1955—611no—
262#1787—612no—
263#1787—612no—
264#1763—611no—
265C2×C61—44no—
266C2×C61#243511yescharacteristic
267C2×C61#300411yescharacteristic
268C2×C61#301311yescharacteristic
269C2×C61#331212yes—
270C2×C61#332012yes—
271C23×Q8—612no—
272D4×Q8—1224no—
273D4×Q8—1224no—
274C2×C4⋊Q8—611no—
275C2×C4⋊Q8—1221no—
276C2×#125C2×D6111yescharacteristic
277C2×C22⋊Q8—612no—
278C2×C22⋊Q8—1222no—
279C2×C4×Q8—611no—
280C2×C4×D4—1221no—
281C2×C4×D4C2×D6111yescharacteristic
282#491—1224no—
283#479—1224no—
284C2×#107—611no—
285C2×#106—611no—
286C2×C4⋊C8—611no—
287C2×#103—611no—
288C2×#103—611no—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×C22⋊C8—612no—
295C2×C22⋊C8—612no—
296C2×#97—611no—
297#400—21no—
298#394—21no—
299#388—22no—
300#380C2×D6112yes—
301#380C2×D6112yes—
302C4×C4⋊C4C2×D6111yescharacteristic
303#340—612no—
304C61#2589411yescharacteristic, complemented, Sylow
305C22×Q8—611no—
306C22×Q8—612no—
307C22×Q8—1224no—
308C22×Q8—1224no—
309C22×Q8—1224no—
310C22×Q8—1224no—
311C22×Q8—1222no—
312C22×Q8—2442no—
313C22×Q8—2442no—
314C22×D4C22⋊D6111yescharacteristic
315C23×C4—1222no—
316C23×C4C22×D6112yes—
317C22×C8—612no—
318C22×C8—611no—
319C22×C8—611no—
320C4⋊Q8—614no—
321C4⋊Q8—1228no—
322C4⋊Q8—1222no—
323#125—22no—
324#125—22no—
325C22⋊Q8—12216no—
326C22⋊Q8—1224no—
327C22⋊Q8—1224no—
328C4×Q8—614no—
329C4×Q8—1222no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—1228no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C4⋊C4C22×D6111yescharacteristic
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C4⋊C4C22×D6112yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4C22⋊D6112yes—
348C2×C42—21no—
349C2×C42—1221no—
350C2×C42C22×D6111yescharacteristic
351#107—1222no—
352#106—1222no—
353C4⋊C8—1222no—
354#103—1224no—
355#103—1222no—
356#103—1222no—
357#103—1224no—
358#103—1224no—
359#103—2442no—
360#103—2442no—
361C22⋊C8—2442no—
362C22⋊C8—2442no—
363#97—1222no—
364#95—21no—
365#95—22no—
366#95—22no—
367C24C2×C22⋊D6112yes—
368C2×Q8—612no—
369C2×Q8—614no—
370C2×Q8—614no—
371C2×Q8—1228no—
372C2×Q8—1221no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—2444no—
380C2×Q8—2444no—
381C2×Q8—2448no—
382C2×Q8—2448no—
383C2×Q8—2448no—
384C2×Q8—2448no—
385C2×D4—24no—
386C2×D4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—1221no—
397C22×C4—1228no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—2444no—
401C22×C4—2444no—
402C22×C4C2×C22⋊D6112yes—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406C2×C8—1222no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—612no—
423C4⋊C4—1228no—
424C4⋊C4—1221no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—1228no—
432C42—81no—
433C42—1224no—
434C42D4×D6112yes—
435C42D4×D6112yes—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23#1451980714yes—
443C23C22×C22⋊D6111yescharacteristic
444Q8—614no—
445Q8—1222no—
446Q8—1224no—
447Q8—1224no—
448Q8—1224no—
449Q8—24416no—
450Q8—2442no—
451Q8—2442no—
452D4—28no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—1222no—
472C2×C4—1224no—
473C2×C4—1222no—
474C2×C4—1224no—
475C2×C4—24416no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4#1451974512yes—
481C2×C4#1451974512yes—
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4#41×D6112yes—
485C2×C4#1451982312yes—
486C8—2441no—
487C8—2441no—
488C8—2442no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C2×#1451974511yescharacteristic
497C22#1684040012yes—
498C22C2×#1451980712yes—
499C22#1684057311yescharacteristic
500C22#1684057711yescharacteristic
501C4—22no—
502C4—22no—
503C4—42no—
504C4—81no—
505C4—82no—
506C4—81no—
507C4—1224no—
508C4—2442no—
509C4—2444no—
510C4#124×D6111yescharacteristic
511C4#1684033011yescharacteristic
512C4#1684040512yes—
513C2—44no—
514C2#2144946711yescharacteristic
515C2#2145505211yescharacteristic
516C2#2145513111yescharacteristic
517C2#2145786412yes—
518C2#2145798412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2562C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2562D61
hypocenterC61#25894
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2562

Complements

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

NH ≅ G/Ntype
2. C2×#16840132C2semidirect
3. C2×#16840066C2semidirect
4. #21447022C2semidirect
5. #21447015C2semidirect
6. #21446742C2semidirect
8. #21445753C2semidirect
9. #21445753C2semidirect
10. #21445652C2semidirect
304. C61#25894semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
C2×#168401322
C2×#168400662
#214470222
#214470152
#214467422
#214466182
#214457532
#214456522
C61×#25622
#2589461