Finite Group Databasebeta

← #29571625

Subgroups — #29571625

Order 15616.

782 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840480C211yesmaximal, characteristic
2#21445918C211yesmaximal, characteristic, complemented
3C4×#14519691C211yesmaximal, characteristic
4#21444340C211yesmaximal, characteristic, complemented
5#21444338C211yesmaximal, characteristic, complemented
6#21444155C211yesmaximal, characteristic
7C61×#1826C211yesmaximal, characteristic, complemented
8#16840480—22no—
9#16840480—22no—
10#16840480—22no—
11#16840480—22no—
12C2×#14519776C411yescharacteristic, complemented
13C2×#14519771—21no—
14C2×#14519770—21no—
15C2×#14519763—21no—
16C2×#14519757C411yescharacteristic, complemented
17C2×#14519756C2211yescharacteristic
18C2×C4×Dic122—21no—
19C2×#14519742C411yescharacteristic, complemented
20C2×C4×C22⋊D61C2211yescharacteristic
21C2×#14519735C411yescharacteristic, complemented
22C2×#14519691—21no—
23#16839726C2211yescharacteristic
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—21no—
26#16839521—22no—
27#16839500—22no—
28#16839498—22no—
29#16839487—21no—
30#16839485—21no—
31C2×#117×C61C2211yescharacteristic
32C61×#407C2211yescharacteristic
33C4×C22⋊C4×C61C2211yescharacteristic
34C61×#339—22no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C22×C22⋊D61D411yescharacteristic
38C22×C4×D61C2×C411yescharacteristic
39C2×#11758194—21no—
40#14519776—22no—
41#14519776—22no—
42#14519771—24no—
43#14519771—24no—
44#14519770—24no—
45#14519770—24no—
46#14519763—24no—
47#14519763—24no—
48#14519757—22no—
49#14519757—22no—
50#14519756—21no—
51#14519756—21no—
52#14519756D412yes—
53#14519756D412yes—
54C4×Dic122—24no—
55C4×Dic122—24no—
56#14519742—22no—
57#14519742—22no—
58C4×C22⋊D61—21no—
59C4×C22⋊D61—21no—
60C4×C22⋊D61—21no—
61C4×C22⋊D61—21no—
62#14519735—22no—
63#14519735—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×#11758196D411yescharacteristic
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61D411yescharacteristic
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198D411yescharacteristic
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2×C411yescharacteristic
80#14519691—24no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83#14519684—21no—
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88#117×C61D412yes—
89#117×C61D412yes—
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C4⋊C4×C61C2×C411yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C42×C61—21no—
95C2×C42×C61—21no—
96C22⋊C8×C61—21no—
97C22⋊C8×C61—21no—
98#95×C61—21no—
99C24×C61C2×D411yescharacteristic
100C23×D61C22⋊C411yescharacteristic
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—42no—
104C22×C4×C61—41no—
105C22×C4×C61C22×C411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—21no—
112C2×C22⋊D61—21no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—41no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22⋊C411yescharacteristic
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—42no—
124C2×Dic122—24no—
125C2×Dic122—21no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122—42no—
129C2×Dic122C22⋊C412yes—
130M16×C61—42no—
131C2×C8×C61—41no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61C22⋊C412yes—
135C4⋊C4×C61C22⋊C412yes—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—41no—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61—42no—
143C22⋊Dic61—22no—
144C22⋊Dic61—22no—
145C22⋊Dic61—21no—
146C22⋊Dic61—21no—
147C22⋊Dic61—22no—
148C22⋊Dic61—41no—
149#11758198—24no—
150#11758198—24no—
151#11758198—21no—
152#11758198—21no—
153C4⋊Dic61—24no—
154C4⋊Dic61—21no—
155C4⋊Dic61C22⋊C412yes—
156#11758196—24no—
157#11758196—24no—
158#11758196—22no—
159#11758196—22no—
160#11758196—21no—
161#11758196—21no—
162C4×Dic61—24no—
163C4×Dic61—24no—
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—42no—
167#11758194—42no—
168C2×#92418—41no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61C2×C22⋊C411yescharacteristic
174C23×C61#12011yescharacteristic
175C23×C61#12011yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—41no—
180C22×D61—41no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—81no—
194C2×C4×C61—81no—
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C22⋊D61—42no—
200C22⋊D61—42no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—81no—
214C2×Dic61—81no—
215C2×Dic61#10412yes—
216C4×D61—42no—
217C4×D61—42no—
218Dic122—48no—
219Dic122—42no—
220Dic122—42no—
221C8×C61—81no—
222#92418—81no—
223#8365—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#37911yescharacteristic
235C22×C61C2×#10411yescharacteristic
236C22×C61#42111yescharacteristic
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—81no—
241C2×D61—81no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61#41012yes—
249Dic61—22no—
250Dic61—41no—
251Dic61—42no—
252Dic61—41no—
253Dic61—42no—
254Dic61—82no—
255C2×#556—611no—
256#1826—611no—
257#1826D6111yescharacteristic, complemented
258C4×#99—611no—
259#1508—611no—
260#1508—611no—
261#1465—611no—
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—41no—
265C2×C61—41no—
266C2×C61#178211yescharacteristic
267C2×C61#178911yescharacteristic
268C2×C61#179611yescharacteristic
269D61—81no—
270#556—1222no—
271#556—1222no—
272#556—1222no—
273#556—1222no—
274C2×C4⋊Q8—611no—
275C2×#126—1221no—
276C2×#126—1221no—
277C2×#124—1221no—
278C2×C22⋊Q8—611no—
279C2×C22⋊Q8—611no—
280C2×C4×Q8—611no—
281C2×C4×Q8—1221no—
282C2×C4×D4—611no—
283C2×#117—611no—
284C2×#117C2×D6111yescharacteristic
285C2×#99—1221no—
286#407—611no—
287#407C2×D6111yescharacteristic
288C4×C22⋊C4—611no—
289C4×C22⋊C4C2×D6111yescharacteristic
290#342—1221no—
291#339—22no—
292#339—1222no—
293#328—1222no—
294#328—1222no—
295#324—1221no—
296#323—1221no—
297C61#836511yescharacteristic, complemented, Sylow
298C22×Q8—611no—
299C22×Q8—1221no—
300C22×D4—611no—
301C23×C4—611no—
302C23×C4C22×D6111yescharacteristic
303C2×M16—21no—
304C2×M16—1221no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307#126—1224no—
308#126—1224no—
309#126—1224no—
310#126—1224no—
311#124—1224no—
312#124—1224no—
313C22⋊Q8—612no—
314C22⋊Q8—612no—
315C22⋊Q8—1222no—
316C22⋊Q8—1221no—
317C22⋊Q8—1221no—
318C22⋊Q8—1222no—
319C4×Q8—1224no—
320C4×Q8—1224no—
321C4×Q8—1222no—
322C4×Q8—1222no—
323C4×D4—1221no—
324C4×D4—1221no—
325C4×D4—1221no—
326C4×D4—1221no—
327#117—21no—
328#117—21no—
329#117—1222no—
330#117—1222no—
331#117D24412yes—
332#117C22⋊D6112yes—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4C4×D6111yescharacteristic
340C2×C4⋊C4C4×D6111yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—611no—
344C2×C22⋊C4—611no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—611no—
352C2×C42—1221no—
353C2×C42—1221no—
354C2×C42—1221no—
355#99—1224no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—1221no—
359C22⋊C8—1221no—
360#95—21no—
361#95—1221no—
362C24—611no—
363C24D4×D6111yescharacteristic
364C2×Q8—612no—
365C2×Q8—1224no—
366C2×Q8—1221no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×Q8—2442no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—2441no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—2442no—
386C22×C4—2441no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4C2×D24411yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393M16—42no—
394M16—2442no—
395C2×C8—41no—
396C2×C8—2441no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—612no—
400C4⋊C4—1224no—
401C4⋊C4—1224no—
402C4⋊C4—1224no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—1222no—
408C4⋊C4#1175819812yes—
409C4⋊C4#1175819812yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1221no—
417C22⋊C4—1222no—
418C22⋊C4—1221no—
419C22⋊C4—1222no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1222no—
423C22⋊C4—2441no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—1224no—
428C42—1224no—
429C42—1221no—
430C42—1221no—
431C42—2442no—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—1221no—
437C23—1221no—
438C23—1221no—
439C23—2441no—
440C23—2441no—
441C23C22⋊C4×D6111yescharacteristic
442C23#1451974511yescharacteristic
443C23#1451974711yescharacteristic
444Q8—2442no—
445Q8—2448no—
446Q8—2442no—
447D4—2442no—
448D4—2442no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—612no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1222no—
468C2×C4—1221no—
469C2×C4—1222no—
470C2×C4—4881no—
471C2×C4—4881no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4C22⋊C4×D6111yescharacteristic
481C2×C4C2×#1175819811yescharacteristic
482C2×C4#1451974511yescharacteristic
483C2×C4#1451974711yescharacteristic
484C8—81no—
485C8—4881no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—81no—
495C22—81no—
496C22—4881no—
497C22—4881no—
498C22—2441no—
499C22—2441no—
500C22—2441no—
501C22#1683961711yescharacteristic
502C22#104×D6111yescharacteristic
503C22#1683977711yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4—1222no—
511C4—4882no—
512C4—2441no—
513C4—2442no—
514C4—2442no—
515C4—2441no—
516C4#1683973512yes—
517C2—21no—
518C2—21no—
519C2—41no—
520C2—41no—
521C2—4881no—
522C2#2144573611yescharacteristic
523C2#2144576111yescharacteristic
524C2#2144579311yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1826C2
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC23×C4×C61C23
hypercenter#1826D61
hypocenterC61#8365
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
2. #21445918C2semidirect
4. #21444340C2semidirect
5. #21444338C2semidirect
7. C61×#1826C2semidirect
12. C2×#14519776C4semidirect
16. C2×#14519757C4semidirect
19. C2×#14519742C4semidirect
21. C2×#14519735C4semidirect
257. #1826D61semidirect
297. C61#8365semidirect

Maximal subgroups

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

subgroupindex
C2×#168404802
#214459182
C4×#145196912
#214443402
#214443382
#214441552
C61×#18262
#836561