Finite Group Databasebeta

← #29710004

Subgroups — #29710004

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519751C211yesmaximal, characteristic
2#21448226C211yesmaximal, characteristic, complemented
3#21447378C211yesmaximal, characteristic, complemented
4#21447346C211yesmaximal, characteristic, complemented
5#21447292C211yesmaximal, characteristic, complemented
6#21447242C212yesmaximal, complemented
7#21447224C211yesmaximal, characteristic, complemented
8#21447002C212yesmaximal
9C4×#14519711C212yesmaximal, complemented
10C4×#14519699C212yesmaximal, complemented
11C4×C22⋊C8×C61C211yesmaximal, characteristic, complemented
12C2×#14519751C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×C4×C22⋊D61—21no—
15C2×#14519736—21no—
16C2×#14519711C2212yes—
17C2×#14519711C2212yes—
18C2×#14519699C2212yes—
19C2×#14519699C2212yes—
20#16839664C2211yescharacteristic
21#16839660C2211yescharacteristic
22#16839653—21no—
23#16839641—21no—
24#16839632C2211yescharacteristic
25#16839632C2211yescharacteristic
26#16839628—21no—
27#16839622—21no—
28#16839622C2212yes—
29#16839622C2211yescharacteristic
30C4×C4⋊Dic61—21no—
31C4×C4⋊Dic61C2212yes—
32C4×C4⋊Dic61C2211yescharacteristic
33C4×#11758198—21no—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C22×C42×C61C2211yescharacteristic
39C2×C22⋊C8×C61C2212yes—
40C2×C4×C8×C61C2212yes—
41C61×#336C2211yescharacteristic
42C61×#336C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C22×C4×D61—21no—
46#14519751—22no—
47#14519751—22no—
48C4×D244—24no—
49C4×C22⋊D61—44no—
50#14519736—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61—21no—
69#14519711—24no—
70#14519711—24no—
71#14519699—24no—
72#14519699—24no—
73#14519684—21no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684—21no—
78C23×C4×C61D411yescharacteristic
79C23×C4×C61D411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C22×C8×C61C2312yes—
82C22×C8×C61C2312yes—
83C2×C42×C61—24no—
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88C22⋊C8×C61—24no—
89C4×C8×C61—24no—
90C24×C61C2×D411yescharacteristic
91C23×D61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—41no—
101C22×C4×C61D814yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C2×C22⋊D61—24no—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—42no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—21no—
119C2×D244#4112yes—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×C4×D61—42no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61#4114yes—
129C42×C61—22no—
130C42×C61—22no—
131C42×C61—22no—
132C42×C61#4114yes—
133C22⋊Dic61—44no—
134C22⋊Dic61—42no—
135#11758198—24no—
136#11758198—42no—
137#11758198—42no—
138#11758198—42no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—42no—
147C4⋊Dic61—42no—
148C4⋊Dic61—42no—
149C4⋊Dic61—42no—
150C4⋊Dic61#4112yes—
151#11758196—42no—
152C4×Dic61—44no—
153C4×Dic61—42no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61C2×D812yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—28no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—44no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61#12312yes—
183C2×C4×C61#12312yes—
184C2×C4×C61C2×D812yes—
185C2×C4×C61C2×D812yes—
186C2×C4×C61#13512yes—
187C2×C4×C61#13712yes—
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4112yes—
191C2×C4×C61C2×#4112yes—
192C22⋊D61—48no—
193C2×Dic61—22no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208D244—24no—
209D244—42no—
210C4×D61—84no—
211C8×C61—24no—
212C8×C61—42no—
213#23754—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61C2×#12311yescharacteristic
222C22×C61#52511yescharacteristic
223C22×C61#53211yescharacteristic
224C22×C61C22×D811yescharacteristic
225C22×C61C2×#13511yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C22×C61#57511yescharacteristic
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—22no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—41no—
239C4×C61#48014yes—
240C4×C61#52912yes—
241C4×C61#52912yes—
242Dic61—42no—
243Dic61—81no—
244Dic61—82no—
245Dic61—82no—
246Dic61—81no—
247C4×#121—611no—
248#2285—611no—
249#2099—611no—
250#2093—611no—
251#2083—611no—
252#2075—612no—
253#2071—611no—
254#2033—612no—
255C4×#107—612no—
256C4×#102—612no—
257C4×C22⋊C8D6111yescharacteristic, complemented
258C2×C61—42no—
259C2×C61#239511yescharacteristic
260C2×C61C2×#48011yescharacteristic
261C2×C61#309111yescharacteristic
262C2×C61#310011yescharacteristic
263C2×C61#310711yescharacteristic
264C2×C61#316911yescharacteristic
265C2×C61#317211yescharacteristic
266D61—82no—
267C2×#121—611no—
268C2×C4×D4—611no—
269C2×C4×D4—1221no—
270C2×#117—1221no—
271C22×C42C2×D6111yescharacteristic
272C2×#107—612no—
273C2×#107—612no—
274C2×#102—612no—
275C2×#102—612no—
276C2×C22⋊C8C2×D6112yes—
277C2×C4×C8C2×D6112yes—
278#390—611no—
279#389—611no—
280#388—1221no—
281#386—1221no—
282#384—611no—
283#384—611no—
284#382—1221no—
285#380—611no—
286#380—612no—
287#380—1221no—
288C4×C4⋊C4—612no—
289C4×C4⋊C4—611no—
290C4×C4⋊C4—1221no—
291C4×C22⋊C4—1221no—
292#340—612no—
293#340—612no—
294#340—612no—
295#340—612no—
296#336C2×D6111yescharacteristic
297#336C2×D6111yescharacteristic
298C61#2375411yescharacteristic, complemented, Sylow
299C22×D4—611no—
300C22×D4—1221no—
301C23×C4—1221no—
302C23×C4D24411yescharacteristic
303C23×C4D24411yescharacteristic
304C23×C4C22×D6111yescharacteristic
305C22×C8C22×D6112yes—
306C22×C8C22×D6112yes—
307#121—1222no—
308#121—1222no—
309C4×D4—1224no—
310C4×D4—2444no—
311#117—2444no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1221no—
328C2×C42—24no—
329C2×C42—1222no—
330C2×C42—1221no—
331C2×C42D24411yescharacteristic
332C2×C42D24411yescharacteristic
333C2×C42C22×D6111yescharacteristic
334C2×C42C22×D6111yescharacteristic
335#107—1224no—
336#107—1224no—
337#102—1224no—
338#102—1224no—
339C22⋊C8—24no—
340C4×C8—24no—
341#95—1221no—
342#95—1221no—
343#95—1222no—
344#95—1221no—
345#95—1222no—
346C24—1221no—
347C24C2×D24411yescharacteristic
348C2×D4—612no—
349C2×D4—1224no—
350C2×D4—1221no—
351C2×D4—1222no—
352C2×D4—1222no—
353C2×D4—2442no—
354C2×D4—2442no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1222no—
367C22×C4—1222no—
368C22×C4—1221no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4D48814yes—
373C22×C4C2×D24411yescharacteristic
374C22×C4C2×D24411yescharacteristic
375C22×C4C2×D24411yescharacteristic
376C22×C4C2×D24411yescharacteristic
377C22×C4C2×D24411yescharacteristic
378C22×C4C23×D6111yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8#1175822214yes—
385C4⋊C4—612no—
386C4⋊C4—1221no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—1222no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4—2442no—
394C4⋊C4—2442no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4—2442no—
398C22⋊C4—1224no—
399C22⋊C4—2444no—
400C22⋊C4—2442no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C22⋊C4—2442no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—2444no—
408C42—2442no—
409C42#1175822414yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—1222no—
417C23—2441no—
418C23—2441no—
419C23—2441no—
420C23—2441no—
421C23—2441no—
422C23—2441no—
423C23C2×D48812yes—
424C23C22×D24411yescharacteristic
425D4—1224no—
426D4—2448no—
427D4—2442no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—28no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—1222no—
442C2×C4—4884no—
443C2×C4—2441no—
444C2×C4—2442no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2441no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4#1451976112yes—
458C2×C4#1451976112yes—
459C2×C4C2×D48812yes—
460C2×C4C2×D48812yes—
461C2×C4#1451979912yes—
462C2×C4#1451981012yes—
463C2×C4#41×D6111yescharacteristic
464C2×C4#41×D6111yescharacteristic
465C2×C4C2×#1175822412yes—
466C2×C4C2×#1175822212yes—
467C8—24no—
468C8—42no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—4881no—
477C22—4881no—
478C22—4881no—
479C22—4881no—
480C22—2442no—
481C22—2442no—
482C22—2442no—
483C22C2×#1451976111yescharacteristic
484C22#1684027411yescharacteristic
485C22#1684031211yescharacteristic
486C22C22×D48811yescharacteristic
487C22C2×#1451979911yescharacteristic
488C22C2×#1451981011yescharacteristic
489C22#1684056211yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—4881no—
495C4—4882no—
496C4—4881no—
497C4—4882no—
498C4—2442no—
499C4#1684007614yes—
500C4#1684029212yes—
501C4#1684029412yes—
502C2—42no—
503C2—4882no—
504C2#2144907911yescharacteristic
505C2C2×#1684007611yescharacteristic
506C2#2145573411yescharacteristic
507C2#2145582011yescharacteristic
508C2#2145589111yescharacteristic
509C2#2145639811yescharacteristic
510C2#2145642511yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊C8×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC4×C22⋊C8D61
hypocenterC61#23754
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #21448226C2semidirect
3. #21447378C2semidirect
4. #21447346C2semidirect
5. #21447292C2semidirect
6. #21447242C2semidirect
7. #21447224C2semidirect
9. C4×#14519711C2semidirect
10. C4×#14519699C2semidirect
11. C4×C22⋊C8×C61C2semidirect
257. C4×C22⋊C8D61semidirect
298. C61#23754semidirect

Maximal subgroups

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

subgroupindex
C4×#145197512
#214482262
#214473782
#214473462
#214472922
#214472422
#214472242
#214470022
C4×#145197112
C4×#145196992
C4×C22⋊C8×C612
#2375461