Finite Group Databasebeta

← #29730129

Subgroups — #29730129

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic, complemented
2#21450992C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447252C212yesmaximal, complemented
5#21447009C212yesmaximal, complemented
6#21446780C211yesmaximal, characteristic, complemented
7#21446583C211yesmaximal, characteristic, complemented
8#21446247C212yesmaximal, complemented
9#21445844C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×#14519748—21no—
15C2×C4×D244C2211yescharacteristic
16#16840053—24no—
17C2×#14519709C2212yes—
18C2×#14519709C2212yes—
19C2×#14519699C2211yescharacteristic
20C2×#14519699C2211yescharacteristic
21C2×#14519699C2212yes—
22#16839678—22no—
23#16839674—22no—
24#16839665—21no—
25#16839632C2211yescharacteristic
26#16839627—21no—
27#16839622C2212yes—
28#16839622C2212yes—
29#16839622C2212yes—
30#16839619—21no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37C23×C8×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C8×D4×C61—24no—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61—21no—
47#14519777—22no—
48#14519777—22no—
49#14519751—24no—
50#14519751—42no—
51#14519748—44no—
52C4×D244—22no—
53C4×D244—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×#11758198—21no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68#14519709—24no—
69#14519709—42no—
70#14519699—22no—
71#14519699—24no—
72#14519699—22no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76#14519684—22no—
77C22×D4×C61D411yescharacteristic
78C23×C4×C61C2312yes—
79C22×C8×C61—24no—
80C22×C8×C61—24no—
81C22×C8×C61—42no—
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C22×C8×C61C2311yescharacteristic
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C22⋊C4×C61D412yes—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61D414yescomplemented
91C22⋊C8×C61—24no—
92C4×C8×C61—22no—
93C24×C61C2×D412yes—
94C23×D61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—22no—
97C2×D4×C61SD1614yes—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C2×C22⋊D61—44no—
109C2×C22⋊D61—44no—
110C2×C22⋊D61—82no—
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244#4112yes—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—42no—
126C2×Dic122—81no—
127C2×Dic122—81no—
128C2×C8×C61—21no—
129C2×C8×C61—22no—
130C2×C8×C61—21no—
131C2×C8×C61—22no—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—48no—
136C2×C8×C61C2×D414yes—
137C2×C8×C61#4112yes—
138C4⋊C4×C61—22no—
139C22⋊C4×C61—24no—
140C42×C61C2×D412yes—
141C42×C61#4112yes—
142C22⋊Dic61—44no—
143C22⋊Dic61—82no—
144C22⋊Dic61—82no—
145#11758198—44no—
146#11758198—42no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—42no—
158C4⋊Dic61—42no—
159C4⋊Dic61—81no—
160C4×Dic61—42no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61C2×SD1614yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176D4×C61—28no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61#12112yes—
187C2×C4×C61#12112yes—
188C2×C4×C61C2×SD1612yes—
189C2×C4×C61#13512yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61H(5,𝔽2)11yescharacteristic
195C2×C4×C612−1+411yescharacteristic
196C22⋊D61—88no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—84no—
217Dic122—84no—
218C8×C61—22no—
219C8×C61—24no—
220C8×C61—41no—
221#25339—611nomaximal, Sylow
222C22×C61—24no—
223C22×C61—24no—
224C22×C61—24no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61C2×#12111yescharacteristic
230C22×C61#54611yescharacteristic
231C22×C61#54711yescharacteristic
232C22×C61C22×SD1611yescharacteristic
233C22×C61C2×#13511yescharacteristic
234C22×C61#57611yescharacteristic
235C22×C61#57711yescharacteristic
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—22no—
244C4×C61—42no—
245C4×C61#46514yes—
246C4×C61#52911yescharacteristic
247C4×C61#54211yescharacteristic
248C4×C61#54611yescharacteristic
249C4×C61#54811yescharacteristic
250Dic61—81no—
251Dic61—82no—
252Dic61—81no—
253Dic61—81no—
254Dic61—82no—
255C2×#476—611no—
256C2×C8×D4D6111yescharacteristic, complemented
257#2563—611no—
258#2537—611no—
259#2077—612no—
260#2034—612no—
261#1990—611no—
262#1948—611no—
263#1893—612no—
264#1807—612no—
265C4×#102—611no—
266C2×C61—44no—
267C2×C61#242711yescharacteristic
268C2×C61C2×#46511yescharacteristic
269C2×C61#306211yescharacteristic
270C2×C61#329511yescharacteristic
271C2×C61#330311yescharacteristic
272C2×C61#333511yescharacteristic
273C2×C61#334111yescharacteristic
274D61—82no—
275C23×C8C2×D6112yes—
276C2×C4⋊Q8—611no—
277C2×#121—612no—
278C2×#121—1221no—
279C2×C4×D4—611no—
280C2×C4×D4C2×D6111yescharacteristic
281#476—1224no—
282C8×D4—24no—
283C2×#106—612no—
284C2×#106—612no—
285C2×C4⋊C8C2×D6111yescharacteristic
286C2×#102—611no—
287C2×#102—612no—
288C2×#102—611no—
289C2×C22⋊C8C2×D6112yes—
290C2×C4×C8C2×D6111yescharacteristic
291#394—1222no—
292#393—1222no—
293#390—1221no—
294#384—611no—
295#382—1221no—
296#380—612no—
297#380—612no—
298#380—612no—
299#380—1221no—
300C4×C4⋊C4—611no—
301#340—611no—
302#340—612no—
303#340—611no—
304#340—612no—
305#340—612no—
306C61#2533911yescharacteristic, complemented, Sylow
307C22×Q8—1221no—
308C22×D4—611no—
309C22×D4—1222no—
310C22×D4D24411yescharacteristic
311C23×C4—1221no—
312C23×C4C22×D6112yes—
313C22×C8—24no—
314C22×C8—24no—
315C22×C8—42no—
316C22×C8C22×D6111yescharacteristic
317C22×C8C22×D6111yescharacteristic
318C22×C8C22×D6112yes—
319C4⋊Q8—1222no—
320C4⋊Q8—1222no—
321#121—1224no—
322#121—2442no—
323#121—2444no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—1222no—
327C4×D4—1222no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4D24411yescharacteristic
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4D24412yes—
343C2×C42—1221no—
344C2×C42C22×D6111yescharacteristic
345#106—1224no—
346#106—2442no—
347C4⋊C8C22⋊D6114yescomplemented
348#102—1222no—
349#102—1222no—
350#102—1224no—
351C22⋊C8—24no—
352C4×C8—22no—
353#95—1221no—
354#95—1222no—
355#95—1222no—
356#95—1222no—
357C24—1221no—
358C24C2×D24412yes—
359C2×Q8—4881no—
360C2×Q8—4881no—
361C2×Q8—2442no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—612no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—4882no—
371C2×D4—2444no—
372C2×D4—2444no—
373C2×D4#1175819014yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4C2×D24412yes—
389C22×C4C2×D24411yescharacteristic
390C22×C4C2×D24411yescharacteristic
391C22×C4C23×D6111yescharacteristic
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—48no—
400C2×C8#1175822412yes—
401C2×C8C2×C22⋊D6114yes—
402C4⋊C4—22no—
403C4⋊C4—1222no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1221no—
413C4⋊C4—4881no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C22⋊C4—24no—
417C22⋊C4—4882no—
418C22⋊C4—4882no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2442no—
422C42—2442no—
423C42#1175822212yes—
424C42C2×C22⋊D6112yes—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—1222no—
432C23—2441no—
433C23—2441no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23C2×#1175819014yes—
439C23C22×D24411yescharacteristic
440Q8—4884no—
441D4—28no—
442D4—1222no—
443D4—1222no—
444D4—4888no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—4884no—
457C2×C4—2441no—
458C2×C4—2442no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4#1451975112yes—
473C2×C4#1451975112yes—
474C2×C4C2×#1175819012yes—
475C2×C4#1451979912yes—
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4#41×D6111yescharacteristic
478C2×C4C2×#1175822411yescharacteristic
479C2×C4C2×#1175822211yescharacteristic
480C2×C4#1451982211yescharacteristic
481C2×C4#1451982411yescharacteristic
482C8—22no—
483C8—24no—
484C8—41no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975111yescharacteristic
500C22#1684040711yescharacteristic
501C22#1684041611yescharacteristic
502C22C22×#1175819011yescharacteristic
503C22C2×#1451979911yescharacteristic
504C22#1684056711yescharacteristic
505C22#1684057411yescharacteristic
506C4—22no—
507C4—42no—
508C4—4881no—
509C4—4881no—
510C4—4881no—
511C4—4882no—
512C4—4882no—
513C4#1683999114yes—
514C4#1684029411yescharacteristic
515C4#1684037711yescharacteristic
516C4#1684040611yescharacteristic
517C4#1684041911yescharacteristic
518C2—44no—
519C2—4882no—
520C2#2144939011yescharacteristic
521C2C2×#1683999111yescharacteristic
522C2#2145549911yescharacteristic
523C2#2145764411yescharacteristic
524C2#2145774811yescharacteristic
525C2#2145817711yescharacteristic
526C2#2145824211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×C8×D4D61
hypocenterC61#25339
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840053C2semidirect
2. #21450992C2semidirect
4. #21447252C2semidirect
5. #21447009C2semidirect
6. #21446780C2semidirect
7. #21446583C2semidirect
8. #21446247C2semidirect
9. #21445844C2semidirect
10. C4×#14519699C2semidirect
11. C2×C8×D4×C61C2semidirect
90. C4⋊C8×C61D4semidirect
256. C2×C8×D4D61semidirect
306. C61#25339semidirect
347. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
#214509922
#214507322
#214472522
#214470092
#214467802
#214465832
#214462472
#214458442
C4×#145196992
C2×C8×D4×C612
#2533961