Finite Group Databasebeta

← #29721575

Subgroups — #29721575

Order 15616.

946 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×#16839985C212yesmaximal
2C2×#16839973C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450814C211yesmaximal, characteristic
5C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
6#21447199C212yesmaximal, complemented
7#21447061C212yesmaximal
8#21446621C211yesmaximal, characteristic, complemented
9#21446159C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×Dic244C2212yes—
13C22×#11758190C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519765C2212yes—
16C2×C4×Dic122C2212yes—
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16839985—28no—
20#16839973—24no—
21C4×#11758190—24no—
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2212yes—
29#16839689—22no—
30#16839668C2212yes—
31#16839660C2212yes—
32#16839659—22no—
33#16839632C2211yescharacteristic
34#16839631—21no—
35#16839619—21no—
36#16839619—21no—
37C4⋊C4×Dic61—21no—
38#16839525C2212yes—
39C2×C4⋊C8×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#336C2212yes—
44C22×Dic122C2312yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×Dic244—24no—
49C2×Dic244—24no—
50C2×Dic244—24no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C2×#11758190D414yes—
54#14519777—22no—
55#14519777—22no—
56#14519765—24no—
57#14519765—42no—
58C4×Dic122—22no—
59C4×Dic122—24no—
60C4×Dic122—42no—
61C4×Dic122D414yes—
62C4×D244—22no—
63C4×D244—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—22no—
79#14519709—22no—
80#14519702—22no—
81#14519702—24no—
82#14519702—24no—
83#14519702—42no—
84#14519699—24no—
85#14519699—22no—
86#14519684—22no—
87#14519684—22no—
88#14519684—22no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2312yes—
91C22×C8×C61C2311yescharacteristic
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C42×C61C2312yes—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—24no—
97C4⋊C8×C61—22no—
98C4×C8×C61—22no—
99#95×C61D412yes—
100C23×D61—21no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244C2×D412yes—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122C2×D412yes—
130C2×Dic122C2×D412yes—
131C2×Dic122#4112yes—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61C2×D412yes—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147#11758198—44no—
148#11758198—44no—
149#11758198—42no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157C4⋊Dic61—42no—
158C4⋊Dic61—81no—
159C4⋊Dic61C2×D412yes—
160#11758196—24no—
161#11758196—44no—
162#11758196—82no—
163C4×Dic61—24no—
164C4×Dic61—44no—
165Dic244—48no—
166#11758190—28no—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61#12112yes—
184C2×C4×C61#12112yes—
185C2×C4×C61#13512yes—
186C2×C4×C61#13712yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)11yescharacteristic
192C2×C4×C612−1+411yescharacteristic
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208D244—22no—
209D244—22no—
210D244—41no—
211D244—41no—
212C4×D61—84no—
213Dic122—22no—
214Dic122—22no—
215Dic122—22no—
216Dic122—42no—
217Dic122—42no—
218Dic122—42no—
219Dic122—42no—
220Dic122#12114yes—
221C8×C61—22no—
222C8×C61—42no—
223C8×C61—41no—
224#24729—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61D4211yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61C2×#13511yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57711yescharacteristic
231C22×C61#57811yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—22no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61#46414yescomplemented
243C4×C61C2×#12111yescharacteristic
244C4×C61#53711yescharacteristic
245C4×C61#54711yescharacteristic
246C4×C61#55311yescharacteristic
247Dic61—24no—
248Dic61—42no—
249Dic61—82no—
250Dic61—82no—
251Dic61—81no—
252C2×#464—612no—
253C2×#462—611no—
254C2×C4×SD16—611no—
255#2547—611no—
256Q8×C4⋊C4—611no—
257#2066—612no—
258#2042—612no—
259#1956—611no—
260#1939D6111yescharacteristic, complemented
261#1878—612no—
262#1802—611no—
263C2×C61#241611yescharacteristic
264C2×C61C2×#46411yescharacteristic
265C2×C61#304611yescharacteristic
266C2×C61#327611yescharacteristic
267C2×C61#328411yescharacteristic
268C2×C61#336411yescharacteristic
269C2×C61#336911yescharacteristic
270D61—82no—
271C22×Q16—612no—
272C22×SD16—611no—
273C2×C4⋊Q8—611no—
274C2×#124—612no—
275C2×C4×Q8—611no—
276C2×C4×Q8—612no—
277C2×C4×D4—611no—
278#464—1228no—
279#462—1224no—
280C4×SD16—1224no—
281C2×#106—611no—
282C2×C4⋊C8C2×D6111yescharacteristic
283C2×C4⋊C8C2×D6112yes—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—611no—
287C2×#103—612no—
288C2×#102—612no—
289C2×#102—611no—
290C2×C4×C8C2×D6111yescharacteristic
291#397—1222no—
292#391—612no—
293#389—612no—
294#389—1222no—
295#384—611no—
296#384—1221no—
297#380—1221no—
298#380—1221no—
299C4×C4⋊C4—1221no—
300C4×C4⋊C4C2×D6111yescharacteristic
301#340—612no—
302#336C2×D6112yes—
303C61#2472911yescharacteristic, complemented, Sylow
304C22×Q8—611no—
305C22×Q8—612no—
306C22×D4—611no—
307C23×C4—1221no—
308C2×Q16—1224no—
309C2×Q16—1224no—
310C2×Q16—1224no—
311C2×SD16—614no—
312C2×SD16—1222no—
313C2×SD16—1222no—
314C22×C8C22×D6112yes—
315C22×C8C22×D6111yescharacteristic
316C22×C8C22×D6111yescharacteristic
317C4⋊Q8—1222no—
318C4⋊Q8—1222no—
319#124—1224no—
320#124—2442no—
321C4×Q8—614no—
322C4×Q8—1224no—
323C4×Q8—1222no—
324C4×Q8—2442no—
325C4×D4—1222no—
326C4×D4—1222no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4D24411yescharacteristic
338C2×C4⋊C4D24411yescharacteristic
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C42—1221no—
343C2×C42—1222no—
344C2×C42C22×D6112yes—
345C2×C42C22×D6111yescharacteristic
346#106—1222no—
347C4⋊C8—24no—
348C4⋊C8—22no—
349#103—1224no—
350#103—1222no—
351#103—1224no—
352#103—2442no—
353#102—1222no—
354#102—1224no—
355C4×C8—22no—
356#95—1222no—
357#95—1222no—
358#95—1222no—
359#95D24412yes—
360C24—1221no—
361C2×Q8—612no—
362C2×Q8—612no—
363C2×Q8—612no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×D4—612no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4C2×D24412yes—
388C22×C4C2×D24412yes—
389C22×C4C2×D24411yescharacteristic
390C22×C4C2×D24411yescharacteristic
391C22×C4C23×D6111yescharacteristic
392Q16—2448no—
393SD16—1228no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8D4×D6112yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—612no—
406C4⋊C4—1221no—
407C4⋊C4—1224no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—4882no—
413C4⋊C4—4881no—
414C4⋊C4—2444no—
415C4⋊C4—2442no—
416C4⋊C4—2442no—
417C4⋊C4—2442no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C42—22no—
422C42—22no—
423C42—1224no—
424C42—2444no—
425C42D4×D6112yes—
426C42#1175822612yes—
427C23—1222no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23C22×D24411yescharacteristic
435Q8—614no—
436Q8—1222no—
437Q8—1222no—
438Q8—1222no—
439Q8—2442no—
440Q8—2442no—
441Q8—2442no—
442Q8—2442no—
443D4—1222no—
444D4—1222no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—4884no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451975212yes—
472C2×C4#1451975212yes—
473C2×C4#1451979912yes—
474C2×C4#1451981012yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4#41×D6111yescharacteristic
478C2×C4C2×#1175822611yescharacteristic
479C2×C4#1451982311yescharacteristic
480C2×C4#1451982511yescharacteristic
481C8—22no—
482C8—42no—
483C8—41no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×#1451975211yescharacteristic
492C22D4×D24411yescharacteristic
493C22#1684042411yescharacteristic
494C22C2×#1451979911yescharacteristic
495C22C2×#1451981011yescharacteristic
496C22#1684057411yescharacteristic
497C22#1684057911yescharacteristic
498C4—22no—
499C4—42no—
500C4—42no—
501C4—1224no—
502C4—4882no—
503C4—4882no—
504C4—4881no—
505C4—2442no—
506C4#1683998514yescomplemented
507C4#121×D6111yescharacteristic
508C4#1684034211yescharacteristic
509C4#1684041511yescharacteristic
510C4#1684046511yescharacteristic
511C2—4882no—
512C2#2144925711yescharacteristic
513C2C2×#1683998511yescharacteristic
514C2#2145537111yescharacteristic
515C2#2145746711yescharacteristic
516C2#2145755111yescharacteristic
517C2#2145850611yescharacteristic
518C2#2145856111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1939D61
hypocenterC61#24729
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. C2×#16839973C2semidirect
5. C4⋊C4×Dic122C2semidirect
6. #21447199C2semidirect
8. #21446621C2semidirect
9. #21446159C2semidirect
11. C61×#1939C2semidirect
242. C4×C61#464semidirect
260. #1939D61semidirect
303. C61#24729semidirect
506. C4#16839985semidirect

Maximal subgroups

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

subgroupindex
C2×#168399852
C2×#168399732
C2×C4×#117581902
#214508142
C4⋊C4×Dic1222
#214471992
#214470612
#214466212
#214461592
#214458202
C61×#19392
#2472961