Finite Group Databasebeta

← #29721571

Subgroups — #29721571

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×#16839984C212yesmaximal
2C2×#16839972C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic
4#21450814C211yesmaximal, characteristic
5#21447197C212yesmaximal, complemented
6#21447060C212yesmaximal
7#21446620C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446160C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic
11Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×#11758201C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519765C2212yes—
16C2×C4×D244C2211yescharacteristic
17#16839984—28no—
18#16839972—24no—
19C4×#11758201—24no—
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2212yes—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839689—22no—
31#16839668C2212yes—
32#16839619—21no—
33#16839619—21no—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C4×Q8×C61C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C61×#389—22no—
39C61×#389C2212yes—
40C61×#384—21no—
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758202—24no—
48C2×#11758202—24no—
49C2×#11758202—24no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×#11758201D414yes—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56#14519777—22no—
57#14519777—22no—
58#14519765—24no—
59#14519765—42no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758198—21no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68#14519708—22no—
69C4⋊#92418—22no—
70C4⋊#92418—24no—
71#14519701—22no—
72#14519700—24no—
73#14519700—24no—
74#14519700—42no—
75#14519697—24no—
76#14519697—22no—
77C4×#92418—22no—
78#14519684—22no—
79#14519684—22no—
80C22×Q8×C61C2311yescharacteristic
81C4×Q8×C61—22no—
82C4×Q8×C61—24no—
83C4×Q8×C61—42no—
84C4×Q8×C61D414yes—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C4⋊C4×C61D411yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61—22no—
96C2×C42×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#95×C61—22no—
99#95×C61D412yes—
100C23×D61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61C2×D412yes—
105C2×Q8×C61C2×D412yes—
106C2×Q8×C61#4112yes—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—22no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C2×D412yes—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—44no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61—81no—
141C4⋊C4×C61—82no—
142C4⋊C4×C61C2×D412yes—
143C42×C61—24no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—44no—
147C42×C61C2×D412yes—
148C42×C61#4112yes—
149#11758202—48no—
150#11758201—28no—
151#11758198—44no—
152#11758198—44no—
153#11758198—42no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—42no—
157C4⋊Dic61—42no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418C2×D412yes—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175Q8×C61—22no—
176Q8×C61—22no—
177Q8×C61—22no—
178Q8×C61#12114yes—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61#12112yes—
197C2×C4×C61#12112yes—
198C2×C4×C61#13512yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61H(5,𝔽2)11yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—84no—
217Dic122—42no—
218Dic122—42no—
219Dic122—42no—
220Dic122—42no—
221#92418—22no—
222#92418—42no—
223#92418—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—24no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—82no—
245C4×C61#46414yescomplemented
246C4×C61C2×#12111yescharacteristic
247C4×C61#53711yescharacteristic
248C4×C61#54711yescharacteristic
249C4×C61#55311yescharacteristic
250Dic61—82no—
251Dic61—81no—
252C2×#464—612no—
253C2×#462—611no—
254C2×C4×SD16—611no—
255#2547—611no—
256Q8×C4⋊C4D6111yescharacteristic, complemented
257#2066—612no—
258#2042—612no—
259#1956—611no—
260#1939—611no—
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×Q8C2×D6112yes—
276C2×C4×Q8C2×D6111yescharacteristic
277C2×C4×D4—611no—
278#464—1228no—
279#462—1224no—
280C4×SD16—1224no—
281C2×#106—611no—
282C2×C4⋊C8—611no—
283C2×C4⋊C8—612no—
284C2×#103—611no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103—612no—
288C2×#102—612no—
289C2×#102—611no—
290C2×C4×C8—611no—
291#397—1222no—
292#391—612no—
293#389—22no—
294#389C2×D6112yes—
295#384—21no—
296#384C2×D6111yescharacteristic
297#380—1221no—
298#380—1221no—
299C4×C4⋊C4—21no—
300C4×C4⋊C4C2×D6111yescharacteristic
301#340—612no—
302#336—612no—
303C61#2472911yescharacteristic, complemented, Sylow
304C22×Q8—612no—
305C22×Q8C22×D6111yescharacteristic
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×C8—611no—
315C22×C8—612no—
316C22×C8—611no—
317C4⋊Q8—1222no—
318C4⋊Q8—1222no—
319#124—1224no—
320#124—2442no—
321C4×Q8—22no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8D24414yes—
325C4×D4—1222no—
326C4×D4—1222no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C4⋊C4C22⋊D6111yescharacteristic
337C2×C4⋊C4C22×D6112yes—
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42C22×D6112yes—
345C2×C42C22×D6111yescharacteristic
346#106—1222no—
347C4⋊C8—1224no—
348C4⋊C8—1222no—
349#103—1224no—
350#103—1222no—
351#103—1224no—
352#103—2442no—
353#102—1224no—
354#102—1222no—
355C4×C8—1222no—
356#95—22no—
357#95—1222no—
358#95—1222no—
359#95C22⋊D6112yes—
360C24—1221no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—1222no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8C2×D24412yes—
371C2×Q8D4×D6112yes—
372C2×Q8#1175822612yes—
373C2×D4—612no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4C2×C22⋊D6112yes—
388C22×C4C2×C22⋊D6112yes—
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392Q16—2448no—
393SD16—1228no—
394C2×C8—612no—
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—1222no—
398C2×C8—1221no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1221no—
402C2×C8—1221no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—81no—
412C4⋊C4—82no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—2442no—
416C4⋊C4—2442no—
417C4⋊C4C2×D24412yes—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—44no—
425C42C2×D24412yes—
426C42#1175822212yes—
427C23—1222no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23C22×C22⋊D6111yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—22no—
438Q8—2442no—
439Q8—2442no—
440Q8—2442no—
441Q8—2442no—
442Q8#1451975214yes—
443D4—1222no—
444D4—1222no—
445D4—2441no—
446D4—2441no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—4884no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4#1451975112yes—
472C2×C4#1451975112yes—
473C2×C4#1451979712yes—
474C2×C4#1451980812yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×D24411yescharacteristic
477C2×C4C2×#1175822211yescharacteristic
478C2×C4C2×#1175822611yescharacteristic
479C2×C4#1451982211yescharacteristic
480C2×C4#1451982511yescharacteristic
481C8—1222no—
482C8—2442no—
483C8—2441no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×#1451975111yescharacteristic
492C22D4×C22⋊D6111yescharacteristic
493C22#1684042311yescharacteristic
494C22C2×#1451979711yescharacteristic
495C22C2×#1451980811yescharacteristic
496C22#1684057311yescharacteristic
497C22#1684057711yescharacteristic
498C4—22no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—4882no—
505C4—4881no—
506C4#1683998414yescomplemented
507C4C2×#1451975211yescharacteristic
508C4#1684034311yescharacteristic
509C4#1684041311yescharacteristic
510C4#1684046011yescharacteristic
511C2—4882no—
512C2#2144925311yescharacteristic
513C2C2×#1683998411yescharacteristic
514C2#2145536911yescharacteristic
515C2#2145746511yescharacteristic
516C2#2145754911yescharacteristic
517C2#2145850411yescharacteristic
518C2#2145855911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterQ8×C4⋊C4D61
hypocenterC61#24729
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839972C2semidirect
5. #21447197C2semidirect
7. #21446620C2semidirect
8. #21446538C2semidirect
9. #21446160C2semidirect
11. Q8×C4⋊C4×C61C2semidirect
245. C4×C61#464semidirect
256. Q8×C4⋊C4D61semidirect
303. C61#24729semidirect
506. C4#16839984semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
C2×#168399722
C2×C4×#117582012
#214508142
#214471972
#214470602
#214466202
#214465382
#214461602
#214458162
Q8×C4⋊C4×C612
#2472961