Finite Group Databasebeta

← #29699175

Subgroups — #29699175

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758201C211yesmaximal, characteristic
2#21448766C211yesmaximal, characteristic
3#21446515C212yesmaximal
4#21446394C212yesmaximal, complemented
5#21446174C212yesmaximal
6#21445946C211yesmaximal, characteristic, complemented
7#21445824C211yesmaximal, characteristic, complemented
8#21445816C211yesmaximal, characteristic
9#21445688C211yesmaximal, characteristic, complemented
10#21445649C212yesmaximal, complemented
11Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×#14519735—21no—
15C4×#11758201C418yescomplemented
16C2×#14519708C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519701C2211yescharacteristic
20C2×#14519697C2212yes—
21C2×#14519697C2211yescharacteristic
22C2×#14519688C2212yes—
23C2×C4×#92418C2211yescharacteristic
24#16839664C2212yes—
25#16839654—22no—
26#16839632C2211yescharacteristic
27#16839619—21no—
28C4×C4⋊Dic61C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×C4×Q8×C61C2212yes—
35C2×C4×Q8×C61C2211yescharacteristic
36C61×#389—22no—
37C61×#389C2212yes—
38C61×#384—21no—
39C61×#384C2211yescharacteristic
40C4×C4⋊C4×C61—21no—
41C4×C4⋊C4×C61C2211yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C4×D61—21no—
44C2×#11758201—22no—
45C2×#11758201—22no—
46C2×#11758201C2×C414yes—
47C22×#92418C2312yes—
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50C4×D244—22no—
51C4×D244C2×C414yes—
52#14519735—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—21no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—22no—
61#14519708C2×C414yes—
62C4⋊#92418—22no—
63#14519701—24no—
64#14519701C2×C414yes—
65#14519697—42no—
66#14519697C2×C414yes—
67#14519688—42no—
68C4×#92418C2×C414yes—
69#14519684—22no—
70C22×Q8×C61C2311yescharacteristic
71C4×Q8×C61—22no—
72C4×Q8×C61—24no—
73C4×Q8×C61—42no—
74C4×Q8×C61C2×C414yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61C2312yes—
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61—22no—
86C2×C42×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88#95×C61—22no—
89#95×C61D412yes—
90C23×D61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—21no—
94C2×Q8×C61C22×C412yes—
95C2×Q8×C61#4112yes—
96C2×Q8×C61#4112yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—22no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244C22×C412yes—
113C2×C4×D61—24no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—44no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—81no—
125C4⋊C4×C61—82no—
126C4⋊C4×C61C22×C412yes—
127C42×C61—24no—
128C42×C61—22no—
129C42×C61—22no—
130C42×C61—44no—
131C42×C61C22×C412yes—
132C42×C61#4112yes—
133#11758201—28no—
134#11758198—24no—
135#11758198—44no—
136#11758198—82no—
137C4⋊Dic61—21no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—42no—
141C4⋊Dic61—81no—
142C4⋊Dic61C22×C412yes—
143C4×Dic61—44no—
144C2×#92418—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418—22no—
151C2×#92418C22×C412yes—
152C2×#92418C22×C412yes—
153C23×C61C22×D411yescharacteristic
154C22×D61—22no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161Q8×C61—22no—
162Q8×C61—22no—
163Q8×C61—22no—
164Q8×C61#11714yescomplemented
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61C4×D412yes—
183C2×C4×C61C4×D412yes—
184C2×C4×C61AGL(1,ℤ/8)12yes—
185C2×C4×C61#13712yes—
186C2×C4×C61C23×C411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61H(5,𝔽2)11yescharacteristic
191C2×C4×C612−1+411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198D244—22no—
199D244—22no—
200D244—41no—
201D244—41no—
202C4×D61—48no—
203#92418—22no—
204#92418—22no—
205#92418—42no—
206#22759—611nomaximal, Sylow
207C22×C61C2×C4×D411yescharacteristic
208C22×C61#54611yescharacteristic
209C22×C61#54811yescharacteristic
210C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C22×C61C2×#13711yescharacteristic
212C22×C61#57711yescharacteristic
213C22×C61#57711yescharacteristic
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C4×C61—24no—
222C4×C61—22no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61—82no—
227C4×C61#44014yes—
228C4×C61C2×#11711yescharacteristic
229C4×C61#52011yescharacteristic
230C4×C61#55211yescharacteristic
231C4×C61#55311yescharacteristic
232Dic61—42no—
233Dic61—82no—
234C2×C4×SD16—611no—
235#2354—611no—
236Q8×C4⋊C4D6111yescharacteristic, complemented
237#1936—612no—
238#1919—612no—
239#1880—612no—
240#1834—611no—
241#1803—611no—
242#1802—611no—
243#1770—611no—
244#1763—612no—
245C2×C61#235211yescharacteristic
246C2×C61C2×#44011yescharacteristic
247C2×C61#295211yescharacteristic
248C2×C61#332911yescharacteristic
249C2×C61#333211yescharacteristic
250C2×C61#336811yescharacteristic
251C2×C61#336911yescharacteristic
252D61—82no—
253C22×SD16—611no—
254C2×C4×Q8C2×D6112yes—
255C2×C4×Q8C2×D6111yescharacteristic
256C2×C4×D4—611no—
257C2×#117—1221no—
258C4×SD16—618no—
259C2×#106—611no—
260C2×C4⋊C8—611no—
261C2×#103—611no—
262C2×#103—612no—
263C2×#102—611no—
264C2×#102—612no—
265C2×#97—612no—
266C2×C4×C8—611no—
267#390—612no—
268#389—22no—
269#389C2×D6112yes—
270#388—1222no—
271#384—21no—
272#384—611no—
273#384C2×D6111yescharacteristic
274#380—1221no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—612no—
277C4×C4⋊C4C2×D6111yescharacteristic
278#340—612no—
279#340—612no—
280#340—612no—
281#340—612no—
282#336—612no—
283C61#2275911yescharacteristic, complemented, Sylow
284C22×Q8C22×D6111yescharacteristic
285C22×D4—611no—
286C23×C4—1221no—
287C2×SD16—614no—
288C2×SD16—1222no—
289C2×SD16—1222no—
290C22×C8—611no—
291C22×C8—611no—
292C22×C8—612no—
293C4×Q8—22no—
294C4×Q8—24no—
295C4×Q8—42no—
296C4×Q8C4×D6114yes—
297C4×D4—614no—
298C4×D4—1222no—
299#117—2444no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4C22⋊D6111yescharacteristic
310C2×C4⋊C4C22⋊D6111yescharacteristic
311C2×C4⋊C4C22×D6112yes—
312C2×C4⋊C4C22×D6111yescharacteristic
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—1222no—
315C2×C22⋊C4—1222no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—1222no—
319C2×C42C22×D6112yes—
320C2×C42C22×D6111yescharacteristic
321#106—614no—
322C4⋊C8—1222no—
323#103—614no—
324#103—1224no—
325#102—614no—
326#102—2442no—
327#97—2442no—
328C4×C8—614no—
329#95—22no—
330#95—1222no—
331#95C22⋊D6112yes—
332C24—1221no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C2×C4×D6112yes—
337C2×Q8#1175822412yes—
338C2×Q8#1175822612yes—
339C2×D4—612no—
340C2×D4—1221no—
341C2×D4—1221no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1221no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—1221no—
349C22×C4—1224no—
350C22×C4—1222no—
351C22×C4—2442no—
352C22×C4—2442no—
353C22×C4C2×C22⋊D6112yes—
354C22×C4C2×C22⋊D6112yes—
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357C22×C4C23×D6111yescharacteristic
358SD16—1228no—
359C2×C8—612no—
360C2×C8—612no—
361C2×C8—1221no—
362C2×C8—1221no—
363C2×C8—1221no—
364C2×C8—1221no—
365C2×C8—1222no—
366C2×C8—1222no—
367C2×C8—1222no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4—612no—
379C4⋊C4—1221no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—4881no—
383C4⋊C4—2442no—
384C4⋊C4C2×C4×D6112yes—
385C22⋊C4—1224no—
386C22⋊C4—4882no—
387C22⋊C4—2444no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—44no—
392C42—2444no—
393C42C2×C4×D6112yes—
394C42#1175822212yes—
395C23—1222no—
396C23—2441no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23C22×C22⋊D6111yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—22no—
406Q8#1451973514yescomplemented
407D4—1222no—
408D4—1222no—
409D4—2441no—
410D4—2441no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—1222no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2448no—
435C2×C4C4×C22⋊D6112yes—
436C2×C4C4×C22⋊D6112yes—
437C2×C4#1451980212yes—
438C2×C4#1451980812yes—
439C2×C4C22×C4×D6111yescharacteristic
440C2×C4C2×#1175822411yescharacteristic
441C2×C4C2×#1175822211yescharacteristic
442C2×C4C2×#1175822611yescharacteristic
443C2×C4#1451982211yescharacteristic
444C2×C4#1451982511yescharacteristic
445C8—1222no—
446C8—1222no—
447C8—2442no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22—2442no—
453C22—2442no—
454C22—2442no—
455C22C2×C4×C22⋊D6111yescharacteristic
456C22#1684040611yescharacteristic
457C22#1684042311yescharacteristic
458C22C2×#1451980211yescharacteristic
459C22C2×#1451980811yescharacteristic
460C22#1684057011yescharacteristic
461C22#1684057311yescharacteristic
462C4—24no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—82no—
468C4—4882no—
469C4—2442no—
470C4#1683986014yes—
471C4C2×#1451973511yescharacteristic
472C4#1684024311yescharacteristic
473C4#1684044711yescharacteristic
474C4#1684046011yescharacteristic
475C2—4882no—
476C2#2144874411yescharacteristic
477C2C2×#1683986011yescharacteristic
478C2#2145470011yescharacteristic
479C2#2145811311yescharacteristic
480C2#2145814611yescharacteristic
481C2#2145854811yescharacteristic
482C2#2145855911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterQ8×C4⋊C4D61
hypocenterC61#22759
LCSG ▷ C2×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
4. #21446394C2semidirect
6. #21445946C2semidirect
7. #21445824C2semidirect
9. #21445688C2semidirect
10. #21445649C2semidirect
11. Q8×C4⋊C4×C61C2semidirect
15. C4×#11758201C4semidirect
164. Q8×C61#117semidirect
236. Q8×C4⋊C4D61semidirect
283. C61#22759semidirect
406. Q8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582012
#214487662
#214465152
#214463942
#214461742
#214459462
#214458242
#214458162
#214456882
#214456492
Q8×C4⋊C4×C612
#2275961