Finite Group Databasebeta

← #29663959

Subgroups — #29663959

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456333C211yesmaximal, characteristic, complemented
2#21455982C212yesmaximal
3C8×#11758222C211yesmaximal, characteristic
4#21445224C211yesmaximal, characteristic, complemented
5#21445219C211yesmaximal, characteristic, complemented
6#21445214C211yesmaximal, characteristic, complemented
7#21445209C211yesmaximal, characteristic, complemented
8#21445202C212yesmaximal, complemented
9#21445195C212yesmaximal, complemented
10#21444530C212yesmaximal, complemented
11C61×#1568C211yesmaximal, characteristic, complemented
12#16840319C2212yes—
13C4×#11758222C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840164—21no—
17#16840164—21no—
18#16840160C2212yes—
19#16840104—22no—
20#16840058C2212yes—
21#16840053C2212yes—
22#16839973C2212yes—
23#16839968C2212yes—
24#16839926—22no—
25C8×Dic122C2211yescharacteristic
26C8×Dic122C2211yescharacteristic
27#16839870—22no—
28#16839864—22no—
29C8×D244C2211yescharacteristic
30C8×D244C2211yescharacteristic
31C8×C22⋊D61—21no—
32C8×C22⋊D61—21no—
33#16839815C2211yescharacteristic
34#16839813—21no—
35#16839800—21no—
36#16839800—21no—
37C4×C8×D61—21no—
38#16839510C2212yes—
39#16839510C2212yes—
40#16839509C2212yes—
41#16839509C2212yes—
42C61×#432C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#334C2212yes—
47C2×#11758222D411yescharacteristic
48C2×Dic244—22no—
49C2×#11758190—22no—
50C2×C8×D61—21no—
51C2×C8×D61—21no—
52#14519777C2312yes—
53#14519777C2312yes—
54#14519769—22no—
55#14519756—22no—
56#14519756—22no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×Dic122D411yescharacteristic
60C4×Dic122C2311yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×C22⋊D61—21no—
64C4×C22⋊D61—21no—
65#14519736—22no—
66#14519736D411yescharacteristic
67#14519734—21no—
68C42×D61—21no—
69#14519711—21no—
70#14519711—21no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519709—22no—
75#14519709—22no—
76#14519709—22no—
77C4⋊#92418C2311yescharacteristic
78C4⋊#92418C2311yescharacteristic
79#14519706—21no—
80#14519706—21no—
81#14519702—22no—
82#14519702—22no—
83#14519702—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418—21no—
89C8×Dic61—21no—
90C8×Dic61—21no—
91C22×C8×C61D411yescharacteristic
92C22×C8×C61D411yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61C2312yes—
95C4⋊C8×C61C2312yes—
96C22⋊C8×C61—22no—
97#97×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106#11758222—22no—
107#11758222—22no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×Dic122—22no—
113C2×Dic122—22no—
114C2×Dic122C2×D411yescharacteristic
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127C22⋊Dic61—21no—
128C22⋊Dic61—42no—
129#11758198—21no—
130#11758198—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61C2×D411yescharacteristic
136C4⋊Dic61C2×D411yescharacteristic
137#11758196—21no—
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—42no—
145C2×#92418—21no—
146C2×#92418—21no—
147Dic244—44no—
148#11758190—44no—
149C8×D61—42no—
150C8×D61—42no—
151C23×C61#12111yescharacteristic
152C22×D61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C22⋊D61—24no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170D244—21no—
171D244#13512yes—
172C4×D61—24no—
173C4×D61—42no—
174C4×D61—41no—
175C4×D61—41no—
176Dic122—21no—
177Dic122—42no—
178Dic122—42no—
179Dic122—82no—
180Dic122#13512yes—
181C8×C61—21no—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—42no—
185C8×C61#13712yes—
186C8×C61#13712yes—
187#92418—41no—
188#92418—41no—
189#19273—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61C2×#12111yescharacteristic
194C2×D61—22no—
195C2×D61—41no—
196C2×D61—41no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61C2×#13511yescharacteristic
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×#13711yescharacteristic
204C4×C61C2×#13711yescharacteristic
205Dic61—22no—
206Dic61—41no—
207Dic61—41no—
208Dic61—82no—
209Dic61—82no—
210#3160—611no—
211#3116—612no—
212C8×#41—611no—
213#1676—611no—
214#1675—611no—
215#1674—611no—
216#1673—611no—
217#1672—612no—
218#1671—612no—
219#1568D6111yescharacteristic, complemented
220#1554—612no—
221C2×C61—41no—
222C2×C61#303911yescharacteristic
223C2×C61#305011yescharacteristic
224C2×C61#306711yescharacteristic
225D61—42no—
226#533—612no—
227C4×#41—611no—
228#500—611no—
229#500—611no—
230#499—1221no—
231#499—1221no—
232#498—612no—
233#485—1222no—
234#477—612no—
235#476—612no—
236#462—612no—
237#461—612no—
238#452—1222no—
239C8×Q8—611no—
240C8×Q8—611no—
241#441—1222no—
242#440—1222no—
243C8×D4—611no—
244C8×D4—611no—
245C8×D4—1221no—
246C8×D4—1221no—
247#432C2×D6112yes—
248#431—611no—
249#431—1221no—
250#427—1221no—
251#427—1221no—
252C2×C4×C8—1221no—
253C2×C4×C8C2×D6111yescharacteristic
254#335C2×D6111yescharacteristic
255#335C2×D6111yescharacteristic
256#334C2×D6112yes—
257#333—612no—
258#333—612no—
259#332—612no—
260#332—612no—
261C61#1927311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263C2×Q16—1222no—
264C2×SD16—1222no—
265C22×C8—1221no—
266C22×C8—1221no—
267C22×C8D24411yescharacteristic
268C22×C8D24411yescharacteristic
269C4⋊Q8—612no—
270C4⋊Q8—612no—
271#125—1222no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C4×Q8—611no—
275C4×Q8—611no—
276C4×Q8—1222no—
277C4×Q8—1222no—
278C4×D4—611no—
279C4×D4—611no—
280C4×D4—1221no—
281C4×D4—1221no—
282#117—611no—
283#117—1221no—
284#117—1222no—
285C2×C42—1221no—
286C2×C42C22×D6111yescharacteristic
287#107—1222no—
288#107—1221no—
289#107—1221no—
290#107—1221no—
291#107—1221no—
292#106—1222no—
293#106—1222no—
294#106—1222no—
295C4⋊C8—611no—
296C4⋊C8—611no—
297C4⋊C8—1221no—
298C4⋊C8—1221no—
299C4⋊C8C22×D6112yes—
300C4⋊C8C22×D6112yes—
301#103—1222no—
302#103—1222no—
303#103—1222no—
304#102—1222no—
305C22⋊C8—22no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#97—22no—
310C4×C8—1221no—
311C4×C8—1221no—
312C4×C8—1221no—
313C4×C8D24411yescharacteristic
314C4×C8D24411yescharacteristic
315C4×C8C22×D6111yescharacteristic
316C4×C8C22×D6111yescharacteristic
317#41—1222no—
318#41—1222no—
319C2×Q8—611no—
320C2×Q8—1222no—
321C2×Q8—1222no—
322C2×D4—611no—
323C2×D4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4C2×D24411yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4#1175822611yescharacteristic
330Q16—2444no—
331SD16—2444no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—1221no—
337C2×C8—1221no—
338C2×C8—2442no—
339C2×C8—2442no—
340C2×C8C2×D24411yescharacteristic
341C2×C8C2×D24411yescharacteristic
342C2×C8C2×D24411yescharacteristic
343C2×C8C2×D24411yescharacteristic
344C4⋊C4—611no—
345C4⋊C4—611no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—1222no—
351C4⋊C4—1222no—
352C4⋊C4—2442no—
353C4⋊C4—2442no—
354C22⋊C4—1221no—
355C22⋊C4—1221no—
356C22⋊C4—1221no—
357C22⋊C4—2442no—
358C42—1221no—
359C42—1221no—
360C42—1221no—
361C42—2442no—
362C42C2×D24411yescharacteristic
363C42D4×D6111yescharacteristic
364C42#1175822611yescharacteristic
365C42C23×D6111yescharacteristic
366C23—1221no—
367C23#1451975211yescharacteristic
368Q8—612no—
369Q8—1221no—
370Q8—4882no—
371Q8—2442no—
372Q8—2442no—
373D4—612no—
374D4—1224no—
375D4—1221no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—1224no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4#1451975211yescharacteristic
391C2×C4#1451975211yescharacteristic
392C2×C4#1451975211yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C22×D24411yescharacteristic
395C2×C4C2×#1175822611yescharacteristic
396C8—21no—
397C8—21no—
398C8—42no—
399C8—42no—
400C8—2441no—
401C8—2441no—
402C8#1451981012yes—
403C8#1451981012yes—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—1222no—
408C22—2441no—
409C22—2441no—
410C22C2×#1451975211yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—1222no—
416C4—4882no—
417C4—4882no—
418C4—2441no—
419C4—2441no—
420C4#135×D6111yescharacteristic
421C4#135×D6111yescharacteristic
422C4C2×#1451981011yescharacteristic
423C4C2×#1451981011yescharacteristic
424C2—41no—
425C2—2442no—
426C2#2145530711yescharacteristic
427C2#2145540111yescharacteristic
428C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1568C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1568D61
hypocenterC61#19273
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1568

Complements

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

NH ≅ G/Ntype
1. #21456333C2semidirect
4. #21445224C2semidirect
5. #21445219C2semidirect
6. #21445214C2semidirect
7. #21445209C2semidirect
8. #21445202C2semidirect
9. #21445195C2semidirect
10. #21444530C2semidirect
11. C61×#1568C2semidirect
219. #1568D61semidirect
261. C61#19273semidirect

Maximal subgroups

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

subgroupindex
#214563332
#214559822
C8×#117582222
#214452242
#214452192
#214452142
#214452092
#214452022
#214451952
#214445302
C61×#15682
#1927361