Finite Group Databasebeta

← #29575601

Subgroups — #29575601

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839819C211yesmaximal, characteristic, complemented
2#21454150C211yesmaximal, characteristic
3C2×#16839495C212yesmaximal
4#21443698C212yesmaximal, complemented
5C61×#1434C211yesmaximal, characteristic, complemented
6C2×#14519760—21no—
7C2×#14519751C2211yescharacteristic
8#16839819—24no—
9C2×C4⋊#92418—21no—
10C2×C22⋊#92418C2212yes—
11C2×#14519688—21no—
12#16839664C411yescharacteristic, complemented
13#16839664C411yescharacteristic, complemented
14#16839660C411yescharacteristic, complemented
15#16839660C411yescharacteristic, complemented
16#16839653—21no—
17#16839653—21no—
18#16839641—21no—
19#16839622C2211yescharacteristic
20#16839495—24no—
21C22×C42×C61C2211yescharacteristic
22C2×#95×C61—21no—
23C2×#95×C61—21no—
24C2×#95×C61C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×#92418—22no—
28#14519760—44no—
29#14519751—22no—
30#14519751—22no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×C22⋊Dic61—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C4⋊#92418—24no—
43C22⋊#92418—22no—
44C22⋊#92418D414yes—
45#14519688—24no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C23×C4×C61—22no—
50C23×C4×C61D411yescharacteristic
51C23×C4×C61D411yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C2×C42×C61—22no—
54C2×C42×C61—22no—
55C2×C42×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59#95×C61—24no—
60#95×C61—22no—
61#95×C61—22no—
62#95×C61—42no—
63C24×C61C2×D411yescharacteristic
64C23×D61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—21no—
68C22×C4×C61—22no—
69C22×C4×C61—21no—
70C22×C4×C61—22no—
71C22×C4×C61—24no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C22×C4×C61#4112yes—
89C2×C22⋊D61—42no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—41no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C2×D244—22no—
97C2×D244—41no—
98C2×D244—41no—
99C42×C61—22no—
100C42×C61—22no—
101C42×C61—21no—
102C42×C61—21no—
103C42×C61—21no—
104C42×C61—21no—
105C22⋊Dic61—42no—
106#11758198—42no—
107#11758198—42no—
108#11758198—42no—
109#11758198—42no—
110#11758198—41no—
111#11758198—41no—
112C4⋊Dic61—22no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115#11758196—42no—
116#11758196—42no—
117C2×#92418—24no—
118C2×#92418—42no—
119C2×#92418—42no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61C2×C22⋊C411yescharacteristic
125C23×C61#12112yes—
126C23×C61AGL(1,ℤ/8)11yescharacteristic
127C23×C61AGL(1,ℤ/8)12yes—
128C23×C61#13711yescharacteristic
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—44no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—44no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—82no—
160C2×C4×C61—82no—
161C2×C4×C61—82no—
162C2×C4×C61—82no—
163C2×C4×C61#10412yes—
164C2×C4×C61C4×D412yes—
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12311yescharacteristic
167C22⋊D61—82no—
168C22⋊D61—82no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178D244—82no—
179#92418—44no—
180#9118—611nomaximal, Sylow
181C22×C61—22no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—22no—
189C22×C61#38611yescharacteristic
190C22×C61#41011yescharacteristic
191C22×C61#41011yescharacteristic
192C22×C61#41711yescharacteristic
193C22×C61#41811yescharacteristic
194C22×C61C2×#10411yescharacteristic
195C22×C61#42111yescharacteristic
196C22×C61#46812yes—
197C22×C61#46912yes—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C4×C61—22no—
206C4×C61—21no—
207C4×C61—22no—
208C4×C61—21no—
209C4×C61—41no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61—84no—
213Dic61—81no—
214Dic61—81no—
215Dic61—81no—
216C2×#432—611no—
217#2854—611no—
218C2×#327—612no—
219#1434D6111yescharacteristic, complemented
220#1294—612no—
221C2×C61—22no—
222C2×C61—22no—
223C2×C61#150511yescharacteristic
224C2×C61#150911yescharacteristic
225C2×C61#151211yescharacteristic
226C2×C61#151411yescharacteristic
227C2×C61#187111yescharacteristic
228C2×C61#189511yescharacteristic
229C2×C61#189911yescharacteristic
230D61—161no—
231C2×#123—1221no—
232C2×#121—611no—
233C22×C42C2×D6111yescharacteristic
234#432—1224no—
235C2×C4⋊C8—1221no—
236C2×C22⋊C8—612no—
237C2×#97—1221no—
238#390—611no—
239#390—611no—
240#389—611no—
241#389—611no—
242#388—1221no—
243#388—1221no—
244#386—1221no—
245#380—611no—
246C2×#95—21no—
247C2×#95—21no—
248C2×#95C2×D6112yes—
249#327—1224no—
250C61#911811yescharacteristic, complemented, Sylow
251C22×D4—611no—
252C22×D4—1221no—
253C23×C4—22no—
254C23×C4C22⋊D6111yescharacteristic
255C23×C4C22⋊D6111yescharacteristic
256C23×C4C22×D6111yescharacteristic
257C22×C8—1222no—
258#123—2444no—
259#121—1222no—
260#121—1222no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C4×D6111yescharacteristic
275C2×C42C4×D6111yescharacteristic
276C2×C42D24411yescharacteristic
277C2×C42D24411yescharacteristic
278C4⋊C8—1224no—
279C22⋊C8—614no—
280C22⋊C8—1222no—
281#97—1224no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—42no—
286#95—1221no—
287#95—1221no—
288#95—1221no—
289C24—1221no—
290C24C2×C22⋊D6111yescharacteristic
291C2×D4—1222no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2441no—
295C2×D4—2442no—
296C2×D4—2442no—
297C2×D4—2441no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4#1175819811yescharacteristic
318C22×C4#1175819811yescharacteristic
319C22×C4#1175819811yescharacteristic
320C22×C4#1175819811yescharacteristic
321C22×C4C2×C4×D6111yescharacteristic
322C22×C4C2×D24411yescharacteristic
323C22×C4D4×D6112yes—
324C22×C4#1175822412yes—
325C2×C8—1224no—
326C2×C8—2442no—
327C2×C8—2442no—
328C4⋊C4—1222no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C4⋊C4—2442no—
332C4⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C22⋊C4—2442no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—2441no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23C2×#1175819811yescharacteristic
358C23#1451974912yes—
359C23#1451980212yes—
360C23#1451980311yescharacteristic
361C23#1451980811yescharacteristic
362D4—4882no—
363D4—4882no—
364D4—4882no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4#1451970512yes—
402C2×C4#1451973712yes—
403C2×C4#1451974711yescharacteristic
404C2×C4#1451976111yescharacteristic
405C8—2444no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—4881no—
415C22—4881no—
416C22—4881no—
417C22—4881no—
418C22—4881no—
419C22—4881no—
420C22—4881no—
421C22#1683963911yescharacteristic
422C22#1683973511yescharacteristic
423C22#1683973511yescharacteristic
424C22#1683976111yescharacteristic
425C22#1683976611yescharacteristic
426C22C2×#1451970511yescharacteristic
427C22#1683978011yescharacteristic
428C22#1684001012yes—
429C22#1684001312yes—
430C4—21no—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—4881no—
439C4—4881no—
440C4—4881no—
441C2—22no—
442C2—22no—
443C2—9761no—
444C2#2144432611yescharacteristic
445C2#2144434211yescharacteristic
446C2#2144435511yescharacteristic
447C2#2144436511yescharacteristic
448C2#2144611911yescharacteristic
449C2#2144625511yescharacteristic
450C2#2144627511yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1434C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1434D61
hypocenterC61#9118
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#16839819C2semidirect
4. #21443698C2semidirect
5. C61×#1434C2semidirect
12. #16839664C4semidirect
13. #16839664C4semidirect
14. #16839660C4semidirect
15. #16839660C4semidirect
219. #1434D61semidirect
250. C61#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#168398192
#214541502
C2×#168394952
#214436982
C61×#14342
#911861