Finite Group Databasebeta

← #29575604

Subgroups — #29575604

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
2C2×#16839496C212yesmaximal
3#21444046C211yesmaximal, characteristic, complemented
4#21443697C212yesmaximal, complemented
5C61×#2854C211yesmaximal, characteristic
6#16839819—24no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418C2212yes—
9C2×#14519688—21no—
10C2×#14519684—21no—
11C2×#14519684—21no—
12C2×#14519684C2212yes—
13#16839496—24no—
14C2×#123×C61—21no—
15C2×#121×C61C2211yescharacteristic
16C22×C42×C61C2211yescharacteristic
17C61×#390C411yescharacteristic, complemented
18C61×#390C411yescharacteristic, complemented
19C61×#389C411yescharacteristic, complemented
20C61×#389C411yescharacteristic, complemented
21C61×#388—21no—
22C61×#388—21no—
23C61×#386—21no—
24C61×#380C2211yescharacteristic
25C23×Dic61—22no—
26C22×#92418—22no—
27C4⋊#92418—24no—
28C22⋊#92418—22no—
29C22⋊#92418D414yes—
30#14519688—24no—
31#14519684—24no—
32#14519684—22no—
33#14519684—22no—
34#14519684—42no—
35C22×D4×C61—21no—
36C22×D4×C61C2×C411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61C2311yescharacteristic
40#123×C61—44no—
41#121×C61—22no—
42#121×C61—22no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C42×C61—22no—
55C2×C42×C61—22no—
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61C2×C411yescharacteristic
58C2×C42×C61D411yescharacteristic
59C2×C42×C61D411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—21no—
63C24×C61—21no—
64C24×C61C2×D411yescharacteristic
65C2×D4×C61—22no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D412yes—
91C22×C4×C61#4112yes—
92C22×Dic61—24no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—42no—
105C22⋊C4×C61—42no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—42no—
108C22⋊C4×C61—42no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—22no—
112C42×C61—22no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C2×#92418—24no—
118C2×#92418—42no—
119C2×#92418—42no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61#12112yes—
133C23×C61AGL(1,ℤ/8)11yescharacteristic
134C23×C61AGL(1,ℤ/8)12yes—
135C23×C61#13711yescharacteristic
136D4×C61—82no—
137D4×C61—82no—
138D4×C61—82no—
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—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—44no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61#10412yes—
169C2×C4×C61C4×D412yes—
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12311yescharacteristic
172C2×Dic61—44no—
173C2×Dic61—44no—
174C2×Dic61—44no—
175C2×Dic61—82no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178C2×Dic61—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—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61#38611yescharacteristic
197C22×C61#41011yescharacteristic
198C22×C61#41011yescharacteristic
199C22×C61#41711yescharacteristic
200C22×C61#41811yescharacteristic
201C22×C61C2×#10411yescharacteristic
202C22×C61#42111yescharacteristic
203C22×C61#46812yes—
204C22×C61#46912yes—
205C4×C61—22no—
206C4×C61—21no—
207C4×C61—22no—
208C4×C61—21no—
209C4×C61—41no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61—81no—
213C4×C61—81no—
214C4×C61—81no—
215Dic61—84no—
216C2×#432—611no—
217#2854D6111yescharacteristic
218C2×#327—612no—
219#1434—611no—
220#1294—612no—
221C2×C61—22no—
222C2×C61—22no—
223C2×C61—161no—
224C2×C61#150511yescharacteristic
225C2×C61#150911yescharacteristic
226C2×C61#151211yescharacteristic
227C2×C61#151411yescharacteristic
228C2×C61#187111yescharacteristic
229C2×C61#189511yescharacteristic
230C2×C61#189911yescharacteristic
231C2×#123—21no—
232C2×#121C2×D6111yescharacteristic
233C22×C42C2×D6111yescharacteristic
234#432—1224no—
235C2×C4⋊C8—1221no—
236C2×C22⋊C8—612no—
237C2×#97—1221no—
238#390Dic6111yescharacteristic, complemented
239#390Dic6111yescharacteristic, complemented
240#389Dic6111yescharacteristic, complemented
241#389Dic6111yescharacteristic, complemented
242#388—21no—
243#388—21no—
244#386—21no—
245#380C2×D6111yescharacteristic
246C2×#95—612no—
247C2×#95—1221no—
248C2×#95—1221no—
249#327—1224no—
250C61#911811yescharacteristic, complemented, Sylow
251C22×D4—21no—
252C22×D4C2×Dic6111yescharacteristic
253C23×C4—1222no—
254C23×C4C22⋊D6111yescharacteristic
255C23×C4C22⋊D6111yescharacteristic
256C23×C4C22×D6111yescharacteristic
257C22×C8—1222no—
258#123—44no—
259#121—22no—
260#121—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Dic6111yescharacteristic
264C2×C4⋊C4C2×Dic6111yescharacteristic
265C2×C4⋊C4C2×Dic6111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C2×Dic6111yescharacteristic
275C2×C42C2×Dic6111yescharacteristic
276C2×C42C22⋊D6111yescharacteristic
277C2×C42C22⋊D6111yescharacteristic
278C4⋊C8—1224no—
279C22⋊C8—614no—
280C22⋊C8—1222no—
281#97—1224no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—1224no—
286#95—1222no—
287#95—1222no—
288#95—2442no—
289C24—21no—
290C24C2×C22⋊D6111yescharacteristic
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—41no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1224no—
311C22×C4—2442no—
312C22×C4—2442no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4C22⋊Dic6111yescharacteristic
318C22×C4C22⋊Dic6111yescharacteristic
319C22×C4C22⋊Dic6111yescharacteristic
320C22×C4C22⋊Dic6111yescharacteristic
321C22×C4D4×D6112yes—
322C22×C4#1175822412yes—
323C22×C4C22×Dic6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325C2×C8—1224no—
326C2×C8—2442no—
327C2×C8—2442no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C2×C22⋊Dic6111yescharacteristic
358C23#1451974912yes—
359C23#1451980211yescharacteristic
360C23#1451980312yes—
361C23#1451980711yescharacteristic
362D4—82no—
363D4—82no—
364D4—82no—
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—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—4882no—
395C2×C4—4882no—
396C2×C4—4882no—
397C2×C4—4882no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4—2444no—
401C2×C4#1451970412yes—
402C2×C4D4×Dic6112yes—
403C2×C4#1451974511yescharacteristic
404C2×C4#1451975811yescharacteristic
405C8—2444no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22#1683964211yescharacteristic
422C22#1683973611yescharacteristic
423C22#1683973611yescharacteristic
424C22#1683976311yescharacteristic
425C22#1683976711yescharacteristic
426C22C2×#1451970411yescharacteristic
427C22#1683977911yescharacteristic
428C22#1684000612yes—
429C22#1684001312yes—
430C4—21no—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—4884no—
441C2—22no—
442C2—22no—
443C2—161no—
444C2#2144432711yescharacteristic
445C2#2144434411yescharacteristic
446C2#2144435711yescharacteristic
447C2#2144436611yescharacteristic
448C2#2144612211yescharacteristic
449C2#2144625811yescharacteristic
450C2#2144627811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2854C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#2854D61
hypocenterC61#9118
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#16839819C2semidirect
3. #21444046C2semidirect
4. #21443697C2semidirect
17. C61×#390C4semidirect
18. C61×#390C4semidirect
19. C61×#389C4semidirect
20. C61×#389C4semidirect
238. #390Dic61semidirect
239. #390Dic61semidirect
240. #389Dic61semidirect
241. #389Dic61semidirect
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
C2×#168394962
#214440462
#214436972
C61×#28542
#911861