Finite Group Databasebeta

← #29574766

Subgroups — #29574766

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic
2#21448766C211yesmaximal, characteristic
3#21446767C211yesmaximal, characteristic, complemented
4#21446621C211yesmaximal, characteristic, complemented
5C2×#16839491C211yesmaximal, characteristic
6#21443703C211yesmaximal, characteristic, complemented
7C61×#1279C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10C2×#14519735—21no—
11#16840053—22no—
12#16840053C414yescomplemented
13C2×#14519709—21no—
14C2×C4⋊#92418C2211yescharacteristic
15C2×#14519702—22no—
16C2×#14519699—21no—
17#16839664—21no—
18#16839654—22no—
19#16839632C2211yescharacteristic
20#16839619—21no—
21C4×C4⋊Dic61—21no—
22#16839524—21no—
23#16839524—21no—
24#16839491—22no—
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4×C8×C61C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C22×Dic122—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×#92418—21no—
32#14519777—21no—
33#14519777—21no—
34#14519777—21no—
35#14519777C2×C412yes—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244C2×C412yes—
39C4×D244D412yes—
40#14519735—42no—
41#14519735—42no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×C4⋊Dic61D412yes—
48C2×#11758198—21no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×C4×Dic61—22no—
52#14519709—24no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55#14519702—44no—
56#14519699—24no—
57#14519684—22no—
58C22×C8×C61—21no—
59C22×C8×C61—21no—
60C2×C4⋊C4×C61D411yescharacteristic
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C42×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61C2×C412yes—
66C4×C8×C61—21no—
67C4×C8×C61—21no—
68#95×C61—21no—
69C23×D61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—22no—
77C22×Dic61—41no—
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244C22⋊C412yes—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×Dic122—22no—
91C2×Dic122—42no—
92C2×Dic122—42no—
93C2×Dic122—41no—
94C2×C8×C61—22no—
95C2×C8×C61—22no—
96C2×C8×C61—41no—
97C2×C8×C61—41no—
98C2×C8×C61—41no—
99C2×C8×C61—41no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61SD1612yes—
104C42×C61—41no—
105C42×C61—41no—
106C42×C61C22×C411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61#4111yescharacteristic
109C42×C61#4111yescharacteristic
110#11758198—22no—
111#11758198—22no—
112#11758198—44no—
113#11758198—82no—
114C4⋊Dic61—21no—
115C4⋊Dic61—22no—
116C4⋊Dic61—22no—
117C4⋊Dic61—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123C4⋊Dic61—42no—
124C4⋊Dic61C22⋊C412yes—
125C4×Dic61—44no—
126C2×#92418—41no—
127C2×#92418—41no—
128C2×#92418—41no—
129C23×C61#12011yescharacteristic
130C22×D61—22no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61C4×D411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12311yescharacteristic
151C2×C4×C61C2×SD1611yescharacteristic
152C2×C4×C61#13511yescharacteristic
153C2×C4×C61#13511yescharacteristic
154C2×C4×C61#13511yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61#13711yescharacteristic
157C2×C4×C61#13711yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—81no—
167C2×Dic61—81no—
168D244—22no—
169D244—22no—
170D244—41no—
171D244—41no—
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—42no—
175C4×D61—42no—
176Dic122—44no—
177Dic122—82no—
178C8×C61—42no—
179C8×C61—42no—
180#92418—81no—
181#8976—611nomaximal, Sylow
182C22×C61#38611yescharacteristic
183C22×C61#45011yescharacteristic
184C22×C61#45211yescharacteristic
185C22×C61#45211yescharacteristic
186C22×C61#45211yescharacteristic
187C22×C61#45311yescharacteristic
188C22×C61#45411yescharacteristic
189C2×D61—42no—
190C2×D61—42no—
191C2×D61—42no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—42no—
199C4×C61—81no—
200C4×C61#41611yescharacteristic
201C4×C61#41911yescharacteristic
202C4×C61C4×SD1611yescharacteristic
203C4×C61#44011yescharacteristic
204C4×C61#46111yescharacteristic
205C4×C61#46311yescharacteristic
206C4×C61#48511yescharacteristic
207C4×C61#48511yescharacteristic
208Dic61—42no—
209Dic61—82no—
210Dic61—82no—
211C2×#476—611no—
212#2354—611no—
213#1987—611no—
214#1956—611no—
215C2×D4⋊C8—611no—
216#1296—611no—
217#1279D6111yescharacteristic, complemented
218C2×C61#163111yescharacteristic
219C2×C61#163411yescharacteristic
220C2×C61#163511yescharacteristic
221C2×C61#163811yescharacteristic
222C2×C61#187311yescharacteristic
223C2×C61#187711yescharacteristic
224C2×C61#188311yescharacteristic
225D61—82no—
226C2×C4⋊Q8—611no—
227C2×C4×D4—611no—
228C2×#117—1221no—
229#476—614no—
230#476—1222no—
231C2×#106—1221no—
232C2×C4⋊C8—611no—
233C2×C4⋊C8C2×D6111yescharacteristic
234C2×#103—1222no—
235C2×#102—1221no—
236C2×C4×C8C2×D6111yescharacteristic
237#390—1221no—
238#388—1222no—
239#384—611no—
240#380—1221no—
241C4×C4⋊C4—1221no—
242C4×C4⋊C4C2×D6111yescharacteristic
243#340—1221no—
244#340—1221no—
245D4⋊C8—1222no—
246C61#897611yescharacteristic, complemented, Sylow
247C22×Q8—1221no—
248C22×D4—611no—
249C23×C4—1221no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—1221no—
253C4⋊Q8—612no—
254C4⋊Q8—1221no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4×D4—612no—
258C4×D4—612no—
259C4×D4—1221no—
260C4×D4—1221no—
261#117—2442no—
262#117—2442no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4D24411yescharacteristic
270C2×C4⋊C4D24411yescharacteristic
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C42—21no—
275C2×C42—1222no—
276C2×C42C22×D6111yescharacteristic
277#106—1224no—
278C4⋊C8—21no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8C4×D6112yes—
282#103—2444no—
283#102—1224no—
284C4×C8—21no—
285C4×C8—21no—
286#95—21no—
287#95—1222no—
288C24—1221no—
289C2×Q8—1222no—
290C2×Q8—2442no—
291C2×Q8—2442no—
292C2×Q8—2441no—
293C2×D4—612no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—1221no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—1221no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—1222no—
305C22×C4—2441no—
306C22×C4—2441no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4—2441no—
310C22×C4C2×D24411yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—2441no—
320C2×C8—2441no—
321C2×C8—2441no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—612no—
326C4⋊C4—1221no—
327C4⋊C4—1222no—
328C4⋊C4—1222no—
329C4⋊C4—2442no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C4⋊C4#1175819012yes—
337C22⋊C4—1222no—
338C22⋊C4—1222no—
339C22⋊C4—4882no—
340C22⋊C4—2444no—
341C42—41no—
342C42—41no—
343C42—2444no—
344C42C2×C4×D6111yescharacteristic
345C42#1175822211yescharacteristic
346C42D4×D6111yescharacteristic
347C42#1175822611yescharacteristic
348C23—1222no—
349C23—2441no—
350C23—2441no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23#1451974711yescharacteristic
356Q8—4882no—
357Q8—2444no—
358D4—1222no—
359D4—1222no—
360D4—2441no—
361D4—2441no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—1222no—
371C2×C4—4881no—
372C2×C4—4881no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4C22⋊C4×D6111yescharacteristic
384C2×C4#1451973911yescharacteristic
385C2×C4C4×D24411yescharacteristic
386C2×C4#1451975011yescharacteristic
387C2×C4#1451975211yescharacteristic
388C2×C4#1451976211yescharacteristic
389C2×C4C2×#1175819011yescharacteristic
390C2×C4#1451979611yescharacteristic
391C2×C4#1451979811yescharacteristic
392C2×C4#1451979911yescharacteristic
393C2×C4#1451980511yescharacteristic
394C2×C4#1451980611yescharacteristic
395C2×C4#1451980911yescharacteristic
396C2×C4#1451981011yescharacteristic
397C8—42no—
398C8—42no—
399C8—4881no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—4881no—
404C22—2442no—
405C22—2442no—
406C22—2442no—
407C22#1683964011yescharacteristic
408C22#1683991511yescharacteristic
409C22#1683992511yescharacteristic
410C22#1683992611yescharacteristic
411C22#1683992711yescharacteristic
412C22#1683993111yescharacteristic
413C22#1683993611yescharacteristic
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—4882no—
419C4—4882no—
420C4—2442no—
421C4#1683975511yescharacteristic
422C4#1683976911yescharacteristic
423C4C4×#1175819011yescharacteristic
424C4#1683986411yescharacteristic
425C4#1683996811yescharacteristic
426C4#1683997811yescharacteristic
427C4#1684010511yescharacteristic
428C4#1684010611yescharacteristic
429C2—4882no—
430C2#2144494811yescharacteristic
431C2#2144496611yescharacteristic
432C2#2144497211yescharacteristic
433C2#2144499411yescharacteristic
434C2#2144613011yescharacteristic
435C2#2144615211yescharacteristic
436C2#2144619611yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1279C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1279D61
hypocenterC61#8976
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #21446767C2semidirect
4. #21446621C2semidirect
6. #21443703C2semidirect
7. C61×#1279C2semidirect
12. #16840053C4semidirect
217. #1279D61semidirect
246. C61#8976semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
#214487662
#214467672
#214466212
C2×#168394912
#214437032
C61×#12792
#897661