Finite Group Databasebeta

← #29569290

Subgroups — #29569290

Order 15616.

516 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21452183C211yesmaximal, characteristic
2#21447383C211yesmaximal, characteristic, complemented
3#21447352C211yesmaximal, characteristic, complemented
4#21447351C211yesmaximal, characteristic, complemented
5#21443641C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic, complemented
7C61×#1270C211yesmaximal, characteristic, complemented
8C2×#14519761—21no—
9C2×#14519749—21no—
10C2×#14519709—21no—
11C2×#14519708—21no—
12C2×#14519699—21no—
13C2×#14519698—21no—
14#16839684—21no—
15#16839678—21no—
16#16839671—21no—
17#16839664C2211yescharacteristic
18#16839660C2211yescharacteristic
19#16839637—21no—
20#16839632C2211yescharacteristic
21#16839525—21no—
22#16839525—21no—
23#16839517C2211yescharacteristic
24C61×#390C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×D244D411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C23×Dic61—21no—
31C22×#92418—21no—
32#14519761—42no—
33#14519761—42no—
34#14519749—42no—
35#14519749—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—21no—
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49#14519709—42no—
50#14519708—42no—
51#14519699—42no—
52#14519698—42no—
53#14519684—21no—
54#14519684—21no—
55C22×D4×C61D411yescharacteristic
56C22×C8×C61—21no—
57C22×C8×C61—21no—
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C24×C61—21no—
63C23×D61—21no—
64C2×D4×C61—21no—
65C2×D4×C61—41no—
66C2×D4×C61—41no—
67C2×D4×C61SD1612yes—
68C22×C4×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—81no—
76C2×C22⋊D61—81no—
77C2×C22⋊D61—81no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—42no—
82C22×Dic61—42no—
83C22×Dic61—41no—
84C22×Dic61—81no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—41no—
88C2×D244—41no—
89C2×C8×C61—22no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101C22⋊Dic61—42no—
102#11758198—42no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106#11758198—41no—
107#11758198—81no—
108C4⋊Dic61—21no—
109C4⋊Dic61—42no—
110C4⋊Dic61—41no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4⋊Dic61—42no—
117C4⋊Dic61SD1612yes—
118#11758196—41no—
119#11758196—41no—
120C2×#92418—22no—
121C2×#92418—41no—
122C2×#92418—41no—
123C23×C61—22no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61#12011yescharacteristic
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138D4×C61—42no—
139D4×C61—81no—
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—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61C2×SD1611yescharacteristic
150C2×C4×C61C2×SD1611yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61#13711yescharacteristic
155C22⋊D61—84no—
156C22⋊D61—84no—
157C2×Dic61—22no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—84no—
167C2×Dic61—81no—
168C2×Dic61—81no—
169C2×Dic61—81no—
170D244—42no—
171D244—81no—
172C8×C61—42no—
173C8×C61—81no—
174#92418—42no—
175#7825—611nomaximal, Sylow
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#45011yescharacteristic
184C22×C61#45011yescharacteristic
185C22×C61#45211yescharacteristic
186C22×C61#45311yescharacteristic
187C22×C61#45311yescharacteristic
188C22×C61#45411yescharacteristic
189C22×C61#45811yescharacteristic
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204Dic61—42no—
205Dic61—81no—
206Dic61—81no—
207Dic61—161no—
208#2682—611no—
209#2100—611no—
210#2094—611no—
211#2094—611no—
212#1271—611no—
213#1270—611no—
214#1270D6111yescharacteristic, complemented
215C2×C61—82no—
216C2×C61#161311yescharacteristic
217C2×C61#161611yescharacteristic
218C2×C61#161811yescharacteristic
219C2×C61#162911yescharacteristic
220C2×C61#163111yescharacteristic
221C2×C61#163511yescharacteristic
222C2×C61#165611yescharacteristic
223D61—161no—
224C2×#123—1221no—
225C2×#121—1221no—
226C2×#106—1221no—
227C2×#106—1221no—
228C2×#102—1221no—
229C2×#102—1221no—
230#396—1221no—
231#394—1221no—
232#392—1221no—
233#390—611no—
234#390C2×D6111yescharacteristic
235#389—611no—
236#385—1221no—
237#384—611no—
238#340—1221no—
239#340—1221no—
240#336—611no—
241#336C2×D6111yescharacteristic
242#336C2×D6111yescharacteristic
243C61#782511yescharacteristic, complemented, Sylow
244C22×D4—611no—
245C22×D4—1221no—
246C22×D4—1221no—
247C22×D4D24411yescharacteristic
248C23×C4—1221no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—1221no—
252#123—2442no—
253#123—2442no—
254#121—2442no—
255#121—2442no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4D24411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42C22×D6111yescharacteristic
273#106—2442no—
274#106—2442no—
275#102—2442no—
276#102—2442no—
277#95—1221no—
278#95—1221no—
279C24—21no—
280C24—1221no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—4881no—
287C2×D4—4881no—
288C2×D4—4881no—
289C2×D4—2441no—
290C2×D4—2441no—
291C2×D4—2442no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4#1175819012yes—
295C22×C4—21no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—4881no—
300C22×C4—2442no—
301C22×C4—2441no—
302C22×C4—2442no—
303C22×C4C2×D24411yescharacteristic
304C22×C4D4×D6111yescharacteristic
305C22×C4D4×D6111yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—1222no—
313C2×C8—2441no—
314C2×C8—2441no—
315C4⋊C4—41no—
316C4⋊C4—612no—
317C4⋊C4—1221no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2442no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—4881no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—41no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23#1451974711yescharacteristic
355D4—42no—
356D4—81no—
357D4—4881no—
358D4—4884no—
359D4—4884no—
360D4—2442no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—1222no—
371C2×C4—4881no—
372C2×C4—4881no—
373C2×C4—4881no—
374C2×C4—4884no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4SD16×D6111yescharacteristic
384C2×C4C2×#1175819011yescharacteristic
385C2×C4#1451979611yescharacteristic
386C2×C4#1451980111yescharacteristic
387C2×C4#1451980511yescharacteristic
388C2×C4#1451980611yescharacteristic
389C8—42no—
390C8—81no—
391C8—2442no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22#1683991411yescharacteristic
407C22#1683991511yescharacteristic
408C22#1683992511yescharacteristic
409C22#1683993111yescharacteristic
410C22#1683993211yescharacteristic
411C22#1683993611yescharacteristic
412C22#1683995311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—9761no—
421C4—4881no—
422C4—4881no—
423C4—2442no—
424C2—82no—
425C2—9761no—
426C2#2144484111yescharacteristic
427C2#2144485711yescharacteristic
428C2#2144486811yescharacteristic
429C2#2144493711yescharacteristic
430C2#2144494711yescharacteristic
431C2#2144497211yescharacteristic
432C2#2144511311yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61#120
prime radical#21452183C2
hypercenter#1270D61
hypocenterC61#7825
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21447383C2semidirect
3. #21447352C2semidirect
4. #21447351C2semidirect
5. #21443641C2semidirect
6. #21443639C2semidirect
7. C61×#1270C2semidirect
214. #1270D61semidirect
243. C61#7825semidirect

Maximal subgroups

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

subgroupindex
#214521832
#214473832
#214473522
#214473512
#214436412
#214436392
C61×#12702
#782561