Finite Group Databasebeta

← #29568826

Subgroups — #29568826

Order 15616.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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#21452190C211yesmaximal, characteristic
2#21447153C211yesmaximal, characteristic, complemented
3#21447152C211yesmaximal, characteristic, complemented
4#21446904C211yesmaximal, characteristic
5#21443641C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic
7C61×#1271C211yesmaximal, characteristic, complemented
8C22×#11758190—21no—
9C2×#14519777C2211yescharacteristic
10C2×#14519757—21no—
11C2×#14519755—21no—
12C2×#14519709—21no—
13C2×#14519708—21no—
14C2×#14519702—21no—
15#16839684—21no—
16#16839684—21no—
17#16839683—21no—
18#16839664C2211yescharacteristic
19#16839660C2211yescharacteristic
20#16839640—21no—
21#16839525—21no—
22#16839524—21no—
23#16839517C2211yescharacteristic
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758190—42no—
32C2×#11758190—42no—
33C2×#11758190—42no—
34C22×#92418—21no—
35#14519777—42no—
36#14519757—42no—
37#14519757—42no—
38#14519755—42no—
39#14519755—42no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53#14519709—42no—
54#14519708—42no—
55#14519702—42no—
56#14519684—21no—
57#14519684—21no—
58C22×C8×C61—21no—
59C22×C8×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C4⋊C4×C61D411yescharacteristic
63C2×C42×C61C2311yescharacteristic
64#95×C61—21no—
65C23×D61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—41no—
75C2×D244—21no—
76C2×D244—41no—
77C2×D244—41no—
78C2×D244SD1612yes—
79C2×C4×D61—42no—
80C2×C4×D61—42no—
81C2×C4×D61—81no—
82C2×Dic122—22no—
83C2×Dic122—41no—
84C2×Dic122—41no—
85C2×Dic122—41no—
86C2×Dic122—41no—
87C2×Dic122—42no—
88C2×Dic122—42no—
89C2×Dic122—41no—
90C2×Dic122—81no—
91C2×C8×C61—22no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61SD1612yescomplemented
102C42×C61—41no—
103#11758198—42no—
104#11758198—42no—
105#11758198—41no—
106#11758198—41no—
107#11758198—41no—
108#11758198—41no—
109#11758198—81no—
110C4⋊Dic61—21no—
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113C4⋊Dic61—42no—
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4⋊Dic61SD1612yescomplemented
119#11758196—42no—
120#11758196—41no—
121#11758196—41no—
122C2×#92418—22no—
123C2×#92418—41no—
124C2×#92418—41no—
125#11758190—48no—
126C23×C61#12011yescharacteristic
127C22×D61—22no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61C2×SD1611yescharacteristic
148C2×C4×C61C2×SD1611yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—81no—
163C2×Dic61—81no—
164D244—42no—
165D244—81no—
166C4×D61—84no—
167Dic122—42no—
168Dic122—42no—
169Dic122—84no—
170Dic122—81no—
171Dic122—81no—
172C8×C61—42no—
173C8×C61—42no—
174#92418—42no—
175#7754—611nomaximal, Sylow
176C22×C61#44811yescharacteristic
177C22×C61#45011yescharacteristic
178C22×C61#45011yescharacteristic
179C22×C61#45311yescharacteristic
180C22×C61#45611yescharacteristic
181C22×C61#45611yescharacteristic
182C22×C61B(4,𝔽2)11yescharacteristic
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—42no—
197C4×C61—81no—
198Dic61—42no—
199Dic61—82no—
200Dic61—81no—
201Dic61—81no—
202#2683—611no—
203#2058—611no—
204#2058—611no—
205#2013—611no—
206#1271—611no—
207#1271D6111yescharacteristic, complemented
208#1270—611no—
209C2×C61#159611yescharacteristic
210C2×C61#160411yescharacteristic
211C2×C61#160411yescharacteristic
212C2×C61#162911yescharacteristic
213C2×C61#165111yescharacteristic
214C2×C61#165111yescharacteristic
215C2×C61#166011yescharacteristic
216D61—82no—
217C22×SD16—1221no—
218C2×C4⋊Q8—611no—
219C2×C22⋊Q8—1221no—
220C2×C22⋊Q8—1221no—
221C2×#106—1221no—
222C2×#106—1221no—
223C2×#103—1221no—
224#396—1221no—
225#396—1221no—
226#395—1221no—
227#390—611no—
228#389—611no—
229#389C2×D6111yescharacteristic
230#386—1221no—
231#340—1221no—
232#340—1221no—
233#336—611no—
234#336C2×D6111yescharacteristic
235#336C2×D6111yescharacteristic
236C61#775411yescharacteristic, complemented, Sylow
237C22×Q8—1221no—
238C22×Q8—1221no—
239C22×D4—611no—
240C23×C4—1221no—
241C2×SD16—2442no—
242C2×SD16—2442no—
243C2×SD16—2442no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—1221no—
247C4⋊Q8—2442no—
248C22⋊Q8—2442no—
249C22⋊Q8—2442no—
250C22⋊Q8—2442no—
251C22⋊Q8—2442no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4D24411yescharacteristic
262C2×C4⋊C4D24411yescharacteristic
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C42C22×D6111yescharacteristic
269#106—2442no—
270#106—2442no—
271#103—2442no—
272#95—21no—
273#95—1221no—
274#95—1221no—
275C24—1221no—
276C2×Q8—1222no—
277C2×Q8—4881no—
278C2×Q8—2441no—
279C2×Q8—2441no—
280C2×Q8—2442no—
281C2×Q8—2441no—
282C2×Q8—2442no—
283C2×Q8—2441no—
284C2×Q8—2441no—
285C2×D4—612no—
286C2×D4—1221no—
287C2×D4—2441no—
288C2×D4—2441no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—4881no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4—2441no—
298C22×C4C2×D24411yescharacteristic
299C22×C4D4×D6111yescharacteristic
300C22×C4D4×D6111yescharacteristic
301SD16—2448no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—1222no—
309C2×C8—2441no—
310C2×C8—2441no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—612no—
316C4⋊C4—1222no—
317C4⋊C4—1222no—
318C4⋊C4—1221no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2442no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2442no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4#1175819012yescomplemented
328C22⋊C4—4881no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C42—41no—
336C23—1222no—
337C23—2441no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23#1451974711yescharacteristic
344Q8—4881no—
345Q8—4881no—
346Q8—4884no—
347Q8—2442no—
348Q8—2442no—
349D4—4881no—
350D4—2442no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1222no—
364C2×C4—4884no—
365C2×C4—4881no—
366C2×C4—4881no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4SD16×D6111yescharacteristic
377C2×C4SD16×D6111yescharacteristic
378C2×C4C2×#1175819011yescharacteristic
379C2×C4#1451980111yescharacteristic
380C2×C4#1451980511yescharacteristic
381C2×C4#1451980911yescharacteristic
382C8—42no—
383C8—42no—
384C8—2442no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22—2442no—
390C22—2442no—
391C22—2442no—
392C22#1683990211yescharacteristic
393C22#1683991411yescharacteristic
394C22#1683991511yescharacteristic
395C22#1683993111yescharacteristic
396C22#1683994611yescharacteristic
397C22#1683994711yescharacteristic
398C22#1683995011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—42no—
406C4—81no—
407C4—4881no—
408C4—4882no—
409C4—4881no—
410C4—2442no—
411C2—4882no—
412C2#2144474311yescharacteristic
413C2#2144478911yescharacteristic
414C2#2144479011yescharacteristic
415C2#2144493711yescharacteristic
416C2#2144508311yescharacteristic
417C2#2144508511yescharacteristic
418C2#2144513211yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61#120
prime radicalC22×D244D4
hypercenter#1271D61
hypocenterC61#7754
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
2. #21447153C2semidirect
3. #21447152C2semidirect
5. #21443641C2semidirect
7. C61×#1271C2semidirect
101. C4⋊C4×C61SD16semidirect
118. C4⋊Dic61SD16semidirect
207. #1271D61semidirect
236. C61#7754semidirect
327. C4⋊C4#11758190semidirect

Maximal subgroups

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

subgroupindex
#214521902
#214471532
#214471522
#214469042
#214436412
#214436392
C61×#12712
#775461