Finite Group Databasebeta

← #29576930

Subgroups — #29576930

Order 15616.

574 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455942C211yesmaximal, characteristic
2#21448182C211yesmaximal, characteristic, complemented
3#21446738C211yesmaximal, characteristic
4#21444326C211yesmaximal, characteristic
5#21444267C211yesmaximal, characteristic, complemented
6#21443676C211yesmaximal, characteristic, complemented
7C61×#1300C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×Dic122—21no—
10C2×C4×Dic122C2211yescharacteristic
11#16840076—21no—
12#16840058C412yescomplemented
13#16839963C412yescomplemented
14#16839903—21no—
15#16839864—22no—
16#16839819C2211yescharacteristic
17C2×#14519703—22no—
18C4×C4⋊Dic61—21no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839613—22no—
21#16839613—21no—
22#16839607—21no—
23#16839495—21no—
24#16839495—21no—
25#16839494—22no—
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C2×#11758222—21no—
30C22×Dic122D411yescharacteristic
31C2×#11758190—22no—
32#14519777C2×C411yescharacteristic
33#14519774C2×C411yescharacteristic
34#14519765—21no—
35#14519751—21no—
36#14519751—21no—
37C4×Dic122—28no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122C2×C412yes—
41C4×Dic122D412yes—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×#11758196—21no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61D412yes—
52#14519711—22no—
53C4⋊#92418—21no—
54#14519703—44no—
55#14519702—22no—
56#14519699—22no—
57C22⋊#92418—21no—
58#14519688—21no—
59#14519684—22no—
60#14519684—21no—
61C2×M16×C61—21no—
62C2×C42×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61C2×C412yes—
65C22⋊C8×C61—21no—
66#97×C61—21no—
67#97×C61—21no—
68C4×C8×C61—21no—
69#95×C61—21no—
70#95×C61D411yescharacteristic
71#95×C61D411yescharacteristic
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—41no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—22no—
81#11758222—42no—
82#11758222—42no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244—21no—
86C2×C4×D61—41no—
87C2×Dic122—21no—
88C2×Dic122—24no—
89C2×Dic122—21no—
90C2×Dic122C22⋊C412yes—
91M16×C61—22no—
92M16×C61—41no—
93C2×C8×C61—22no—
94C2×C8×C61—41no—
95C42×C61—24no—
96C42×C61C22×C411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99C42×C61#4111yescharacteristic
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—24no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61C22⋊C412yes—
107#11758196—28no—
108#11758196—22no—
109#11758196—22no—
110C4×Dic61—28no—
111C4×Dic61—22no—
112C4×Dic61—22no—
113C4×Dic61—42no—
114C4×Dic61—82no—
115C2×#92418—41no—
116#11758190—44no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12311yescharacteristic
135C22⋊D61—81no—
136C22⋊D61—81no—
137C2×Dic61—24no—
138C2×Dic61—22no—
139C2×Dic61—24no—
140C2×Dic61—22no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146D244—42no—
147D244—42no—
148D244—81no—
149C4×D61—82no—
150Dic122—24no—
151Dic122—22no—
152Dic122—42no—
153Dic122#10414yescomplemented
154C8×C61—42no—
155C8×C61—42no—
156#92418—81no—
157#9326—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45311yescharacteristic
162C22×C61#45411yescharacteristic
163C2×D61—81no—
164C2×D61—81no—
165C2×D61—81no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—42no—
172C4×C61—42no—
173C4×C61C2×#10412yes—
174Dic61—24no—
175Dic61—44no—
176Dic61—42no—
177Dic61—81no—
178#3112—611no—
179#2276—611no—
180#1980—611no—
181#1505—611no—
182#1491—611no—
183#1300D6111yescharacteristic, complemented
184#1285—611no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#188311yescharacteristic
188C2×C61#189611yescharacteristic
189C2×C61#190511yescharacteristic
190D61—161no—
191#532—611no—
192C2×C4×Q8—611no—
193C2×C4×Q8—1221no—
194#480—1221no—
195#477—612no—
196#460—612no—
197#448—1221no—
198#440—1222no—
199#432—611no—
200#432C2×D6111yescharacteristic
201C2×#104—1222no—
202C4×M16C2×D6111yescharacteristic
203C4×C4⋊C4—611no—
204C4×C4⋊C4—1222no—
205C4×C4⋊C4—1221no—
206C4×C4⋊C4—1221no—
207#376—1221no—
208#376C2×D6111yescharacteristic
209#327—1221no—
210#327—1221no—
211Q8⋊C8—1222no—
212C61#932611yescharacteristic, complemented, Sylow
213C2×#41—1221no—
214C22×Q8—611no—
215C2×SD16—1222no—
216C2×M16—21no—
217C4⋊Q8—611no—
218#127—611no—
219#124—1221no—
220#121—1221no—
221#121—1221no—
222C4×Q8—612no—
223C4×Q8—612no—
224C4×Q8—1228no—
225C4×Q8—1221no—
226C4×Q8—1221no—
227C4×D4—1221no—
228C4×D4—1221no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—1222no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C42—21no—
234C2×C42—612no—
235C2×C42—1222no—
236C2×C42—1221no—
237C2×C42—1222no—
238C2×C42C22×D6111yescharacteristic
239#107—1222no—
240C4⋊C8—1221no—
241C4⋊C8C4×D6112yes—
242#104—2444no—
243#103—1222no—
244#102—1222no—
245C22⋊C8—21no—
246C22⋊C8—1221no—
247#97—21no—
248#97—21no—
249#97—1221no—
250C4×C8—21no—
251#95—21no—
252#95—1222no—
253#95—1221no—
254#95D24411yescharacteristic
255#95D24411yescharacteristic
256#41—2442no—
257#41—2442no—
258C2×Q8—612no—
259C2×Q8—1224no—
260C2×Q8—1221no—
261C2×Q8—1221no—
262C2×D4—1221no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—2441no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1221no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—2441no—
272C22×C4C2×D24411yescharacteristic
273C22×C4D4×D6111yescharacteristic
274C22×C4D4×D6111yescharacteristic
275SD16—2444no—
276M16—22no—
277M16—41no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—2441no—
281C4⋊C4—612no—
282C4⋊C4—1228no—
283C4⋊C4—1224no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—1221no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C42—24no—
292C42—1228no—
293C42—1222no—
294C42—1222no—
295C42—4882no—
296C42—2442no—
297C42C2×C4×D6111yescharacteristic
298C42#1175822211yescharacteristic
299C42D4×D6111yescharacteristic
300C42#1175822611yescharacteristic
301C23—2441no—
302C23#1451974711yescharacteristic
303Q8—614no—
304Q8—1224no—
305Q8—1222no—
306Q8—2442no—
307D4—4881no—
308D4—4881no—
309D4—4881no—
310D4—2442no—
311D4—2442no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1224no—
323C2×C4—1222no—
324C2×C4—1224no—
325C2×C4—1222no—
326C2×C4—4882no—
327C2×C4—2441no—
328C2×C4—2442no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2442no—
332C2×C4C22⋊C4×D6111yescharacteristic
333C2×C4#1451973911yescharacteristic
334C2×C4C4×D24411yescharacteristic
335C2×C4#1451975011yescharacteristic
336C2×C4#1451975211yescharacteristic
337C2×C4#1451976211yescharacteristic
338C8—42no—
339C8—42no—
340C8—4881no—
341C22—21no—
342C22—21no—
343C22—4881no—
344C22—4881no—
345C22—4881no—
346C22#1683964011yescharacteristic
347C22#1683993111yescharacteristic
348C22#1683993611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—1224no—
357C4—4881no—
358C4—2444no—
359C4—2442no—
360C4#104×D6112yes—
361C2—21no—
362C2—21no—
363C2—9761no—
364C2#2144619611yescharacteristic
365C2#2144626111yescharacteristic
366C2#2144631211yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1300C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1300D61
hypocenterC61#9326
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #21448182C2semidirect
5. #21444267C2semidirect
6. #21443676C2semidirect
7. C61×#1300C2semidirect
12. #16840058C4semidirect
13. #16839963C4semidirect
153. Dic122#104semidirect
183. #1300D61semidirect
212. C61#9326semidirect

Maximal subgroups

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

subgroupindex
#214559422
#214481822
#214467382
#214443262
#214442672
#214436762
C61×#13002
#932661