Finite Group Databasebeta

← #29575926

Subgroups — #29575926

Order 15616.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#21455626C211yesmaximal, characteristic
2#21448739C211yesmaximal, characteristic
3#21444384C211yesmaximal, characteristic
4#21444317C211yesmaximal, characteristic
5#21444215C211yesmaximal, characteristic
6#21443679C211yesmaximal, characteristic
7C61×#1303C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×C4×Dic122—21no—
10C2×C4×Dic122C2211yescharacteristic
11#16840070—22no—
12#16840058C412yescomplemented
13#16840048C412yescomplemented
14#16839815C2211yescharacteristic
15#16839804—21no—
16#16839772—21no—
17#16839660—21no—
18#16839657—22no—
19#16839628C2211yescharacteristic
20#16839626—21no—
21#16839613—21no—
22#16839500—22no—
23#16839498—21no—
24#16839498—21no—
25#16839494—22no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C22×Dic122D411yescharacteristic
30#14519777C2×C411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519769—21no—
33#14519756—21no—
34#14519756—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×Dic122—42no—
40C4×Dic122—42no—
41C4×Dic122C2×C412yes—
42C4×Dic122D412yes—
43#14519736—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—22no—
48C2×#11758196—22no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
51#14519711—22no—
52#14519709—22no—
53C4⋊#92418—21no—
54#14519702—22no—
55#14519702—22no—
56C22⋊#92418—21no—
57C4×#92418—21no—
58#14519684—22no—
59#14519684—21no—
60#14519684D412yes—
61C2×M16×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61C2311yescharacteristic
65C4⋊C8×C61—21no—
66C4⋊C8×C61—21no—
67C4⋊C8×C61C2×C412yes—
68C22⋊C8×C61—21no—
69C4×C8×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
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—22no—
80C2×Dic122—21no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122C22⋊C412yes—
86C2×Dic122SD1612yes—
87C2×Dic122Q1612yes—
88M16×C61—41no—
89M16×C61—41no—
90C2×C8×C61—22no—
91C2×C8×C61—41no—
92C4⋊C4×C61—42no—
93C42×C61—41no—
94C42×C61—41no—
95C42×C61C22×C411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61#4111yescharacteristic
98C42×C61#4111yescharacteristic
99C22⋊Dic61—41no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—41no—
106C4⋊Dic61—41no—
107C4⋊Dic61C22⋊C412yes—
108#11758196—22no—
109#11758196—22no—
110#11758196—44no—
111#11758196—44no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—22no—
115C4×Dic61—22no—
116C4×Dic61—44no—
117C4×Dic61—41no—
118C2×#92418—41no—
119C23×C61#12011yescharacteristic
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12311yescharacteristic
134C2×C4×C61C2×SD1611yescharacteristic
135C2×C4×C61C2×Q1611yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148Dic122—22no—
149Dic122—22no—
150Dic122—22no—
151Dic122—81no—
152Dic122—81no—
153Dic122#10314yes—
154C8×C61—42no—
155C8×C61—81no—
156#92418—81no—
157#9176—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45011yescharacteristic
162C22×C61#45111yescharacteristic
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—42no—
168C4×C61—81no—
169C4×C61C2×#10311yescharacteristic
170C4×C61#41711yescharacteristic
171C4×C61#48311yescharacteristic
172C4×C61#48411yescharacteristic
173Dic61—24no—
174Dic61—42no—
175Dic61—82no—
176Dic61—81no—
177Dic61—81no—
178#3078—611no—
179#2351—611no—
180#1518—611no—
181#1503—611no—
182#1479—611no—
183#1303D6111yescharacteristic
184#1286—611no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#187211yescharacteristic
188C2×C61#190511yescharacteristic
189C2×C61#190611yescharacteristic
190#533—611no—
191C2×C4×Q8—611no—
192C2×C4×Q8—1221no—
193#479—1222no—
194#477—612no—
195#475—612no—
196#431—611no—
197#431C2×D6111yescharacteristic
198#428—1221no—
199#423C2×D6111yescharacteristic
200#419—1221no—
201#389—1222no—
202#389—1221no—
203#382—611no—
204#382—1221no—
205C4×C4⋊C4—1221no—
206C4×C4⋊C4C2×D6111yescharacteristic
207#328—1222no—
208#328—1221no—
209#328—1221no—
210Q8⋊C8—1222no—
211C61#917611yescharacteristic, complemented, Sylow
212C22×Q8—611no—
213C2×M16—21no—
214C4⋊Q8—611no—
215C4⋊Q8—611no—
216#125—1221no—
217C22⋊Q8—1221no—
218C22⋊Q8—1221no—
219C4×Q8—612no—
220C4×Q8—612no—
221C4×Q8—1221no—
222C4×Q8—1221no—
223C4×Q8—1221no—
224C4×Q8—1221no—
225C4×Q8—2442no—
226C4×Q8—2442no—
227#117—1221no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4—1222no—
233C2×C4⋊C4—1221no—
234C2×C42—21no—
235C2×C42—1221no—
236C2×C42—1222no—
237C2×C42C22×D6111yescharacteristic
238#107—1222no—
239#106—1222no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—1221no—
243C4⋊C8C4×D6112yes—
244#103—1222no—
245#103—1222no—
246C22⋊C8—21no—
247C22⋊C8—1221no—
248C4×C8—21no—
249C4×C8—1221no—
250#95—612no—
251#95—1222no—
252#95—1221no—
253#95D24411yescharacteristic
254#95D24411yescharacteristic
255C2×Q8—612no—
256C2×Q8—612no—
257C2×Q8—612no—
258C2×Q8—1221no—
259C2×Q8—1221no—
260C2×Q8—1221no—
261C2×Q8—1221no—
262C2×Q8—1221no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1222no—
268C22×C4C2×D24411yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4D4×D6111yescharacteristic
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—2441no—
276C4⋊C4—42no—
277C4⋊C4—612no—
278C4⋊C4—1222no—
279C4⋊C4—1222no—
280C4⋊C4—1221no—
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C4⋊C4—1221no—
284C4⋊C4—1221no—
285C4⋊C4—2444no—
286C4⋊C4—2444no—
287C4⋊C4—2441no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C22⋊C4—2441no—
292C42—41no—
293C42—41no—
294C42—1222no—
295C42—1222no—
296C42—2444no—
297C42—2441no—
298C42C2×C4×D6111yescharacteristic
299C42#1175822211yescharacteristic
300C42D4×D6111yescharacteristic
301C42#1175822611yescharacteristic
302C23#1451974711yescharacteristic
303Q8—614no—
304Q8—1222no—
305Q8—1222no—
306Q8—1222no—
307Q8—4881no—
308Q8—4881no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4C22⋊C4×D6111yescharacteristic
328C2×C4#1451973911yescharacteristic
329C2×C4C4×D24411yescharacteristic
330C2×C4#1451975011yescharacteristic
331C2×C4#1451975211yescharacteristic
332C2×C4#1451976211yescharacteristic
333C2×C4SD16×D6111yescharacteristic
334C2×C4Q16×D6111yescharacteristic
335C2×C4#1451980111yescharacteristic
336C2×C4#1451980611yescharacteristic
337C8—42no—
338C8—81no—
339C8—4881no—
340C22—21no—
341C22—21no—
342C22#1683964011yescharacteristic
343C22#1683991411yescharacteristic
344C22#1683991911yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4—1224no—
352C4—4882no—
353C4—4881no—
354C4—4881no—
355C4—2442no—
356C4#103×D6111yescharacteristic
357C4#1683976011yescharacteristic
358C4#1684009411yescharacteristic
359C4#1684009911yescharacteristic
360C2—21no—
361C2—21no—
362C2#2144612511yescharacteristic
363C2#2144631211yescharacteristic
364C2#2144631911yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1303C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1303D61
hypocenterC61#9176
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
12. #16840058C4semidirect
13. #16840048C4semidirect
211. C61#9176semidirect

Maximal subgroups

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

subgroupindex
#214556262
#214487392
#214443842
#214443172
#214442152
#214436792
C61×#13032
#917661