Finite Group Databasebeta

← #29576718

Subgroups — #29576718

Order 15616.

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 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#21456965C211yesmaximal, characteristic
2#21448538C211yesmaximal, characteristic
3#21446800C211yesmaximal, characteristic
4#21444490C211yesmaximal, characteristic
5#21444384C211yesmaximal, characteristic
6#21443678C211yesmaximal, characteristic
7C61×#1274C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×#14519777C2211yescharacteristic
10C2×C4×Dic122—21no—
11#16840160C412yescomplemented
12#16840113C412yescomplemented
13#16840093—22no—
14#16840048—22no—
15#16839786C2211yescharacteristic
16#16839668—21no—
17#16839647—22no—
18#16839647—22no—
19#16839628C2211yescharacteristic
20#16839522—21no—
21#16839522—21no—
22#16839507—22no—
23#16839500—22no—
24C61×#431C2211yescharacteristic
25C61×#431C2211yescharacteristic
26C61×#376C2211yescharacteristic
27C22×Dic122—21no—
28C2×#11758194—21no—
29#14519777—21no—
30#14519777—21no—
31#14519777C2×C411yescharacteristic
32#14519777C2×C411yescharacteristic
33#14519777C2×C412yes—
34#14519777D412yes—
35#14519769—21no—
36#14519756—21no—
37#14519756—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122—42no—
41C4×Dic122—42no—
42#14519736—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D412yes—
46C2×#11758196—22no—
47C2×C4×Dic61—22no—
48#14519709—24no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51#14519702—24no—
52#14519684—24no—
53#14519684—21no—
54#14519684D412yes—
55C2×C42×C61D411yescharacteristic
56C2×C42×C61D411yescharacteristic
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—21no—
59C4⋊C8×C61—21no—
60C22⋊C8×C61—21no—
61C22⋊C8×C61—21no—
62C4×C8×C61—21no—
63C4×C8×C61C2×C412yes—
64#95×C61—21no—
65#95×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—22no—
72C22×Dic61—22no—
73C2×Dic122—22no—
74C2×Dic122—22no—
75C2×Dic122—21no—
76C2×Dic122—21no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×C8×C61—22no—
82C2×C8×C61—41no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61SD1612yes—
87C42×C61C22×C411yescharacteristic
88C42×C61C2×D411yescharacteristic
89C42×C61#4111yescharacteristic
90C42×C61#4111yescharacteristic
91C22⋊Dic61—41no—
92C4⋊Dic61—22no—
93C4⋊Dic61—22no—
94C4⋊Dic61—22no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61—41no—
100C4⋊Dic61—41no—
101C4⋊Dic61C22⋊C414yes—
102#11758196—44no—
103#11758196—41no—
104#11758196—41no—
105C4×Dic61—44no—
106C4×Dic61—41no—
107#11758194—41no—
108#11758194—41no—
109C2×#92418—41no—
110C23×C61#12011yescharacteristic
111C2×C4×C61—22no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×C4×C61C4×D411yescharacteristic
122C2×C4×C61#12111yescharacteristic
123C2×C4×C61#12111yescharacteristic
124C2×C4×C61#12311yescharacteristic
125C2×C4×C61C2×SD1611yescharacteristic
126C2×C4×C61#13511yescharacteristic
127C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C61#13711yescharacteristic
129C2×Dic61—24no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—42no—
137Dic122—42no—
138Dic122—42no—
139Dic122—42no—
140Dic122—42no—
141Dic122—42no—
142Dic122—81no—
143C8×C61—42no—
144C8×C61—81no—
145#92418—81no—
146#9296—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#38611yescharacteristic
150C22×C61#45011yescharacteristic
151C22×C61#45211yescharacteristic
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—42no—
157C4×C61—81no—
158C4×C61C4×SD1611yescharacteristic
159C4×C61#44011yescharacteristic
160C4×C61#46111yescharacteristic
161C4×C61#46311yescharacteristic
162Dic61—44no—
163Dic61—82no—
164Dic61—81no—
165Dic61—81no—
166#3229—611no—
167#2327—611no—
168#1995—611no—
169#1546—611no—
170#1518—611no—
171#1286—611no—
172#1274D6111yescharacteristic
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#187711yescharacteristic
176C2×C61#190511yescharacteristic
177C2×C61#191211yescharacteristic
178#533—611no—
179C2×C4⋊Q8—611no—
180C2×C4×Q8—1221no—
181#498—612no—
182#487—612no—
183#483—1222no—
184#475—1222no—
185#431C2×D6111yescharacteristic
186#431C2×D6111yescharacteristic
187#423—611no—
188#391—1221no—
189#387—1222no—
190#387—1222no—
191#382—611no—
192#376C2×D6111yescharacteristic
193#339—1221no—
194#339—1221no—
195#332—1222no—
196#328—1222no—
197C61#929611yescharacteristic, complemented, Sylow
198C22×Q8—1221no—
199C2×M16—1221no—
200C4⋊Q8—612no—
201C4⋊Q8—611no—
202C4⋊Q8—611no—
203C4⋊Q8—612no—
204C4⋊Q8—1221no—
205C4⋊Q8—1221no—
206#125—1221no—
207C22⋊Q8—1221no—
208C22⋊Q8—1221no—
209C4×Q8—1221no—
210C4×Q8—1221no—
211C4×Q8—2442no—
212C4×Q8—2442no—
213#117—1221no—
214C2×C4⋊C4—612no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—1222no—
217C2×C4⋊C4—1221no—
218C2×C42—1222no—
219C2×C42D24411yescharacteristic
220C2×C42D24411yescharacteristic
221C2×C42C22×D6111yescharacteristic
222#106—1224no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—1221no—
226C4⋊C8—1221no—
227#103—1224no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C4×C8—21no—
231C4×C8C4×D6112yes—
232#95—21no—
233#95—21no—
234#95—612no—
235#95—1224no—
236#95—1221no—
237C2×Q8—1222no—
238C2×Q8—1222no—
239C2×Q8—1221no—
240C2×Q8—1221no—
241C2×Q8—2441no—
242C2×Q8—2441no—
243C2×Q8—2441no—
244C2×Q8—2441no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4C2×D24411yescharacteristic
250C22×C4D4×D6111yescharacteristic
251C22×C4D4×D6111yescharacteristic
252M16—2441no—
253M16—2441no—
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—2441no—
257C4⋊C4—614no—
258C4⋊C4—1222no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
261C4⋊C4—1221no—
262C4⋊C4—1221no—
263C4⋊C4—1221no—
264C4⋊C4—1221no—
265C4⋊C4—2444no—
266C4⋊C4—2441no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C22⋊C4—2441no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—2444no—
275C42—2441no—
276C42#1175819012yes—
277C42C2×C4×D6111yescharacteristic
278C42#1175822211yescharacteristic
279C42D4×D6111yescharacteristic
280C42#1175822611yescharacteristic
281C23#1451974711yescharacteristic
282Q8—4881no—
283Q8—2442no—
284Q8—2442no—
285Q8—2442no—
286Q8—2442no—
287Q8—2442no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—1224no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4C22⋊C4×D6111yescharacteristic
305C2×C4#1451973911yescharacteristic
306C2×C4C4×D24411yescharacteristic
307C2×C4#1451975011yescharacteristic
308C2×C4#1451975211yescharacteristic
309C2×C4#1451976211yescharacteristic
310C2×C4C2×#1175819011yescharacteristic
311C2×C4#1451979911yescharacteristic
312C2×C4#1451980511yescharacteristic
313C2×C4#1451981011yescharacteristic
314C8—42no—
315C8—81no—
316C8—4881no—
317C22—21no—
318C22—21no—
319C22#1683964011yescharacteristic
320C22#1683991511yescharacteristic
321C22#1683992611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—81no—
328C4—4882no—
329C4—4881no—
330C4—4881no—
331C4—2444no—
332C4C4×#1175819011yescharacteristic
333C4#1683986411yescharacteristic
334C4#1683996811yescharacteristic
335C4#1683997811yescharacteristic
336C2—21no—
337C2—21no—
338C2#2144615211yescharacteristic
339C2#2144631411yescharacteristic
340C2#2144635511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1274C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1274D61
hypocenterC61#9296
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
11. #16840160C4semidirect
12. #16840113C4semidirect
197. C61#9296semidirect

Maximal subgroups

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

subgroupindex
#214569652
#214485382
#214468002
#214444902
#214443842
#214436782
C61×#12742
#929661