Finite Group Databasebeta

← #29583105

Subgroups — #29583105

Order 15616.

416 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 93 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#21454195C211yesmaximal, characteristic
2#21446795C211yesmaximal, characteristic
3#21444384C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic
5#21443679C211yesmaximal, characteristic
6#21443651C211yesmaximal, characteristic
7C61×#1286C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×#14519776—21no—
10#16839815C2211yescharacteristic
11#16839666—22no—
12#16839666—22no—
13#16839628C2211yescharacteristic
14#16839627—21no—
15C42×Dic61C2211yescharacteristic
16#16839523—21no—
17#16839523—21no—
18#16839500—22no—
19#16839498—22no—
20C61×#431C2211yescharacteristic
21C61×#423C2211yescharacteristic
22C61×#382C2211yescharacteristic
23C22×Dic122—22no—
24#14519777Q811yescharacteristic, complemented
25#14519777Q811yescharacteristic, complemented
26#14519776—42no—
27#14519776—42no—
28#14519769—21no—
29#14519756—21no—
30#14519756—21no—
31C4×Dic122—21no—
32C4×Dic122—21no—
33#14519736—21no—
34C2×C4⋊Dic61—21no—
35C2×#11758196—22no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C4⋊#92418—21no—
41C22⋊#92418—21no—
42C4×#92418—21no—
43#14519684—24no—
44#14519684—21no—
45#14519684D412yes—
46C2×M16×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C42×C61C2311yescharacteristic
49C4⋊C8×C61—21no—
50C4⋊C8×C61—21no—
51C4⋊C8×C61—21no—
52C22⋊C8×C61—21no—
53C4×C8×C61—21no—
54#95×C61—21no—
55#95×C61D412yes—
56C22×C4×C61—22no—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—22no—
63C2×Dic122—21no—
64C2×Dic122—21no—
65C2×Dic122—44no—
66C2×Dic122—44no—
67C2×Dic122—44no—
68M16×C61—41no—
69M16×C61—41no—
70C2×C8×C61—41no—
71C2×C8×C61—41no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C42×C61C2×Q811yescharacteristic
75C42×C61#4111yescharacteristic
76C42×C61#4111yescharacteristic
77C42×C61#4111yescharacteristic
78C22⋊Dic61—41no—
79C4⋊Dic61—21no—
80C4⋊Dic61—21no—
81C4⋊Dic61—21no—
82C4⋊Dic61—21no—
83C4⋊Dic61—41no—
84C4⋊Dic61—41no—
85#11758196—44no—
86#11758196—41no—
87#11758196—41no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C4×Dic61—42no—
91C4×Dic61—42no—
92C4×Dic61—41no—
93C4×Dic61SD1612yes—
94C4×Dic61Q1612yes—
95C2×#92418—41no—
96C23×C61#12011yescharacteristic
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61C22⋊Q811yescharacteristic
103C2×C4×C61C22⋊Q811yescharacteristic
104C2×C4×C61C22⋊Q811yescharacteristic
105C2×C4×C61#12411yescharacteristic
106C2×C4×C61#12411yescharacteristic
107C2×C4×C61#12411yescharacteristic
108C2×C4×C61C2×SD1611yescharacteristic
109C2×C4×C61C2×Q1611yescharacteristic
110C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C61#13711yescharacteristic
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123Dic122—88no—
124Dic122—81no—
125Dic122—81no—
126C8×C61—81no—
127C8×C61—81no—
128#92418—81no—
129#10363—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61#39311yescharacteristic
133C22×C61#45011yescharacteristic
134C22×C61#45111yescharacteristic
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—82no—
140C4×C61#47511yescharacteristic
141C4×C61#47711yescharacteristic
142C4×C61#48711yescharacteristic
143C4×C61#48911yescharacteristic
144Dic61—24no—
145Dic61—42no—
146Dic61—82no—
147Dic61—81no—
148Dic61—81no—
149#2862—611no—
150#1994—611no—
151#1518—611no—
152#1518—611no—
153#1286—611no—
154#1286D6111yescharacteristic
155#1276—611no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61#202411yescharacteristic
159C2×C61#203111yescharacteristic
160C2×C61#203111yescharacteristic
161#533—611no—
162C2×C4⋊Q8—1221no—
163#431—611no—
164#431C2×D6111yescharacteristic
165#423C2×D6111yescharacteristic
166#391—1222no—
167#391—1222no—
168#382—611no—
169#382—1221no—
170#382C2×D6111yescharacteristic
171C43—611no—
172#339—1221no—
173#339—1221no—
174#328—1222no—
175#328—1222no—
176C61#1036311yescharacteristic, complemented, Sylow
177C22×Q8—1222no—
178C2×M16—21no—
179C4⋊Q8—611no—
180C4⋊Q8—611no—
181C4⋊Q8—2442no—
182C4⋊Q8—2442no—
183#125—1221no—
184C22⋊Q8—1221no—
185C22⋊Q8—1221no—
186C4×Q8—1221no—
187C4×Q8—1221no—
188#117—1221no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1221no—
192C2×C42—611no—
193C2×C42—611no—
194C2×C42—1221no—
195C2×C42—1221no—
196C2×C42C22×D6111yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—1221no—
201C22⋊C8—21no—
202C22⋊C8—1221no—
203C4×C8—21no—
204C4×C8—1221no—
205#95—21no—
206#95—612no—
207#95—1224no—
208#95—1221no—
209#95D24412yes—
210C2×Q8—1221no—
211C2×Q8—1221no—
212C2×Q8—2444no—
213C2×Q8—2444no—
214C2×Q8—2444no—
215C22×C4—22no—
216C22×C4—1221no—
217C22×C4—1221no—
218C22×C4—1222no—
219C22×C4C2×D24411yescharacteristic
220C22×C4D4×D6111yescharacteristic
221C22×C4D4×D6111yescharacteristic
222M16—41no—
223M16—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—2441no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—1221no—
230C4⋊C4—1221no—
231C4⋊C4—1221no—
232C4⋊C4—1221no—
233C4⋊C4—2444no—
234C4⋊C4—2441no—
235C4⋊C4—2441no—
236C4⋊C4—2441no—
237C4⋊C4—2441no—
238C22⋊C4—2441no—
239C42—612no—
240C42—612no—
241C42—1221no—
242C42—1221no—
243C42—2442no—
244C42—2442no—
245C42—2441no—
246C42#1175822211yescharacteristic
247C42#1175822211yescharacteristic
248C42#1175822411yescharacteristic
249C42Q8×D6111yescharacteristic
250C23#1451974711yescharacteristic
251Q8—4881no—
252Q8—4881no—
253Q8—4888no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—1222no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4#1451975511yescharacteristic
271C2×C4#1451975511yescharacteristic
272C2×C4#1451975711yescharacteristic
273C2×C4#1451976311yescharacteristic
274C2×C4#1451976311yescharacteristic
275C2×C4#1451976511yescharacteristic
276C2×C4SD16×D6111yescharacteristic
277C2×C4Q16×D6111yescharacteristic
278C2×C4#1451980111yescharacteristic
279C2×C4#1451980611yescharacteristic
280C8—81no—
281C8—81no—
282C8—4881no—
283C22—21no—
284C22—21no—
285C22#1683967511yescharacteristic
286C22#1683991411yescharacteristic
287C22#1683991911yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—82no—
293C4—1224no—
294C4—4882no—
295C4—4881no—
296C4—4881no—
297C4—2442no—
298C4#1684004611yescharacteristic
299C4#1684005611yescharacteristic
300C4#1684011111yescharacteristic
301C4#1684012011yescharacteristic
302C2—21no—
303C2—21no—
304C2#2144695911yescharacteristic
305C2#2144699011yescharacteristic
306C2#2144699411yescharacteristic
subgroupG/H
centerC22#16839675
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#393
prime radicalC23×C61#120
hypercenter#1286D61
hypocenterC61#10363
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
24. #14519777Q8semidirect
25. #14519777Q8semidirect
176. C61#10363semidirect

Maximal subgroups

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

subgroupindex
#214541952
#214467952
#214443842
#214443822
#214436792
#214436512
C61×#12862
#1036361