Finite Group Databasebeta

← #29598339

Subgroups — #29598339

Order 15616.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#21450804C211yesmaximal, characteristic
2C2×#16839560C214yesmaximal
3#21445815C211yesmaximal, characteristic
4C4×#101×C61C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic
6C2×#14519775C2212yes—
7#16839786—21no—
8C2×#14519696C2212yes—
9#16839692—21no—
10#16839692—21no—
11#16839667C412yes—
12#16839660—21no—
13#16839660—21no—
14C4×C4⋊Dic61C411yescharacteristic
15#16839560—28no—
16#16839531—21no—
17C2×C4×Q8×C61C2211yescharacteristic
18C2×#101×C61C2212yes—
19C4×M16×C61—21no—
20C61×#343—21no—
21C22×Dic122C2×C412yes—
22C2×#11758194—22no—
23#14519777—41no—
24#14519777—41no—
25#14519775—24no—
26#14519775—24no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×C4×Dic61C2×C412yes—
33C4⋊#92418—41no—
34C4⋊#92418—41no—
35#14519696—24no—
36#14519684—21no—
37#14519684—22no—
38#14519684—22no—
39C22×Q8×C61C2311yescharacteristic
40C2×M16×C61—22no—
41C4×Q8×C61—41no—
42C4×Q8×C61—41no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61D411yescharacteristic
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C42×C61C2×C411yescharacteristic
47C2×C42×C61D411yescharacteristic
48C2×C42×C61D411yescharacteristic
49#101×C61—22no—
50#101×C61D414yes—
51#97×C61—41no—
52C4×C8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61C2×D412yes—
56C2×Q8×C61#4112yes—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×Dic61—22no—
65C22×Dic61—22no—
66C2×Dic122—24no—
67C2×Dic122—42no—
68C2×Dic122—42no—
69M16×C61—24no—
70M16×C61—42no—
71C2×C8×C61—42no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C4⋊C4×C61—41no—
75C42×C61—21no—
76C42×C61—21no—
77C42×C61—21no—
78C42×C61—21no—
79C42×C61—21no—
80C42×C61—21no—
81C4⋊Dic61—24no—
82C4⋊Dic61—42no—
83C4⋊Dic61—41no—
84C4⋊Dic61—41no—
85#11758196—44no—
86#11758196—42no—
87#11758196—42no—
88C4×Dic61—24no—
89#11758194—42no—
90#11758194—42no—
91C2×#92418—42no—
92C23×C61C2×C22⋊C411yescharacteristic
93Q8×C61—42no—
94Q8×C61—42no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61#9911yescharacteristic
109C2×C4×C61#9911yescharacteristic
110C2×C4×C61C4×D412yes—
111C2×C4×C61#12112yes—
112C2×C4×C61#12411yescharacteristic
113C2×C4×C61#12711yescharacteristic
114C2×Dic61—24no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120Dic122—84no—
121C8×C61—44no—
122#92418—82no—
123#13227—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61#39011yescharacteristic
127C22×C61C2×#9911yescharacteristic
128C22×C61#41011yescharacteristic
129C4×C61—21no—
130C4×C61—21no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—21no—
134C4×C61—41no—
135C4×C61—41no—
136C4×C61—41no—
137C4×C61#35612yescomplemented
138Dic61—44no—
139Dic61—82no—
140#2546—611no—
141C2×#356—614no—
142#1801—611no—
143C4×#101D6111yescharacteristic
144C2×C61—21no—
145C2×C61—21no—
146C2×C61#196411yescharacteristic
147C2×C61C2×#35611yescharacteristic
148C2×C61#213511yescharacteristic
149C2×C4⋊Q8—612no—
150C2×C4⋊Q8—611no—
151C2×C4×Q8C2×D6111yescharacteristic
152#423—1221no—
153C2×#101—612no—
154C2×#101C2×D6112yes—
155C4×M16—21no—
156#398—1221no—
157#398—1221no—
158#391—612no—
159#389—1221no—
160#389—1221no—
161C4×C4⋊C4—611no—
162#356—1228no—
163#343—21no—
164#343—1221no—
165C61#1322711yescharacteristic, complemented, Sylow
166C22×Q8—612no—
167C22×Q8C22×D6111yescharacteristic
168C2×M16—22no—
169C2×M16—1222no—
170C4⋊Q8—1224no—
171C4⋊Q8—1224no—
172C4⋊Q8—2441no—
173C4⋊Q8—2441no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1221no—
177C2×C4⋊C4—1221no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—1222no—
180C2×C4⋊C4—1222no—
181C2×C4⋊C4C4×D6111yescharacteristic
182C2×C4⋊C4D24411yescharacteristic
183C2×C4⋊C4C22⋊D6111yescharacteristic
184C2×C42—612no—
185C2×C42C4×D6111yescharacteristic
186C2×C42D24411yescharacteristic
187C2×C42C22⋊D6111yescharacteristic
188C4⋊C8—2441no—
189C4⋊C8—2441no—
190#101—22no—
191#101—1224no—
192#101D24414yes—
193#97—41no—
194C4×C8—41no—
195#95—1221no—
196#95—1222no—
197#95—1222no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—1224no—
201C2×Q8—2442no—
202C2×Q8—2442no—
203C2×Q8C2×D24412yes—
204C2×Q8#1175822212yes—
205C22×C4—1222no—
206C22×C4—1222no—
207C22×C4#1175819811yescharacteristic
208C22×C4#1175819811yescharacteristic
209C22×C4#1175819811yescharacteristic
210C22×C4#1175819811yescharacteristic
211C22×C4C2×C4×D6111yescharacteristic
212C22×C4C2×D24411yescharacteristic
213C22×C4C2×C22⋊D6111yescharacteristic
214M16—24no—
215M16—42no—
216M16—2442no—
217M16—2442no—
218C2×C8—42no—
219C2×C8—2442no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—1224no—
224C4⋊C4—2442no—
225C4⋊C4—2441no—
226C4⋊C4—2441no—
227C4⋊C4—2444no—
228C4⋊C4—2442no—
229C4⋊C4—2442no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1224no—
237C23C2×#1175819811yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—4884no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1224no—
255C2×C4—2442no—
256C2×C4—2442no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4—2442no—
260C2×C4#1451969111yescharacteristic
261C2×C4#1451969111yescharacteristic
262C2×C4C4×D24412yes—
263C2×C4#1451975112yes—
264C2×C4#1451976511yescharacteristic
265C2×C4#1451977411yescharacteristic
266C8—44no—
267C8—4882no—
268C22—21no—
269C22—21no—
270C22#1683966411yescharacteristic
271C22C2×#1451969111yescharacteristic
272C22#1683973511yescharacteristic
273C4—21no—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—4882no—
282C4—2444no—
283C4#1683956012yescomplemented
284C2—21no—
285C2—21no—
286C2#2144665511yescharacteristic
287C2C2×#1683956011yescharacteristic
288C2#2144751311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC4×#101×C61C2
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC22×Q8×C61C23
hypercenterC4×#101D61
hypocenterC61#13227
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
137. C4×C61#356semidirect
165. C61#13227semidirect
283. C4#16839560semidirect

Maximal subgroups

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

subgroupindex
#214508042
C2×#168395602
#214458152
C4×#101×C612
#1322761