Finite Group Databasebeta

← #29711955

Subgroups — #29711955

Order 15616.

1286 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1286 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#16840367C211yesmaximal, characteristic
2C2×#16839903C212yesmaximal, complemented
3#21448403C211yesmaximal, characteristic, complemented
4#21446971C212yesmaximal, complemented
5#21446909C212yesmaximal, complemented
6#21446152C214yesmaximal, complemented
7#21446139C212yesmaximal, complemented
8C61×#1818C211yesmaximal, characteristic, complemented
9C23×Dic122C2211yescharacteristic
10C22×#11758190C2214yes—
11#16840367—28no—
12C2×#14519774C2211yescharacteristic
13C2×#14519769C2211yescharacteristic
14C2×#14519762—22no—
15C2×#14519751C2212yes—
16C2×#14519750—22no—
17C2×C4×D244C2212yes—
18C2×#14519736C2211yescharacteristic
19#16839903—28no—
20C2×#14519702C2214yes—
21C2×#14519699C2214yes—
22#16839668C2212yes—
23#16839668C2212yes—
24#16839667—22no—
25#16839623—22no—
26#16839525C2214yes—
27C2×#117×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2212yes—
29C61×#336C2212yes—
30C61×#336C2212yes—
31C22×Dic122—21no—
32C22×Dic122—24no—
33C22×Dic122—24no—
34C22×Dic122—42no—
35C22×Dic122C2312yes—
36C22×D244—22no—
37C22×D244C2312yes—
38C22×C22⋊D61—22no—
39C23×Dic61—21no—
40C22×C4×D61—22no—
41C2×#11758190—28no—
42C2×#11758190—28no—
43C2×#11758190—44no—
44#14519774—24no—
45#14519769—24no—
46#14519762—216no—
47#14519751—24no—
48#14519751D418yes—
49#14519750—216no—
50C4×D244—24no—
51C4×D244—24no—
52#14519736—42no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×C22⋊Dic61—21no—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×C4×Dic61—21no—
60#14519702—28no—
61#14519699—44no—
62#14519684—24no—
63#14519684—24no—
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61C2314yes—
66#117×C61—24no—
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61D412yes—
69C2×C42×C61C2311yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C22⋊C8×C61—24no—
72C24×C61C2×D411yescharacteristic
73C23×D61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D414yes—
80C22×C4×C61C2411yescharacteristic
81C2×C22⋊D61—28no—
82C2×C22⋊D61—84no—
83C22×Dic61—24no—
84C22×Dic61—22no—
85C22×Dic61—22no—
86C22×Dic61—22no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C2×D244—24no—
90C2×D244—24no—
91C2×D244—22no—
92C2×D244—28no—
93C2×D244—84no—
94C2×D244C2×D414yes—
95C2×C4×D61—28no—
96C2×C4×D61—84no—
97C2×Dic122—24no—
98C2×Dic122—24no—
99C2×Dic122—24no—
100C2×Dic122—22no—
101C2×Dic122—416no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122#4114yes—
105C2×C8×C61—24no—
106C2×C8×C61—24no—
107C2×C8×C61—24no—
108C4⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C42×C61—22no—
111C42×C61—22no—
112C22⋊Dic61—42no—
113#11758198—28no—
114#11758198—28no—
115C4⋊Dic61—22no—
116C4⋊Dic61—24no—
117C4⋊Dic61C2×D414yes—
118#11758196—28no—
119C4×Dic61—42no—
120#11758190—416no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—24no—
124C23×C61#12012yes—
125C23×C61C22×D411yescharacteristic
126C22×D61—24no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—44no—
130C22×D61—44no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—28no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—24no—
137C2×C4×C61—24no—
138C2×C4×C61#12012yes—
139C2×C4×C61C22×D412yes—
140C2×C4×C61C2×#4112yes—
141C2×C4×C61H(5,𝔽2)12yes—
142C22⋊D61—88no—
143C2×Dic61—24no—
144C2×Dic61—24no—
145C2×Dic61—48no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—44no—
149C2×Dic61—42no—
150C2×Dic61—44no—
151D244—28no—
152D244—44no—
153D244—88no—
154C4×D61—88no—
155Dic122—28no—
156Dic122—48no—
157Dic122—44no—
158C8×C61—44no—
159#23940—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—24no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61C2×#12011yescharacteristic
165C22×C61#52511yescharacteristic
166C22×C61#53511yescharacteristic
167C22×C61#57714yes—
168C2×D61—44no—
169C2×D61—44no—
170C2×D61—44no—
171C2×D61—84no—
172C2×D61—84no—
173C4×C61—22no—
174C4×C61—42no—
175C4×C61—42no—
176C4×C61#54614yes—
177Dic61—44no—
178Dic61—44no—
179Dic61—82no—
180C2×#540—611no—
181C2×#448—612no—
182#2310—611no—
183#2026—612no—
184#2014—612no—
185#1877—614no—
186#1875—612no—
187#1818D6111yescharacteristic, complemented
188C2×C61—42no—
189C2×C61#240411yescharacteristic
190C2×C61#300912yes—
191C2×C61#312612yes—
192C2×C61#322112yes—
193D61—84no—
194C23×Q8—611no—
195C22×SD16—614no—
196#540—1228no—
197C2×#127—611no—
198C2×#125—611no—
199C2×#123—1222no—
200C2×#121—612no—
201C2×#121—1222no—
202C2×C4×D4—612no—
203C2×#117—611no—
204C2×#117C2×D6111yescharacteristic
205#448—1228no—
206C2×#103—614no—
207C2×#102—614no—
208C2×C22⋊C8C2×D6112yes—
209#391—612no—
210#391—612no—
211#391—1222no—
212#381—1222no—
213#340—614no—
214#336C2×D6112yes—
215#336C2×D6112yes—
216C61#2394011yescharacteristic, complemented, Sylow
217C22×Q8—612no—
218C22×Q8—1224no—
219C22×Q8—1221no—
220C22×Q8—1224no—
221C22×Q8—2442no—
222C22×D4—612no—
223C22×D4—1222no—
224C22×D4—1222no—
225C23×C4—1221no—
226C23×C4—1222no—
227C23×C4C22×D6111yescharacteristic
228C2×SD16—1228no—
229C2×SD16—1228no—
230C2×SD16—2444no—
231C22×C8C22×D6114yes—
232#127—1224no—
233#125—1224no—
234#123—12216no—
235#121—618no—
236#121—12216no—
237#121—1224no—
238C4×D4—1224no—
239C4×D4—1224no—
240#117—24no—
241#117—2442no—
242C2×C4⋊C4—612no—
243C2×C4⋊C4—612no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4D24412yes—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4D24412yes—
250C2×C42—1221no—
251C2×C42C22×D6111yescharacteristic
252C2×C42C22×D6111yescharacteristic
253#103—1228no—
254#102—2444no—
255C22⋊C8—24no—
256#95—1224no—
257#95—1224no—
258C24—1222no—
259C24C2×D24411yescharacteristic
260C2×Q8—614no—
261C2×Q8—1224no—
262C2×Q8—1222no—
263C2×Q8—1224no—
264C2×Q8—1224no—
265C2×Q8—24416no—
266C2×Q8—2442no—
267C2×Q8—2442no—
268C2×D4—614no—
269C2×D4—1228no—
270C2×D4—1228no—
271C2×D4—1222no—
272C2×D4—1224no—
273C2×D4—1224no—
274C2×D4—4884no—
275C2×D4—4884no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—1228no—
279C22×C4—1224no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—4884no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4C2×D24412yes—
287C22×C4C2×D24412yes—
288C22×C4C2×D24411yescharacteristic
289C22×C4D4×D6114yes—
290C22×C4C23×D6111yescharacteristic
291SD16—24416no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—24no—
296C4⋊C4—614no—
297C4⋊C4—1228no—
298C4⋊C4—1222no—
299C4⋊C4—1224no—
300C22⋊C4—24no—
301C22⋊C4—1228no—
302C22⋊C4—1228no—
303C22⋊C4—2442no—
304C42—22no—
305C42—22no—
306C42—2442no—
307C23—21no—
308C23—24no—
309C23—21no—
310C23—1224no—
311C23—2442no—
312C23—2444no—
313C23—2444no—
314C23—2442no—
315C23#1451974712yes—
316C23C22×D24411yescharacteristic
317Q8—1228no—
318Q8—2448no—
319Q8—2444no—
320D4—1228no—
321D4—4888no—
322D4—4888no—
323D4—2444no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—1224no—
332C2×C4—1224no—
333C2×C4—4888no—
334C2×C4—2444no—
335C2×C4—2442no—
336C2×C4—2448no—
337C2×C4—2442no—
338C2×C4—2444no—
339C2×C4—2442no—
340C2×C4#1451974712yes—
341C2×C4C2×D4×D6112yes—
342C2×C4#41×D6112yes—
343C2×C4#1451982312yes—
344C8—44no—
345C22—22no—
346C22—24no—
347C22—42no—
348C22—42no—
349C22—4884no—
350C22—4884no—
351C22—2444no—
352C22—2444no—
353C22—2444no—
354C22C2×#1451974711yescharacteristic
355C22#1684027411yescharacteristic
356C22#1684033111yescharacteristic
357C22#1684057414yes—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—4882no—
362C4—2444no—
363C4—2444no—
364C4#1684040414yes—
365C2—42no—
366C2—4884no—
367C2#2144914011yescharacteristic
368C2#2145509812yes—
369C2#2145611112yes—
370C2#2145688212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1818C2
socleC23×C61C22×D4
prime radicalC2×#16840367C2
hypercenter#1818D61
hypocenterC61#23940
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. C2×#16839903C2semidirect
3. #21448403C2semidirect
4. #21446971C2semidirect
5. #21446909C2semidirect
6. #21446152C2semidirect
7. #21446139C2semidirect
8. C61×#1818C2semidirect
187. #1818D61semidirect
216. C61#23940semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
C2×#168399032
#214484032
#214469712
#214469092
#214461522
#214461392
C61×#18182
#2394061