Finite Group Databasebeta

← #29689306

Subgroups — #29689306

Order 15616.

1386 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 235 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×#16840326C211yesmaximal, characteristic
2#21454454C212yesmaximal, complemented
3#21446667C214yesmaximal, complemented
4#21446657C214yesmaximal, complemented
5#21445747C211yesmaximal, characteristic, complemented
6#21445739C212yesmaximal, complemented
7C61×#1785C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9#16840326—28no—
10C2×#14519773C2212yes—
11C2×#14519766C2212yes—
12C2×#14519751C414yes—
13C2×#14519745C414yes—
14C2×#14519736C2211yescharacteristic
15#16839820—28no—
16#16839785C2214yescomplemented
17C2×#14519696C2212yescomplemented
18C2×#14519694C2212yescomplemented
19C2×#14519693C2214yes—
20C2×#14519693C2214yes—
21#16839726C2212yes—
22M16×Dic61C2214yescomplemented
23C22×#41×C61C2211yescharacteristic
24C2×#101×C61C2212yescomplemented
25C2×#100×C61C2212yescomplemented
26C61×#407C2212yes—
27C24×D61C2×C412yes—
28C22×D244—24no—
29C22×D244—42no—
30C22×D244—42no—
31C22×D244C2×C412yes—
32C22×D244C2312yes—
33C22×C22⋊D61C2×C418yes—
34C2×#11758194C2314yes—
35#14519773—24no—
36#14519773—24no—
37#14519766—24no—
38#14519766—24no—
39#14519751—216no—
40#14519745—216no—
41#14519736D418yes—
42C2×C4⋊Dic61C2×C412yes—
43C2×C22⋊Dic61C2×C412yes—
44C2×#11758198C2×C418yes—
45C2×C4×Dic61C2312yes—
46#14519706—28no—
47#14519696—42no—
48#14519694—42no—
49#14519693—44no—
50#14519693—44no—
51C22⋊#92418—24no—
52#14519687—24no—
53C8×Dic61—24no—
54C2×#41×C61—24no—
55C2×#41×C61—42no—
56C2×#41×C61—42no—
57C22×Q8×C61C2311yescharacteristic
58C22×D4×C61C2×C412yes—
59C22×D4×C61C2311yescharacteristic
60C23×C4×C61C2×C412yes—
61C23×C4×C61C2311yescharacteristic
62C2×M16×C61C2314yes—
63#101×C61—42no—
64#100×C61—42no—
65C22⋊C8×C61—24no—
66C24×C61C22×C412yes—
67C24×C61C22×C411yescharacteristic
68C23×D61—24no—
69C23×D61—42no—
70C23×D61—42no—
71C23×D61—84no—
72C23×D61C22×C414yes—
73#41×C61—416no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61—22no—
77C2×D4×C61—24no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—42no—
82C2×D4×C61—42no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—24no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61C22×C412yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C2×D414yes—
91C22×C4×C61C2411yescharacteristic
92C2×C22⋊D61—216no—
93C2×C22⋊D61—88no—
94C22×Dic61C22×C414yes—
95C2×D244—24no—
96C2×D244—22no—
97C2×D244—22no—
98C2×D244—24no—
99C2×D244—24no—
100C2×D244—42no—
101C2×D244—42no—
102C2×D244—416no—
103M16×C61—24no—
104M16×C61—24no—
105C2×C8×C61—24no—
106C22⋊Dic61C22⋊C418yes—
107#11758198—216no—
108C4⋊Dic61C22⋊C418yes—
109C4×Dic61C2×D414yes—
110C4×Dic61C2×D414yes—
111#11758194—24no—
112#11758194—24no—
113C2×#92418—24no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—22no—
119C23×C61—42no—
120C23×C61—44no—
121C23×C61C2×C22⋊C412yes—
122C23×C61C23×C411yescharacteristic
123C22×D61—24no—
124C22×D61—24no—
125C22×D61—24no—
126C22×D61—48no—
127C22×D61—48no—
128C22×D61—88no—
129C22×D61—88no—
130C22×D61—88no—
131Q8×C61—44no—
132D4×C61—48no—
133D4×C61—44no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—28no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—48no—
141C2×C4×C61C2×C22⋊C412yes—
142C2×C4×C61C22×D412yes—
143C2×C4×C61H(5,𝔽2)12yes—
144C2×C4×C61H(5,𝔽2)12yes—
145C22⋊D61—816no—
146C2×Dic61—44no—
147C2×Dic61C2×C22⋊C418yes—
148D244—48no—
149D244—44no—
150D244—44no—
151C8×C61—44no—
152#92418—44no—
153#21801—611nomaximal, Sylow
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—24no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—42no—
160C22×C61—44no—
161C22×C61—82no—
162C22×C61—82no—
163C22×C61C22×C22⋊C411yescharacteristic
164C22×C61#51811yescharacteristic
165C22×C61#51811yescharacteristic
166C2×D61—44no—
167C2×D61—48no—
168C2×D61—816no—
169C2×D61—84no—
170C2×D61—84no—
171C4×C61—22no—
172C4×C61—42no—
173C4×C61#53514yes—
174Dic61—28no—
175C2×#535—611no—
176#2921—612no—
177#1967—614no—
178#1965—614no—
179#1785—611no—
180#1785D6111yescharacteristic, complemented
181#1783—612no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—42no—
185C2×C61—82no—
186C2×C61#230911yescharacteristic
187C2×C61#288812yes—
188D61—84no—
189C22×#41C2×D6111yescharacteristic
190C23×D4—611no—
191#535—1228no—
192C2×#127—612no—
193C2×#124—612no—
194C2×#121—614no—
195C2×#120—614no—
196C2×#117—611no—
197#433—1228no—
198#423—614no—
199C2×#101—612no—
200C2×#101C2×D6112yescomplemented
201C2×#100—612no—
202C2×#100—614no—
203C2×#100—614no—
204C2×#100C2×D6112yescomplemented
205#407—612no—
206#407C2×D6112yes—
207C4×M16—614no—
208C61#2180111yescharacteristic, complemented, Sylow
209C25—612no—
210C2×#41—24no—
211C2×#41—42no—
212C2×#41—42no—
213C22×Q8C22×D6111yescharacteristic
214C22×D4—618no—
215C22×D4—612no—
216C22×D4—612no—
217C22×D4—1224no—
218C22×D4—2442no—
219C22×D4—2442no—
220C22×D4C4×D6112yes—
221C22×D4C22×D6111yescharacteristic
222C23×C4C4×D6112yes—
223C23×C4C22×D6111yescharacteristic
224C2×M16—614no—
225C2×M16C22×D6114yes—
226#127—1224no—
227#127—1224no—
228#124—1224no—
229#124—1224no—
230#121—12216no—
231#120—12216no—
232#117—618no—
233C2×C4⋊C4—612no—
234C2×C22⋊C4—612no—
235C2×C22⋊C4—618no—
236C2×C42—612no—
237C4⋊C8—1228no—
238#101—42no—
239#101—2442no—
240#100—42no—
241#100—2444no—
242#100—2444no—
243#100—2442no—
244C22⋊C8—24no—
245C22⋊C8—1224no—
246#97—1224no—
247C4×C8—1224no—
248C24—614no—
249C24—1224no—
250C24—4884no—
251C24—2442no—
252C24—2442no—
253C24C2×C4×D6112yes—
254C24C2×C4×D6111yescharacteristic
255#41—416no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—12216no—
266C2×D4—1222no—
267C2×D4—1222no—
268C2×D4—1224no—
269C2×D4—1224no—
270C2×D4—1224no—
271C2×D4—4888no—
272C2×D4—2442no—
273C2×D4—2442no—
274C2×D4—24416no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—614no—
281C22×C4C2×C4×D6112yes—
282C22×C4C2×C4×D6111yescharacteristic
283C22×C4D4×D6114yes—
284C22×C4C23×D6111yescharacteristic
285M16—24no—
286M16—24no—
287M16—1224no—
288M16—1224no—
289C2×C8—24no—
290C2×C8—1224no—
291C4⋊C4—618no—
292C22⋊C4—618no—
293C22⋊C4—12216no—
294C42—614no—
295C42—614no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—44no—
303C23—1224no—
304C23—1224no—
305C23—1224no—
306C23—4888no—
307C23—4888no—
308C23—4888no—
309C23—2448no—
310C23—2448no—
311C23C22⋊C4×D6112yes—
312C23C22×C4×D6111yescharacteristic
313Q8—44no—
314D4—48no—
315D4—44no—
316D4—48816no—
317D4—2448no—
318D4—2444no—
319D4—2444no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—618no—
328C2×C4—2444no—
329C2×C4C22⋊C4×D6112yes—
330C2×C4C2×D4×D6112yes—
331C2×C4#1451982212yes—
332C2×C4#1451982212yes—
333C8—44no—
334C8—2444no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—82no—
343C22—82no—
344C22—4884no—
345C22—4884no—
346C22—48816no—
347C22—2444no—
348C22—2448no—
349C22C2×C22⋊C4×D6111yescharacteristic
350C22#1684023511yescharacteristic
351C22#1684023511yescharacteristic
352C4—22no—
353C4—42no—
354C4—1228no—
355C4#1684032714yes—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—82no—
360C2—4884no—
361C2#2144840011yescharacteristic
362C2#2145429612yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1785C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenter#1785D61
hypocenterC61#21801
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #21454454C2semidirect
3. #21446667C2semidirect
4. #21446657C2semidirect
5. #21445747C2semidirect
6. #21445739C2semidirect
7. C61×#1785C2semidirect
16. #16839785C22semidirect
17. C2×#14519696C22semidirect
18. C2×#14519694C22semidirect
22. M16×Dic61C22semidirect
24. C2×#101×C61C22semidirect
25. C2×#100×C61C22semidirect
180. #1785D61semidirect
200. C2×#101C2×D61semidirect
204. C2×#100C2×D61semidirect
208. C61#21801semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
#214544542
#214466672
#214466572
#214457472
#214457392
C61×#17852
#2180161