Finite Group Databasebeta

← #29572518

Subgroups — #29572518

Order 15616.

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

normal subgroups only

This group has 754 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
1C2×#16840383C211yesmaximal, characteristic
2#21445902C211yesmaximal, characteristic, complemented
3#21444351C212yesmaximal, complemented
4#21444347C212yesmaximal
5C61×#1820C211yesmaximal, characteristic, complemented
6#16840383—24no—
7#16840383—24no—
8C2×#14519777C411yescharacteristic, complemented
9C2×#14519772—21no—
10C2×#14519765C411yescharacteristic, complemented
11C2×#14519763—21no—
12C2×#14519759—21no—
13C2×#14519756C2211yescharacteristic
14C2×#14519755—21no—
15C2×#14519751C2211yescharacteristic
16C2×C4×Dic122C411yescharacteristic, complemented
17C2×C4×D244C411yescharacteristic, complemented
18#16839726C2212yes—
19#16839531—21no—
20#16839531—21no—
21#16839498—24no—
22#16839495—24no—
23C2×#117×C61C2211yescharacteristic
24C61×#381C2212yes—
25C61×#342—21no—
26C61×#342—21no—
27C22×Dic122—21no—
28C22×Dic122C2×C411yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C4×D61—21no—
32C2×#11758194—22no—
33#14519777—22no—
34#14519777—22no—
35#14519772—28no—
36#14519765—22no—
37#14519765—22no—
38#14519763—28no—
39#14519759—28no—
40#14519756—22no—
41#14519756D414yes—
42#14519755—28no—
43#14519751—22no—
44#14519751D414yes—
45C4×Dic122—22no—
46C4×Dic122—22no—
47C4×D244—22no—
48C4×D244—22no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×C22⋊Dic61—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×C4×Dic61—21no—
60C22⋊#92418—22no—
61C22⋊#92418—22no—
62C23×C4×C61C2311yescharacteristic
63#117×C61—22no—
64#117×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61D411yescharacteristic
70C2×C42×C61C2×C411yescharacteristic
71C2×C42×C61C2×C411yescharacteristic
72#95×C61—22no—
73#95×C61—22no—
74C24×C61C2×D411yescharacteristic
75C23×D61—21no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D412yes—
86C2×C22⋊D61—24no—
87C2×C22⋊D61—82no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C2×D244—21no—
92C2×D244—42no—
93C2×D244C22⋊C412yes—
94C2×C4×D61—24no—
95C2×C4×D61—82no—
96C2×Dic122—21no—
97C2×Dic122—24no—
98C2×Dic122—42no—
99C2×Dic122—82no—
100C2×Dic122C22⋊C412yes—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—44no—
106C42×C61—21no—
107C42×C61—21no—
108C42×C61—21no—
109C42×C61—21no—
110C22⋊Dic61—24no—
111#11758198—24no—
112#11758198—24no—
113#11758198—24no—
114#11758198—24no—
115C4⋊Dic61—24no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61C22⋊C412yes—
119C4⋊Dic61C22⋊C412yes—
120#11758196—24no—
121#11758196—24no—
122C4×Dic61—24no—
123#11758194—44no—
124C2×#92418—42no—
125C23×C61—22no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61#12012yes—
130C22×D61—22no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—82no—
147C2×C4×C61—82no—
148C2×C4×C61—82no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C2×C4×C61#12012yes—
152C22⋊D61—84no—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159D244—42no—
160D244—42no—
161C4×D61—84no—
162Dic122—42no—
163Dic122—42no—
164Dic122—84no—
165#92418—82no—
166#8519—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—22no—
170C22×C61—22no—
171C22×C61—22no—
172C22×C61—82no—
173C22×C61#37911yescharacteristic
174C22×C61#41012yes—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61—84no—
179C4×C61—22no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—41no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—162no—
187Dic61—42no—
188Dic61—42no—
189Dic61—42no—
190C2×#542—611no—
191#1822—611no—
192#1820D6111yescharacteristic, complemented
193#1511—612no—
194#1510—612no—
195C2×C61—22no—
196C2×C61—42no—
197C2×C61#178211yescharacteristic
198C2×C61#179511yescharacteristic
199C2×C61#179611yescharacteristic
200D61—82no—
201#542—1224no—
202#542—1224no—
203C2×C4⋊Q8—611no—
204C2×#126—1221no—
205C2×#124—611no—
206C2×#124—1221no—
207C2×#123—1221no—
208C2×C22⋊Q8—611no—
209C2×C22⋊Q8—1221no—
210C2×#121—611no—
211C2×C4×Q8—611no—
212C2×C4×D4—611no—
213C2×#117C2×D6111yescharacteristic
214#407—612no—
215#381C2×D6112yes—
216#343—1221no—
217#343—1221no—
218#342—21no—
219#342—21no—
220#328—1224no—
221#327—1224no—
222C61#851911yescharacteristic, complemented, Sylow
223C22×Q8—611no—
224C22×Q8—1221no—
225C22×D4—611no—
226C22×D4—1221no—
227C23×C4—1221no—
228C23×C4C22×D6111yescharacteristic
229C2×M16—1222no—
230C4⋊Q8—1222no—
231C4⋊Q8—1222no—
232#126—1228no—
233#124—1222no—
234#124—1222no—
235#124—1228no—
236#123—1228no—
237C22⋊Q8—614no—
238C22⋊Q8—1228no—
239C22⋊Q8—1222no—
240#121—614no—
241#121—1222no—
242C4×Q8—1222no—
243C4×Q8—1222no—
244C4×D4—1222no—
245C4×D4—1222no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4D24411yescharacteristic
254C2×C4⋊C4D24411yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4C22⋊D6111yescharacteristic
262C2×C22⋊C4C22⋊D6111yescharacteristic
263C2×C42—1221no—
264C2×C42C4×D6111yescharacteristic
265C2×C42C4×D6111yescharacteristic
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268#95—22no—
269#95—22no—
270C24—1221no—
271C24C2×C22⋊D6111yescharacteristic
272C2×Q8—612no—
273C2×Q8—1224no—
274C2×Q8—1221no—
275C2×Q8—4882no—
276C2×Q8—2442no—
277C2×D4—612no—
278C2×D4—1224no—
279C2×D4—1221no—
280C2×D4—4882no—
281C2×D4—2442no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—1224no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—4882no—
289C22×C4#1175819811yescharacteristic
290C22×C4#1175819811yescharacteristic
291C22×C4#1175819811yescharacteristic
292C22×C4#1175819811yescharacteristic
293C22×C4C2×C4×D6111yescharacteristic
294C22×C4C2×D24411yescharacteristic
295C22×C4D4×D6112yes—
296C22×C4D4×D6112yes—
297M16—2444no—
298C2×C8—2442no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—612no—
302C4⋊C4—612no—
303C4⋊C4—1224no—
304C4⋊C4—1224no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1224no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—44no—
311C22⋊C4—1224no—
312C22⋊C4—1224no—
313C22⋊C4—1224no—
314C22⋊C4—1224no—
315C22⋊C4—1224no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—1224no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—1222no—
325C23—2442no—
326C23—2442no—
327C23—2442no—
328C23C2×#1175819811yescharacteristic
329C23#1451974512yes—
330Q8—4884no—
331Q8—2442no—
332Q8—2442no—
333D4—4884no—
334D4—2442no—
335D4—2442no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—1222no—
352C2×C4—1222no—
353C2×C4—1222no—
354C2×C4—4884no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4C22⋊C4×D6111yescharacteristic
359C2×C4C22⋊C4×D6111yescharacteristic
360C2×C4#1451974712yes—
361C8—4882no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—82no—
368C22—4884no—
369C22—2442no—
370C22—2442no—
371C22—2442no—
372C22#1683961711yescharacteristic
373C22#1683973512yes—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—162no—
382C4—2442no—
383C4—2442no—
384C4—2442no—
385C2—22no—
386C2—42no—
387C2—4882no—
388C2#2144573611yescharacteristic
389C2#2144578711yescharacteristic
390C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1820C2
socleC22×C61#379
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1820D61
hypocenterC61#8519
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #21445902C2semidirect
3. #21444351C2semidirect
5. C61×#1820C2semidirect
8. C2×#14519777C4semidirect
10. C2×#14519765C4semidirect
16. C2×C4×Dic122C4semidirect
17. C2×C4×D244C4semidirect
192. #1820D61semidirect
222. C61#8519semidirect

Maximal subgroups

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

subgroupindex
C2×#168403832
#214459022
#214443512
#214443472
C61×#18202
#851961