Finite Group Databasebeta

← #29572505

Subgroups — #29572505

Order 15616.

1168 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 115 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#21445747C211yesmaximal, characteristic, complemented
3#21444347C214yesmaximal, complemented
4C61×#1782C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic, complemented
6#16840326—28no—
7C2×#14519773—21no—
8C2×#14519766—21no—
9C2×#14519751C412yescomplemented
10C2×#14519751C2212yes—
11C2×#14519745—21no—
12C2×#14519745—21no—
13C2×#14519736C411yescharacteristic, complemented
14C2×#14519696—21no—
15C2×#14519694—21no—
16#16839726C2212yescomplemented
17#16839495—28no—
18C22×#41×C61C2211yescharacteristic
19C2×#99×C61—21no—
20C2×#99×C61—21no—
21C61×#381C2212yescomplemented
22C24×D61—21no—
23C22×D244—21no—
24C22×D244—24no—
25C22×D244—42no—
26C22×D244—42no—
27C22×D244C2×C412yes—
28C22×C22⋊D61—22no—
29C22×C22⋊D61—22no—
30C2×#11758194—22no—
31#14519773—28no—
32#14519766—28no—
33#14519751—24no—
34#14519751—24no—
35#14519751—24no—
36#14519751D418yes—
37#14519745—28no—
38#14519745—28no—
39#14519736—24no—
40C2×C4⋊Dic61C2×C412yes—
41C2×C22⋊Dic61—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×C4×Dic61—21no—
45#14519696—42no—
46#14519694—42no—
47C22⋊#92418—24no—
48C2×#41×C61—24no—
49C2×#41×C61—42no—
50C2×#41×C61—42no—
51C22×Q8×C61D411yescharacteristic
52C22×D4×C61D411yescharacteristic
53C22×D4×C61D411yescharacteristic
54C22×D4×C61D411yescharacteristic
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C2×C22⋊C4×C61—22no—
59#99×C61—42no—
60#99×C61—42no—
61#95×C61—24no—
62C24×C61C22⋊C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C23×D61—22no—
66C23×D61—24no—
67C23×D61—42no—
68C23×D61—42no—
69C23×D61—84no—
70#41×C61—416no—
71C2×Q8×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—41no—
74C2×D4×C61—22no—
75C2×D4×C61—22no—
76C2×D4×C61—22no—
77C2×D4×C61—22no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61—42no—
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22⋊C411yescharacteristic
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D414yes—
97C2×C22⋊D61—28no—
98C2×C22⋊D61—28no—
99C2×C22⋊D61—84no—
100C2×C22⋊D61—84no—
101C22×Dic61—22no—
102C2×D244—24no—
103C2×D244—22no—
104C2×D244—416no—
105C2×D244—42no—
106C2×D244—42no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244C22⋊C414yes—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊Dic61—24no—
113#11758198—28no—
114#11758198—28no—
115C4⋊Dic61—22no—
116C4⋊Dic61C22⋊C414yes—
117C4×Dic61—24no—
118#11758194—42no—
119#11758194—42no—
120C2×#92418—42no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61#9912yes—
132C23×C61C2×C22⋊C411yescharacteristic
133C23×C61#12012yes—
134C22×D61—24no—
135C22×D61—48no—
136C22×D61—44no—
137C22×D61—44no—
138C22×D61—42no—
139C22×D61—42no—
140C22×D61—88no—
141C22×D61—88no—
142C22×D61—88no—
143Q8×C61—44no—
144D4×C61—44no—
145D4×C61—44no—
146D4×C61—44no—
147C2×C4×C61—28no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—82no—
160C2×C4×C61—84no—
161C2×C4×C61#9912yes—
162C2×C4×C61C2×C22⋊C412yes—
163C2×C4×C61#12012yes—
164C22⋊D61—88no—
165C22⋊D61—88no—
166C2×Dic61—24no—
167C2×Dic61—44no—
168D244—44no—
169D244—44no—
170D244—48no—
171#92418—82no—
172#8516—611nomaximal, Sylow
173C22×C61—22no—
174C22×C61—24no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—41no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61#37911yescharacteristic
187C22×C61C2×#9911yescharacteristic
188C22×C61C2×#9911yescharacteristic
189C2×D61—44no—
190C2×D61—44no—
191C2×D61—44no—
192C2×D61—84no—
193C2×D61—84no—
194C2×D61—816no—
195C4×C61—22no—
196C4×C61—22no—
197C4×C61—22no—
198C4×C61—41no—
199C4×C61—162no—
200Dic61—44no—
201C2×#535—611no—
202#1785—611no—
203#1782D6111yescharacteristic, complemented
204#1510—614no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61—42no—
208C2×C61—42no—
209C2×C61—81no—
210C2×C61#178111yescharacteristic
211C2×C61#179512yes—
212D61—84no—
213C22×#41C2×D6111yescharacteristic
214C23×D4—611no—
215#535—1228no—
216C2×#127—1221no—
217C2×#124—1221no—
218C2×#121—612no—
219C2×#121—612no—
220C2×#120—1221no—
221C2×#120—1221no—
222C2×#117—611no—
223C2×#101—1221no—
224C2×#100—1221no—
225C2×#99—21no—
226C2×#99—21no—
227#407—612no—
228#381C2×D6112yescomplemented
229#327—1228no—
230C61#851611yescharacteristic, complemented, Sylow
231C25—1221no—
232C2×#41—24no—
233C2×#41—42no—
234C2×#41—42no—
235C22×Q8D24411yescharacteristic
236C22×D4—612no—
237C22×D4—1224no—
238C22×D4—1222no—
239C22×D4—1222no—
240C22×D4—1221no—
241C22×D4—2442no—
242C22×D4—2442no—
243C22×D4D24411yescharacteristic
244C22×D4C22⋊D6111yescharacteristic
245C22×D4C22⋊D6111yescharacteristic
246C23×C4C4×D6111yescharacteristic
247C23×C4C4×D6111yescharacteristic
248C23×C4C22×D6111yescharacteristic
249C2×M16—1222no—
250#127—1228no—
251#124—1228no—
252#121—618no—
253#121—1224no—
254#121—1224no—
255#121—1224no—
256#120—1228no—
257#120—1228no—
258#117—1224no—
259C2×C4⋊C4—612no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4—1222no—
264C2×C42—1221no—
265#101—2442no—
266#100—2442no—
267#99—42no—
268#99—42no—
269C22⋊C8—1224no—
270#95—24no—
271C24—1224no—
272C24—1222no—
273C24—4884no—
274C24—2442no—
275C24—2442no—
276C24#1175819811yescharacteristic
277C24#1175819811yescharacteristic
278C24C2×C22⋊D6111yescharacteristic
279#41—416no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—614no—
293C2×D4—1224no—
294C2×D4—1228no—
295C2×D4—1228no—
296C2×D4—1222no—
297C2×D4—4884no—
298C2×D4—4884no—
299C2×D4—2442no—
300C2×D4—2442no—
301C2×D4—24416no—
302C2×D4—2442no—
303C2×D4—2442no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—1222no—
314C22×C4#1175819811yescharacteristic
315C22×C4#1175819811yescharacteristic
316C22×C4C2×C4×D6111yescharacteristic
317C22×C4C2×D24411yescharacteristic
318C22×C4D4×D6114yes—
319M16—2442no—
320M16—2442no—
321C2×C8—2442no—
322C4⋊C4—614no—
323C4⋊C4—1222no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—1228no—
327C22⋊C4—1224no—
328C22⋊C4—1228no—
329C42—1224no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—1224no—
341C23—4888no—
342C23—4888no—
343C23—4888no—
344C23—2444no—
345C23—2444no—
346C23—2448no—
347C23—2442no—
348C23—2442no—
349C23#1451969112yes—
350C23C2×#1175819811yescharacteristic
351C23#1451974512yes—
352Q8—44no—
353D4—44no—
354D4—44no—
355D4—44no—
356D4—4888no—
357D4—4888no—
358D4—2444no—
359D4—2444no—
360D4—2448no—
361C2×C4—28no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—1224no—
376C2×C4—2444no—
377C2×C4#1451969112yes—
378C2×C4C22⋊C4×D6112yes—
379C2×C4#1451974712yes—
380C8—4882no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—4884no—
395C22—4884no—
396C22—48816no—
397C22—2444no—
398C22—2444no—
399C22—2444no—
400C22#1683961711yescharacteristic
401C22C2×#1451969111yescharacteristic
402C22C2×#1451969111yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—162no—
408C4—2444no—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—42no—
413C2—81no—
414C2—4884no—
415C2#2144573311yescharacteristic
416C2#2144578712yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1782C2
socleC22×C61#379
prime radicalC2×#16840326C2
cosocleC23×C4×C61C23
hypercenter#1782D61
hypocenterC61#8516
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #21445747C2semidirect
3. #21444347C2semidirect
4. C61×#1782C2semidirect
5. C23×D244C4semidirect
9. C2×#14519751C4semidirect
13. C2×#14519736C4semidirect
16. #16839726C22semidirect
21. C61×#381C22semidirect
203. #1782D61semidirect
228. #381C2×D61semidirect
230. C61#8516semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
#214457472
#214443472
C61×#17822
#851661