Finite Group Databasebeta

← #29571629

Subgroups — #29571629

Order 15616.

1068 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16840421C211yesmaximal, characteristic
2#21445764C212yesmaximal, complemented
3#21444338C212yesmaximal, complemented
4#21444157C211yesmaximal, characteristic, complemented
5C4×#99×C61C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C23×Dic122C411yescharacteristic, complemented
8#16840421—24no—
9#16840421—24no—
10C2×#14519765C411yescharacteristic, complemented
11C2×#14519763—21no—
12C2×#14519756C2211yescharacteristic
13C2×#14519755—21no—
14C2×#14519751C411yescharacteristic, complemented
15C2×#14519749—21no—
16C2×C4×Dic122C411yescharacteristic, complemented
17C2×#14519737—21no—
18C2×#14519705—24no—
19#16839726C2212yes—
20#16839498—24no—
21#16839488—21no—
22#16839486—21no—
23C2×C4×D4×C61C2211yescharacteristic
24C2×#99×C61—21no—
25C4×C22⋊C4×C61C2212yes—
26C61×#342—21no—
27C2×#11758222—22no—
28C2×#11758222—44no—
29C2×#11758222D414yes—
30C22×Dic122—21no—
31C22×Dic122—22no—
32C22×Dic122—22no—
33C22×Dic122—44no—
34C22×Dic122C2×C411yescharacteristic
35C22×Dic122C2×C411yescharacteristic
36C22×D244C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C22⋊D61—21no—
39C23×Dic61—21no—
40C22×C4×D61—21no—
41C2×#11758194—22no—
42#14519765—22no—
43#14519765—22no—
44#14519763—28no—
45#14519756—22no—
46#14519756D414yes—
47#14519755—28no—
48#14519751—22no—
49#14519751—22no—
50#14519749—28no—
51C4×Dic122—22no—
52C4×Dic122—22no—
53#14519737—28no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×C22⋊Dic61—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×C4×Dic61—21no—
62#14519705—48no—
63C22⋊#92418—22no—
64C22⋊#92418—22no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61D411yescharacteristic
74C2×C42×C61—22no—
75C2×C42×C61C2×C411yescharacteristic
76#99×C61—24no—
77#95×C61—22no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61—41no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—24no—
97C2×C22⋊D61—24no—
98C2×C22⋊D61—44no—
99C2×C22⋊D61—42no—
100C2×C22⋊D61—42no—
101C22×Dic61—21no—
102C22×Dic61—24no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—44no—
106#11758222—28no—
107#11758222—44no—
108#11758222—48no—
109C2×D244—21no—
110C2×D244—42no—
111C2×D244C22⋊C412yes—
112C2×C4×D61—24no—
113C2×C4×D61—44no—
114C2×Dic122—21no—
115C2×Dic122—24no—
116C2×Dic122—24no—
117C2×Dic122—21no—
118C2×Dic122—48no—
119C2×Dic122—44no—
120C2×Dic122—44no—
121C2×Dic122—44no—
122C2×Dic122—42no—
123C2×Dic122C22⋊C412yes—
124C2×Dic122C22⋊C412yes—
125C4⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—42no—
130C42×C61—21no—
131C42×C61—21no—
132C42×C61—44no—
133C22⋊Dic61—24no—
134#11758198—24no—
135#11758198—24no—
136#11758198—24no—
137C4⋊Dic61—21no—
138C4⋊Dic61C22⋊C412yes—
139#11758196—24no—
140#11758196—24no—
141C4×Dic61—24no—
142#11758194—44no—
143C2×#92418—42no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61#10412yescomplemented
153C23×C61C2×C22⋊C411yescharacteristic
154C23×C61#12012yes—
155C22×D61—22no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—41no—
159C22×D61—41no—
160D4×C61—44no—
161C2×C4×C61—24no—
162C2×C4×C61—22no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—44no—
175C2×C4×C61—82no—
176C2×C4×C61—82no—
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61#12012yes—
180C22⋊D61—48no—
181C22⋊D61—48no—
182C2×Dic61—24no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—48no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190D244—24no—
191D244—42no—
192C4×D61—48no—
193Dic122—24no—
194Dic122—22no—
195Dic122—48no—
196Dic122—42no—
197Dic122—42no—
198Dic122#9914yescomplemented
199#92418—82no—
200#8366—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—41no—
210C22×C61—41no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#37911yescharacteristic
214C22×C61C2×#10411yescharacteristic
215C22×C61#42111yescharacteristic
216C2×D61—42no—
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—82no—
220C2×D61—82no—
221C4×C61—22no—
222C4×C61—21no—
223C4×C61—41no—
224C4×C61—41no—
225C4×C61—41no—
226C4×C61—84no—
227C4×C61C2×#9912yes—
228Dic61—24no—
229Dic61—42no—
230Dic61—42no—
231C2×#548—611no—
232#1789—612no—
233C4×#99D6111yescharacteristic, complemented
234#1508—612no—
235#1465—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61—42no—
240C2×C61#178111yescharacteristic
241C2×C61#178911yescharacteristic
242C2×C61#179611yescharacteristic
243D61—82no—
244C22×#41—611no—
245C23×Q8—611no—
246#548—1224no—
247#548—1224no—
248C2×#124—611no—
249C2×#124—1221no—
250C2×C22⋊Q8—611no—
251C2×C22⋊Q8—1221no—
252C2×#121—611no—
253C2×#121—1221no—
254C2×C4×Q8—611no—
255C2×C4×D4—1221no—
256C2×C4×D4C2×D6111yescharacteristic
257C2×#104—1224no—
258C2×#99—21no—
259#407—612no—
260C4×C22⋊C4C2×D6112yes—
261#342—21no—
262#328—1224no—
263#324—1221no—
264#323—1221no—
265C61#836611yescharacteristic, complemented, Sylow
266C2×#41—614no—
267C2×#41—1222no—
268C2×#41—2444no—
269C22×Q8—611no—
270C22×Q8—611no—
271C22×Q8—1222no—
272C22×Q8—1222no—
273C22×Q8—1221no—
274C22×Q8—2444no—
275C22×D4—611no—
276C22×D4—1221no—
277C22×D4—1221no—
278C22×D4C22⋊D6111yescharacteristic
279C23×C4—1221no—
280C23×C4—1221no—
281C23×C4C4×D6111yescharacteristic
282C23×C4C22×D6111yescharacteristic
283C2×M16—1222no—
284#124—1222no—
285#124—1222no—
286#124—1228no—
287C22⋊Q8—614no—
288C22⋊Q8—1228no—
289C22⋊Q8—1222no—
290#121—1222no—
291#121—1222no—
292#121—1228no—
293C4×Q8—1222no—
294C4×Q8—1222no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—1228no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4D24411yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4D24411yescharacteristic
308C2×C22⋊C4C22⋊D6111yescharacteristic
309C2×C42—22no—
310C2×C42—1221no—
311C2×C42C4×D6111yescharacteristic
312#104—2448no—
313#99—24no—
314C22⋊C8—1222no—
315C22⋊C8—1222no—
316#95—22no—
317C24—1221no—
318C24#1175819811yescharacteristic
319C24C2×C22⋊D6111yescharacteristic
320#41—1228no—
321#41—2444no—
322#41—2448no—
323C2×Q8—612no—
324C2×Q8—612no—
325C2×Q8—1224no—
326C2×Q8—1224no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×Q8—2442no—
330C2×Q8—2444no—
331C2×Q8—2444no—
332C2×Q8—2448no—
333C2×Q8—2444no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—612no—
338C2×D4—1224no—
339C2×D4—1224no—
340C2×D4—1221no—
341C2×D4—2442no—
342C2×D4—2444no—
343C2×D4—2442no—
344C2×D4—2442no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—1221no—
351C22×C4—1224no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1224no—
355C22×C4—2444no—
356C22×C4—2444no—
357C22×C4#1175819811yescharacteristic
358C22×C4#1175819811yescharacteristic
359C22×C4#1175819811yescharacteristic
360C22×C4C2×C4×D6111yescharacteristic
361C22×C4C2×D24411yescharacteristic
362C22×C4D4×D6112yes—
363C22×C4D4×D6112yes—
364M16—2444no—
365C2×C8—2442no—
366C4⋊C4—22no—
367C4⋊C4—612no—
368C4⋊C4—1224no—
369C4⋊C4—1221no—
370C4⋊C4—1224no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—1224no—
376C22⋊C4—1224no—
377C22⋊C4—1224no—
378C22⋊C4—1224no—
379C42—21no—
380C42—21no—
381C42—44no—
382C42—1224no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—1222no—
392C23—2442no—
393C23—2442no—
394C23—2441no—
395C23—2441no—
396C23#1451970512yescomplemented
397C23C2×#1175819811yescharacteristic
398C23#1451974512yes—
399Q8—614no—
400Q8—1224no—
401Q8—1222no—
402Q8—2442no—
403Q8—2448no—
404Q8—2442no—
405D4—44no—
406D4—1224no—
407D4—2442no—
408D4—2448no—
409D4—2448no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—1224no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—2448no—
431C2×C4—2448no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4C22⋊C4×D6111yescharacteristic
436C2×C4C22⋊C4×D6111yescharacteristic
437C2×C4#1451974712yes—
438C8—4882no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—42no—
446C22—42no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22—81no—
451C22—4882no—
452C22—4882no—
453C22—2442no—
454C22—2442no—
455C22—2442no—
456C22#1683961711yescharacteristic
457C22C2×#1451970511yescharacteristic
458C22#1683978011yescharacteristic
459C4—22no—
460C4—21no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—84no—
465C4—1224no—
466C4—2442no—
467C4—2442no—
468C4#99×D6112yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—4882no—
474C2#2144573211yescharacteristic
475C2#2144576411yescharacteristic
476C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC4×#99×C61C2
socleC22×C61#379
prime radicalC2×#16840421C2
cosocleC23×C4×C61C23
hypercenterC4×#99D61
hypocenterC61#8366
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21445764C2semidirect
3. #21444338C2semidirect
4. #21444157C2semidirect
5. C4×#99×C61C2semidirect
7. C23×Dic122C4semidirect
10. C2×#14519765C4semidirect
14. C2×#14519751C4semidirect
16. C2×C4×Dic122C4semidirect
152. C23×C61#104semidirect
198. Dic122#99semidirect
233. C4×#99D61semidirect
265. C61#8366semidirect
396. C23#14519705semidirect

Maximal subgroups

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

subgroupindex
C2×#168404212
#214457642
#214443382
#214441572
C4×#99×C612
#836661