Finite Group Databasebeta

← #29595501

Subgroups — #29595501

Order 15616.

1206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#14519764C211yesmaximal, characteristic
2C2×#16839556C214yesmaximal, complemented
3#21445734C211yesmaximal, characteristic
4C61×#1781C211yesmaximal, characteristic
5C23×Dic122C411yescharacteristic, complemented
6C2×#14519764—28no—
7C2×#14519764—21no—
8C2×#14519764—21no—
9C2×#14519764C412yescomplemented
10C2×#14519764C2212yes—
11C22×C22⋊Dic61—21no—
12C22×C22⋊Dic61—21no—
13C22×C4×Dic61C411yescharacteristic, complemented
14C2×#14519692—21no—
15C2×#14519692C2212yes—
16#16839618—21no—
17#16839556—28no—
18C23×D4×C61C2211yescharacteristic
19C2×#99×C61—21no—
20C2×#99×C61C2212yes—
21C61×#379—21no—
22C22×Dic122—21no—
23C22×Dic122—24no—
24C22×Dic122—42no—
25C22×Dic122—42no—
26C22×Dic122C2×C412yes—
27C23×Dic61—21no—
28C23×Dic61—21no—
29#14519764—28no—
30#14519764—28no—
31#14519764—24no—
32#14519764—24no—
33#14519764—24no—
34#14519764D418yes—
35C2×C22⋊Dic61—28no—
36C2×C22⋊Dic61—28no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—42no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—24no—
45C2×C4×Dic61C2×C412yes—
46#14519692—24no—
47#14519692—42no—
48C25×C61D411yescharacteristic
49C25×C61D411yescharacteristic
50C22×D4×C61—24no—
51C22×D4×C61—42no—
52C22×D4×C61—42no—
53C22×D4×C61C2×C411yescharacteristic
54C22×D4×C61D411yescharacteristic
55C22×D4×C61D411yescharacteristic
56C22×D4×C61C2311yescharacteristic
57C23×C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—42no—
60#99×C61—24no—
61#99×C61—42no—
62C24×C61—22no—
63C24×C61—22no—
64C24×C61—22no—
65C24×C61—22no—
66C24×C61—41no—
67C24×C61—41no—
68C24×C61—44no—
69C24×C61—44no—
70C24×C61C22⋊C411yescharacteristic
71C24×C61C22⋊C411yescharacteristic
72C24×C61C22⋊C411yescharacteristic
73C24×C61C2×D411yescharacteristic
74C24×C61C2×D411yescharacteristic
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—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—416no—
87C2×D4×C61C2×D414yes—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—42no—
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22×C411yescharacteristic
94C22×Dic61—24no—
95C22×Dic61—24no—
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—44no—
99C22×Dic61—42no—
100C22×Dic61—42no—
101C22×Dic61—42no—
102C2×Dic122—24no—
103C2×Dic122—22no—
104C2×Dic122—42no—
105C2×Dic122—416no—
106C2×Dic122—42no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122C22⋊C414yescomplemented
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—84no—
113C22⋊Dic61—28no—
114C22⋊Dic61—28no—
115C22⋊Dic61—28no—
116C22⋊Dic61—28no—
117C22⋊Dic61—42no—
118C22⋊Dic61—42no—
119C22⋊Dic61—84no—
120C4×Dic61—24no—
121C4×Dic61—22no—
122C4×Dic61C22⋊C414yescomplemented
123C23×C61—24no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—44no—
137C23×C61—44no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—48no—
143C23×C61—48no—
144C23×C61—44no—
145C23×C61—44no—
146C23×C61—82no—
147C23×C61—82no—
148C23×C61—84no—
149C23×C61—84no—
150C23×C61#9912yes—
151C23×C61C2×C22⋊C411yescharacteristic
152C23×C61#12012yes—
153C23×C61#12012yes—
154D4×C61—44no—
155D4×C61—44no—
156D4×C61—44no—
157D4×C61—44no—
158C2×C4×C61—28no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—82no—
164C2×C4×C61—84no—
165C2×C4×C61#9912yes—
166C2×C4×C61C2×C22⋊C412yes—
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—48no—
170C2×Dic61—48no—
171C2×Dic61—44no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—82no—
175C2×Dic61—84no—
176Dic122—48no—
177Dic122—44no—
178Dic122—44no—
179#12694—611nomaximal, Sylow
180C22×C61—24no—
181C22×C61—24no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—88no—
200C22×C61—88no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61#35414yescomplemented
204C22×C61#37911yescharacteristic
205C22×C61C2×#9911yescharacteristic
206C22×C61C2×#9911yescharacteristic
207C4×C61—22no—
208C4×C61—22no—
209C4×C61—162no—
210Dic61—44no—
211Dic61—44no—
212Dic61—162no—
213C22×#124—611no—
214C2×#354—614no—
215#1781—611no—
216#1781D6111yescharacteristic
217C2×C61—21no—
218C2×C61—24no—
219C2×C61—21no—
220C2×C61—42no—
221C2×C61—42no—
222C2×C61—81no—
223C2×C61—81no—
224C2×C61#178111yescharacteristic
225C2×C61C2×#35412yes—
226C23×Q8—611no—
227C23×D4C2×D6111yescharacteristic
228C2×#124—612no—
229C2×#124—612no—
230C2×#124—1228no—
231C2×#124—1221no—
232C2×#124—1221no—
233C22×C22⋊C4—1221no—
234C22×C22⋊C4—1221no—
235C22×C42—611no—
236C2×#99—21no—
237C2×#99—612no—
238C2×#99—1221no—
239C2×#99C2×D6112yes—
240#379—21no—
241#379—1221no—
242#354—1228no—
243C61#1269411yescharacteristic, complemented, Sylow
244C25D24411yescharacteristic
245C25C22⋊D6111yescharacteristic
246C22×Q8—612no—
247C22×Q8—1224no—
248C22×Q8—1221no—
249C22×Q8—2442no—
250C22×Q8—2442no—
251C22×D4—24no—
252C22×D4—42no—
253C22×D4—42no—
254C22×D4C4×D6111yescharacteristic
255C22×D4D24411yescharacteristic
256C22×D4C22⋊D6111yescharacteristic
257C22×D4C22×D6111yescharacteristic
258C23×C4—1221no—
259C23×C4—1221no—
260C23×C4C4×D6111yescharacteristic
261#124—618no—
262#124—1224no—
263#124—1224no—
264#124—1228no—
265#124—1228no—
266#124—1224no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—1228no—
270C2×C22⋊C4—1228no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—1222no—
275C2×C22⋊C4—1222no—
276C2×C22⋊C4—2442no—
277C2×C42—612no—
278C2×C42—1224no—
279C2×C42—1221no—
280#99—24no—
281#99—42no—
282#99—1224no—
283#99—2442no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—44no—
291C24—44no—
292C24#1175819811yescharacteristic
293C24#1175819811yescharacteristic
294C24#1175819811yescharacteristic
295C24C2×D24411yescharacteristic
296C24C2×C22⋊D6111yescharacteristic
297C2×Q8—614no—
298C2×Q8—1224no—
299C2×Q8—1222no—
300C2×Q8—24416no—
301C2×Q8—2442no—
302C2×Q8—2442no—
303C2×Q8—2442no—
304C2×Q8—2442no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—416no—
317C2×D4D4×D6114yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—1224no—
323C22×C4—1224no—
324C22×C4—1222no—
325C22×C4—1222no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4—2444no—
329C22×C4—2442no—
330C22×C4#1175819811yescharacteristic
331C22×C4C2×C4×D6111yescharacteristic
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—84no—
335C22⋊C4—1228no—
336C22⋊C4—1228no—
337C22⋊C4—1228no—
338C22⋊C4—1228no—
339C22⋊C4—4884no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C42—614no—
343C42—1224no—
344C42—1222no—
345C23—21no—
346C23—24no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—48no—
365C23—48no—
366C23—44no—
367C23—44no—
368C23—82no—
369C23—82no—
370C23—84no—
371C23—84no—
372C23#1451969112yes—
373C23C2×#1175819811yescharacteristic
374C23#1451974512yes—
375C23#1451974712yes—
376Q8—2444no—
377Q8—2448no—
378Q8—2444no—
379D4—44no—
380D4—44no—
381D4—44no—
382D4—44no—
383C2×C4—28no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—1224no—
391C2×C4—1224no—
392C2×C4—4882no—
393C2×C4—4884no—
394C2×C4—2448no—
395C2×C4—2444no—
396C2×C4—2448no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4#1451969112yes—
400C2×C4C22⋊C4×D6112yes—
401C22—24no—
402C22—24no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—81no—
419C22—81no—
420C22—88no—
421C22—88no—
422C22—81no—
423C22—81no—
424C22#1683955614yescomplemented
425C22#1683961711yescharacteristic
426C22C2×#1451969111yescharacteristic
427C22C2×#1451969111yescharacteristic
428C4—22no—
429C4—22no—
430C4—162no—
431C4—9762no—
432C4—2444no—
433C4—2444no—
434C2—21no—
435C2—21no—
436C2—24no—
437C2—42no—
438C2—42no—
439C2—81no—
440C2—81no—
441C2#2144573311yescharacteristic
442C2C2×#1683955612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1781C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenter#1781D61
hypocenterC61#12694
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. C2×#16839556C2semidirect
5. C23×Dic122C4semidirect
9. C2×#14519764C4semidirect
13. C22×C4×Dic61C4semidirect
109. C2×Dic122C22⋊C4semidirect
122. C4×Dic61C22⋊C4semidirect
203. C22×C61#354semidirect
243. C61#12694semidirect
424. C22#16839556semidirect

Maximal subgroups

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

subgroupindex
C22×#145197642
C2×#168395562
#214457342
C61×#17812
#1269461