Finite Group Databasebeta

← #29712468

Subgroups — #29712468

Order 15616.

1474 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 187 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
1#21459508C211yesmaximal, characteristic
2#21454111C212yesmaximal, complemented
3#21446986C212yesmaximal, complemented
4#21446950C212yesmaximal, complemented
5#21446272C212yesmaximal, complemented
6#21446229C214yesmaximal, complemented
7#21445893C211yesmaximal, characteristic, complemented
8C2×C61×#381C211yesmaximal, characteristic, complemented
9#16840438—24no—
10#16840402—24no—
11#16840388C2211yescharacteristic
12#16840350—22no—
13#16840324C2211yescharacteristic
14#16840273—22no—
15C2×#14519760C2214yes—
16C2×#14519746C2212yes—
17#16840235C2212yes—
18C2×#14519691C2214yes—
19#16839641—24no—
20#16839635—24no—
21#16839625C2214yes—
22#16839618C2212yes—
23#16839529C2212yes—
24#16839529C2212yes—
25#16839528C2214yes—
26C24×C4×C61C2211yescharacteristic
27C22×C22⋊C4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2211yescharacteristic
29C61×#381—22no—
30C61×#381C2212yes—
31C61×#381C2212yes—
32C2×#95×C61—22no—
33C2×#11758222—22no—
34C22×C22⋊D61D414yes—
35C22×C22⋊D61C2312yes—
36#14519770—24no—
37#14519763—24no—
38#14519761—22no—
39#14519760—24no—
40#14519760—28no—
41#14519760—44no—
42#14519759—24no—
43#14519756—22no—
44#14519756—22no—
45#14519754—24no—
46#14519751—22no—
47#14519751—22no—
48#14519750—24no—
49#14519747—22no—
50#14519746—22no—
51#14519746—22no—
52#14519746—24no—
53C4×C22⋊D61—24no—
54C4×C22⋊D61—24no—
55#14519737—24no—
56#14519736—22no—
57#14519733—22no—
58C2×#11758196—24no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61D414yes—
61C2×C22⋊Dic61C2314yes—
62C2×C22⋊Dic61C2312yes—
63C2×#11758198—24no—
64C22⋊C4×D61—22no—
65#14519691—44no—
66#14519684—28no—
67C25×C61C2311yescharacteristic
68C23×C4×C61—21no—
69C23×C4×C61—21no—
70C23×C4×C61—24no—
71C23×C4×C61—24no—
72C23×C4×C61—21no—
73C23×C4×C61—42no—
74C23×C4×C61D412yes—
75C23×C4×C61C2312yes—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—42no—
79C2×C22⋊C4×C61D412yes—
80C2×C22⋊C4×C61C2312yes—
81C2×C22⋊C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61C2311yescharacteristic
83#95×C61—24no—
84#95×C61—24no—
85C24×C61—22no—
86C24×C61—22no—
87C24×C61—44no—
88C24×C61C2×D411yescharacteristic
89C24×C61C2×D414yes—
90C24×C61C2×D411yescharacteristic
91C24×C61C2411yescharacteristic
92C23×D61C2×D412yes—
93C22×C4×C61—24no—
94C22×C4×C61—24no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—24no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—416no—
103C22×C4×C61—42no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61—44no—
108C22×C4×C61—42no—
109C22×C4×C61—44no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61#4114yescomplemented
113C2×C22⋊D61—24no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—22no—
116C2×C22⋊D61—24no—
117C2×C22⋊D61—24no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—48no—
120C22×Dic61—24no—
121C22×Dic61C2×D412yes—
122C22×Dic61C2×D414yes—
123#11758222—84no—
124C2×D244—22no—
125C2×C4×D61—24no—
126C2×Dic122—22no—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—42no—
132C22⋊Dic61—24no—
133C22⋊Dic61—24no—
134C22⋊Dic61—22no—
135C22⋊Dic61—22no—
136C22⋊Dic61—24no—
137C22⋊Dic61—24no—
138C22⋊Dic61—48no—
139#11758198—24no—
140#11758198—24no—
141#11758198—48no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144#11758196—24no—
145#11758196—24no—
146#11758196—48no—
147C4×Dic61—24no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—24no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—28no—
154C23×C61—28no—
155C23×C61—44no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—48no—
159C23×C61—48no—
160C23×C61#12012yes—
161C23×C61#12012yes—
162C23×C61C22×D411yescharacteristic
163C23×C61C22×D412yes—
164C23×C61C2×#4112yes—
165C23×C61H(5,𝔽2)12yes—
166C22×D61—22no—
167C22×D61—24no—
168C22×D61—44no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—24no—
172C2×C4×C61—28no—
173C2×C4×C61—48no—
174C2×C4×C61—48no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—44no—
178C2×C4×C61—44no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—44no—
182C2×C4×C61—42no—
183C2×C4×C61—82no—
184C2×C4×C61—84no—
185C22⋊D61—42no—
186C22⋊D61—42no—
187C22⋊D61—44no—
188C22⋊D61—48no—
189C22⋊D61—84no—
190C2×Dic61—22no—
191C2×Dic61—24no—
192C2×Dic61—24no—
193C2×Dic61—44no—
194C2×Dic61—48no—
195C2×Dic61—44no—
196D244—82no—
197C4×D61—84no—
198Dic122—82no—
199#23985—611nomaximal, Sylow
200C22×C61—24no—
201C22×C61—24no—
202C22×C61—22no—
203C22×C61—24no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—24no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—416no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61#45814yes—
214C22×C61C2×#12011yescharacteristic
215C22×C61#52511yescharacteristic
216C22×C61#53511yescharacteristic
217C22×C61#54614yes—
218C2×D61—42no—
219C2×D61—44no—
220C2×D61—84no—
221C4×C61—44no—
222C4×C61—42no—
223C4×C61—42no—
224C4×C61—82no—
225Dic61—42no—
226Dic61—44no—
227Dic61—84no—
228#3490—611no—
229#2849—612no—
230C2×#381D6111yescharacteristic, complemented
231#2030—612no—
232#2022—612no—
233#1898—612no—
234#1890—614no—
235#1819—611no—
236C2×C61—24no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—24no—
240C2×C61—42no—
241C2×C61#240411yescharacteristic
242C2×C61C2×#45812yes—
243D61—82no—
244C24×C4C2×D6111yescharacteristic
245#551—1224no—
246#546—1224no—
247#543—611no—
248#538—1222no—
249#534—611no—
250#525—1222no—
251C2×#123—614no—
252C2×#120—612no—
253#518—612no—
254C22×C22⋊C4C2×D6111yescharacteristic
255C22×C22⋊C4C2×D6111yescharacteristic
256C2×#99—614no—
257#386—1224no—
258#385—1224no—
259#381—22no—
260#381—614no—
261#381C2×D6112yes—
262#381C2×D6112yes—
263#379—612no—
264C2×#95—22no—
265#342—612no—
266#342—612no—
267#342—614no—
268C61#2398511yescharacteristic, complemented, Sylow
269C25C22×D6111yescharacteristic
270C2×#41—1222no—
271C22×D4—612no—
272C22×D4—614no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—24no—
276C23×C4—24no—
277C23×C4—21no—
278C23×C4—42no—
279C23×C4C22⋊D6112yes—
280C23×C4C22×D6112yes—
281#126—1224no—
282#124—1224no—
283#123—1222no—
284#123—1224no—
285#123—1224no—
286#123—1228no—
287#123—2444no—
288C22⋊Q8—1224no—
289C22⋊Q8—1222no—
290C22⋊Q8—1222no—
291#121—1224no—
292#121—1222no—
293#121—1222no—
294#120—1222no—
295#120—1224no—
296#120—1222no—
297#120—1222no—
298C4×D4—1224no—
299C4×D4—1224no—
300C4×D4—1224no—
301#117—1222no—
302#117—1222no—
303C2×C4⋊C4—1224no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—614no—
308C2×C22⋊C4—612no—
309C2×C22⋊C4—614no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1224no—
312C2×C22⋊C4—1224no—
313C2×C22⋊C4C22⋊D6112yes—
314C2×C22⋊C4C22×D6112yes—
315C2×C22⋊C4C22×D6111yescharacteristic
316C2×C22⋊C4C22×D6111yescharacteristic
317#99—2444no—
318#95—24no—
319#95—24no—
320#95—1228no—
321C24—22no—
322C24—22no—
323C24—44no—
324C24—612no—
325C24D4×D6114yes—
326C24C2×C22⋊D6111yescharacteristic
327C24C2×C22⋊D6111yescharacteristic
328C24C23×D6111yescharacteristic
329#41—4884no—
330C2×Q8—1222no—
331C2×D4—1222no—
332C2×D4—1224no—
333C2×D4—1222no—
334C2×D4—1222no—
335C2×D4—1224no—
336C2×D4—1224no—
337C2×D4—1224no—
338C2×D4—2448no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—416no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—612no—
357C22×C4—614no—
358C22×C4—1224no—
359C22×C4—1224no—
360C22×C4#1175822214yescomplemented
361C22×C4C2×C22⋊D6112yes—
362C22×C4C2×C22⋊D6112yes—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1222no—
366C4⋊C4—1222no—
367C4⋊C4—2448no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—1224no—
374C22⋊C4—1224no—
375C22⋊C4—1222no—
376C22⋊C4—1222no—
377C22⋊C4—1224no—
378C22⋊C4—1224no—
379C22⋊C4—1224no—
380C22⋊C4—1224no—
381C22⋊C4—2448no—
382C22⋊C4—2448no—
383C42—1224no—
384C23—24no—
385C23—21no—
386C23—21no—
387C23—28no—
388C23—21no—
389C23—21no—
390C23—28no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—48no—
395C23—48no—
396C23—1224no—
397C23—1222no—
398C23—2444no—
399C23#1451974512yes—
400C23#1451974512yes—
401C23C2×D4×D6112yes—
402C23C22×C22⋊D6111yescharacteristic
403C23C2×#1175822212yes—
404C23#1451982212yes—
405Q8—4882no—
406D4—4882no—
407D4—4884no—
408D4—2448no—
409D4—2442no—
410D4—2442no—
411D4—2444no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—1222no—
429C2×C4—1224no—
430C2×C4—1224no—
431C2×C4—4884no—
432C2×C4—2448no—
433C2×C4—2444no—
434C2×C4—2444no—
435C22—24no—
436C22—24no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—22no—
441C22—24no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—416no—
446C22—42no—
447C22—42no—
448C22—4884no—
449C22—2444no—
450C22—2442no—
451C22#1683995114yes—
452C22C2×#1451974511yescharacteristic
453C22#1684027311yescharacteristic
454C22#1684032911yescharacteristic
455C22#1684040214yes—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—82no—
460C4—4884no—
461C4—2444no—
462C4—2442no—
463C2—21no—
464C2—24no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2—4882no—
469C2#2144913811yescharacteristic
470C2C2×#1683995112yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC2×C61×#381C2
socleC22×C61C2×#120
prime radical#21459508C2
hypercenterC2×#381D61
hypocenterC61#23985
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #21454111C2semidirect
3. #21446986C2semidirect
4. #21446950C2semidirect
5. #21446272C2semidirect
6. #21446229C2semidirect
7. #21445893C2semidirect
8. C2×C61×#381C2semidirect
112. C22×C4×C61#41semidirect
230. C2×#381D61semidirect
268. C61#23985semidirect
360. C22×C4#11758222semidirect

Maximal subgroups

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

subgroupindex
#214595082
#214541112
#214469862
#214469502
#214462722
#214462292
#214458932
C2×C61×#3812
#2398561