Finite Group Databasebeta

← #29737048

Subgroups — #29737048

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450305C212yesmaximal, complemented
2#21450127C211yesmaximal, characteristic
3#21446985C214yesmaximal, complemented
4#21446805C212yesmaximal, complemented
5#21446232C214yesmaximal
6C4×#14519692C211yesmaximal, characteristic
7C61×#3501C211yesmaximal, characteristic
8C2×#14519758C2212yes—
9C22×C4⋊Dic61C2211yescharacteristic
10C2×#14519692C2212yes—
11#16839701—24no—
12#16839692—21no—
13#16839687—21no—
14#16839669—22no—
15#16839660—21no—
16#16839627—22no—
17#16839623C2218yes—
18#16839623C2212yes—
19#16839622—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2212yes—
22#16839529C2212yes—
23#16839529C2218yes—
24C61×#554—24no—
25C61×#554—22no—
26D4×Q8×C61—22no—
27C61×#544—21no—
28C61×#543—21no—
29C61×#543C2212yes—
30C61×#533—21no—
31C2×C22⋊Q8×C61—21no—
32C2×C22⋊Q8×C61C2212yes—
33C61×#518C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C23×Dic61C2312yes—
36#14519758—28no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—21no—
42C2×#11758196—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61C2314yes—
45C2×C22⋊Dic61C2312yes—
46C2×C4×Dic61—22no—
47#14519692—24no—
48#14519684—21no—
49#14519684—22no—
50#14519684—22no—
51#14519684—22no—
52#14519684—28no—
53#14519684—28no—
54C22×Q8×C61—21no—
55C22×Q8×C61D412yes—
56C22×D4×C61C2311yescharacteristic
57C23×C4×C61D411yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—24no—
62#125×C61—22no—
63#125×C61—24no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—24no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—22no—
76C4×Q8×C61—22no—
77C4×Q8×C61—22no—
78C4×D4×C61—22no—
79C4×D4×C61—24no—
80C4×D4×C61—22no—
81C4×D4×C61—24no—
82#117×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C4⋊C4×C61D414yes—
87C2×C4⋊C4×C61D412yes—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61D411yescharacteristic
90C2×C22⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61C2314yes—
92C2×C42×C61D411yescharacteristic
93C24×C61C2×D411yescharacteristic
94C24×C61C2411yescharacteristic
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—44no—
100C2×Q8×C61—82no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61#4114yes—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D414yes—
115C22×C4×C61C2×D414yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—24no—
122C22×Dic61—22no—
123C22×Dic61—24no—
124C22×Dic61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—24no—
136C22⋊C4×C61—24no—
137C22⋊C4×C61—24no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—22no—
141C42×C61—24no—
142C42×C61—21no—
143C42×C61—21no—
144C22⋊Dic61—24no—
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147C22⋊Dic61—44no—
148C4⋊Dic61—42no—
149C4⋊Dic61—41no—
150C4⋊Dic61—42no—
151C4⋊Dic61—41no—
152#11758196—44no—
153C4×Dic61—44no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61#12412yes—
161C23×C61C22×D411yescharacteristic
162C23×C61C22×D412yes—
163C23×C61C2×#4112yes—
164C23×C612−1+412yes—
165Q8×C61—44no—
166Q8×C61—82no—
167Q8×C61—82no—
168Q8×C61—82no—
169D4×C61—28no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—24no—
173C2×C4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—24no—
176C2×C4×C61—24no—
177C2×C4×C61—21no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—44no—
183C2×C4×C61—41no—
184C2×C4×C61—44no—
185C2×C4×C61—81no—
186C2×C4×C61—82no—
187C2×C4×C61—82no—
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12012yes—
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12412yes—
192C2×Dic61—44no—
193C2×Dic61—42no—
194C2×Dic61—44no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—44no—
201C2×Dic61—48no—
202#25948—611nomaximal, Sylow
203C22×C61—22no—
204C22×C61—21no—
205C22×C61—22no—
206C22×C61—21no—
207C22×C61—22no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—81no—
214C22×C61C2×#12011yescharacteristic
215C22×C61C2×#12411yescharacteristic
216C22×C61#53611yescharacteristic
217C22×C61#54814yes—
218C4×C61—22no—
219C4×C61—21no—
220C4×C61—22no—
221C4×C61—41no—
222C4×C61—44no—
223C4×C61—44no—
224C4×C61—41no—
225C4×C61#45812yes—
226Dic61—82no—
227Dic61—84no—
228Dic61—82no—
229#3501D6111yescharacteristic
230#2503—612no—
231#2486—611no—
232#2030—614no—
233#1996—612no—
234#1890—614no—
235C4×#99—611no—
236C2×C61—21no—
237C2×C61—24no—
238C2×C61—21no—
239C2×C61—42no—
240C2×C61#243611yescharacteristic
241C2×C61C2×#45811yescharacteristic
242C2×C61#302811yescharacteristic
243#554—22no—
244#554—24no—
245D4×Q8—22no—
246#544—21no—
247#543—21no—
248#543C2×D6112yes—
249#533—21no—
250C2×#123—612no—
251C2×C22⋊Q8—21no—
252C2×C22⋊Q8C2×D6112yes—
253#518C2×D6112yes—
254C2×C4×D4C2×D6111yescharacteristic
255C22×C4⋊C4—611no—
256C2×#99—612no—
257#400—1224no—
258#398—1221no—
259#397—1221no—
260#391—1222no—
261#389—1221no—
262#382—1222no—
263#381—618no—
264#381—612no—
265#380—611no—
266#380—1221no—
267C4×C22⋊C4—612no—
268#342—618no—
269#342—612no—
270C61#2594811yescharacteristic, complemented, Sylow
271C22×Q8—21no—
272C22×Q8C22⋊D6112yes—
273C22×D4C22×D6111yescharacteristic
274C23×C4—612no—
275C23×C4C22⋊D6111yescharacteristic
276C23×C4C22×D6111yescharacteristic
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—24no—
280#125—22no—
281#125—24no—
282#123—1228no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—24no—
301#117—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1222no—
306C2×C4⋊C4—1221no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4C22⋊D6111yescharacteristic
311C2×C4⋊C4C22⋊D6112yes—
312C2×C4⋊C4C22⋊D6114yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—612no—
315C2×C22⋊C4—614no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4C22⋊D6111yescharacteristic
318C2×C22⋊C4C22×D6111yescharacteristic
319C2×C22⋊C4C22×D6114yes—
320C2×C42—1222no—
321C2×C42C22⋊D6111yescharacteristic
322#99—1224no—
323#95—1222no—
324#95—1221no—
325#95—1222no—
326#95—1228no—
327#95—1222no—
328#95—1228no—
329C24C2×C22⋊D6111yescharacteristic
330C24C23×D6111yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4#1175822614yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—1224no—
347C22×C4—1224no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1224no—
351C22×C4—1224no—
352C22×C4C2×C22⋊D6112yes—
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C2×C22⋊D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6114yes—
357C22×C4C2×C22⋊D6114yes—
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6112yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—2442no—
373C4⋊C4—2441no—
374C4⋊C4—2442no—
375C4⋊C4—2441no—
376C4⋊C4—2444no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—1224no—
383C22⋊C4—1224no—
384C22⋊C4—1224no—
385C22⋊C4—2444no—
386C42—24no—
387C42—21no—
388C42—21no—
389C42—2444no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#1451976612yes—
397C23C22×C22⋊D6111yescharacteristic
398C23C22×C22⋊D6112yes—
399C23C2×#1175822612yes—
400C23#1451982512yes—
401Q8—44no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405D4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—2444no—
425C2×C4—2442no—
426C2×C4—2444no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2444no—
433C2×C4—2448no—
434C2×C4#1451974611yescharacteristic
435C2×C4#1451974612yes—
436C2×C4#1451974611yescharacteristic
437C2×C4#1451976612yes—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22C2×#1451974611yescharacteristic
450C22C2×#1451976611yescharacteristic
451C22#1684033611yescharacteristic
452C22#1684042314yes—
453C4—22no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—44no—
458C4—44no—
459C4—41no—
460C4—4882no—
461C4—4884no—
462C4—4882no—
463C4#1683995212yes—
464C2—21no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2#2144947311yescharacteristic
469C2C2×#1683995211yescharacteristic
470C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#25948
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21450305C2semidirect
3. #21446985C2semidirect
4. #21446805C2semidirect
270. C61#25948semidirect

Maximal subgroups

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

subgroupindex
#214503052
#214501272
#214469852
#214468052
#214462322
C4×#145196922
C61×#35012
#2594861