Finite Group Databasebeta

← #29736156

Subgroups — #29736156

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449464C212yesmaximal, complemented
2#21449184C211yesmaximal, characteristic
3#21447171C214yesmaximal, complemented
4#21446805C212yesmaximal, complemented
5#21446100C214yesmaximal
6C4×#14519692C211yesmaximal, characteristic
7C61×#3501C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C2×#14519775—22no—
10C2×#14519756—21no—
11C2×#14519692C2212yes—
12#16839682—21no—
13#16839682—24no—
14#16839674—21no—
15#16839668—21no—
16#16839667—22no—
17#16839642C2212yes—
18#16839634C2212yes—
19#16839634C2218yes—
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
35C22×Dic122—21no—
36C22×Dic122—22no—
37C22×Dic122—22no—
38C22×Dic122—21no—
39C22×Dic122—44no—
40C23×Dic61C2312yes—
41#14519775—48no—
42#14519756—44no—
43C2×C4⋊Dic61—21no—
44C2×#11758196—22no—
45C2×#11758196—28no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61C2312yes—
48C2×C22⋊Dic61C2314yes—
49C2×C4×Dic61—22no—
50#14519692Q818yescomplemented
51#14519684—22no—
52#14519684—21no—
53#14519684—28no—
54#14519684—22no—
55#14519684—22no—
56C22×Q8×C61—21no—
57C22×Q8×C61D412yes—
58C22×D4×C61C2311yescharacteristic
59C23×C4×C61D411yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—24no—
64#125×C61—22no—
65#125×C61—24no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—24no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—24no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—22no—
78C4×Q8×C61—22no—
79C4×Q8×C61—22no—
80C4×D4×C61—22no—
81C4×D4×C61—24no—
82C4×D4×C61—22no—
83C4×D4×C61—24no—
84#117×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C4⋊C4×C61D414yes—
89C2×C4⋊C4×C61D412yes—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C22⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61C2314yes—
94C2×C42×C61D411yescharacteristic
95C24×C61C2×D411yescharacteristic
96C24×C61C2411yescharacteristic
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—44no—
102C2×Q8×C61—82no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61C2×Q814yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D414yes—
117C22×C4×C61C2×D414yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×Dic61—22no—
122C22×Dic61—24no—
123C22×Dic61—24no—
124C22×Dic61—22no—
125C22×Dic61—44no—
126C2×Dic122—24no—
127C2×Dic122—28no—
128C2×Dic122—48no—
129C2×Dic122—44no—
130C2×Dic122—44no—
131C2×Dic122—42no—
132C2×Dic122—41no—
133C2×Dic122—42no—
134C2×Dic122—41no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—24no—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—22no—
151C42×C61—24no—
152C42×C61—21no—
153C42×C61—21no—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156C22⋊Dic61—44no—
157C22⋊Dic61C2×Q818yescomplemented
158C4⋊Dic61—42no—
159#11758196—44no—
160#11758196—416no—
161C4×Dic61—44no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61C4⋊Q812yescomplemented
169C23×C61C22×D411yescharacteristic
170C23×C61C22×D412yes—
171C23×C61C22×Q812yes—
172C23×C612−1+412yes—
173Q8×C61—44no—
174Q8×C61—82no—
175Q8×C61—82no—
176Q8×C61—82no—
177D4×C61—28no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—21no—
183C2×C4×C61—24no—
184C2×C4×C61—24no—
185C2×C4×C61—21no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—44no—
191C2×C4×C61—44no—
192C2×C4×C61—41no—
193C2×C4×C61—81no—
194C2×C4×C61—82no—
195C2×C4×C61—82no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12012yes—
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61C4⋊Q812yescomplemented
200C2×Dic61—28no—
201C2×Dic61—24no—
202C2×Dic61—28no—
203C2×Dic61—44no—
204C2×Dic61—44no—
205C2×Dic61—42no—
206C2×Dic61—44no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211Dic122—44no—
212Dic122—48no—
213Dic122—44no—
214#25852—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—21no—
217C22×C61—22no—
218C22×C61—21no—
219C22×C61—22no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—81no—
226C22×C61C2×#12011yescharacteristic
227C22×C61C2×C4⋊Q811yescharacteristic
228C22×C61#53611yescharacteristic
229C22×C61D4×Q814yes—
230C4×C61—22no—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—41no—
234C4×C61—44no—
235C4×C61—44no—
236C4×C61—41no—
237C4×C61B(4,𝔽2)12yes—
238Dic61—48no—
239Dic61—44no—
240Dic61—82no—
241#3501D6111yescharacteristic
242#2434—612no—
243#2409—611no—
244#2062—614no—
245#1996—612no—
246#1867—614no—
247C4×#99—611no—
248C2×C61—24no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—42no—
252C2×C61#243411yescharacteristic
253C2×C61C2×B(4,𝔽2)11yescharacteristic
254C2×C61#302811yescharacteristic
255C23×Q8—611no—
256#554—22no—
257#554—24no—
258D4×Q8—22no—
259#544—21no—
260#543—21no—
261#543C2×D6112yes—
262#533—21no—
263C2×C4⋊Q8—1222no—
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8—1221no—
266C2×C22⋊Q8C2×D6112yes—
267#518C2×D6112yes—
268C2×C4×D4C2×D6111yescharacteristic
269C2×#99—612no—
270#395—1221no—
271#395—1224no—
272#393—1221no—
273#391—1222no—
274#391—1221no—
275#386—612no—
276#385—612no—
277#385—618no—
278#380—611no—
279C4×C22⋊C4—612no—
280#342—612no—
281#342—618no—
282C61#2585211yescharacteristic, complemented, Sylow
283C22×Q8—21no—
284C22×Q8—1221no—
285C22×Q8—1222no—
286C22×Q8—1221no—
287C22×Q8—1222no—
288C22×Q8—2444no—
289C22×Q8C22⋊D6112yes—
290C22×D4C22×D6111yescharacteristic
291C23×C4—612no—
292C23×C4C22⋊D6111yescharacteristic
293C23×C4C22×D6111yescharacteristic
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—24no—
297C4⋊Q8—2448no—
298#125—22no—
299#125—24no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—24no—
311C22⋊Q8—24no—
312C22⋊Q8—2444no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—24no—
318C4×D4—24no—
319#117—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1228no—
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C4⋊C4C22⋊D6112yes—
327C2×C4⋊C4C22⋊D6114yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—614no—
330C2×C22⋊C4—612no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4C22⋊D6111yescharacteristic
333C2×C22⋊C4C22×D6111yescharacteristic
334C2×C22⋊C4C22×D6114yes—
335C2×C42—1222no—
336C2×C42C22⋊D6111yescharacteristic
337#99—618no—
338#95—1222no—
339#95—1221no—
340#95—1222no—
341#95—1228no—
342#95—1222no—
343C24C2×C22⋊D6111yescharacteristic
344C24C23×D6111yescharacteristic
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—82no—
351C2×Q8—1224no—
352C2×Q8—1228no—
353C2×Q8—2444no—
354C2×Q8—2442no—
355C2×Q8—2444no—
356C2×Q8—2448no—
357C2×Q8—2441no—
358C2×Q8—2442no—
359C2×Q8—2441no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4Q8×D6114yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—1224no—
370C22×C4—1222no—
371C22×C4—1224no—
372C22×C4—1222no—
373C22×C4—2444no—
374C22×C4C2×C22⋊D6112yes—
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378C22×C4C2×C22⋊D6114yes—
379C22×C4C2×C22⋊D6114yes—
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6112yes—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—2442no—
395C4⋊C4—2444no—
396C4⋊C4—24416no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—22no—
402C22⋊C4—618no—
403C22⋊C4—1224no—
404C22⋊C4—1224no—
405C22⋊C4—2444no—
406C42—24no—
407C42—21no—
408C42—21no—
409C42—2444no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#1451977512yescomplemented
417C23C22×C22⋊D6111yescharacteristic
418C23C22×C22⋊D6112yes—
419C23C2×Q8×D6112yes—
420C23#1451982512yes—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—2444no—
426Q8—2448no—
427Q8—2444no—
428D4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—1224no—
448C2×C4—1228no—
449C2×C4—1228no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2444no—
454C2×C4—2444no—
455C2×C4—2444no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4#1451974611yescharacteristic
459C2×C4#1451974612yes—
460C2×C4#1451974611yescharacteristic
461C2×C4#1451977512yescomplemented
462C22—22no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22C2×#1451974611yescharacteristic
474C22C2×#1451977511yescharacteristic
475C22#1684033611yescharacteristic
476C22Q8×C22⋊D6114yes—
477C4—21no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—44no—
482C4—44no—
483C4—41no—
484C4—4882no—
485C4—2448no—
486C4—2444no—
487C4#1683994912yes—
488C2—21no—
489C2—24no—
490C2—21no—
491C2—42no—
492C2#2144945911yescharacteristic
493C2C2×#1683994911yescharacteristic
494C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#25852
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. #21449464C2semidirect
3. #21447171C2semidirect
4. #21446805C2semidirect
50. #14519692Q8semidirect
157. C22⋊Dic61C2×Q8semidirect
168. C23×C61C4⋊Q8semidirect
199. C2×C4×C61C4⋊Q8semidirect
282. C61#25852semidirect
416. C23#14519775semidirect
461. C2×C4#14519775semidirect

Maximal subgroups

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

subgroupindex
#214494642
#214491842
#214471712
#214468052
#214461002
C4×#145196922
C61×#35012
#2585261