Finite Group Databasebeta

← #29719019

Subgroups — #29719019

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839918C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic
3#21450304C211yesmaximal, characteristic
4#21447237C212yesmaximal, complemented
5#21447141C212yesmaximal, complemented
6#21446240C211yesmaximal, characteristic
7#21446160C212yesmaximal, complemented
8#21445999C211yesmaximal, characteristic
9#21445903C211yesmaximal, characteristic
10#21445834C212yesmaximal, complemented
11Q8×C22⋊C4×C61C211yesmaximal, characteristic
12C22×#11758202C2211yescharacteristic
13C22×#11758202C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17#16839918—24no—
18C2×C4⋊#92418C2212yes—
19C2×#14519701—24no—
20C2×#14519701C2212yes—
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2212yes—
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2212yes—
25C2×C22⋊#92418C2211yescharacteristic
26#16839702—22no—
27#16839668C2212yes—
28#16839628C2212yes—
29#16839623—21no—
30#16839623—21no—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C23×Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2212yes—
37C61×#391—22no—
38C61×#391C2212yes—
39C61×#380—21no—
40C61×#380C2211yescharacteristic
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C22×Dic122C2311yescharacteristic
44C23×Dic61—21no—
45C2×#11758202—22no—
46C2×#11758202—22no—
47C2×#11758202—24no—
48C2×#11758202—42no—
49C22×#92418—22no—
50C22×#92418—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54#14519756—22no—
55#14519756—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C4⋊#92418—42no—
64#14519701—24no—
65#14519701—24no—
66#14519701—22no—
67#14519701—22no—
68#14519700—42no—
69C22⋊#92418—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684—22no—
73#14519684—22no—
74C22×Q8×C61—21no—
75C22×Q8×C61—22no—
76C22×Q8×C61—24no—
77C22×Q8×C61—42no—
78C22×Q8×C61D414yes—
79C22×Q8×C61C2311yescharacteristic
80C22×Q8×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C4×Q8×C61—24no—
84C4×Q8×C61—42no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61C2312yes—
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61D411yescharacteristic
90C2×C42×C61—22no—
91C2×C42×C61C2312yes—
92#95×C61—22no—
93#95×C61—22no—
94#95×C61—21no—
95#95×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C2×Q8×C61—24no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—22no—
101C2×Q8×C61—28no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—44no—
105C2×Q8×C61—44no—
106C2×Q8×C61—48no—
107C2×Q8×C61C2×D412yes—
108C2×Q8×C61C2×D412yes—
109C2×Q8×C61#4114yes—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—24no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—44no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—44no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—42no—
139C42×C61—22no—
140C42×C61—22no—
141C42×C61—44no—
142C22⋊Dic61—42no—
143#11758202—44no—
144#11758202—44no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—42no—
150#11758196—42no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—44no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61#12112yes—
162C23×C61#13512yes—
163C23×C61#13712yes—
164C23×C61C22×D411yescharacteristic
165Q8×C61—22no—
166Q8×C61—24no—
167Q8×C61—22no—
168Q8×C61—22no—
169Q8×C61—48no—
170Q8×C61—42no—
171Q8×C61#12314yes—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—48no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61#12112yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4112yes—
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61H(5,𝔽2)12yes—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—84no—
203Dic122—41no—
204Dic122—41no—
205Dic122—42no—
206#92418—41no—
207#92418—42no—
208#92418—41no—
209#24529—611nomaximal, Sylow
210C22×C61—22no—
211C22×C61—22no—
212C22×C61—24no—
213C22×C61—24no—
214C22×C61#46414yes—
215C22×C61C2×#12111yescharacteristic
216C22×C61#54612yes—
217C22×C61C2×#13511yescharacteristic
218C22×C61C2×#13711yescharacteristic
219C22×C61#57812yes—
220C4×C61—22no—
221C4×C61—24no—
222C4×C61—42no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—82no—
226C4×C61C2×#12311yescharacteristic
227C4×C61#53411yescharacteristic
228C4×C61#55112yes—
229Dic61—81no—
230Dic61—82no—
231Dic61—81no—
232C2×#451—611no—
233C22×#103—611no—
234#2503—611no—
235Q8×C22⋊C4D6111yescharacteristic
236#2074—612no—
237#2056—612no—
238#1892—611no—
239#1878—612no—
240#1850—611no—
241#1823—611no—
242#1805—612no—
243C2×C61—24no—
244C2×C61#241211yescharacteristic
245C2×C61C2×#46411yescharacteristic
246C2×C61#304411yescharacteristic
247C2×C61#330012yes—
248C2×C61#331212yes—
249C23×Q8C2×D6111yescharacteristic
250C22×Q16—611no—
251C22×Q16—611no—
252C23×C8—611no—
253C2×C22⋊Q8—611no—
254C2×C4×Q8C2×D6112yes—
255C22×C4⋊C4—611no—
256#451—1224no—
257C2×C4⋊C8—612no—
258C2×#103—612no—
259C2×#103—611no—
260C2×#103—612no—
261C2×#103—611no—
262C2×#103—612no—
263C2×#103—1224no—
264C2×C22⋊C8—611no—
265#400—1222no—
266#391—22no—
267#391—612no—
268#391C2×D6112yes—
269#382—612no—
270#381—1221no—
271#381—1221no—
272#380—21no—
273#380C2×D6111yescharacteristic
274C4×C22⋊C4—21no—
275C4×C22⋊C4C2×D6111yescharacteristic
276#340—612no—
277#340—612no—
278#340—612no—
279#336—612no—
280C61#2452911yescharacteristic, complemented, Sylow
281C22×Q8—22no—
282C22×Q8—21no—
283C22×Q8—24no—
284C22×Q8—42no—
285C22×Q8—611no—
286C22×Q8D24414yes—
287C22×Q8C22×D6111yescharacteristic
288C22×Q8C22×D6111yescharacteristic
289C23×C4—21no—
290C23×C4—1221no—
291C23×C4C22×D6111yescharacteristic
292C2×Q16—1222no—
293C2×Q16—1222no—
294C2×Q16—1224no—
295C2×Q16—2442no—
296C22×C8—612no—
297C22×C8—611no—
298C22×C8—611no—
299C22×C8—1222no—
300C22×C8—2442no—
301C22⋊Q8—1222no—
302C22⋊Q8—1222no—
303C4×Q8—24no—
304C4×Q8—42no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4C22×D6112yes—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4C22⋊D6111yescharacteristic
316C2×C22⋊C4C22⋊D6111yescharacteristic
317C2×C42—22no—
318C2×C42C22×D6112yes—
319C4⋊C8—2442no—
320#103—1224no—
321#103—1224no—
322#103—1222no—
323#103—1222no—
324#103—2442no—
325C22⋊C8—1222no—
326#95—22no—
327#95—22no—
328#95—21no—
329#95—1222no—
330#95—1222no—
331#95—1222no—
332#95—1222no—
333#95C22⋊D6112yes—
334C24C2×C22⋊D6111yescharacteristic
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—28no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—1221no—
346C2×Q8—1222no—
347C2×Q8—1221no—
348C2×Q8—1222no—
349C2×Q8C2×D24412yes—
350C2×Q8C2×D24412yes—
351C2×Q8#1175822414yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1222no—
362C22×C4—2442no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4C2×D24412yes—
366C22×C4#1175822212yes—
367C22×C4C2×C22⋊D6112yes—
368C22×C4C2×C22⋊D6112yes—
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C23×D6111yescharacteristic
371Q16—2444no—
372Q16—2444no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—1222no—
379C2×C8—1222no—
380C2×C8—2444no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—2442no—
389C4⋊C4—2442no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C22⋊C4—2442no—
394C42—22no—
395C42—22no—
396C42—44no—
397C23—22no—
398C23—22no—
399C23#1451975112yes—
400C23#1451979712yes—
401C23#1451980812yes—
402C23C22×C22⋊D6111yescharacteristic
403Q8—22no—
404Q8—24no—
405Q8—22no—
406Q8—22no—
407Q8—48no—
408Q8—42no—
409Q8—2441no—
410Q8—2441no—
411Q8—2442no—
412Q8#1451976114yes—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—4884no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4#1451975112yes—
440C2×C4C22×D24411yescharacteristic
441C2×C4C2×#1175822412yes—
442C2×C4C2×#1175822211yescharacteristic
443C2×C4#1451982212yes—
444C8—2441no—
445C8—2442no—
446C8—2441no—
447C22—24no—
448C22—24no—
449C22—22no—
450C22—22no—
451C22#1683998414yes—
452C22C2×#1451975111yescharacteristic
453C22#1684040612yes—
454C22C2×#1451979711yescharacteristic
455C22C2×#1451980811yescharacteristic
456C22#1684057712yes—
457C4—22no—
458C4—24no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—82no—
463C4—4882no—
464C4—4881no—
465C4—4881no—
466C4C2×#1451976111yescharacteristic
467C4#1684032411yescharacteristic
468C4#1684043812yes—
469C2—24no—
470C2#2144920911yescharacteristic
471C2C2×#1683998411yescharacteristic
472C2#2145534911yescharacteristic
473C2#2145770612yes—
474C2#2145787012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterQ8×C22⋊C4D61
hypocenterC61#24529
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #21447237C2semidirect
5. #21447141C2semidirect
7. #21446160C2semidirect
10. #21445834C2semidirect
280. C61#24529semidirect

Maximal subgroups

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

subgroupindex
C2×#168399182
C22×#145197012
#214503042
#214472372
#214471412
#214462402
#214461602
#214459992
#214459032
#214458342
Q8×C22⋊C4×C612
#2452961