Finite Group Databasebeta

← #29586241

Subgroups — #29586241

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839634C211yesmaximal, characteristic
2#21448172C211yesmaximal, characteristic
3C2×#16839528C212yesmaximal
4#21444041C212yesmaximal
5C2×C61×#342C211yesmaximal, characteristic
6C24×Dic61C2211yescharacteristic
7C22×#11758196—21no—
8C22×C22⋊Dic61C2211yescharacteristic
9C22×C22⋊Dic61C2212yes—
10#16839634—22no—
11#16839634C414yescomplemented
12C22⋊C4×Dic61—24no—
13C4×C22⋊Dic61—21no—
14C4×C22⋊Dic61—21no—
15C2×#14519684—21no—
16C2×#14519684—24no—
17#16839528C418yescomplemented
18C22×C22⋊C4×C61C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C61×#342—22no—
21C23×Dic61—21no—
22C23×Dic61—21no—
23C23×Dic61—22no—
24C23×Dic61—21no—
25C23×Dic61—22no—
26C23×Dic61—44no—
27C23×Dic61C2×C412yes—
28C23×Dic61D414yes—
29C2×#11758196—24no—
30C2×#11758196—44no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61C2×C414yes—
38C2×C22⋊Dic61C2×C414yes—
39C2×C22⋊Dic61C2×C412yes—
40C2×C22⋊Dic61Q814yes—
41C2×C4×Dic61—22no—
42C2×C4×Dic61—22no—
43#14519684—22no—
44#14519684—22no—
45#14519684—24no—
46#14519684—48no—
47C25×C61C2311yescharacteristic
48C23×C4×C61—22no—
49C23×C4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61—22no—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61C2×C414yes—
59C2×C22⋊C4×C61C2×C412yes—
60C2×C22⋊C4×C61D412yes—
61C24×C61—42no—
62C24×C61—42no—
63C24×C61C22×C411yescharacteristic
64C24×C61C22×C412yes—
65C24×C61C2×D411yescharacteristic
66C24×C61C2×D412yes—
67C24×C61C2×D411yescharacteristic
68C24×C61C2×Q812yes—
69C24×C61#4112yes—
70C24×C61#4112yes—
71C24×C61#4112yes—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—24no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—44no—
80C22×C4×C61—44no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61C22⋊C412yes—
84C22×C4×C61C22⋊C412yes—
85C22×Dic61—28no—
86C22×Dic61—24no—
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—24no—
90C22×Dic61—24no—
91C22×Dic61—24no—
92C22×Dic61—24no—
93C22×Dic61—44no—
94C22×Dic61—44no—
95C22×Dic61—48no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61—44no—
99C22×Dic61—42no—
100C22×Dic61—41no—
101C22×Dic61—42no—
102C22×Dic61—41no—
103C22×Dic61—44no—
104C22×Dic61—44no—
105C22×Dic61C4⋊C418yescomplemented
106C22⋊C4×C61—24no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊Dic61—22no—
113C22⋊Dic61—24no—
114C22⋊Dic61—24no—
115C22⋊Dic61—24no—
116C22⋊Dic61—24no—
117C22⋊Dic61—24no—
118C22⋊Dic61C4214yescomplemented
119#11758196—48no—
120C4×Dic61—82no—
121C4×Dic61—82no—
122C23×C61—24no—
123C23×C61—22no—
124C23×C61—24no—
125C23×C61—22no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—42no—
129C23×C61—44no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—44no—
133C23×C61—44no—
134C23×C61—44no—
135C23×C61—42no—
136C23×C61C2×C4211yescharacteristic
137C23×C61C2×C22⋊C411yescharacteristic
138C23×C61C2×C4⋊C412yes—
139C23×C61#11711yescharacteristic
140C23×C61#11712yes—
141C23×C61C4×D412yes—
142C23×C61C4×D412yes—
143C23×C61C4×D412yes—
144C23×C61C4×D412yes—
145C23×C61C4×Q812yes—
146C23×C61C4×Q812yes—
147C23×C61#12011yescharacteristic
148C23×C61#12112yes—
149C23×C61C22⋊Q812yes—
150C23×C61C22⋊Q812yes—
151C23×C61#12312yes—
152C23×C61#12311yescharacteristic
153C23×C61#12411yescharacteristic
154C23×C61#12512yes—
155C23×C61#12612yes—
156C23×C61C4⋊Q811yescharacteristic
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—48no—
164C2×C4×C61—44no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61—84no—
168C2×C4×C61—82no—
169C2×C4×C61—82no—
170C2×Dic61—24no—
171C2×Dic61—28no—
172C2×Dic61—24no—
173C2×Dic61—28no—
174C2×Dic61—42no—
175C2×Dic61—48no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—44no—
179C2×Dic61—48no—
180C2×Dic61—44no—
181C2×Dic61—42no—
182C2×Dic61—44no—
183C2×Dic61—44no—
184C2×Dic61—44no—
185C2×Dic61—48no—
186C2×Dic61—84no—
187#10918—611nomaximal, Sylow
188C22×C61—24no—
189C22×C61—24no—
190C22×C61—24no—
191C22×C61—24no—
192C22×C61—24no—
193C22×C61—24no—
194C22×C61—21no—
195C22×C61—22no—
196C22×C61—22no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—22no—
200C22×C61—21no—
201C22×C61—22no—
202C22×C61—41no—
203C22×C61—42no—
204C22×C61—41no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—41no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61C4×C22⋊C411yescharacteristic
213C22×C61C4×C4⋊C412yes—
214C22×C61#38512yes—
215C22×C61#38611yescharacteristic
216C22×C61#38712yes—
217C22×C61#38712yes—
218C22×C61#38812yes—
219C22×C61#38912yes—
220C22×C61#39111yescharacteristic
221C22×C61B(4,𝔽2)12yes—
222C22×C61#45812yes—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—84no—
226Dic61—44no—
227Dic61—48no—
228Dic61—44no—
229C2×#385—611no—
230#2273—611no—
231C2×#342—612no—
232C2×#342D6111yescharacteristic
233C22⋊#95—612no—
234C2×C61—21no—
235C2×C61—22no—
236C2×C61—22no—
237C2×C61—22no—
238C2×C61—22no—
239C2×C61—22no—
240C2×C61—21no—
241C2×C61#143611yescharacteristic
242C2×C61#186714yes—
243C2×C61#188911yescharacteristic
244C2×C61#189011yescharacteristic
245C24×C4—611no—
246C22×C4⋊C4—1221no—
247C22×C22⋊C4—611no—
248C22×C22⋊C4—612no—
249C22×C22⋊C4C2×D6112yes—
250C22×C22⋊C4C2×D6111yescharacteristic
251#385—614no—
252#385—1222no—
253C4×C22⋊C4—1224no—
254C4×C22⋊C4—1221no—
255C4×C22⋊C4—1221no—
256C2×#95—1221no—
257C2×#95—1224no—
258#342—22no—
259#342—618no—
260C61#1091811yescharacteristic, complemented, Sylow
261C25C22×D6111yescharacteristic
262C23×C4—22no—
263C23×C4—614no—
264C23×C4—612no—
265C23×C4—1221no—
266C23×C4—1221no—
267C23×C4—1222no—
268C23×C4—1222no—
269C23×C4—1221no—
270C23×C4—2444no—
271C23×C4D24411yescharacteristic
272C23×C4D24411yescharacteristic
273C2×C4⋊C4—1224no—
274C2×C4⋊C4—2444no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—614no—
283C2×C22⋊C4—612no—
284C2×C22⋊C4—614no—
285C2×C22⋊C4—614no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4—1222no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4C4×D6114yes—
293C2×C22⋊C4C4×D6112yes—
294C2×C22⋊C4C22⋊D6112yes—
295C2×C42—1222no—
296C2×C42—1222no—
297#95—1222no—
298#95—1222no—
299#95—1224no—
300#95—2448no—
301C24—42no—
302C24—42no—
303C24C2×C4×D6111yescharacteristic
304C24C2×C4×D6112yes—
305C24C2×D24411yescharacteristic
306C24#1175822212yes—
307C24D4×D6112yes—
308C24#1175822412yes—
309C24#1175822412yes—
310C24Q8×D6112yes—
311C24C2×C22⋊D6111yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—618no—
324C22×C4—1224no—
325C22×C4—1228no—
326C22×C4—1224no—
327C22×C4—1222no—
328C22×C4—1222no—
329C22×C4—1224no—
330C22×C4—1224no—
331C22×C4—1224no—
332C22×C4—2444no—
333C22×C4—2444no—
334C22×C4—2448no—
335C22×C4—2444no—
336C22×C4—2444no—
337C22×C4—2442no—
338C22×C4—2442no—
339C22×C4—2444no—
340C22×C4—2442no—
341C22×C4—2441no—
342C22×C4—2442no—
343C22×C4—2441no—
344C22×C4#1175819812yes—
345C22×C4#1175819812yes—
346C4⋊C4—2448no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—614no—
354C22⋊C4—1222no—
355C22⋊C4—1224no—
356C22⋊C4—1224no—
357C22⋊C4—1224no—
358C22⋊C4—1224no—
359C22⋊C4—1224no—
360C42—4882no—
361C42—4882no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—44no—
371C23—44no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—41no—
376C23C42×D6111yescharacteristic
377C23C2×#1175819811yescharacteristic
378C23C4⋊C4×D6112yes—
379C23#1451973312yes—
380C23#1451973411yescharacteristic
381C23#1451973712yes—
382C23#1451973712yes—
383C23C4×C22⋊D6112yes—
384C23C4×D24412yes—
385C23#1451974212yes—
386C23#1451974212yes—
387C23#1451974711yescharacteristic
388C23#1451974912yes—
389C23#1451975512yes—
390C23#1451975712yes—
391C23#1451975912yes—
392C23#1451976111yescharacteristic
393C23#1451976411yescharacteristic
394C23#1451976712yes—
395C23#1451977012yes—
396C23#1451977511yescharacteristic
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—84no—
410C2×C4—1224no—
411C2×C4—1228no—
412C2×C4—1224no—
413C2×C4—1228no—
414C2×C4—4884no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2448no—
418C2×C4—2442no—
419C2×C4—2448no—
420C2×C4—2444no—
421C2×C4—2444no—
422C2×C4—2444no—
423C2×C4—2444no—
424C2×C4—2444no—
425C2×C4—2442no—
426C2×C4—2448no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—21no—
434C22—22no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—41no—
442C22—42no—
443C22—41no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—41no—
451C22C4×#1175819811yescharacteristic
452C22#1683961312yes—
453C22#1683963612yes—
454C22#1683963911yescharacteristic
455C22#1683964412yes—
456C22#1683964712yes—
457C22#1683965612yes—
458C22#1683965712yes—
459C22#1683966711yescharacteristic
460C22#1683995012yes—
461C22#1683995312yes—
462C4—42no—
463C4—42no—
464C4—84no—
465C4—2444no—
466C4—2444no—
467C4—2448no—
468C2—22no—
469C2—21no—
470C2—22no—
471C2—22no—
472C2—22no—
473C2—22no—
474C2—21no—
475C2#2144405411yescharacteristic
476C2#2144610114yes—
477C2#2144622511yescharacteristic
478C2#2144623011yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC23×C61C2×C42
FrattiniC25C22×D61
FittingC2×C61×#342C2
socleC23×C61#120
prime radical, cosocleC25×C61C23
hypercenterC2×#342D61
hypocenterC61#10918
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
11. #16839634C4semidirect
17. #16839528C4semidirect
105. C22×Dic61C4⋊C4semidirect
118. C22⋊Dic61C42semidirect
260. C61#10918semidirect

Maximal subgroups

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

subgroupindex
C2×#168396342
#214481722
C2×#168395282
#214440412
C2×C61×#3422
#1091861