Finite Group Databasebeta

← #29573391

Subgroups — #29573391

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839622C211yesmaximal, characteristic
2#21444042C211yesmaximal, characteristic
3#21443699C214yesmaximal, complemented
4C61×#1779C211yesmaximal, characteristic
5C22×C4⋊Dic61C411yescharacteristic, complemented
6C22×C4⋊Dic61C411yescharacteristic, complemented
7C22×C22⋊Dic61—21no—
8C22×C22⋊Dic61—21no—
9#16839622—21no—
10#16839622—21no—
11#16839622C412yescomplemented
12#16839622C2212yes—
13C2×#14519684—21no—
14C2×#14519684—21no—
15C2×#14519684C2212yes—
16C24×C4×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C2×C22⋊C8×C61—21no—
19C2×C22⋊C8×C61C2212yes—
20C23×Dic61—21no—
21C23×Dic61—21no—
22C23×Dic61—22no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—42no—
26C2×C4⋊Dic61—42no—
27C2×C4⋊Dic61C2×C412yes—
28C2×C4⋊Dic61C2×C412yes—
29C2×C22⋊Dic61—24no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—42no—
33C2×C22⋊Dic61—44no—
34C2×C22⋊Dic61—42no—
35#14519684—22no—
36#14519684—22no—
37#14519684—24no—
38#14519684—42no—
39C25×C61D411yescharacteristic
40C23×C4×C61—24no—
41C23×C4×C61—42no—
42C23×C4×C61—42no—
43C23×C4×C61C2×C411yescharacteristic
44C23×C4×C61C2×C411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C22×C8×C61—22no—
50C22⋊C8×C61—24no—
51C22⋊C8×C61—42no—
52C22⋊C8×C61—42no—
53C24×C61—22no—
54C24×C61—22no—
55C24×C61—22no—
56C24×C61—22no—
57C24×C61—41no—
58C24×C61—41no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C24×C61SD1612yes—
62C24×C61Q1612yes—
63C24×C61C2×D411yescharacteristic
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61—416no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4114yes—
87C22×Dic61—24no—
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—42no—
91C22×Dic61—42no—
92C22×Dic61—42no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61—42no—
101C22×Dic61—42no—
102C2×C8×C61—42no—
103C2×C8×C61—42no—
104C2×C8×C61—42no—
105C22⋊Dic61—48no—
106C22⋊Dic61—44no—
107C22⋊Dic61—44no—
108C4⋊Dic61—42no—
109C4⋊Dic61—42no—
110C4⋊Dic61—42no—
111C4⋊Dic61—42no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—24no—
121C23×C61—24no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—24no—
129C23×C61—24no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—44no—
133C23×C61—44no—
134C23×C61—44no—
135C23×C61—44no—
136C23×C61#9912yes—
137C23×C61#10312yes—
138C23×C61#10312yes—
139C23×C61#10412yes—
140C23×C61C2×C22⋊C411yescharacteristic
141C23×C61#12312yes—
142C23×C61C2×SD1611yescharacteristic
143C23×C61C2×Q1611yescharacteristic
144C23×C61AGL(1,ℤ/8)12yes—
145C2×C4×C61—28no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—44no—
156C2×C4×C61—44no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—44no—
160C2×C4×C61#9912yes—
161C2×C4×C61#11712yes—
162C2×C4×C61#12312yes—
163C2×Dic61—44no—
164C2×Dic61—44no—
165C2×Dic61—44no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—82no—
173C2×Dic61—82no—
174C2×Dic61—82no—
175C2×Dic61—84no—
176C2×Dic61—84no—
177C2×Dic61—82no—
178C8×C61—82no—
179#8713—611nomaximal, Sylow
180C22×C61—22no—
181C22×C61—22no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—22no—
188C22×C61—22no—
189C22×C61—22no—
190C22×C61—22no—
191C22×C61—21no—
192C22×C61—22no—
193C22×C61—21no—
194C22×C61—22no—
195C22×C61—22no—
196C22×C61—22no—
197C22×C61—48no—
198C22×C61—48no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61#32814yes—
204C22×C61#38111yescharacteristic
205C22×C61C2×#9911yescharacteristic
206C22×C61C2×#9911yescharacteristic
207C22×C61C2×#10311yescharacteristic
208C22×C61#41811yescharacteristic
209C22×C61C2×#10411yescharacteristic
210C22×C61#42111yescharacteristic
211C22×C61#48112yes—
212C22×C61#48412yes—
213C4×C61—22no—
214C4×C61—22no—
215C4×C61—22no—
216C4×C61—41no—
217Dic61—84no—
218Dic61—82no—
219Dic61—82no—
220C2×#380—611no—
221#1779D6111yescharacteristic
222C22⋊#95—611no—
223#1294—614no—
224C2×C61—22no—
225C2×C61—24no—
226C2×C61—22no—
227C2×C61—42no—
228C2×C61C2×#32811yescharacteristic
229C2×C61#150711yescharacteristic
230C2×C61#151011yescharacteristic
231C2×C61#152011yescharacteristic
232C2×C61#181911yescharacteristic
233C2×C61#182411yescharacteristic
234C2×C61#182611yescharacteristic
235C24×C4C2×D6111yescharacteristic
236C22×C4⋊C4—611no—
237C22×C4⋊C4—611no—
238C22×C22⋊C4—1221no—
239C22×C22⋊C4—1221no—
240C2×C22⋊C8—21no—
241C2×C22⋊C8—21no—
242C2×C22⋊C8C2×D6112yes—
243#380—612no—
244#380—612no—
245#380—1221no—
246#380—1221no—
247C2×#95—612no—
248C2×#95—1221no—
249C2×#95—1221no—
250C61#871311yescharacteristic, complemented, Sylow
251C25D24411yescharacteristic
252C23×C4—24no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—1221no—
256C23×C4—1221no—
257C23×C4—1222no—
258C23×C4C4×D6111yescharacteristic
259C23×C4C4×D6111yescharacteristic
260C23×C4D24411yescharacteristic
261C23×C4C22⋊D6111yescharacteristic
262C23×C4C22⋊D6111yescharacteristic
263C23×C4C22×D6111yescharacteristic
264C22×C8—22no—
265C2×C4⋊C4—612no—
266C2×C4⋊C4—612no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—2442no—
270C2×C4⋊C4—2442no—
271C2×C22⋊C4—1224no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—2442no—
275C2×C22⋊C4—2444no—
276C2×C22⋊C4—2442no—
277C22⋊C8—24no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#95—1222no—
281#95—1222no—
282#95—1224no—
283#95—2442no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24#1175819012yes—
291C24Dic24412yes—
292C24#1175819811yescharacteristic
293C24#1175819811yescharacteristic
294C24C2×D24411yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—416no—
313C22×C4—1224no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4#1175819811yescharacteristic
329C22×C4#1175819811yescharacteristic
330C22×C4C2×C4×D6111yescharacteristic
331C22×C4#1175822414yes—
332C22×C4C2×C22⋊D6111yescharacteristic
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—2442no—
337C4⋊C4—2442no—
338C4⋊C4—2442no—
339C4⋊C4—2442no—
340C22⋊C4—2448no—
341C22⋊C4—2444no—
342C22⋊C4—2444no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—24no—
354C23—24no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—24no—
359C23—24no—
360C23—21no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23#1451969112yes—
368C23#1451970212yes—
369C23#1451970212yes—
370C23#1451970312yes—
371C23C2×#1175819811yescharacteristic
372C23#1451976112yes—
373C23C2×#1175819011yescharacteristic
374C23C2×Dic24411yescharacteristic
375C23#1451980512yes—
376C2×C4—28no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—4882no—
394C2×C4—4884no—
395C2×C4—4884no—
396C2×C4—4882no—
397C2×C4—2444no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4#1451969112yes—
407C2×C4#1451973312yes—
408C2×C4#1451975812yes—
409C8—82no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—48no—
428C22—48no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#1683950014yes—
434C22#1683962311yescharacteristic
435C22C2×#1451969111yescharacteristic
436C22C2×#1451969111yescharacteristic
437C22C2×#1451970211yescharacteristic
438C22#1683976811yescharacteristic
439C22C2×#1451970311yescharacteristic
440C22#1683977811yescharacteristic
441C22#1684008112yes—
442C22#1684009812yes—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—4884no—
448C4—4882no—
449C4—4882no—
450C2—22no—
451C2—24no—
452C2—22no—
453C2—42no—
454C2C2×#1683950011yescharacteristic
455C2#2144433611yescharacteristic
456C2#2144434911yescharacteristic
457C2#2144439211yescharacteristic
458C2#2144589111yescharacteristic
459C2#2144591011yescharacteristic
460C2#2144591611yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1779C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenter#1779D61
hypocenterC61#8713
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #21443699C2semidirect
5. C22×C4⋊Dic61C4semidirect
6. C22×C4⋊Dic61C4semidirect
11. #16839622C4semidirect
250. C61#8713semidirect

Maximal subgroups

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

subgroupindex
C2×#168396222
#214440422
#214436992
C61×#17792
#871361