Finite Group Databasebeta

← #29579145

Subgroups — #29579145

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445799C211yesmaximal, characteristic, complemented
2C2×#16839499C212yesmaximal, complemented
3C2×#16839496C212yesmaximal
4C4×#14519684C211yesmaximal, characteristic, complemented
5C61×#2850C211yesmaximal, characteristic
6C22×C4×Dic61—21no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418C2212yes—
9C2×#14519684—21no—
10C2×#14519684C2212yes—
11#16839517—21no—
12#16839499—24no—
13#16839496—24no—
14C2×#124×C61C411yescharacteristic, complemented
15C2×C22⋊Q8×C61C2211yescharacteristic
16C2×#121×C61C2211yescharacteristic
17C22×C42×C61C2211yescharacteristic
18C61×#391C411yescharacteristic, complemented
19C61×#390C411yescharacteristic, complemented
20C61×#388—21no—
21C61×#388—21no—
22C61×#386—21no—
23C61×#386—21no—
24C61×#382C411yescharacteristic, complemented
25C23×Dic61—22no—
26C22×#92418—22no—
27C2×C4×Dic61—24no—
28C2×C4×Dic61—42no—
29C2×C4×Dic61—42no—
30C4⋊#92418—42no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33#14519684—24no—
34#14519684—22no—
35#14519684D414yes—
36C22×Q8×C61C2×C411yescharacteristic
37C22×D4×C61—21no—
38C22×D4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42#124×C61—42no—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61—22no—
45#121×C61—22no—
46#121×C61—22no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C42×C61—42no—
57C2×C42×C61C2×C411yescharacteristic
58C2×C42×C61C2×C411yescharacteristic
59C2×C42×C61D411yescharacteristic
60C2×C42×C61D411yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63#95×C61—21no—
64#95×C61—21no—
65C24×C61—21no—
66C24×C61C2×D411yescharacteristic
67C2×Q8×C61—22no—
68C2×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×D4×C61—22no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—42no—
74C2×D4×C61—42no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22×C411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96C22×C4×C61#4112yes—
97C22×Dic61—24no—
98C22×Dic61—24no—
99C22×Dic61—24no—
100C22×Dic61—42no—
101C22×Dic61—42no—
102C22×Dic61—42no—
103C22×Dic61—42no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—42no—
109C22⋊C4×C61—42no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61—22no—
116C42×C61—22no—
117C42×C61—41no—
118C42×C61—41no—
119C4×Dic61—48no—
120C2×#92418—42no—
121C2×#92418—42no—
122C2×#92418—42no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61#12112yes—
136C23×C61#13512yes—
137C23×C61AGL(1,ℤ/8)11yescharacteristic
138C23×C61#13711yescharacteristic
139Q8×C61—82no—
140D4×C61—82no—
141D4×C61—82no—
142D4×C61—82no—
143C2×C4×C61—24no—
144C2×C4×C61—24no—
145C2×C4×C61—21no—
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—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—44no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61#9912yes—
171C2×C4×C61#10412yes—
172C2×C4×C61#10412yes—
173C2×C4×C61C4×D412yes—
174C2×C4×C61#12411yescharacteristic
175C2×C4×C61#12711yescharacteristic
176C2×Dic61—28no—
177C2×Dic61—44no—
178C2×Dic61—44no—
179C2×Dic61—44no—
180C2×Dic61—44no—
181C2×Dic61—44no—
182C2×Dic61—44no—
183#92418—82no—
184#9713—611nomaximal, Sylow
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—22no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#39011yescharacteristic
201C22×C61C2×#9911yescharacteristic
202C22×C61#41011yescharacteristic
203C22×C61#41611yescharacteristic
204C22×C61#41711yescharacteristic
205C22×C61C2×#10411yescharacteristic
206C22×C61C2×#10411yescharacteristic
207C22×C61#46312yes—
208C22×C61#46412yes—
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—22no—
213C4×C61—22no—
214C4×C61—22no—
215C4×C61—41no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61—81no—
219Dic61—48no—
220#2850D6111yescharacteristic
221#1797—611no—
222C2×#328—612no—
223C2×#327—612no—
224C4×#95—611no—
225C2×C61—22no—
226C2×C61—22no—
227C2×C61—161no—
228C2×C61#150011yescharacteristic
229C2×C61#150111yescharacteristic
230C2×C61#150411yescharacteristic
231C2×C61#150911yescharacteristic
232C2×C61#196211yescharacteristic
233C2×C61#196411yescharacteristic
234C2×C61#198311yescharacteristic
235C2×#124Dic6111yescharacteristic, complemented
236C2×C22⋊Q8C2×D6111yescharacteristic
237C2×#121C2×D6111yescharacteristic
238C22×C42—1221no—
239C22×C42C2×D6111yescharacteristic
240C2×C4⋊C8—1221no—
241C2×C22⋊C8—612no—
242#391Dic6111yescharacteristic, complemented
243#390Dic6111yescharacteristic, complemented
244#388—21no—
245#388—21no—
246#386—21no—
247#386—21no—
248#382Dic6111yescharacteristic, complemented
249C2×#95—612no—
250C2×#95—1221no—
251#336—1221no—
252#328—1224no—
253#327—1224no—
254C61#971311yescharacteristic, complemented, Sylow
255C22×Q8C2×Dic6111yescharacteristic
256C22×D4—21no—
257C22×D4C2×Dic6111yescharacteristic
258C23×C4—1222no—
259C23×C4C22⋊D6111yescharacteristic
260C23×C4C22⋊D6111yescharacteristic
261C23×C4C22×D6111yescharacteristic
262C22×C8—1222no—
263#124—42no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C2×Dic6111yescharacteristic
270C2×C4⋊C4C2×Dic6111yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—42no—
278C2×C42—1224no—
279C2×C42—2442no—
280C2×C42—2442no—
281C2×C42C2×Dic6111yescharacteristic
282C2×C42C2×Dic6111yescharacteristic
283C2×C42C22⋊D6111yescharacteristic
284C2×C42C22⋊D6111yescharacteristic
285C4⋊C8—2442no—
286C22⋊C8—1222no—
287C22⋊C8—1222no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—614no—
293#95—1224no—
294#95—1222no—
295C24—21no—
296C24C2×C22⋊D6111yescharacteristic
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—1224no—
320C22×C4—1224no—
321C22×C4—1224no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4C22⋊Dic6111yescharacteristic
327C22×C4C22⋊Dic6111yescharacteristic
328C22×C4C22⋊Dic6111yescharacteristic
329C22×C4C22⋊Dic6111yescharacteristic
330C22×C4D4×D6112yes—
331C22×C4#1175822412yes—
332C22×C4C22×Dic6111yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334C2×C8—2442no—
335C2×C8—2442no—
336C2×C8—2442no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—2448no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C2×C22⋊Dic6111yescharacteristic
365C23#1451974812yes—
366C23#1451979712yes—
367C23#1451980211yescharacteristic
368C23#1451980711yescharacteristic
369Q8—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
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—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—1228no—
401C2×C4—2444no—
402C2×C4—2444no—
403C2×C4—2444no—
404C2×C4—2444no—
405C2×C4—2444no—
406C2×C4—2444no—
407C2×C4#1451969212yes—
408C2×C4#1451970412yes—
409C2×C4#1451970412yes—
410C2×C4D4×Dic6112yes—
411C2×C4#1451976411yescharacteristic
412C2×C4#1451977311yescharacteristic
413C8—4882no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#1683966511yescharacteristic
430C22C2×#1451969211yescharacteristic
431C22#1683973611yescharacteristic
432C22#1683975811yescharacteristic
433C22#1683976311yescharacteristic
434C22C2×#1451970411yescharacteristic
435C22C2×#1451970411yescharacteristic
436C22#1683997412yes—
437C22#1683998112yes—
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—2448no—
449C2—22no—
450C2—22no—
451C2—161no—
452C2#2144430511yescharacteristic
453C2#2144431011yescharacteristic
454C2#2144432311yescharacteristic
455C2#2144434411yescharacteristic
456C2#2144664811yescharacteristic
457C2#2144665611yescharacteristic
458C2#2144675211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2850C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#2850D61
hypocenterC61#9713
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #21445799C2semidirect
2. C2×#16839499C2semidirect
4. C4×#14519684C2semidirect
14. C2×#124×C61C4semidirect
18. C61×#391C4semidirect
19. C61×#390C4semidirect
24. C61×#382C4semidirect
235. C2×#124Dic61semidirect
242. #391Dic61semidirect
243. #390Dic61semidirect
248. #382Dic61semidirect
254. C61#9713semidirect

Maximal subgroups

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

subgroupindex
#214457992
C2×#168394992
C2×#168394962
C4×#145196842
C61×#28502
#971361