Finite Group Databasebeta

← #29610560

Subgroups — #29610560

Order 15616.

1094 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
1#21444087C211yesmaximal, characteristic
2#21444081C212yesmaximal
3#21444072C212yesmaximal
4#21444053C212yesmaximal
5#21444053C212yesmaximal
6#21444051C211yesmaximal, characteristic
7#21444044C212yesmaximal
8#21444044C212yesmaximal
9C2×C61×#400C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C4×Dic61C2212yes—
12C22×C4×Dic61C2211yescharacteristic
13C2×#14519684C2212yes—
14C2×#14519684C2212yes—
15C2×#14519684C2212yes—
16C2×#14519684C2212yes—
17C2×#14519684C2212yes—
18C2×#14519684C2211yescharacteristic
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2211yescharacteristic
26C22×C4⋊C4×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C61×#400C418yescomplemented
29C2×#95×C61C2211yescharacteristic
30C2×#95×C61C2212yes—
31C2×#95×C61C2212yes—
32C23×Dic61C2312yes—
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C2×C4⋊Dic61—24no—
37C2×C4⋊Dic61—24no—
38C2×C4⋊Dic61—42no—
39C2×C4×Dic61—24no—
40C2×C4×Dic61—24no—
41C2×C4×Dic61—42no—
42C2×C4×Dic61—42no—
43C2×C4×Dic61—41no—
44#14519684—24no—
45#14519684—24no—
46#14519684—24no—
47#14519684—24no—
48#14519684—24no—
49#14519684—41no—
50#14519684—42no—
51#14519684—42no—
52#14519684—42no—
53#14519684—42no—
54#14519684—42no—
55#14519684—41no—
56#14519684—41no—
57C23×C4×C61C2312yes—
58C23×C4×C61C2311yescharacteristic
59C23×C4×C61C2312yes—
60C23×C4×C61C2311yescharacteristic
61C23×C4×C61C2311yescharacteristic
62C2×C4⋊C4×C61—41no—
63C2×C4⋊C4×C61—41no—
64C2×C4⋊C4×C61—41no—
65C2×C4⋊C4×C61—41no—
66C2×C4⋊C4×C61C2×C414yes—
67C2×C4⋊C4×C61C2×C414yes—
68#95×C61C2×C414yes—
69#95×C61C2×C418yes—
70#95×C61C2×C418yes—
71C24×C61C2411yescharacteristic
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61C22×C414yes—
97C22×C4×C61C22×C414yes—
98C22×C4×C61C22×C412yes—
99C22×C4×C61C22×C412yes—
100C22×C4×C61C22×C412yes—
101C22×Dic61—22no—
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—22no—
105C22×Dic61—22no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61#4114yes—
127C22×Dic61#4114yes—
128C22×Dic61#4114yes—
129C4⋊C4×C61—44no—
130C4⋊C4×C61—44no—
131C4⋊Dic61—48no—
132C4×Dic61—48no—
133C4×Dic61—44no—
134C23×C61C23×C411yescharacteristic
135C23×C61C2×#4112yes—
136C23×C61C2×#4112yes—
137C23×C61C2×#4112yes—
138C23×C61H(5,𝔽2)11yescharacteristic
139C23×C61H(5,𝔽2)11yescharacteristic
140C23×C61H(5,𝔽2)12yes—
141C23×C612−1+412yes—
142C23×C612−1+412yes—
143C2×C4×C61—22no—
144C2×C4×C61—24no—
145C2×C4×C61—24no—
146C2×C4×C61—24no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—24no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—24no—
154C2×C4×C61—24no—
155C2×C4×C61—24no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×Dic61—24no—
173C2×Dic61—24no—
174C2×Dic61—24no—
175C2×Dic61—24no—
176C2×Dic61—24no—
177C2×Dic61—24no—
178C2×Dic61—24no—
179C2×Dic61—24no—
180C2×Dic61—24no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200#14612—611nomaximal, Sylow
201C22×C61C4×#4111yescharacteristic
202C22×C61C4×#4112yes—
203C22×C61#51811yescharacteristic
204C22×C61#51911yescharacteristic
205C22×C61#52012yes—
206C22×C61#53912yes—
207C22×C61#53912yes—
208C22×C61#53912yes—
209C22×C61#54112yes—
210C22×C61#54112yes—
211C22×C61#54212yes—
212C22×C61#55112yes—
213C22×C61#55112yes—
214C22×C61#55212yes—
215C22×C61#55312yes—
216C22×C61#55511yescharacteristic
217C22×C61#55511yescharacteristic
218C22×C61#55612yes—
219C22×C61#56011yescharacteristic
220C22×C61#56212yes—
221C22×C61#56311yescharacteristic
222C4×C61—42no—
223C4×C61—44no—
224C4×C61—44no—
225C4×C61—42no—
226C4×C61—42no—
227Dic61—44no—
228Dic61—44no—
229Dic61—44no—
230Dic61—82no—
231C2×#400D6111yescharacteristic
232#1445—611no—
233#1443—612no—
234#1441—612no—
235#1436—612no—
236#1436—612no—
237#1436—611no—
238#1434—612no—
239#1434—612no—
240C2×C61#233211yescharacteristic
241C2×C61#233612yes—
242C2×C61#237211yescharacteristic
243C2×C61#237312yes—
244C2×C61#237611yescharacteristic
245C2×C61#274412yes—
246C2×C61#274612yes—
247C2×C61#275812yes—
248C2×C61#278312yes—
249C22×C4⋊C4—612no—
250C22×C4⋊C4C2×D6111yescharacteristic
251C22×C4⋊C4C2×D6111yescharacteristic
252C22×C42—612no—
253C22×C42—611no—
254#400Dic6118yescomplemented
255C2×#95—612no—
256C2×#95—612no—
257C2×#95—612no—
258C2×#95—612no—
259C2×#95—612no—
260C2×#95—611no—
261C2×#95—612no—
262C2×#95—612no—
263C2×#95—612no—
264C2×#95—612no—
265C2×#95—612no—
266C2×#95—611no—
267C2×#95—611no—
268C2×#95C2×D6111yescharacteristic
269C2×#95C2×D6112yes—
270C2×#95C2×D6112yes—
271C61#1461211yescharacteristic, complemented, Sylow
272C23×C4—612no—
273C23×C4—612no—
274C23×C4—612no—
275C23×C4—612no—
276C23×C4C22×D6112yes—
277C23×C4C22×D6111yescharacteristic
278C23×C4C22×D6111yescharacteristic
279C23×C4C22×D6111yescharacteristic
280C23×C4C22×D6112yes—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—1224no—
286C2×C4⋊C4—1224no—
287C2×C4⋊C4—2442no—
288C2×C4⋊C4C2×Dic6114yes—
289C2×C4⋊C4C2×Dic6114yes—
290C2×C42—1224no—
291C2×C42—1224no—
292C2×C42—2442no—
293C2×C42—2441no—
294C2×C42—2442no—
295#95—1224no—
296#95—1224no—
297#95—1224no—
298#95—1224no—
299#95—1224no—
300#95—2441no—
301#95—2442no—
302#95—2441no—
303#95—2441no—
304#95—2442no—
305#95—2442no—
306#95—2442no—
307#95—2442no—
308#95C2×Dic6114yes—
309#95C2×Dic6118yes—
310#95C2×Dic6118yes—
311C24C23×D6111yescharacteristic
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—614no—
337C22×C4—614no—
338C22×C4—614no—
339C22×C4—1222no—
340C22×C4—1222no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1222no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1222no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4C22×Dic6114yes—
365C22×C4C22×Dic6114yes—
366C22×C4C22×Dic6112yes—
367C22×C4C22×Dic6112yes—
368C22×C4C22×Dic6112yes—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—2448no—
372C42—2448no—
373C42—2444no—
374C23C23×Dic6111yescharacteristic
375C23#41×D6112yes—
376C23#41×D6112yes—
377C23#41×D6112yes—
378C23#1451982211yescharacteristic
379C23#1451982211yescharacteristic
380C23#1451982312yes—
381C23#1451982412yes—
382C23#1451982512yes—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—1224no—
413C2×C4—1224no—
414C2×C4—1224no—
415C2×C4—1224no—
416C2×C4—1224no—
417C2×C4—1224no—
418C2×C4—1224no—
419C2×C4—1224no—
420C2×C4—1224no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C22#41×Dic6111yescharacteristic
441C22#41×Dic6112yes—
442C22#1684023711yescharacteristic
443C22#1684024011yescharacteristic
444C22#1684024512yes—
445C22#1684035812yes—
446C22#1684036012yes—
447C22#1684036012yes—
448C22#1684037112yes—
449C22#1684037112yes—
450C22#1684038212yes—
451C22#1684043712yes—
452C22#1684044212yes—
453C22#1684044812yes—
454C22#1684046312yes—
455C22#1684047411yescharacteristic
456C22#1684047411yescharacteristic
457C22#1684047912yes—
458C22#1684050211yescharacteristic
459C22#1684050912yes—
460C22#1684051511yescharacteristic
461C4—42no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—4882no—
467C4—2444no—
468C4—2444no—
469C4—2444no—
470C2#2144857711yescharacteristic
471C2#2144862512yes—
472C2#2144893111yescharacteristic
473C2#2144894312yes—
474C2#2144896711yescharacteristic
475C2#2145296612yes—
476C2#2145300812yes—
477C2#2145317012yes—
478C2#2145351312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#400C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#400D61
hypocenterC61#14612
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
28. C61×#400C4semidirect
254. #400Dic61semidirect
271. C61#14612semidirect

Maximal subgroups

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

subgroupindex
#214440872
#214440812
#214440722
#214440532
#214440512
#214440442
C2×C61×#4002
#1461261