Finite Group Databasebeta

← #29589261

Subgroups — #29589261

Order 15616.

756 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519745C211yesmaximal, characteristic
2#21444140C211yesmaximal, characteristic, complemented
3#21443766C212yesmaximal, complemented
4#21443746C212yesmaximal, complemented
5C61×#1459C211yesmaximal, characteristic, complemented
6C23×C4×D61C411yescharacteristic
7C2×#14519745C411yescharacteristic
8C2×C4×C22⋊D61—21no—
9#16839815—21no—
10#16839642C411yescharacteristic
11#16839638—21no—
12#16839634C2211yescharacteristic
13#16839633—21no—
14#16839617C2211yescharacteristic
15C4×#11758198—21no—
16C4×C22⋊Dic61C411yescharacteristic
17#16839486—21no—
18#16839486C2212yes—
19C2×C4×D4×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C61×#324—21no—
22C61×#323C2212yes—
23C24×D61C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C23×Dic61C2×C411yescharacteristic
26C22×C4×D61—21no—
27C22×C4×D61—21no—
28C22×C4×D61—44no—
29C22×C4×D61—41no—
30C22×C4×D61—41no—
31#14519745—22no—
32#14519745C814yescomplemented
33C4×C22⋊D61—44no—
34C2×#11758196—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61C2×C411yescharacteristic
37C2×C22⋊Dic61C2×C411yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×C4×Dic61—21no—
42C4⋊#92418—41no—
43C22⋊#92418—22no—
44C4×#92418—41no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C22×C8×C61—41no—
52C4×D4×C61—41no—
53C4×D4×C61—41no—
54C2×C4⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2311yescharacteristic
57C2×C42×C61D411yescharacteristic
58C22⋊C8×C61—22no—
59C22⋊C8×C61—41no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C22×C411yescharacteristic
62C23×D61—21no—
63C23×D61—21no—
64C23×D61—42no—
65C23×D61—42no—
66C23×D61—41no—
67C23×D61—41no—
68C23×D61C2×C812yescomplemented
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61C2×C812yes—
72C2×D4×C61M1612yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61—24no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—42no—
91C22×Dic61—41no—
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C4×D61—42no—
96C2×C4×D61—42no—
97C2×C4×D61—44no—
98C2×C4×D61—44no—
99C2×C4×D61—44no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×C4×D61—44no—
103C2×C8×C61—42no—
104C2×C8×C61—82no—
105C4⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C42×C61—21no—
109C42×C61—21no—
110C22⋊Dic61—21no—
111C22⋊Dic61—42no—
112C22⋊Dic61—41no—
113C22⋊Dic61C2×C812yescomplemented
114#11758198—24no—
115#11758198—42no—
116#11758198—42no—
117#11758196—42no—
118C4×Dic61—42no—
119C2×#92418—42no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61C22⋊C812yes—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61C22×C811yescharacteristic
131C23×C61C2×M1611yescharacteristic
132C22×D61—22no—
133C22×D61—22no—
134C22×D61—22no—
135C22×D61—44no—
136C22×D61—44no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—48no—
142C22×D61—42no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—42no—
148D4×C61—42no—
149D4×C61—42no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61C22⋊C812yes—
164C2×C4×C61#9911yescharacteristic
165C2×C4×C61#9911yescharacteristic
166C22⋊D61—48no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C4×D61—44no—
179C4×D61—48no—
180C4×D61—44no—
181C8×C61—82no—
182#92418—82no—
183#11471—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—22no—
189C22×C61—42no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—42no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—81no—
198C22×C61C2×C22⋊C811yescharacteristic
199C22×C61C2×#9911yescharacteristic
200C22×C61#41311yescharacteristic
201C2×D61—24no—
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—42no—
205C2×D61—44no—
206C2×D61—44no—
207C2×D61—44no—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—42no—
211C2×D61—44no—
212C4×C61—21no—
213C4×C61—22no—
214C4×C61—41no—
215C4×C61—41no—
216C4×C61—41no—
217C4×C61AΣL(1,𝔽16)12yes—
218Dic61—42no—
219Dic61—42no—
220Dic61—81no—
221C4×#120—611no—
222#1460—611no—
223#1459D6111yescharacteristic, complemented
224#1326—612no—
225#1316—612no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—41no—
229C2×C61—41no—
230C2×C61—42no—
231C2×C61#145911yescharacteristic
232C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
233C2×C61#212011yescharacteristic
234D61—44no—
235C24×C4—611no—
236C2×#120—611no—
237C2×C4×D4—1221no—
238C2×C4×D4C2×D6111yescharacteristic
239#431—1221no—
240C2×C22⋊C8—21no—
241#386—611no—
242#386—1221no—
243#385—611no—
244#385—1221no—
245#379—611no—
246C4×C22⋊C4—611no—
247C4×C22⋊C4—1221no—
248#324—21no—
249#323—612no—
250#323—1221no—
251#323C2×D6112yes—
252C61#1147111yescharacteristic, complemented, Sylow
253C25—611no—
254C22×D4—1221no—
255C22×D4C4×D6111yescharacteristic
256C23×C4—611no—
257C23×C4—1221no—
258C23×C4—1221no—
259C23×C4—2441no—
260C23×C4—2441no—
261C23×C4—2444no—
262C23×C4C4×D6111yescharacteristic
263C23×C4C22⋊D6111yescharacteristic
264C22×C8—41no—
265#120—614no—
266#120—1222no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—2444no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4C22⋊D6111yescharacteristic
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4D24411yescharacteristic
279C2×C22⋊C4C22×D6111yescharacteristic
280C2×C42—1221no—
281C2×C42D24411yescharacteristic
282C4⋊C8—2441no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8—1222no—
286C4×C8—2441no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290C24—612no—
291C24—1221no—
292C24—1221no—
293C24—2441no—
294C24—2441no—
295C24—2442no—
296C24—2442no—
297C24#1175819811yescharacteristic
298C24C2×C4×D6111yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—1224no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4C8×D6112yes—
305C2×D4#1175818912yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—1224no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4—2444no—
321C22×C4—2444no—
322C22×C4—2444no—
323C22×C4—2444no—
324C22×C4—2442no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4—2441no—
328C22×C4#1175819811yescharacteristic
329C22×C4#1175819811yescharacteristic
330C22×C4#1175819811yescharacteristic
331C22×C4C2×D24411yescharacteristic
332C22×C4C2×C22⋊D6111yescharacteristic
333C2×C8—42no—
334C2×C8—82no—
335C2×C8—2442no—
336C4⋊C4—41no—
337C4⋊C4—2442no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—612no—
341C22⋊C4—1221no—
342C22⋊C4—1224no—
343C22⋊C4—2442no—
344C22⋊C4—2442no—
345C22⋊C4—2442no—
346C22⋊C4—2441no—
347C42—21no—
348C42—21no—
349C42—2442no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—1222no—
359C23—1222no—
360C23—1222no—
361C23—2442no—
362C23—2441no—
363C23—2441no—
364C23—2442no—
365C23—2444no—
366C23—2444no—
367C23—2442no—
368C23—2442no—
369C23—2442no—
370C23—2448no—
371C23—2442no—
372C23—2441no—
373C23—2441no—
374C23#1451968912yes—
375C23C2×#1175819811yescharacteristic
376C23C2×C8×D6111yescharacteristic
377C23C2×#1175818911yescharacteristic
378D4—42no—
379D4—42no—
380D4—2448no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—1222no—
395C2×C4—1222no—
396C2×C4—2444no—
397C2×C4—2448no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2444no—
407C2×C4—2442no—
408C2×C4#1451968912yes—
409C2×C4#1451969111yescharacteristic
410C2×C4#1451969111yescharacteristic
411C8—82no—
412C8—4882no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—42no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—1224no—
428C22—2442no—
429C22—2442no—
430C22—2442no—
431C22—2444no—
432C22—2444no—
433C22—2444no—
434C22—2444no—
435C22—2442no—
436C22—2442no—
437C22—2442no—
438C22C2×#1451968911yescharacteristic
439C22C2×#1451969111yescharacteristic
440C22#1683974411yescharacteristic
441C4—21no—
442C4—22no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—4881no—
447C4—2442no—
448C4—2442no—
449C4#1683954812yes—
450C2—21no—
451C2—21no—
452C2—41no—
453C2—41no—
454C2—42no—
455C2—2444no—
456C2#2144413511yescharacteristic
457C2C2×#1683954811yescharacteristic
458C2#2144746111yescharacteristic
subgroupG/H
centerC2×C4#14519691
commutatorC23×C61C22×C8
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1459C2
socleC22×C61C2×#99
prime radicalC2×#14519745C4
cosocleC2×C22⋊C4×C61C23
hypercenter#1459D61
hypocenterC61#11471
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21444140C2semidirect
3. #21443766C2semidirect
4. #21443746C2semidirect
5. C61×#1459C2semidirect
32. #14519745C8semidirect
68. C23×D61C2×C8semidirect
113. C22⋊Dic61C2×C8semidirect
223. #1459D61semidirect
252. C61#11471semidirect

Maximal subgroups

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

subgroupindex
C4×#145197452
#214441402
#214437662
#214437462
C61×#14592
#1147161