Finite Group Databasebeta

← #30603972

Subgroups — #30603972

Order 15616.

756 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#21461058C211yesmaximal, characteristic, complemented
2#21461054C211yesmaximal, characteristic, complemented
3#21460764C212yesmaximal, complemented
4#21460743C212yesmaximal, complemented
5C4×#14519746C211yesmaximal, characteristic
6#16840779—21no—
7D61⋊(C22⋊C8)—21no—
8#16840608—21no—
9#16840606—21no—
10#16840605C2212yes—
11#16840605C2212yes—
12C2×#14519746C411yescharacteristic
13C2×C4×C22⋊D61—21no—
14C2×C4×C22⋊D61C2211yescharacteristic
15#16839641—21no—
16#16839641C411yescharacteristic
17#16839635—21no—
18#16839635C2211yescharacteristic
19#16839618C2211yescharacteristic
20C4×C22⋊Dic61—21no—
21C4×C22⋊Dic61C411yescharacteristic
22C24×C4×C61C411yescharacteristic
23C2×D61⋊C8—41no—
24#14519844—41no—
25#14519834—41no—
26C22⋊#92420—22no—
27C22⋊#92420—22no—
28C4×#92420—41no—
29C22×C22⋊D61—21no—
30C22×C22⋊D61C2×C411yescharacteristic
31C22×C4×D61D411yescharacteristic
32#14519746—22no—
33#14519746C814yescomplemented
34C4×C22⋊D61—41no—
35C4×C22⋊D61—41no—
36C4×C22⋊D61—44no—
37C2×#11758196—21no—
38C2×#11758196D411yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61C2×C411yescharacteristic
43C2×C22⋊Dic61C2×C411yescharacteristic
44C2×C22⋊Dic61C2311yescharacteristic
45C2×#11758198—21no—
46C2×#11758198D411yescharacteristic
47C2×C4×Dic61—21no—
48C2×C4×Dic61D411yescharacteristic
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52C25×C61C2×C411yescharacteristic
53C23×C4×C61—21no—
54C23×C4×C61—21no—
55C23×C4×C61—44no—
56C23×C4×C61—41no—
57C23×C4×C61—41no—
58C23×C4×C61C2×C411yescharacteristic
59C23×C4×C61C2×C411yescharacteristic
60C24×C61—21no—
61C24×C61—21no—
62C24×C61—42no—
63C24×C61—41no—
64C24×C61—42no—
65C24×C61—41no—
66C24×C61C2×C812yescomplemented
67C24×C61C22×C411yescharacteristic
68C23×D61C22⋊C411yescharacteristic
69C22×C4×C61—24no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—44no—
80C22×C4×C61—44no—
81C22×C4×C61—44no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—44no—
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C2×C22⋊D61—24no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61C2×C812yes—
94C2×C22⋊D61M1612yes—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C2×D411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—42no—
103C2×#92420—42no—
104C2×#92420—42no—
105D61⋊C8—82no—
106C22⋊Dic61—21no—
107C22⋊Dic61—24no—
108C22⋊Dic61—41no—
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111C22⋊Dic61—41no—
112C22⋊Dic61C2×C812yescomplemented
113#11758198—41no—
114#11758198—42no—
115#11758196—41no—
116#11758196—42no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—42no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—22no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—44no—
128C23×C61—42no—
129C23×C61—41no—
130C23×C61—44no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—42no—
134C23×C61—48no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61C2×C22⋊C411yescharacteristic
141C23×C61C22×C811yescharacteristic
142C23×C61C2×M1611yescharacteristic
143C22×D61—21no—
144C22×D61—21no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61C22⋊C812yes—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—21no—
151C2×C4×C61—44no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—48no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—44no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61#9911yescharacteristic
162C2×C4×C61#9911yescharacteristic
163C22⋊D61—42no—
164C22⋊D61—42no—
165C22⋊D61—48no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61C22⋊C812yes—
179C4×D61—42no—
180C4×D61—81no—
181#92420—82no—
182#92420—82no—
183#11471—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—24no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—44no—
196C22×C61—44no—
197C22×C61—44no—
198C22×C61—44no—
199C22×C61—42no—
200C22×C61—41no—
201C22×C61—42no—
202C22×C61—41no—
203C22×C61—42no—
204C22×C61C2×C22⋊C811yescharacteristic
205C22×C61C2×#9911yescharacteristic
206C22×C61#41311yescharacteristic
207C2×D61—22no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—42no—
211C2×D61—81no—
212C4×C61—21no—
213C4×C61—42no—
214C4×C61—41no—
215C4×C61AΣL(1,𝔽16)12yes—
216Dic61—22no—
217Dic61—42no—
218Dic61—41no—
219Dic61—41no—
220Dic61—81no—
221C4×#120—611no—
222#1460—611no—
223#1459—611no—
224#1326—612no—
225#1316—612no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—44no—
229C2×C61—41no—
230C2×C61—41no—
231C2×C61#145911yescharacteristic
232C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
233C2×C61#212011yescharacteristic
234D61—42no—
235C24×C4#692311yescharacteristic
236C2×#120—611no—
237C2×C4×D4—611no—
238C2×C4×D4—1221no—
239#431—1221no—
240C2×C22⋊C8—1221no—
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—1221no—
249#323—612no—
250#323—612no—
251#323—1221no—
252C61#1147111yescharacteristic, complemented, Sylow
253C25C2×#692311yescharacteristic
254C22×D4—611no—
255C22×D4—1221no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—44no—
259C23×C4—41no—
260C23×C4—41no—
261C23×C4—611no—
262C23×C4C2×#692311yescharacteristic
263C23×C4C2×#692311yescharacteristic
264C22×C8—2441no—
265#120—614no—
266#120—1222no—
267C4×D4—2441no—
268C4×D4—2441no—
269C4×D4—2444no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—611no—
275C2×C22⋊C4—611no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C42—611no—
281C2×C42—1221no—
282C4⋊C8—2441no—
283C22⋊C8—1222no—
284C22⋊C8—1222no—
285C22⋊C8—2441no—
286C4×C8—2441no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290C24—21no—
291C24—21no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24—611no—
297C24D61⋊C812yescomplemented
298C24C22×#692311yescharacteristic
299C2×D4—612no—
300C2×D4—612no—
301C2×D4—1224no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2442no—
305C2×D4—2442no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—2442no—
331C22×C4C22⋊#692311yescharacteristic
332C22×C4C22⋊#692311yescharacteristic
333C2×C8—4882no—
334C2×C8—2442no—
335C2×C8—2442no—
336C4⋊C4—2441no—
337C4⋊C4—2442no—
338C22⋊C4—612no—
339C22⋊C4—1224no—
340C22⋊C4—1221no—
341C22⋊C4—2441no—
342C22⋊C4—2442no—
343C22⋊C4—2441no—
344C22⋊C4—2442no—
345C22⋊C4—2442no—
346C22⋊C4—2441no—
347C42—1221no—
348C42—1221no—
349C42—2442no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—44no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—48no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—612no—
371C23—1221no—
372C23—1221no—
373C23—2441no—
374C23—2441no—
375C23C2×C22⋊#692311yescharacteristic
376C23C2×D61⋊C811yescharacteristic
377C23D61⋊M1611yescharacteristic
378D4—2442no—
379D4—2442no—
380D4—2448no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—612no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1222no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—4881no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4#1451983511yescharacteristic
410C2×C4#1451983511yescharacteristic
411C8—4882no—
412C8—4882no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—42no—
429C22—41no—
430C22—42no—
431C22—41no—
432C22—42no—
433C22—1222no—
434C22—4881no—
435C22—2442no—
436C22—2441no—
437C22—2441no—
438C22D61⋊(C22⋊C8)11yescharacteristic
439C22C2×#1451983511yescharacteristic
440C22#1684074811yescharacteristic
441C4—21no—
442C4—42no—
443C4—41no—
444C4—1222no—
445C4—4881no—
446C4—2442no—
447C4—2441no—
448C4—2441no—
449C4#1684066212yes—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—2442no—
456C2#2146105411yescharacteristic
457C2C2×#1684066211yescharacteristic
458C2#2146207011yescharacteristic
subgroupG/H
centerC2×C4#14519835
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC24×C4×C61C4
socleC22×C61C2×#99
prime radicalC2×#14519746C4
cosocleC2×C22⋊Dic61C23
hypercenterC24×C4#6923
hypocenterC61#11471
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #21461058C2semidirect
2. #21461054C2semidirect
3. #21460764C2semidirect
4. #21460743C2semidirect
33. #14519746C8semidirect
66. C24×C61C2×C8semidirect
112. C22⋊Dic61C2×C8semidirect
252. C61#11471semidirect
297. C24D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214610582
#214610542
#214607642
#214607432
C4×#145197462
#1147161