Finite Group Databasebeta

← #30617811

Subgroups — #30617811

Order 15616.

1538 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 216 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#21462295C212yesmaximal, complemented
2#21461960C214yesmaximal, complemented
3#21461959C214yesmaximal, complemented
4#21461647C212yesmaximal, complemented
5#21461646C212yesmaximal, complemented
6C2×#16840323C211yesmaximal, characteristic
7C2×#14519836C2214yescomplemented
8C2×#14519836C2218yescomplemented
9C2×#14519835C2214yescomplemented
10#16840712—28no—
11#16840704C2214yescomplemented
12#16840703C2214yes—
13C4×C22⋊#6923C2214yescomplemented
14C23×D244C2211yescharacteristic
15C22×D4×D61C2211yescharacteristic
16#16840323—28no—
17C2×#14519762C412yescomplemented
18C2×#14519760C411yescharacteristic, complemented
19C2×#14519752C2212yes—
20C2×#14519750C2212yes—
21C2×#14519747C411yescharacteristic, complemented
22C2×#14519747C412yescomplemented
23C2×#14519745C411yescharacteristic, complemented
24C2×C22⋊C4×D61C2211yescharacteristic
25C2×#123×C61C411yescharacteristic, complemented
26C2×C4⋊#6923—24no—
27C2×C22⋊#6923—24no—
28C2×C22⋊#6923—24no—
29C2×C22⋊#6923C2318yes—
30C2×C4×#6923—24no—
31#14519836—48no—
32#14519836—44no—
33#14519835—44no—
34(C2×#6923)⋊C4—24no—
35(C2×#6923)⋊C4—24no—
36C24×D61C2×C411yescharacteristic
37C24×D61C2×C411yescharacteristic
38C24×D61C2311yescharacteristic
39C22×D244—24no—
40C22×D244—44no—
41C22×D244C2×C412yes—
42C22×D244C2×C412yes—
43C22×D244C2312yes—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61C2×C411yescharacteristic
46C22×C22⋊D61C2×C412yes—
47C22×C22⋊D61C2311yescharacteristic
48C2×D4×D61—24no—
49C2×D4×D61—44no—
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2312yes—
52#14519762—28no—
53#14519760—24no—
54#14519752—24no—
55#14519752—24no—
56#14519750—24no—
57#14519750—24no—
58#14519747—24no—
59#14519747—28no—
60#14519745—24no—
61C2×#11758196C2×C412yes—
62C2×C22⋊Dic61C2×C411yescharacteristic
63C2×#11758198C2×C412yes—
64C2×#11758198C2×C412yes—
65C2×#11758198C2×C412yes—
66C2×#11758198C2×C412yes—
67C22⋊C4×D61D418yes—
68C22×D4×C61C2×C411yescharacteristic
69C23×C4×C61C2×C411yescharacteristic
70#123×C61—24no—
71C2×C4⋊C4×C61C2×C412yes—
72C2×C22⋊C4×C61C2×C412yes—
73C2×C22⋊C4×C61C2×C411yescharacteristic
74C24×C61C22×C411yescharacteristic
75C24×C61C22×C411yescharacteristic
76C23×D61—24no—
77C23×D61—22no—
78C23×D61—22no—
79C23×D61—42no—
80C23×D61—42no—
81C23×D61—82no—
82C23×D61—82no—
83C23×D61—82no—
84C23×D61C22×C412yes—
85C23×D61C22×C411yescharacteristic
86C23×D61C22×C411yescharacteristic
87C23×D61C2×D414yes—
88C23×D61C2411yescharacteristic
89C22×#6923—28no—
90C2×D4×C61—22no—
91C2×D4×C61—42no—
92C22×C4×C61—22no—
93C22×C4×C61—42no—
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C22×C412yes—
96C22×C4×C61C22×C412yes—
97C2×C22⋊D61—22no—
98C2×C22⋊D61—22no—
99C2×C22⋊D61—24no—
100C2×C22⋊D61—24no—
101C2×C22⋊D61—42no—
102C2×C22⋊D61—82no—
103C22×Dic61C22×C411yescharacteristic
104C22×Dic61C22×C412yes—
105D4×D61—416no—
106C2×D244—24no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—24no—
111C2×D244—28no—
112C2×D244—24no—
113C2×D244—44no—
114C2×D244—416no—
115C2×D244—82no—
116C2×C4×D61—22no—
117C2×C4×D61—42no—
118C2×C4×D61—82no—
119C2×C4×D61C2×D414yes—
120C2×C4×D61C2×D414yes—
121C22⋊#6923—28no—
122C22⋊#6923—28no—
123C22⋊#6923—84no—
124C22⋊#6923—84no—
125C4⋊#6923—84no—
126C4×#6923—84no—
127C4⋊C4×C61—24no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61C22⋊C414yes—
130C22⋊Dic61C22⋊C414yes—
131#11758198—24no—
132#11758198—24no—
133#11758198—24no—
134#11758198C22⋊C418yes—
135#11758196—24no—
136C23×C61—22no—
137C23×C61—21no—
138C23×C61—22no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61C2×C22⋊C412yes—
142C23×C61C23×C411yescharacteristic
143C22×D61—22no—
144C22×D61—28no—
145C22×D61—22no—
146C22×D61—21no—
147C22×D61—24no—
148C22×D61—22no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—44no—
152C22×D61—44no—
153C22×D61—44no—
154C22×D61—44no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—84no—
159C22×D61—84no—
160C22×D61—88no—
161C22×D61—88no—
162C22×D61—88no—
163C22×D61C2×C22⋊C412yes—
164C22×D61C22×D412yes—
165C22×D61H(5,𝔽2)14yes—
166C2×#6923—48no—
167C2×#6923—48no—
168C2×#6923—48no—
169D4×C61—44no—
170C2×C4×C61—22no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—42no—
175C2×C4×C61—44no—
176C2×C4×C61C2×C22⋊C414yes—
177C22⋊D61—44no—
178C22⋊D61—44no—
179C22⋊D61—84no—
180C2×Dic61—21no—
181C2×Dic61—22no—
182C2×Dic61—42no—
183C2×Dic61C2×C22⋊C414yes—
184D244—48no—
185D244—44no—
186D244—44no—
187D244—44no—
188D244—84no—
189C4×D61—44no—
190C4×D61—44no—
191C4×D61—44no—
192#21795—611nomaximal, Sylow
193C22×C61—22no—
194C22×C61—22no—
195C22×C61—24no—
196C22×C61—44no—
197C22×C61—41no—
198C22×C61—42no—
199C22×C61—82no—
200C22×C61C22×C22⋊C411yescharacteristic
201C22×C61#51812yes—
202C2×D61—22no—
203C2×D61—22no—
204C2×D61—22no—
205C2×D61—24no—
206C2×D61—44no—
207C2×D61—44no—
208C2×D61—42no—
209C2×D61—41no—
210C2×D61—42no—
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—88no—
214C2×D61—88no—
215C2×D61—88no—
216C2×D61—84no—
217C2×D61—82no—
218C2×D61#53514yes—
219#6923—88no—
220C4×C61—24no—
221C4×C61—41no—
222C4×C61—42no—
223Dic61—24no—
224Dic61—41no—
225C2×#534—611no—
226#2312—612no—
227#1964—614no—
228#1964—614no—
229#1781—612no—
230#1781—612no—
231C2×C61—22no—
232C2×C61—42no—
233C2×C61—81no—
234C2×C61#230911yescharacteristic
235C2×C61#288312yes—
236D61—24no—
237D61—42no—
238D61—82no—
239D61—81no—
240C23×D4—611no—
241C23×D4—611no—
242#534—1228no—
243C2×#123—611no—
244C2×#123—612no—
245C2×#123#692311yescharacteristic, complemented
246C2×#121—612no—
247C2×#121—612no—
248C2×#120—611no—
249C2×#120—612no—
250C2×#120—611no—
251C22×C22⋊C4—611no—
252C2×#99—614no—
253C2×#99—614no—
254C2×#99—618no—
255#388—1228no—
256#380—614no—
257#379—614no—
258C4×C22⋊C4—614no—
259C61#2179511yescharacteristic, complemented, Sylow
260C25—611no—
261C25—611no—
262C25—611no—
263C22×D4—611no—
264C22×D4—612no—
265C22×D4—611no—
266C22×D4—611no—
267C22×D4—612no—
268C22×D4—612no—
269C22×D4—612no—
270C22×D4—1224no—
271C22×D4—1224no—
272C22×D4—2444no—
273C22×D4—2444no—
274C22×D4C2×#692311yescharacteristic
275C23×C4—612no—
276C23×C4—611no—
277C23×C4C2×#692311yescharacteristic
278#123—24no—
279#123—1228no—
280#123—1224no—
281#121—1224no—
282#121—1224no—
283#121—1224no—
284#121—1224no—
285#120—1228no—
286#120—1224no—
287#120—1224no—
288C2×C4⋊C4—612no—
289C2×C4⋊C4—1224no—
290C2×C4⋊C4C2×#692312yes—
291C2×C22⋊C4—611no—
292C2×C22⋊C4—612no—
293C2×C22⋊C4—618no—
294C2×C22⋊C4—612no—
295C2×C22⋊C4—612no—
296C2×C22⋊C4—612no—
297C2×C22⋊C4—618no—
298C2×C22⋊C4—1224no—
299C2×C22⋊C4—1224no—
300C2×C22⋊C4C2×#692312yes—
301C2×C22⋊C4C2×#692311yescharacteristic
302C2×C42—1224no—
303#99—2444no—
304#99—2448no—
305#99—2444no—
306#95—1224no—
307#95—1224no—
308C24—611no—
309C24—614no—
310C24—611no—
311C24—612no—
312C24—611no—
313C24—1224no—
314C24—1222no—
315C24—1222no—
316C24—4882no—
317C24—4882no—
318C24—4882no—
319C24—2442no—
320C24—2442no—
321C24C22×#692311yescharacteristic
322C24C22×#692311yescharacteristic
323C2×D4—22no—
324C2×D4—42no—
325C2×D4—1228no—
326C2×D4—1224no—
327C2×D4—1224no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1224no—
332C2×D4—1224no—
333C2×D4—1222no—
334C2×D4—1222no—
335C2×D4—1224no—
336C2×D4—4882no—
337C2×D4—4882no—
338C2×D4—2444no—
339C2×D4—24416no—
340C2×D4—2442no—
341C2×D4—24416no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—612no—
345C22×C4—614no—
346C22×C4—614no—
347C22×C4—611no—
348C22×C4—1222no—
349C22×C4—1228no—
350C22×C4—4882no—
351C22×C4—2442no—
352C22×C4C22×#692311yescharacteristic
353C22×C4C22×#692312yes—
354C22×C4C22×#692312yes—
355C4⋊C4—24no—
356C4⋊C4—1224no—
357C4⋊C4—4884no—
358C22⋊C4—24no—
359C22⋊C4—614no—
360C22⋊C4—618no—
361C22⋊C4—1224no—
362C22⋊C4—1224no—
363C22⋊C4—1224no—
364C22⋊C4—1228no—
365C22⋊C4—1228no—
366C22⋊C4—4884no—
367C22⋊C4—4884no—
368C22⋊C4C22⋊#692314yes—
369C42—4884no—
370C23—22no—
371C23—21no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—612no—
376C23—614no—
377C23—612no—
378C23—1228no—
379C23—1222no—
380C23—1224no—
381C23—1222no—
382C23—1221no—
383C23—1222no—
384C23—4884no—
385C23—4884no—
386C23—4888no—
387C23—4888no—
388C23—4888no—
389C23—2444no—
390C23—2444no—
391C23—2442no—
392C23—2442no—
393C23—2442no—
394C23—2442no—
395C23—2442no—
396C23—2444no—
397C23—2444no—
398C23C2×C22⋊#692312yes—
399C23C23×#692311yescharacteristic
400D4—44no—
401D4—4884no—
402D4—4884no—
403D4—2444no—
404D4—2444no—
405D4—2444no—
406D4—2448no—
407D4—2444no—
408D4—2444no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—614no—
416C2×C4—1221no—
417C2×C4—1222no—
418C2×C4—2444no—
419C2×C4—2444no—
420C2×C4—2444no—
421C2×C4—2442no—
422C2×C4—2448no—
423C2×C4—2448no—
424C2×C4—2448no—
425C2×C4C2×C22⋊#692314yes—
426C22—22no—
427C22—22no—
428C22—24no—
429C22—44no—
430C22—41no—
431C22—42no—
432C22—82no—
433C22—614no—
434C22—1222no—
435C22—1222no—
436C22—1222no—
437C22—1224no—
438C22—4884no—
439C22—4882no—
440C22—4888no—
441C22—4888no—
442C22—4888no—
443C22—2444no—
444C22—2444no—
445C22—2442no—
446C22—2441no—
447C22—2442no—
448C22—2442no—
449C22—2442no—
450C22C22×C22⋊#692311yescharacteristic
451C22#1684082912yes—
452C4—24no—
453C4—42no—
454C4—41no—
455C4—1224no—
456C4—4888no—
457C4—2441no—
458C2—22no—
459C2—42no—
460C2—81no—
461C2—1224no—
462C2—4882no—
463C2—4881no—
464C2—2442no—
465C2#2146229211yescharacteristic
466C2#2146241912yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840323C2
cosocleC23×D61C24
hypercenterC2×#123#6923
hypocenterC61#21795
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #21462295C2semidirect
2. #21461960C2semidirect
3. #21461959C2semidirect
4. #21461647C2semidirect
5. #21461646C2semidirect
7. C2×#14519836C22semidirect
8. C2×#14519836C22semidirect
9. C2×#14519835C22semidirect
11. #16840704C22semidirect
13. C4×C22⋊#6923C22semidirect
17. C2×#14519762C4semidirect
18. C2×#14519760C4semidirect
21. C2×#14519747C4semidirect
22. C2×#14519747C4semidirect
23. C2×#14519745C4semidirect
25. C2×#123×C61C4semidirect
245. C2×#123#6923semidirect
259. C61#21795semidirect

Maximal subgroups

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

subgroupindex
#214622952
#214619602
#214619592
#214616472
#214616462
C2×#168403232
#2179561