Finite Group Databasebeta

← #30598303

Subgroups — #30598303

Order 15616.

1074 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461660C212yesmaximal, complemented
2C4×#14519836C211yesmaximal, characteristic, complemented
3#21461218C212yesmaximal, complemented
4#21461076C211yesmaximal, characteristic, complemented
5C2×#16840416C211yesmaximal, characteristic
6C2×#14519841—24no—
7C2×#14519836—21no—
8#16840733C2212yes—
9C4×C22⋊#6923C2212yes—
10#16840640—21no—
11#16840615—24no—
12#16840607—21no—
13#16840606—21no—
14C22×#11758224C411yescharacteristic, complemented
15C22×#11758224C2211yescharacteristic
16#16840416—24no—
17#16840416—24no—
18C2×#14519777C411yescharacteristic, complemented
19C2×#14519761—21no—
20C2×#14519761—21no—
21C2×#14519757C411yescharacteristic, complemented
22C2×#14519757C2211yescharacteristic
23C2×#14519751—21no—
24C2×C4×D244C2211yescharacteristic
25C22×C4⋊Dic61—21no—
26C2×C4×D4×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28C2×C22⋊#6923—22no—
29C2×C4×#6923—22no—
30#14519841—48no—
31#14519836—24no—
32#14519834—22no—
33C22⋊#92420—22no—
34(C2×#6923)⋊C4—22no—
35C2×#11758224—22no—
36C2×#11758224—22no—
37C2×#11758224—22no—
38C2×#11758224—44no—
39C2×#11758224—44no—
40C2×#11758224D414yes—
41C22×Dic122C2×C411yescharacteristic
42C22×Dic122C2×C411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C22⋊D61—21no—
45C22×C22⋊D61—21no—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50#14519777—22no—
51#14519777—22no—
52#14519761—28no—
53#14519761—28no—
54#14519757—22no—
55#14519757—22no—
56#14519757—22no—
57#14519757D414yes—
58#14519751—28no—
59C4×D244—22no—
60C4×D244—22no—
61C2×C4⋊Dic61—28no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198D411yescharacteristic
70C2×#11758198D411yescharacteristic
71C22×D4×C61C2×C411yescharacteristic
72C23×C4×C61—21no—
73C4×D4×C61—22no—
74C4×D4×C61—22no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2×C411yescharacteristic
78C24×C61—21no—
79C23×D61C22⋊C411yescharacteristic
80C23×D61C2×D411yescharacteristic
81C22×#6923—42no—
82C2×D4×C61—24no—
83C2×D4×C61—21no—
84C2×D4×C61C22⋊C412yes—
85C22×C4×C61—24no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22×C411yescharacteristic
92C2×C22⋊D61—24no—
93C2×C22⋊D61—24no—
94C2×C22⋊D61—44no—
95C2×C22⋊D61—44no—
96C22×Dic61—24no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—24no—
100C22×Dic61—44no—
101C22×Dic61—44no—
102C22×Dic61C22⋊C411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104#11758224—28no—
105#11758224—44no—
106#11758224—48no—
107#11758224—44no—
108#11758224—48no—
109#11758224—44no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—41no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61C2×D412yes—
118C2×C4×D61C2×D412yes—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—42no—
122C2×Dic122—42no—
123C2×Dic122C22⋊C412yes—
124C2×Dic122C22⋊C412yes—
125C22⋊#6923—24no—
126C22⋊#6923—42no—
127C2×#92420—42no—
128C4×#6923—44no—
129#11758213—44no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61C22⋊C412yes—
132C22⋊C4×C61—24no—
133C42×C61—21no—
134C42×C61—21no—
135#11758198—24no—
136#11758198—24no—
137#11758198—22no—
138#11758198—22no—
139C4⋊Dic61—24no—
140C4⋊Dic61—24no—
141C4⋊Dic61—24no—
142C4⋊Dic61—22no—
143C4⋊Dic61—24no—
144C23×C61—24no—
145C23×C61—22no—
146C23×C61—42no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61C2×C22⋊C411yescharacteristic
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61#12012yes—
160C2×#6923—44no—
161C2×#6923—82no—
162C2×#6923—82no—
163D4×C61—24no—
164D4×C61—22no—
165D4×C61#9914yescomplemented
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—22no—
171C2×C4×C61—48no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#10412yescomplemented
176C2×C4×C61C2×C22⋊C411yescharacteristic
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C22⋊D61—48no—
179C22⋊D61—48no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—21no—
183C2×Dic61—41no—
184C2×Dic61—48no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—48no—
188C2×Dic61—41no—
189C2×Dic61#12012yes—
190D244—44no—
191C4×D61—24no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—42no—
195Dic122—24no—
196Dic122—42no—
197Dic122—42no—
198Dic122—42no—
199#92420—82no—
200#8364—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—24no—
204C22×C61—44no—
205C22×C61—44no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61#37911yescharacteristic
209C22×C61C2×#10411yescharacteristic
210C22×C61#42111yescharacteristic
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—41no—
215C2×D61—41no—
216C2×D61—41no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221#6923—84no—
222C4×C61—21no—
223C4×C61—22no—
224C4×C61—41no—
225C4×C61—42no—
226C4×C61C2×#9912yes—
227Dic61—22no—
228Dic61—42no—
229Dic61—42no—
230Dic61—41no—
231C2×#547—611no—
232#1789—612no—
233C4×#99—611no—
234#1508—612no—
235#1465—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—44no—
239C2×C61#178111yescharacteristic
240C2×C61#178911yescharacteristic
241C2×C61#179611yescharacteristic
242D61—42no—
243D61—81no—
244C22×#41—611no—
245C22×#41—611no—
246#547—1224no—
247#547—1224no—
248C2×C4⋊Q8—611no—
249C2×#123—1221no—
250C2×#123—1221no—
251C2×C22⋊Q8—611no—
252C2×C22⋊Q8—611no—
253C2×#121—1221no—
254C2×C4×D4—611no—
255C2×C4×D4#692311yescharacteristic, complemented
256C22×C4⋊C4—1221no—
257C2×#104—1224no—
258C2×#99—1221no—
259#407—612no—
260C4×C22⋊C4—612no—
261#342—1221no—
262#328—1224no—
263#324—1221no—
264#323—1221no—
265C61#836411yescharacteristic, complemented, Sylow
266C2×#41—614no—
267C2×#41—1222no—
268C2×#41—1222no—
269C2×#41—1222no—
270C2×#41—2444no—
271C2×#41—2444no—
272C22×Q8—611no—
273C22×Q8—611no—
274C22×D4—611no—
275C22×D4—1221no—
276C22×D4—1221no—
277C22×D4C2×#692311yescharacteristic
278C23×C4—21no—
279C23×C4—611no—
280C23×C4—611no—
281C23×C4—1221no—
282C23×C4—1221no—
283C2×M16—1222no—
284C4⋊Q8—1222no—
285C4⋊Q8—1222no—
286#123—1228no—
287#123—1228no—
288C22⋊Q8—614no—
289C22⋊Q8—1222no—
290C22⋊Q8—1222no—
291C22⋊Q8—1222no—
292#121—1228no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—1222no—
296C4×D4—1222no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1228no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4C2×#692311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—611no—
306C2×C22⋊C4—611no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1222no—
310C2×C42—1222no—
311C2×C42C2×#692311yescharacteristic
312#104—2448no—
313#99—1224no—
314C22⋊C8—1222no—
315C22⋊C8—1222no—
316#95—1222no—
317C24—21no—
318C24—611no—
319C24—611no—
320#41—1228no—
321#41—2444no—
322#41—2448no—
323#41—2448no—
324#41—2444no—
325#41—2444no—
326C2×Q8—612no—
327C2×Q8—612no—
328C2×Q8—1221no—
329C2×Q8—1221no—
330C2×Q8—2442no—
331C2×Q8—2442no—
332C2×D4—24no—
333C2×D4—21no—
334C2×D4—1222no—
335C2×D4—1224no—
336C2×D4—1224no—
337C2×D4—1222no—
338C2×D4—2441no—
339C2×D4—2444no—
340C2×D4—2444no—
341C2×D4C22⋊#692312yes—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—611no—
347C22×C4—612no—
348C22×C4—612no—
349C22×C4—611no—
350C22×C4—1224no—
351C22×C4—1224no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—2442no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4—2444no—
360C22×C4—2444no—
361C22×C4C22⋊#692311yescharacteristic
362C22×C4C22⋊#692311yescharacteristic
363C22×C4C22×#692311yescharacteristic
364M16—2444no—
365C2×C8—2442no—
366C4⋊C4—21no—
367C4⋊C4—1222no—
368C4⋊C4—1224no—
369C4⋊C4—1224no—
370C4⋊C4—1224no—
371C4⋊C4—1224no—
372C4⋊C4C22⋊#692312yes—
373C22⋊C4—24no—
374C22⋊C4—1222no—
375C22⋊C4—1222no—
376C22⋊C4—1224no—
377C22⋊C4—1224no—
378C22⋊C4—1224no—
379C22⋊C4—2442no—
380C42—21no—
381C42—21no—
382C42—2444no—
383C23—24no—
384C23—22no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—612no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23—1221no—
393C23—1221no—
394C23—2441no—
395C23—2441no—
396C23—2441no—
397C23—2441no—
398C23C2×C22⋊#692311yescharacteristic
399Q8—1224no—
400Q8—2442no—
401Q8—2442no—
402Q8—2442no—
403D4—24no—
404D4—22no—
405D4—2444no—
406D4—2448no—
407D4—2448no—
408D4#1451983614yescomplemented
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—48no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—612no—
419C2×C4—1224no—
420C2×C4—1221no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—4882no—
424C2×C4—4882no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2448no—
429C2×C4—2444no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2448no—
433C2×C4—2442no—
434C2×C4—2441no—
435C2×C4#1451984112yescomplemented
436C2×C4C2×C22⋊#692311yescharacteristic
437C2×C4C2×C22⋊#692311yescharacteristic
438C8—4882no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—1222no—
447C22—1222no—
448C22—1222no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22—2441no—
454C22—2441no—
455C22—2441no—
456C22#1684070311yescharacteristic
457C22C2×#1451984111yescharacteristic
458C22#1684076011yescharacteristic
459C4—21no—
460C4—22no—
461C4—42no—
462C4—41no—
463C4—1222no—
464C4—4884no—
465C4—2441no—
466C4—2442no—
467C4—2442no—
468C4C2×#1451983612yes—
469C2—21no—
470C2—21no—
471C2—44no—
472C2—4881no—
473C2—2442no—
474C2#2146164711yescharacteristic
475C2#2146166011yescharacteristic
476C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61#379
prime radicalC2×#16840416C2
cosocleC22×C4×D61C23
hypercenterC2×C4×D4#6923
hypocenterC61#8364
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461660C2semidirect
2. C4×#14519836C2semidirect
3. #21461218C2semidirect
4. #21461076C2semidirect
14. C22×#11758224C4semidirect
18. C2×#14519777C4semidirect
21. C2×#14519757C4semidirect
26. C2×C4×D4×C61C4semidirect
165. D4×C61#99semidirect
175. C2×C4×C61#104semidirect
255. C2×C4×D4#6923semidirect
265. C61#8364semidirect
408. D4#14519836semidirect
435. C2×C4#14519841semidirect

Maximal subgroups

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

subgroupindex
#214616602
C4×#145198362
#214612182
#214610762
C2×#168404162
#836461