Finite Group Databasebeta

← #30607689

Subgroups — #30607689

Order 15616.

724 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C2×#16840741C211yesmaximal, characteristic
2C2×#16840664C212yesmaximal, complemented
3#21461655C211yesmaximal, characteristic
4#21460903C212yesmaximal, complemented
5#21449154C211yesmaximal, characteristic
6C2×#14519856—21no—
7C2×#14519838—21no—
8C2×#14519837C2212yes—
9#16840741—24no—
10C2×#14519836—21no—
11C2×#14519835C2212yes—
12#16840734—21no—
13#16840664—24no—
14C22×#11758222C2211yescharacteristic
15C2×#14519760C411yescharacteristic, complemented
16C2×#14519760C2211yescharacteristic
17#16839682—21no—
18#16839679—21no—
19#16839641C411yescharacteristic, complemented
20#16839641C2211yescharacteristic
21#16839633—21no—
22#16839633—21no—
23#16839625C411yescharacteristic, complemented
24C22×C4⋊C4×C61C411yescharacteristic, complemented
25C2×#11758217—22no—
26#14519856—24no—
27#14519856—42no—
28C2×C4⋊#6923—41no—
29C2×C22⋊#6923—22no—
30C2×C4×#6923—41no—
31#14519838—42no—
32#14519837—22no—
33#14519837—22no—
34#14519836—24no—
35#14519835—22no—
36#14519835D414yes—
37#14519834—41no—
38#14519834—41no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—42no—
42C2×#11758222—41no—
43C2×#11758222—41no—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61C2×C411yescharacteristic
47C22×C22⋊D61C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50#14519760—22no—
51#14519760—22no—
52#14519760—42no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61C2×C411yescharacteristic
56C2×C22⋊Dic61C2×C411yescharacteristic
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C23×C4×C61C2×C411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C23×C4×C61C2×C411yescharacteristic
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—41no—
71C2×C4⋊C4×C61—41no—
72C24×C61C22×C411yescharacteristic
73C23×D61C22⋊C411yescharacteristic
74C23×D61C2×D411yescharacteristic
75C22×#6923—42no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61—41no—
83C22×C4×C61—42no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22⋊C411yescharacteristic
89C2×C22⋊D61—22no—
90C2×C22⋊D61—22no—
91C2×C22⋊D61—41no—
92C2×C22⋊D61—41no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61C2×D412yes—
95C2×C22⋊D61#4112yes—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100#11758222—44no—
101#11758222—44no—
102#11758222—44no—
103#11758222—44no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—41no—
107C2×D244—41no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—22no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—22no—
116C2×C4×D61—41no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—41no—
120C22⋊#6923—24no—
121C22⋊#6923—42no—
122#11758217—42no—
123#11758217—42no—
124C2×#92420—42no—
125C4⋊#6923—44no—
126C4×#6923—44no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C22⋊Dic61—22no—
131C22⋊Dic61—41no—
132#11758198—42no—
133#11758198—42no—
134#11758198—42no—
135#11758198—41no—
136#11758198—41no—
137#11758196—42no—
138#11758196—42no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61C2×C22⋊C411yescharacteristic
144C23×C61C4×D412yes—
145C22×D61—21no—
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—41no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61#12011yescharacteristic
155C22×D61#12311yescharacteristic
156C2×#6923—44no—
157C2×#6923—82no—
158C2×#6923—82no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—82no—
175C2×C4×C61#9911yescharacteristic
176C2×C4×C61#9911yescharacteristic
177C22⋊D61—42no—
178C22⋊D61—42no—
179C22⋊D61—44no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61#9912yes—
191C2×Dic61#12112yes—
192D244—42no—
193D244—42no—
194C4×D61—22no—
195C4×D61—22no—
196C4×D61—22no—
197C4×D61—22no—
198C4×D61—42no—
199C4×D61—42no—
200Dic122—44no—
201#92420—82no—
202#13085—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—22no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61#38611yescharacteristic
211C22×C61C2×#9911yescharacteristic
212C22×C61C2×#9911yescharacteristic
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—41no—
216C2×D61—41no—
217C2×D61—41no—
218C2×D61—41no—
219C2×D61—42no—
220C2×D61—41no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61—81no—
224C2×D61—81no—
225#6923—84no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—42no—
229C4×C61—41no—
230C4×C61—81no—
231Dic61—22no—
232Dic61—22no—
233Dic61—81no—
234Dic61—81no—
235C2×#410—611no—
236#2406—611no—
237C2×#352—612no—
238#1785—611no—
239#1390—612no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—42no—
243C2×C61#189811yescharacteristic
244C2×C61#212011yescharacteristic
245C2×C61#212311yescharacteristic
246D61—41no—
247D61—41no—
248D61—81no—
249C22×#41—611no—
250C2×#123—611no—
251C2×#123—611no—
252C2×#117—1221no—
253C22×C4⋊C4#692311yescharacteristic, complemented
254C2×#101—1221no—
255C2×#100—612no—
256#410—1224no—
257C2×#99—612no—
258C2×#99—1221no—
259#407—1221no—
260#395—1221no—
261#394—1221no—
262#386—611no—
263#386—611no—
264#385—1221no—
265#385—1221no—
266#381—611no—
267#352—1224no—
268C61#1308511yescharacteristic, complemented, Sylow
269C2×#41—1222no—
270C2×#41—1222no—
271C2×#41—2441no—
272C2×#41—2441no—
273C2×#41—2442no—
274C22×Q8—611no—
275C22×D4—611no—
276C22×D4—611no—
277C22×D4—611no—
278C23×C4—611no—
279C23×C4—611no—
280C23×C4C2×#692311yescharacteristic
281C23×C4C2×#692311yescharacteristic
282C23×C4C2×#692311yescharacteristic
283C2×M16—1222no—
284#123—1222no—
285#123—1222no—
286#123—2442no—
287#117—1224no—
288#117—2442no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—2441no—
296C2×C22⋊C4—611no—
297C2×C22⋊C4—611no—
298C2×C22⋊C4—1222no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C42—2441no—
304#101—2442no—
305#100—1222no—
306#100—1222no—
307#99—614no—
308#99—1222no—
309#99—1224no—
310C22⋊C8—2441no—
311C22⋊C8—2441no—
312#95—1221no—
313#95—1221no—
314#95—1221no—
315#95—1221no—
316C24—611no—
317C24—611no—
318C24C22×#692311yescharacteristic
319#41—2444no—
320#41—2444no—
321#41—2444no—
322#41—2444no—
323C2×Q8—1222no—
324C2×Q8—1222no—
325C2×Q8—2441no—
326C2×D4—612no—
327C2×D4—612no—
328C2×D4—1221no—
329C2×D4—1221no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2441no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—1222no—
351C22×C4—1222no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—2441no—
361C22×C4—2442no—
362C22×C4C22⋊#692311yescharacteristic
363C22×C4C22⋊#692311yescharacteristic
364M16—2442no—
365M16—2442no—
366C2×C8—2442no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—2442no—
371C4⋊C4—2442no—
372C4⋊C4—2444no—
373C22⋊C4—1224no—
374C22⋊C4—1222no—
375C22⋊C4—2442no—
376C22⋊C4—2442no—
377C22⋊C4—2441no—
378C22⋊C4—2441no—
379C22⋊C4—2442no—
380C22⋊C4—2442no—
381C22⋊C4—2441no—
382C42—2444no—
383C23—21no—
384C23—21no—
385C23—41no—
386C23—41no—
387C23—611no—
388C23—611no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23—1221no—
393C23—1221no—
394C23—2441no—
395C23—2441no—
396C23—2441no—
397C23—2441no—
398C23C2×C22⋊#692311yescharacteristic
399C23D4×#692312yes—
400Q8—2444no—
401D4—2442no—
402D4—2442no—
403D4—2442no—
404D4—2442no—
405D4—2444no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—82no—
422C2×C4—612no—
423C2×C4—612no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—4882no—
433C2×C4—4882no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2444no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4#1451983511yescharacteristic
444C2×C4#1451983511yescharacteristic
445C8—4882no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—41no—
452C22—41no—
453C22—1221no—
454C22—1221no—
455C22—4881no—
456C22—4881no—
457C22—2442no—
458C22—2441no—
459C22—2441no—
460C22—2441no—
461C22—2441no—
462C22—2441no—
463C22—2441no—
464C22—2441no—
465C22#1684071011yescharacteristic
466C22C2×#1451983511yescharacteristic
467C22D61⋊#9911yescharacteristic
468C4—21no—
469C4—21no—
470C4—42no—
471C4—41no—
472C4—81no—
473C4—1222no—
474C4—1222no—
475C4—4881no—
476C4—4881no—
477C4—4884no—
478C2—21no—
479C2—21no—
480C2—42no—
481C2—4881no—
482C2—2441no—
483C2—2441no—
484C2#2146184011yescharacteristic
485C2#2146207011yescharacteristic
486C2#2146208111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×C22⋊D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#13085
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16840664C2semidirect
4. #21460903C2semidirect
15. C2×#14519760C4semidirect
19. #16839641C4semidirect
23. #16839625C4semidirect
24. C22×C4⋊C4×C61C4semidirect
253. C22×C4⋊C4#6923semidirect
268. C61#13085semidirect

Maximal subgroups

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

subgroupindex
C2×#168407412
C2×#168406642
#214616552
#214609032
#214491542
#1308561