Finite Group Databasebeta

← #30599260

Subgroups — #30599260

Order 15616.

868 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461642C211yesmaximal, characteristic
2#21460993C211yesmaximal, characteristic
3#21460692C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21448201C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C2×#14519834—21no—
8C2×#14519834C2212yes—
9C2×C22⋊#92420—21no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4C2212yes—
12C23×C4×D61C2211yescharacteristic
13C2×#14519735C411yescharacteristic, complemented
14C2×#14519735C411yescharacteristic, complemented
15#16839624—21no—
16#16839622C411yescharacteristic, complemented
17#16839619C2211yescharacteristic
18#16839619C2211yescharacteristic
19#16839617—21no—
20C22⋊C4×Dic61—21no—
21#16839609—21no—
22C61×#380C411yescharacteristic, complemented
23C23×#6923—22no—
24C22×#92420—22no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—42no—
27C2×C22⋊#6923—42no—
28#14519834—24no—
29#14519834—42no—
30C22⋊#92420—42no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—42no—
33C24×D61D411yescharacteristic
34C23×Dic61D411yescharacteristic
35C22×C4×D61—24no—
36C22×C4×D61—42no—
37C22×C4×D61—42no—
38C22×C4×D61C2×C411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42#14519735—42no—
43#14519735—42no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×C22⋊Dic61—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56C23×C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60#95×C61—21no—
61C24×C61C22⋊C411yescharacteristic
62C23×D61—22no—
63C23×D61—22no—
64C23×D61—21no—
65C23×D61—21no—
66C23×D61—22no—
67C23×D61—22no—
68C23×D61—41no—
69C23×D61—41no—
70C23×D61C22⋊C411yescharacteristic
71C23×D61C2×D411yescharacteristic
72C22×#6923—24no—
73C22×#6923—42no—
74C22×#6923—42no—
75C22×#6923—42no—
76C22×#6923—42no—
77C22×#6923—42no—
78C22×#6923—42no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×Dic61—22no—
87C22×Dic61—22no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—41no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C2×D411yescharacteristic
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—416no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C4×D61#4114yes—
106C22⋊#6923—48no—
107C2×#92420—42no—
108C2×#92420—42no—
109C2×#92420—42no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—42no—
113C22⋊Dic61—42no—
114#11758198—42no—
115#11758198—42no—
116#11758198—42no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121C4×Dic61—42no—
122C4×Dic61—42no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61#10412yes—
130C23×C61C2×C22⋊C411yescharacteristic
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—24no—
136C22×D61—24no—
137C22×D61—21no—
138C22×D61—21no—
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—44no—
144C22×D61—44no—
145C22×D61—44no—
146C22×D61—44no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61#9912yes—
152C22×D61#12312yes—
153C22×D61#13512yes—
154C22×D61AGL(1,ℤ/8)11yescharacteristic
155C22×D61#13711yescharacteristic
156C2×#6923—44no—
157C2×#6923—44no—
158C2×#6923—44no—
159C2×#6923—82no—
160C2×#6923—82no—
161C2×#6923—82no—
162C2×#6923—82no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—44no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#11712yes—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—44no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61#9912yes—
188C2×Dic61#12312yes—
189C4×D61—28no—
190C4×D61—44no—
191C4×D61—44no—
192C4×D61—44no—
193#92420—82no—
194#8715—611nomaximal, Sylow
195C22×C61—22no—
196C22×C61—22no—
197C22×C61—22no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61#38111yescharacteristic
203C22×C61C2×#9911yescharacteristic
204C22×C61C2×#9911yescharacteristic
205C22×C61#41611yescharacteristic
206C22×C61#41711yescharacteristic
207C22×C61C2×#10411yescharacteristic
208C22×C61#42111yescharacteristic
209C2×D61—22no—
210C2×D61—22no—
211C2×D61—22no—
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—48no—
221C2×D61—48no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61#48212yes—
227C2×D61#48512yes—
228#6923—84no—
229C4×C61—22no—
230C4×C61—41no—
231C4×C61—81no—
232C4×C61—81no—
233Dic61—22no—
234Dic61—22no—
235Dic61—81no—
236Dic61—81no—
237#2280—611no—
238#1779—611no—
239C22⋊#95—611no—
240#1294—612no—
241#1294—612no—
242C2×C61—42no—
243C2×C61#150111yescharacteristic
244C2×C61#150411yescharacteristic
245C2×C61#151311yescharacteristic
246C2×C61#151611yescharacteristic
247C2×C61#181911yescharacteristic
248C2×C61#182511yescharacteristic
249C2×C61#182611yescharacteristic
250D61—22no—
251D61—22no—
252D61—42no—
253C24×C4—611no—
254C2×#117—611no—
255C2×#117—611no—
256C22×C22⋊C4—1221no—
257C2×C22⋊C8—612no—
258C2×C22⋊C8—1221no—
259C2×C22⋊C8—1221no—
260#381—1221no—
261#380—611no—
262#380—611no—
263#380—611no—
264#380#692311yescharacteristic, complemented
265#379—1221no—
266C4×C22⋊C4—1221no—
267C4×C22⋊C4—1221no—
268C2×#95—612no—
269C2×#95—1221no—
270C61#871511yescharacteristic, complemented, Sylow
271C25—611no—
272C23×C4—611no—
273C23×C4—611no—
274C23×C4—611no—
275C23×C4—611no—
276C23×C4—611no—
277C23×C4—1224no—
278C23×C4—1222no—
279C23×C4—2442no—
280C23×C4—2442no—
281C23×C4C2×#692311yescharacteristic
282C22×C8—1222no—
283#117—2442no—
284#117—2442no—
285C2×C4⋊C4—611no—
286C2×C4⋊C4—611no—
287C2×C4⋊C4C2×#692311yescharacteristic
288C2×C4⋊C4C2×#692311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1224no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—2442no—
297C2×C22⋊C4—2442no—
298C2×C42—1221no—
299C2×C42—1221no—
300C22⋊C8—1224no—
301C22⋊C8—2442no—
302C22⋊C8—2442no—
303#95—21no—
304#95—1221no—
305#95—1221no—
306#95—1224no—
307#95—1221no—
308#95—2442no—
309C24—611no—
310C24—611no—
311C24—1221no—
312C24—1222no—
313C24—1221no—
314C24—1222no—
315C24—1222no—
316C24—1222no—
317C24—2441no—
318C24—2441no—
319C24C22⋊#692311yescharacteristic
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—614no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—1222no—
333C22×C4—1222no—
334C22×C4—1222no—
335C22×C4—1222no—
336C22×C4—1222no—
337C22×C4—1224no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—2441no—
341C22×C4—2441no—
342C22×C4—2441no—
343C22×C4—2441no—
344C22×C4—24416no—
345C22×C4—2441no—
346C22×C4—2442no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4—2442no—
352C22×C4C22⋊#692311yescharacteristic
353C22×C4C22×#692311yescharacteristic
354C2×C8—2442no—
355C2×C8—2442no—
356C2×C8—2442no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C22⋊C4—42no—
362C22⋊C4—2442no—
363C22⋊C4—2442no—
364C22⋊C4—2441no—
365C22⋊C4—2441no—
366C22⋊C4—2442no—
367C22⋊C4—2448no—
368C22⋊C4—2442no—
369C42—2442no—
370C42—2442no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—612no—
378C23—612no—
379C23—611no—
380C23—611no—
381C23—612no—
382C23—1224no—
383C23—1224no—
384C23—1221no—
385C23—1221no—
386C23—1221no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23—1221no—
393C23—1221no—
394C23—2444no—
395C23—2444no—
396C23—2442no—
397C23—2442no—
398C23—2444no—
399C23—2444no—
400C23—2442no—
401C23—2442no—
402C23#1451984212yes—
403C23C2×C22⋊#692311yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—612no—
417C2×C4—612no—
418C2×C4—1228no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1221no—
423C2×C4—4882no—
424C2×C4—4882no—
425C2×C4—4882no—
426C2×C4—4882no—
427C2×C4—2444no—
428C2×C4—2444no—
429C2×C4—2444no—
430C2×C4—2444no—
431C2×C4—2444no—
432C2×C4—2444no—
433C2×C4—2444no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4#1451985612yes—
441C8—4882no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—612no—
450C22—612no—
451C22—1222no—
452C22—1222no—
453C22—1221no—
454C22—1221no—
455C22—1222no—
456C22—1221no—
457C22—1221no—
458C22—1222no—
459C22—1222no—
460C22—1222no—
461C22—1222no—
462C22—2448no—
463C22—2448no—
464C22—2441no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22#1684070511yescharacteristic
469C22D61⋊#9911yescharacteristic
470C22D61⋊#9911yescharacteristic
471C22#1684075211yescharacteristic
472C22#1684075311yescharacteristic
473C22C2×#1451984211yescharacteristic
474C22#1684076011yescharacteristic
475C4—22no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4—1222no—
480C4—1222no—
481C4—4884no—
482C4—4881no—
483C4—4881no—
484C2—42no—
485C2—1222no—
486C2—1222no—
487C2—2442no—
488C2#2146119711yescharacteristic
489C2#2146120611yescharacteristic
490C2#2146123511yescharacteristic
491C2#2146124411yescharacteristic
492C2#2146171511yescharacteristic
493C2#2146172411yescharacteristic
494C2#2146172611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#380#6923
hypocenterC61#8715
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #21460692C2semidirect
4. #21460692C2semidirect
13. C2×#14519735C4semidirect
14. C2×#14519735C4semidirect
16. #16839622C4semidirect
22. C61×#380C4semidirect
264. #380#6923semidirect
270. C61#8715semidirect

Maximal subgroups

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

subgroupindex
#214616422
#214609932
#214606922
#214482012
#871561