Finite Group Databasebeta

← #30604738

Subgroups — #30604738

Order 15616.

670 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 88 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×#16840640C211yesmaximal, characteristic, complemented
2C2×#16840606C211yesmaximal, characteristic, complemented
3#21460800C212yesmaximal, complemented
4#21460798C212yesmaximal, complemented
5#21451556C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C2×C22⋊#92420—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606—21no—
11#16840606C2212yes—
12C2×#14519751C411yescharacteristic, complemented
13C2×#14519747—21no—
14C22×#11758198C2211yescharacteristic
15#16839689C411yescharacteristic, complemented
16#16839689C2211yescharacteristic
17#16839679—21no—
18#16839679—21no—
19#16839672C411yescharacteristic, complemented
20#16839672C2211yescharacteristic
21#16839639—21no—
22C61×#380C411yescharacteristic, complemented
23C23×#6923—41no—
24C22×#92420—41no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—41no—
28C2×C22⋊#6923—42no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C24×D61D411yescharacteristic
32C22×D244—21no—
33C22×D244—21no—
34C22×D244C2×C411yescharacteristic, complemented
35C22×C22⋊D61—21no—
36C23×Dic61D411yescharacteristic
37#14519751—41no—
38#14519751—41no—
39#14519747—42no—
40#14519747—42no—
41C2×C4⋊Dic61C2×C411yescharacteristic, complemented
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—41no—
48C2×#11758198—41no—
49C2×#11758198C2×C411yescharacteristic
50C2×#11758198D411yescharacteristic
51C2×#11758198D411yescharacteristic
52C2×#11758198C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55C23×C4×C61C2×C411yescharacteristic
56C2×C4⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58#95×C61C2×C411yescharacteristic, complemented
59#95×C61C2×C411yescharacteristic, complemented
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C23×D61—21no—
63C23×D61—22no—
64C23×D61—21no—
65C23×D61—81no—
66C23×D61—81no—
67C23×D61C22⋊C411yescharacteristic
68C23×D61D812yes—
69C23×D61SD1612yes—
70C23×D61C2×D411yescharacteristic
71C22×#6923—44no—
72C22×#6923—42no—
73C22×#6923—82no—
74C22×#6923—82no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—81no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—81no—
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×D244—41no—
93C2×D244—42no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—81no—
97C2×D244—81no—
98C2×D244—81no—
99C2×D244—81no—
100C2×D244—81no—
101C22⋊#6923—24no—
102C22⋊#6923—44no—
103C22⋊#6923—42no—
104C2×#92420—42no—
105C2×#92420—82no—
106C4⋊C4×C61—81no—
107C22⋊C4×C61—42no—
108#11758198—41no—
109#11758198—41no—
110#11758198—41no—
111#11758198—41no—
112#11758198—42no—
113#11758198—42no—
114#11758198—41no—
115#11758198—41no—
116#11758198—81no—
117#11758198—81no—
118C4⋊Dic61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61#9911yescharacteristic
122C23×C61#9911yescharacteristic
123C23×C61#10212yes—
124C23×C61#10412yes—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—22no—
127C22×D61—22no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—22no—
131C22×D61—22no—
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—81no—
150C22×D61—81no—
151C22×D61—84no—
152C22×D61#10212yes—
153C22×D61C2×D811yescharacteristic
154C22×D61C2×SD1611yescharacteristic
155C2×#6923—44no—
156C2×#6923—88no—
157C2×#6923—82no—
158C2×#6923—82no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—81no—
170C2×C4×C61—81no—
171C22⋊D61—82no—
172C22⋊D61—82no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—81no—
182D244—82no—
183D244—84no—
184D244—82no—
185D244—82no—
186#92420—82no—
187#11692—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61#32712yes—
197C22×C61#32712yes—
198C22×C61C2×#9911yescharacteristic
199C22×C61C2×#10211yescharacteristic
200C22×C61C2×#10411yescharacteristic
201C2×D61—21no—
202C2×D61—22no—
203C2×D61—21no—
204C2×D61—44no—
205C2×D61—44no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—42no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—82no—
216C2×D61—82no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224#6923—84no—
225C4×C61—41no—
226C4×C61—41no—
227C4×C61—81no—
228C4×C61—81no—
229Dic61—41no—
230Dic61—41no—
231Dic61—81no—
232#2617—611no—
233C2×#342—611no—
234C2×#323—611no—
235#1344—612no—
236#1343—612no—
237C2×C61—21no—
238C2×C61—22no—
239C2×C61—21no—
240C2×C61—22no—
241C2×C61C2×#32711yescharacteristic
242C2×C61#212111yescharacteristic
243C2×C61#212711yescharacteristic
244D61—41no—
245D61—42no—
246D61—41no—
247D61—161no—
248C2×#121—611no—
249C2×#120—1221no—
250C22×C22⋊C4—611no—
251C22×C22⋊C4—1221no—
252C2×C22⋊C8—1221no—
253#397—611no—
254#397—611no—
255#394—1221no—
256#394—1221no—
257#392—611no—
258#392—611no—
259#386—1221no—
260#380#692311yescharacteristic, complemented
261#342—612no—
262#342—1221no—
263#323—612no—
264#323—1221no—
265C61#1169211yescharacteristic, complemented, Sylow
266C25—611no—
267C22×D4—611no—
268C22×D4—1221no—
269C22×D4—1221no—
270C22×D4—1221no—
271C23×C4—611no—
272C23×C4—2441no—
273C23×C4C2×#692311yescharacteristic
274C22×C8—2441no—
275#121—2441no—
276#121—2441no—
277#120—2442no—
278#120—2442no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—611no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—611no—
283C2×C22⋊C4—611no—
284C2×C22⋊C4—611no—
285C2×C22⋊C4—611no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4—1224no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C22⋊C4—2441no—
294C2×C22⋊C4—2441no—
295C2×C22⋊C4—2442no—
296C2×C22⋊C4—2441no—
297C22⋊C8—1222no—
298C22⋊C8—2441no—
299#95—1221no—
300#95—1221no—
301#95C2×#692311yescharacteristic, complemented
302#95C2×#692311yescharacteristic, complemented
303C24—612no—
304C24—612no—
305C24—611no—
306C24—611no—
307C24—1221no—
308C24—1222no—
309C24—1221no—
310C24—1221no—
311C24—4881no—
312C24—4881no—
313C24C22⋊#692311yescharacteristic
314C2×D4—4881no—
315C2×D4—4881no—
316C2×D4—4881no—
317C2×D4—4881no—
318C2×D4—4881no—
319C2×D4—2441no—
320C2×D4—2442no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2442no—
326C2×D4—2442no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—81no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—4882no—
338C22×C4—4882no—
339C22×C4—4881no—
340C22×C4—2442no—
341C22×C4—2444no—
342C22×C4C22⋊#692311yescharacteristic
343C22×C4C22×#692311yescharacteristic
344C2×C8—4882no—
345C2×C8—2442no—
346C4⋊C4—81no—
347C4⋊C4—2441no—
348C22⋊C4—42no—
349C22⋊C4—1224no—
350C22⋊C4—4881no—
351C22⋊C4—4881no—
352C22⋊C4—2441no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C22⋊C4—2441no—
356C22⋊C4—2442no—
357C22⋊C4—2442no—
358C22⋊C4—2441no—
359C22⋊C4—2441no—
360C22⋊C4—2442no—
361C22⋊C4—2444no—
362C23—41no—
363C23—41no—
364C23—612no—
365C23—611no—
366C23—611no—
367C23—1221no—
368C23—1221no—
369C23—1222no—
370C23—1222no—
371C23—1221no—
372C23—1221no—
373C23—1222no—
374C23—1222no—
375C23—4881no—
376C23—4881no—
377C23—4884no—
378C23—4882no—
379C23—4882no—
380C23—2441no—
381C23—2441no—
382C23—2441no—
383C23—2441no—
384C23—2441no—
385C23—2441no—
386C23—2441no—
387C23—2441no—
388C23—2442no—
389C23—2442no—
390C23—2441no—
391C23—2441no—
392C23—2441no—
393C23#1451983611yescharacteristic
394C23#1451983611yescharacteristic
395C23#1451983912yes—
396C23#1451984212yes—
397C23C2×C22⋊#692311yescharacteristic
398D4—4882no—
399D4—4882no—
400D4—4884no—
401D4—4882no—
402D4—4882no—
403D4—4882no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—4888no—
419C2×C4—4882no—
420C2×C4—4882no—
421C2×C4—4881no—
422C2×C4—2444no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2441no—
429C8—4882no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—1221no—
439C22—1222no—
440C22—1221no—
441C22—4881no—
442C22—4881no—
443C22—4881no—
444C22—4881no—
445C22—4882no—
446C22—4882no—
447C22—4881no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22—2441no—
453C22—2442no—
454C22—2441no—
455C22—2444no—
456C22—2444no—
457C22—2441no—
458C22—2441no—
459C22—2441no—
460C22—2441no—
461C22#1684061412yes—
462C22#1684061412yes—
463C22C2×#1451983611yescharacteristic
464C22C2×#1451983911yescharacteristic
465C22C2×#1451984211yescharacteristic
466C4—41no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—4884no—
471C4—4881no—
472C4—2441no—
473C4—2441no—
474C2—21no—
475C2—21no—
476C2—22no—
477C2—22no—
478C2—9761no—
479C2—2441no—
480C2—2442no—
481C2—2441no—
482C2C2×#1684061411yescharacteristic
483C2#2146207611yescharacteristic
484C2#2146209611yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC22×C61C2×#99
prime radical#21451556C2
cosocleC2×#11758198C23
hypercenter#380#6923
hypocenterC61#11692
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#16840640C2semidirect
2. C2×#16840606C2semidirect
3. #21460800C2semidirect
4. #21460798C2semidirect
12. C2×#14519751C4semidirect
15. #16839689C4semidirect
19. #16839672C4semidirect
22. C61×#380C4semidirect
34. C22×D244C2×C4semidirect
41. C2×C4⋊Dic61C2×C4semidirect
58. #95×C61C2×C4semidirect
59. #95×C61C2×C4semidirect
260. #380#6923semidirect
265. C61#11692semidirect
301. #95C2×#6923semidirect
302. #95C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406402
C2×#168406062
#214608002
#214607982
#214515562
#1169261