Finite Group Databasebeta

← #29573477

Subgroups — #29573477

Order 15616.

498 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 99 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#21448414C211yesmaximal, characteristic
2C2×#16839485C211yesmaximal, characteristic
3C2×#16839485C211yesmaximal, characteristic
4#21443667C211yesmaximal, characteristic, complemented
5#21443666C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1282C211yesmaximal, characteristic, complemented
8C22×#11758198C2211yescharacteristic
9C2×C22⋊#92418C2211yescharacteristic
10C2×C22⋊#92418C2211yescharacteristic
11#16839653—21no—
12#16839653—21no—
13#16839651—21no—
14#16839651—21no—
15#16839620C411yescharacteristic
16#16839620C411yescharacteristic
17#16839617C411yescharacteristic
18#16839617C411yescharacteristic
19C4×C22⋊Dic61C2211yescharacteristic
20#16839485—21no—
21#16839485—21no—
22#16839485—21no—
23#16839485—21no—
24C2×C22⋊C8×C61C2211yescharacteristic
25C2×C22⋊C8×C61C2211yescharacteristic
26C4×C22⋊C4×C61C2211yescharacteristic
27C24×D61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C22×#92418—21no—
30C22×#92418—21no—
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×C22⋊Dic61C2×C411yescharacteristic
34C2×C22⋊Dic61C2×C411yescharacteristic
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198D411yescharacteristic
44C2×#11758198D411yescharacteristic
45C2×#11758198D411yescharacteristic
46C2×#11758198D411yescharacteristic
47C2×C4×Dic61—21no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55C23×C4×C61C2311yescharacteristic
56C22×C8×C61—21no—
57C22×C8×C61—21no—
58C2×C22⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61—21no—
61C22⋊C8×C61—21no—
62C22⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64C22⋊C8×C61—21no—
65#95×C61—21no—
66C24×C61C22×C411yescharacteristic
67C23×D61—21no—
68C23×D61—21no—
69C23×D61—21no—
70C23×D61—21no—
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—81no—
74C23×D61—81no—
75C23×D61C22⋊C411yescharacteristic
76C23×D61C22⋊C411yescharacteristic
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—81no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×C4×C61#4111yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—81no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C22⋊C411yescharacteristic
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—81no—
96C2×C8×C61—81no—
97C22⋊C4×C61—21no—
98C22⋊C4×C61—21no—
99C22⋊C4×C61—21no—
100C22⋊C4×C61—21no—
101C42×C61—81no—
102C22⋊Dic61—21no—
103C22⋊Dic61—21no—
104C22⋊Dic61—21no—
105C22⋊Dic61—21no—
106#11758198—21no—
107#11758198—21no—
108#11758198—21no—
109#11758198—21no—
110#11758198—21no—
111#11758198—21no—
112#11758198—21no—
113#11758198—21no—
114#11758198—81no—
115#11758198—81no—
116#11758198—81no—
117#11758198—81no—
118#11758196—81no—
119#11758196—81no—
120C4×Dic61—81no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—81no—
124C2×#92418—81no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#11711yescharacteristic
129C23×C61#11711yescharacteristic
130C23×C61#13111yescharacteristic
131C23×C61#13111yescharacteristic
132C23×C61#13111yescharacteristic
133C23×C61#13111yescharacteristic
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—21no—
139C22×D61—21no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—81no—
149C22×D61—81no—
150C22×D61—81no—
151C22×D61—81no—
152C22×D61—82no—
153C22×D61—82no—
154C22×D61—82no—
155C22×D61—82no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
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#12311yescharacteristic
169C2×C4×C61#12311yescharacteristic
170C2×C4×C61#12311yescharacteristic
171C2×C4×C61#12311yescharacteristic
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—81no—
183C2×Dic61—81no—
184C8×C61—81no—
185C8×C61—81no—
186#92418—81no—
187#92418—81no—
188#8736—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61#38111yescharacteristic
200C22×C61#40811yescharacteristic
201C22×C61#40811yescharacteristic
202C22×C61#43311yescharacteristic
203C22×C61#43311yescharacteristic
204C22×C61#43311yescharacteristic
205C22×C61#43311yescharacteristic
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—41no—
211C2×D61—41no—
212C2×D61—82no—
213C2×D61—82no—
214C2×D61—82no—
215C2×D61—82no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—81no—
225C4×C61—81no—
226Dic61—41no—
227Dic61—41no—
228Dic61—81no—
229Dic61—81no—
230#2312—611no—
231C2×#323—611no—
232C2×#323—611no—
233#1282—611no—
234#1282—611no—
235#1282—611no—
236#1282D6111yescharacteristic, complemented
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61#181811yescharacteristic
244C2×C61#182811yescharacteristic
245C2×C61#182911yescharacteristic
246D61—81no—
247D61—81no—
248C22×C22⋊C4—611no—
249C2×C22⋊C8—611no—
250C2×C22⋊C8—611no—
251C2×C22⋊C8C2×D6111yescharacteristic
252C2×C22⋊C8C2×D6111yescharacteristic
253#388—1221no—
254#388—1221no—
255#388—1221no—
256#388—1221no—
257#380—611no—
258#380—611no—
259#379—611no—
260#379—611no—
261C4×C22⋊C4—611no—
262C4×C22⋊C4C2×D6111yescharacteristic
263#323—1221no—
264#323—1221no—
265#323—1221no—
266#323—1221no—
267C61#873611yescharacteristic, complemented, Sylow
268C25—611no—
269C23×C4—611no—
270C23×C4C22×D6111yescharacteristic
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—1221no—
274C22×C8—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C22⋊C4—611no—
278C2×C22⋊C4—611no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—611no—
281C2×C22⋊C4—611no—
282C2×C22⋊C4—611no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4C4×D6111yescharacteristic
292C2×C22⋊C4C4×D6111yescharacteristic
293C2×C42—21no—
294C2×C42—1221no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302C22⋊C8—1221no—
303#95—21no—
304#95—1221no—
305#95—1221no—
306#95—1221no—
307C24—611no—
308C24—611no—
309C24—1221no—
310C24—1221no—
311C24—1221no—
312C24—1221no—
313C24—1221no—
314C24—1221no—
315C24—4881no—
316C24—4881no—
317C24C2×C4×D6111yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—81no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—4881no—
328C22×C4#1175822211yescharacteristic
329C22×C4#1175822211yescharacteristic
330C22×C4D4×D6111yescharacteristic
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4#1175822611yescharacteristic
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—81no—
337C2×C8—81no—
338C2×C8—4881no—
339C2×C8—4881no—
340C2×C8—2441no—
341C2×C8—2441no—
342C4⋊C4—4881no—
343C4⋊C4—4881no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—1221no—
349C22⋊C4—1221no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
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—4881no—
361C22⋊C4—4881no—
362C22⋊C4—4881no—
363C22⋊C4—4881no—
364C42—81no—
365C42—4881no—
366C23—41no—
367C23—41no—
368C23—1221no—
369C23—1221no—
370C23—1221no—
371C23—1221no—
372C23—1221no—
373C23—1221no—
374C23—4881no—
375C23—4881no—
376C23—4881no—
377C23—4881no—
378C23—4882no—
379C23—4882no—
380C23—4882no—
381C23—4882no—
382C23—2441no—
383C23—2441no—
384C23—2441no—
385C23—2441no—
386C23—2441no—
387C23—2441no—
388C23—2441no—
389C23—2441no—
390C23C22⋊C4×D6111yescharacteristic
391C23#1451973411yescharacteristic
392C23#1451973511yescharacteristic
393C23#1451978311yescharacteristic
394C23#1451978311yescharacteristic
395C23#1451978511yescharacteristic
396C23#1451978511yescharacteristic
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—1221no—
410C2×C4—1221no—
411C2×C4—4881no—
412C2×C4—4881no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4#1451975911yescharacteristic
422C2×C4#1451975911yescharacteristic
423C2×C4#1451976211yescharacteristic
424C2×C4#1451976211yescharacteristic
425C8—81no—
426C8—81no—
427C8—4881no—
428C8—4881no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—4881no—
440C22—4881no—
441C22—4881no—
442C22—4881no—
443C22—4882no—
444C22—4882no—
445C22—4882no—
446C22—4882no—
447C22—2441no—
448C22—2441no—
449C22—2441no—
450C22—2441no—
451C22—2441no—
452C22—2441no—
453C22#1683962411yescharacteristic
454C22#1683972711yescharacteristic
455C22#1683972711yescharacteristic
456C22#1683982111yescharacteristic
457C22#1683982211yescharacteristic
458C22#1683982211yescharacteristic
459C22#1683982411yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—81no—
465C4—81no—
466C4—4881no—
467C4—4881no—
468C4—2441no—
469C4—2441no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—4881no—
477C2—4881no—
478C2#2144588911yescharacteristic
479C2#2144592611yescharacteristic
480C2#2144592911yescharacteristic
subgroupG/H
centerC22#16839624
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1282C2
socleC22×C61#381
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1282D61
hypocenterC61#8736
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
4. #21443667C2semidirect
5. #21443666C2semidirect
6. #21443665C2semidirect
7. C61×#1282C2semidirect
236. #1282D61semidirect
267. C61#8736semidirect

Maximal subgroups

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

subgroupindex
#214484142
C2×#168394852
#214436672
#214436662
#214436652
C61×#12822
#873661