Finite Group Databasebeta

← #29598659

Subgroups — #29598659

Order 15616.

612 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 93 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#21449181C211yesmaximal, characteristic
2#21445805C211yesmaximal, characteristic
3#100×Dic61C211yesmaximal, characteristic
4#21443938C211yesmaximal, characteristic
5#21443937C211yesmaximal, characteristic
6#21443936C211yesmaximal, characteristic
7C61×#1390C211yesmaximal, characteristic
8C23×Dic122C411yescharacteristic, complemented
9C2×#14519754—21no—
10C2×D4×Dic61C2211yescharacteristic
11C2×#14519694C2211yescharacteristic
12C2×#14519692C2211yescharacteristic
13M16×Dic61—21no—
14#16839683—21no—
15#16839674—21no—
16#16839666—21no—
17#16839642C2211yescharacteristic
18#16839637C411yescharacteristic, complemented
19#16839635C411yescharacteristic, complemented
20#16839620—21no—
21#16839620C411yescharacteristic, complemented
22#16839530—21no—
23#16839528—21no—
24C2×#100×C61C2211yescharacteristic
25C2×#99×C61C2211yescharacteristic
26C61×#386C2211yescharacteristic
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—44no—
30C22×Dic122—41no—
31C22×Dic122—41no—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61D411yescharacteristic
35C2×#11758194—21no—
36#14519754—44no—
37D4×Dic61—41no—
38D4×Dic61—41no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196—41no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61C2×C411yescharacteristic
47C2×C22⋊Dic61C2×C411yescharacteristic
48C2×C22⋊Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50#14519694—21no—
51#14519694Q812yescomplemented
52#14519692—21no—
53#14519692—21no—
54#14519687—41no—
55C8×Dic61—41no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—41no—
61C22×D4×C61C2311yescharacteristic
62C23×C4×C61C2×C411yescharacteristic
63C2×M16×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61C2×C411yescharacteristic
67#100×C61—21no—
68#100×C61Q812yescomplemented
69#99×C61—21no—
70#99×C61—21no—
71#95×C61—21no—
72C24×C61C22×C411yescharacteristic
73C24×C61C2×D411yescharacteristic
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61C2×Q811yescharacteristic
77C2×D4×C61C2×Q811yescharacteristic
78C2×D4×C61#4111yescharacteristic
79C2×D4×C61#4111yescharacteristic
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—42no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—42no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22⋊C411yescharacteristic
101C2×Dic122—24no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—44no—
107C2×Dic122—44no—
108C2×Dic122—44no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—44no—
112M16×C61—22no—
113M16×C61—41no—
114C2×C8×C61—41no—
115C22⋊C4×C61—42no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121C22⋊Dic61—42no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61—42no—
127#11758196—42no—
128#11758196—42no—
129#11758196—82no—
130C4×Dic61—21no—
131C4×Dic61—21no—
132#11758194—22no—
133#11758194—41no—
134C2×#92418—41no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61C2×C22⋊C411yescharacteristic
144C23×C61C4×Q811yescharacteristic
145C23×C61C4×Q811yescharacteristic
146C23×C61#12411yescharacteristic
147C23×C61C4⋊Q811yescharacteristic
148D4×C61—41no—
149D4×C61—41no—
150D4×C61—41no—
151D4×C61—41no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61C22⋊Q811yescharacteristic
166C2×C4×C61C22⋊Q811yescharacteristic
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—81no—
186C2×Dic61—81no—
187C2×Dic61—81no—
188C2×Dic61—81no—
189C2×Dic61#9911yescharacteristic
190C2×Dic61#9911yescharacteristic
191Dic122—44no—
192Dic122—48no—
193Dic122—44no—
194C8×C61—42no—
195#92418—42no—
196#13293—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—41no—
206C22×C61—41no—
207C22×C61—42no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—81no—
213C22×C61#39111yescharacteristic
214C22×C61C2×#9911yescharacteristic
215C22×C61#41011yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—81no—
220C4×C61—161no—
221Dic61—21no—
222Dic61—42no—
223Dic61—41no—
224Dic61—41no—
225Dic61—41no—
226Dic61—81no—
227Dic61—161no—
228Dic61AΣL(1,𝔽16)12yes—
229#2409—611no—
230#1799—611no—
231C4×#100—611no—
232#1390—611no—
233#1390—611no—
234#1390—611no—
235#1390D6111yescharacteristic
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—41no—
239C2×C61—41no—
240C2×C61—41no—
241C2×C61—41no—
242C2×C61#199611yescharacteristic
243C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
244C2×C61#212311yescharacteristic
245C23×Q8—611no—
246C2×C22⋊Q8—1221no—
247C2×C4×D4—611no—
248C2×#100—611no—
249C2×#100C2×D6111yescharacteristic
250C2×#99—611no—
251C2×#99C2×D6111yescharacteristic
252C4×M16—1221no—
253#395—1221no—
254#393—1221no—
255#391—1221no—
256#386—611no—
257#386C2×D6111yescharacteristic
258#385—611no—
259#385—611no—
260#380—611no—
261#380—1221no—
262#343—1221no—
263#342—1221no—
264C61#1329311yescharacteristic, complemented, Sylow
265C22×Q8—1221no—
266C22×Q8—1221no—
267C22×Q8—2441no—
268C22×Q8—2441no—
269C22×Q8—2444no—
270C22×D4C22×D6111yescharacteristic
271C23×C4—611no—
272C23×C4—611no—
273C23×C4—611no—
274C23×C4C4×D6111yescharacteristic
275C2×M16—21no—
276C2×M16—1221no—
277C22⋊Q8—2444no—
278C4×D4—2441no—
279C4×D4—2441no—
280C2×C4⋊C4—611no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—2441no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—611no—
288C2×C22⋊C4—611no—
289C2×C22⋊C4—611no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4C4×D6111yescharacteristic
293C2×C42—611no—
294#100—21no—
295#100—612no—
296#100—1221no—
297#100Dic12212yescomplemented
298#99—21no—
299#99—21no—
300#99—1221no—
301#99—1221no—
302#97—2441no—
303C4×C8—2441no—
304#95—21no—
305#95—1221no—
306#95—1221no—
307#95—1221no—
308#95—1221no—
309#95—2441no—
310C24C2×C4×D6111yescharacteristic
311C24D4×D6111yescharacteristic
312C2×Q8—1224no—
313C2×Q8—2442no—
314C2×Q8—2441no—
315C2×Q8—2441no—
316C2×Q8—2442no—
317C2×Q8—2442no—
318C2×Q8—2442no—
319C2×Q8—2444no—
320C2×Q8—2444no—
321C2×Q8—2444no—
322C2×Q8—2444no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4C2×Dic12211yescharacteristic
326C2×D4#1175822211yescharacteristic
327C2×D4#1175822411yescharacteristic
328C2×D4Q8×D6111yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—611no—
337C22×C4—611no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2442no—
345C22×C4—2442no—
346C22×C4—2442no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4D4×D6111yescharacteristic
350M16—22no—
351M16—41no—
352M16—1222no—
353M16—2441no—
354C2×C8—41no—
355C2×C8—2441no—
356C4⋊C4—4882no—
357C4⋊C4—2441no—
358C4⋊C4—2442no—
359C4⋊C4—2442no—
360C4⋊C4—2442no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C42—1221no—
372C42—1221no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23C22⋊C4×D6111yescharacteristic
382C23#1451974211yescharacteristic
383C23C4×Dic12211yescharacteristic
384C23#1451976311yescharacteristic
385C23#1451977611yescharacteristic
386Q8—2444no—
387Q8—2444no—
388Q8—2448no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—611no—
407C2×C4—611no—
408C2×C4—1222no—
409C2×C4—1222no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—4881no—
414C2×C4—4881no—
415C2×C4—4881no—
416C2×C4—4881no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2442no—
430C2×C4#1451975411yescharacteristic
431C2×C4#1451975511yescharacteristic
432C8—42no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22#1683966611yescharacteristic
451C22#99×D6111yescharacteristic
452C22#1683973411yescharacteristic
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—161no—
458C4—612no—
459C4—1221no—
460C4—9761no—
461C4—4881no—
462C4—2441no—
463C4—2442no—
464C4—2441no—
465C4—2441no—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2#2144680211yescharacteristic
473C2AΣL(1,𝔽16)×D6111yescharacteristic
474C2#2144747111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1390C2
socleC22×C61C2×#99
prime radical, cosocleC22×D4×C61C23
hypercenter#1390D61
hypocenterC61#13293
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
8. C23×Dic122C4semidirect
18. #16839637C4semidirect
19. #16839635C4semidirect
21. #16839620C4semidirect
51. #14519694Q8semidirect
68. #100×C61Q8semidirect
264. C61#13293semidirect
297. #100Dic122semidirect

Maximal subgroups

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

subgroupindex
#214491812
#214458052
#100×Dic612
#214439382
#214439372
#214439362
C61×#13902
#1329361