Finite Group Databasebeta

← #29568600

Subgroups — #29568600

Order 15616.

570 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 97 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#21454123C211yesmaximal, characteristic
2#21445798C211yesmaximal, characteristic, complemented
3#21445511C211yesmaximal, characteristic, complemented
4#21443645C211yesmaximal, characteristic, complemented
5#21443639C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic, complemented
7C61×#1272C211yesmaximal, characteristic, complemented
8C22×#11758189—21no—
9C2×#14519766C411yescharacteristic
10C2×#14519753C411yescharacteristic
11C2×#14519752C411yescharacteristic
12C2×C42×D61C2211yescharacteristic
13C2×#14519706—21no—
14C2×#14519689—21no—
15#16839664C2211yescharacteristic
16#16839651—21no—
17#16839651—21no—
18#16839640—21no—
19#16839640—21no—
20#16839627C411yescharacteristic
21#16839517C2211yescharacteristic
22#16839517C2211yescharacteristic
23#16839516—21no—
24C61×#391C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×D244—21no—
28C22×D244C2×C411yescharacteristic
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×#11758189—24no—
33C2×#11758189—42no—
34C2×#11758189—42no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519766—42no—
38#14519753—42no—
39#14519752—42no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C42×D61—42no—
52#14519706—42no—
53#14519689—42no—
54#14519684—21no—
55#14519684—21no—
56C22×Q8×C61C2×C411yescharacteristic
57C22×C8×C61—21no—
58C22×C8×C61—21no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2311yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63C23×D61—21no—
64C23×D61C22⋊C411yescharacteristic
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—42no—
83C2×D244—42no—
84C2×D244—42no—
85C2×C4×D61—22no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—22no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—22no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97C2×C8×C61—22no—
98C2×C8×C61—41no—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C4⋊C4×C61—41no—
104C42×C61—41no—
105#11758198—42no—
106#11758198—42no—
107#11758198—41no—
108#11758198—41no—
109#11758198—41no—
110#11758198—41no—
111#11758198—42no—
112#11758198—42no—
113C4⋊Dic61—41no—
114#11758196—42no—
115C4×Dic61—22no—
116C4×Dic61—22no—
117C4×Dic61—41no—
118C2×#92418—22no—
119C2×#92418—41no—
120C2×#92418—41no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124#11758189—48no—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—21no—
131C22×D61—21no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61#10412yes—
140Q8×C61—81no—
141Q8×C61—81no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61#13111yescharacteristic
155C2×C4×C61#13111yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C2×C4×C61#13711yescharacteristic
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61#9912yes—
171C2×Dic61#10012yes—
172D244—81no—
173D244—81no—
174D244—84no—
175C4×D61—24no—
176C4×D61—24no—
177C4×D61—42no—
178C4×D61—42no—
179C4×D61—42no—
180C4×D61—42no—
181C8×C61—42no—
182C8×C61—81no—
183#92418—42no—
184#92418—81no—
185#7710—611nomaximal, Sylow
186C22×C61#40811yescharacteristic
187C22×C61C2×#9911yescharacteristic
188C22×C61C2×#10011yescharacteristic
189C22×C61#41711yescharacteristic
190C22×C61#41711yescharacteristic
191C22×C61C2×#10411yescharacteristic
192C22×C61#42111yescharacteristic
193C2×D61—22no—
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—81no—
215Dic61—22no—
216Dic61—22no—
217Dic61—41no—
218Dic61—81no—
219#2850—611no—
220#1797—611no—
221#1728—611no—
222#1272—611no—
223#1272D6111yescharacteristic, complemented
224#1270—611no—
225#1270—611no—
226C2×C61#147111yescharacteristic
227C2×C61#148311yescharacteristic
228C2×C61#149511yescharacteristic
229C2×C61#150411yescharacteristic
230C2×C61#151311yescharacteristic
231C2×C61#152711yescharacteristic
232C2×C61#156011yescharacteristic
233D61—42no—
234D61—161no—
235C22×M16—1221no—
236C2×#124—611no—
237C2×C22⋊Q8—611no—
238C2×#121—611no—
239C22×C42—611no—
240C2×C4⋊C8—1221no—
241C2×C22⋊C8—1221no—
242#391C2×D6111yescharacteristic
243#390—611no—
244#388—1221no—
245#388—1221no—
246#386—1221no—
247#386—1221no—
248#382—611no—
249#336—611no—
250#336—611no—
251#336—1221no—
252#336C2×D6111yescharacteristic
253#336C2×D6111yescharacteristic
254C61#771011yescharacteristic, complemented, Sylow
255C22×Q8C4×D6111yescharacteristic
256C22×D4—611no—
257C22×D4—1221no—
258C23×C4—611no—
259C23×C4—611no—
260C23×C4—611no—
261C2×M16—1224no—
262C2×M16—2442no—
263C2×M16—2442no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—1221no—
267C22×C8—1221no—
268#124—2442no—
269C22⋊Q8—2442no—
270#121—2442no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4C4×D6111yescharacteristic
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C42—611no—
281C2×C42—611no—
282C2×C42—611no—
283C2×C42—2442no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—2442no—
286C22⋊C8—2442no—
287#95—21no—
288#95—21no—
289#95—1221no—
290#95—1221no—
291C24—611no—
292C24—1221no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×D4—1221no—
298C2×D4—1221no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2442no—
302C2×D4—2442no—
303C2×D4—2442no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4—2441no—
322C22×C4C2×C4×D6111yescharacteristic
323C22×C4D4×D6111yescharacteristic
324C22×C4D4×D6111yescharacteristic
325M16—2448no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—1222no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—41no—
339C4⋊C4—2441no—
340C4⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C22⋊C4—2442no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—41no—
350C42—1222no—
351C42—1222no—
352C42—2441no—
353C23—612no—
354C23—1221no—
355C23—1221no—
356C23—1221no—
357C23—1221no—
358C23—1221no—
359C23—1221no—
360C23—2441no—
361C23—2441no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23C22⋊C4×D6111yescharacteristic
368Q8—81no—
369Q8—81no—
370D4—4881no—
371D4—4881no—
372D4—4884no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—612no—
386C2×C4—612no—
387C2×C4—1224no—
388C2×C4—1224no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451978311yescharacteristic
404C2×C4#1451978511yescharacteristic
405C2×C4#1451980411yescharacteristic
406C2×C4#1451980411yescharacteristic
407C2×C4#1451980911yescharacteristic
408C2×C4#1451980911yescharacteristic
409C8—42no—
410C8—81no—
411C8—4881no—
412C8—2442no—
413C22—1222no—
414C22—1222no—
415C22—1222no—
416C22—4881no—
417C22—4881no—
418C22—4881no—
419C22—4881no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22—2441no—
424C22—2441no—
425C22—2441no—
426C22—2441no—
427C22#1683972711yescharacteristic
428C22#99×D6111yescharacteristic
429C22#100×D6111yescharacteristic
430C22#1683976211yescharacteristic
431C22#1683976211yescharacteristic
432C22#104×D6111yescharacteristic
433C22#1683977711yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—1222no—
443C4—1222no—
444C4—4881no—
445C4—2441no—
446C2—9761no—
447C2—2442no—
448C2#2144417711yescharacteristic
449C2#2144423111yescharacteristic
450C2#2144428311yescharacteristic
451C2#2144432211yescharacteristic
452C2#2144436111yescharacteristic
453C2#2144441511yescharacteristic
454C2#2144455311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7710
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21445798C2semidirect
3. #21445511C2semidirect
4. #21443645C2semidirect
5. #21443639C2semidirect
6. #21443639C2semidirect
7. C61×#1272C2semidirect
223. #1272D61semidirect
254. C61#7710semidirect

Maximal subgroups

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

subgroupindex
#214541232
#214457982
#214455112
#214436452
#214436392
C61×#12722
#771061