Finite Group Databasebeta

← #29580860

Subgroups — #29580860

Order 15616.

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

normal subgroups only

This group has 598 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#21455850C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3#21446651C211yesmaximal, characteristic, complemented
4#21444248C211yesmaximal, characteristic, complemented
5#21444244C211yesmaximal, characteristic, complemented
6#21443648C211yesmaximal, characteristic, complemented
7C61×#2398C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840080—21no—
10#16840075—21no—
11#16840047—21no—
12#16840042—21no—
13#16839962—21no—
14#16839957—21no—
15#16839901—21no—
16#16839896—21no—
17C4×#11758201—22no—
18C4×#11758199—22no—
19#16839815C2211yescharacteristic
20#16839815C2211yescharacteristic
21#16839786C2211yescharacteristic
22C2×#14519705—22no—
23Q8⋊#92418—22no—
24D4⋊#92418—22no—
25C2×C4×Q8×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#400—21no—
28C61×#397—21no—
29C61×#394—21no—
30C61×#393—21no—
31C4×C22⋊C4×C61—22no—
32C61×#376C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—22no—
35C2×#11758199—22no—
36C2×#11758194—21no—
37#14519777D411yescharacteristic
38#14519774D411yescharacteristic
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44#14519710—22no—
45#14519708—22no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50#14519705—44no—
51#14519701—22no—
52#14519698—22no—
53#14519697—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C4×#92418D412yes—
58C4×#92418D412yes—
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—21no—
76C2×C42×C61—21no—
77C2×C42×C61D412yes—
78C2×C42×C61C2311yescharacteristic
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—21no—
82#95×C61—21no—
83C24×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—42no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—42no—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—42no—
97C22×C4×C61—42no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102#11758222—42no—
103#11758222—42no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×C4×D61—41no—
108C2×Dic122—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—44no—
116C22⋊C4×C61—44no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C42×C61—22no—
120C42×C61—22no—
121C42×C61—41no—
122C42×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127#11758201—44no—
128#11758199—44no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—41no—
137C23×C61—22no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61#12011yescharacteristic
143C22×D61—41no—
144Q8×C61—22no—
145Q8×C61—22no—
146Q8×C61—41no—
147Q8×C61—41no—
148D4×C61—22no—
149D4×C61—22no—
150D4×C61—41no—
151D4×C61—41no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12711yescharacteristic
173C2×C4×C61#13512yes—
174C2×C4×C61AGL(1,ℤ/8)12yes—
175C22⋊D61—82no—
176C2×Dic61—41no—
177D244—42no—
178D244—42no—
179D244—42no—
180C4×D61—81no—
181C4×D61—81no—
182Dic122—42no—
183#92418—42no—
184#92418—42no—
185#92418—81no—
186#10013—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—44no—
190C22×C61—42no—
191C22×C61—82no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61#39211yescharacteristic
195C22×C61#44912yes—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—42no—
204C4×C61—82no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61#46312yes—
208C4×C61#49511yescharacteristic
209C4×C61#49611yescharacteristic
210Dic61—81no—
211#3103—611no—
212#3100—611no—
213#2398D6111yescharacteristic, complemented
214#1963—611no—
215#1487—611no—
216#1486—611no—
217#1274—611no—
218C2×C61—22no—
219C2×C61—82no—
220C2×C61#201011yescharacteristic
221C2×C61#201011yescharacteristic
222C2×C61#201211yescharacteristic
223D61—161no—
224#532—611no—
225C2×C4×Q8C2×D6111yescharacteristic
226C2×C4×D4C2×D6111yescharacteristic
227#481—1221no—
228#480—1221no—
229#475—1221no—
230#474—1221no—
231#460—1221no—
232#459—1221no—
233#448—1221no—
234#447—1221no—
235C4×SD16—1222no—
236C4×D8—1222no—
237#431—611no—
238#431—611no—
239#423—611no—
240C2×#104—1222no—
241#400—21no—
242#397—21no—
243#394—21no—
244#393—21no—
245C4×C22⋊C4—22no—
246#376C2×D6111yescharacteristic
247Q8⋊C8—1222no—
248D4⋊C8—1222no—
249C61#1001311yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C22×Q8C22⋊D6111yescharacteristic
252C22×D4C22⋊D6111yescharacteristic
253C23×C4—21no—
254C2×SD16—1222no—
255C2×D8—1222no—
256C2×M16—1221no—
257C4⋊Q8—611no—
258#127—611no—
259#124—1221no—
260#121—1221no—
261#121—1221no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—1221no—
271C4×D4—1221no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C22⋊D6111yescharacteristic
274C2×C4⋊C4C22⋊D6111yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42C22⋊D6112yes—
280C2×C42C22×D6111yescharacteristic
281#107—1222no—
282#106—1222no—
283C4⋊C8—1221no—
284C4⋊C8—1221no—
285C4⋊C8—1221no—
286C4⋊C8—1221no—
287#104—2444no—
288#103—1222no—
289#102—1222no—
290#102—1222no—
291#102—1222no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294C4×C8—612no—
295C4×C8—612no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300C24—21no—
301#41—2442no—
302#41—2442no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—1221no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—2441no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—2441no—
323C22×C4C2×C22⋊D6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326SD16—2444no—
327D8—2444no—
328M16—2442no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—2441no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C42D4×D6111yescharacteristic
353C42#1175822411yescharacteristic
354C23—22no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23—2441no—
360C23#1451974611yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—41no—
364Q8—41no—
365Q8—2442no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—4882no—
371D4—2442no—
372D4—2442no—
373D4—2442no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—4881no—
390C2×C4—4881no—
391C2×C4—2441no—
392C2×C4#1451974511yescharacteristic
393C2×C4#1451974511yescharacteristic
394C2×C4#1451974811yescharacteristic
395C2×C4#1451974911yescharacteristic
396C2×C4#1451974911yescharacteristic
397C2×C4#1451977311yescharacteristic
398C2×C4#1451979712yes—
399C2×C4#1451980212yes—
400C8—4881no—
401C8—2442no—
402C8—2442no—
403C22—21no—
404C22—21no—
405C22—44no—
406C22—42no—
407C22—82no—
408C22—81no—
409C22—81no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22#1683967011yescharacteristic
414C22#1683990712yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—4881no—
424C4#1683997412yes—
425C4#1684014411yescharacteristic
426C4#1684015211yescharacteristic
427C2—22no—
428C2—82no—
429C2—9761no—
430C2#2144688411yescharacteristic
431C2#2144688411yescharacteristic
432C2#2144689611yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2398C2
socleC22×C61#392
prime radical#21455819C2
hypercenter#2398D61
hypocenterC61#10013
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #21455850C2semidirect
3. #21446651C2semidirect
4. #21444248C2semidirect
5. #21444244C2semidirect
6. #21443648C2semidirect
7. C61×#2398C2semidirect
213. #2398D61semidirect
249. C61#10013semidirect

Maximal subgroups

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

subgroupindex
#214558502
#214558192
#214466512
#214442482
#214442442
#214436482
C61×#23982
#1001361