Finite Group Databasebeta

← #29574725

Subgroups — #29574725

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455879C211yesmaximal, characteristic
2#21446646C211yesmaximal, characteristic, complemented
3C4×#14519704C211yesmaximal, characteristic, complemented
4#21444235C211yesmaximal, characteristic, complemented
5#21443708C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#2991C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840086—21no—
10#16840057C411yescharacteristic, complemented
11#16840052C411yescharacteristic, complemented
12#16839962C411yescharacteristic, complemented
13#16839957C411yescharacteristic, complemented
14#16839907—21no—
15C4×#11758201—21no—
16C4×#11758199—21no—
17#16839815C2211yescharacteristic
18C2×#14519704—21no—
19C2×#14519703—22no—
20C4×#11758194C2211yescharacteristic
21C42×Dic61C2211yescharacteristic
22#16839521—21no—
23#16839489—21no—
24#16839489—21no—
25C4×#41×C61C2211yescharacteristic
26C8×Q8×C61—21no—
27C61×#436—22no—
28C61×#436—21no—
29C61×#431—21no—
30C61×#431C2211yescharacteristic
31C61×#423—21no—
32C61×#423C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—21no—
35C2×#11758199—21no—
36C2×#11758194—21no—
37#14519777C2×C411yescharacteristic
38#14519774C2×C411yescharacteristic
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47#14519710—21no—
48#14519708—21no—
49C4⋊#92418C2×C411yescharacteristic
50C4⋊#92418C2×C411yescharacteristic
51#14519704—24no—
52#14519703—44no—
53#14519701—21no—
54#14519698—21no—
55#14519697—21no—
56#14519697—21no—
57C22⋊#92418—21no—
58#14519688—21no—
59C4×#92418—21no—
60C4×#92418—21no—
61C2×#41×C61D411yescharacteristic
62C2×M16×C61—22no—
63C2×M16×C61D412yes—
64C4×Q8×C61C2×C411yescharacteristic
65C4×Q8×C61D411yescharacteristic
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61C2×C411yescharacteristic
69C4×D4×C61D411yescharacteristic
70#117×C61—21no—
71#117×C61D411yescharacteristic
72C2×C42×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—22no—
75C4⋊C8×C61—21no—
76C4⋊C8×C61—21no—
77C4⋊C8×C61D411yescharacteristic
78C4⋊C8×C61D411yescharacteristic
79C22⋊C8×C61—22no—
80C22⋊C8×C61—21no—
81C4×C8×C61—22no—
82C4×C8×C61—21no—
83#41×C61—22no—
84#41×C61—41no—
85C2×Q8×C61C22⋊C411yescharacteristic
86C2×D4×C61—21no—
87C2×D4×C61C22⋊C411yescharacteristic
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—41no—
95C22×Dic61—41no—
96#11758222—42no—
97#11758222—42no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×C4×D61—41no—
102C2×Dic122—21no—
103M16×C61—42no—
104M16×C61—82no—
105C2×C8×C61—21no—
106C2×C8×C61—22no—
107C2×C8×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61C22⋊C411yescharacteristic
111C4⋊C4×C61C22⋊C411yescharacteristic
112C22⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C22⋊C4×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61—21no—
118C42×C61C22×C411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122#11758201—42no—
123#11758199—42no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4×Dic61—22no—
129C4×Dic61—22no—
130C4×Dic61—22no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133C4×Dic61—41no—
134#11758194—22no—
135#11758194—41no—
136C2×#92418—21no—
137C2×#92418—21no—
138C2×#92418—41no—
139C23×C61—21no—
140C23×C61#12111yescharacteristic
141C22×D61—41no—
142Q8×C61—21no—
143Q8×C61#10412yescomplemented
144D4×C61—21no—
145D4×C61—21no—
146D4×C61—42no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12311yescharacteristic
167C22⋊D61—82no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172D244—42no—
173D244—41no—
174D244—41no—
175D244—41no—
176D244—41no—
177C4×D61—81no—
178C4×D61—81no—
179Dic122—42no—
180C8×C61—42no—
181C8×C61—41no—
182C8×C61—41no—
183#92418—42no—
184#92418—41no—
185#92418—41no—
186#8970—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61#38611yescharacteristic
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—21no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—41no—
202C4×C61—41no—
203C4×C61C2×#10411yescharacteristic
204C4×C61#42111yescharacteristic
205C4×C61#44311yescharacteristic
206C4×C61#44411yescharacteristic
207C4×C61#45311yescharacteristic
208C4×C61#45411yescharacteristic
209Dic61—44no—
210Dic61—81no—
211#3106—611no—
212#2991D6111yescharacteristic, complemented
213#1962—611no—
214C4×#104—611no—
215#1484—611no—
216#1299—611no—
217#1276—611no—
218C2×C61—41no—
219C2×C61—81no—
220C2×C61#187111yescharacteristic
221C2×C61#187611yescharacteristic
222C2×C61#188311yescharacteristic
223D61—161no—
224#532—611no—
225C4×#41C2×D6111yescharacteristic
226#482—1221no—
227#477—611no—
228#476—611no—
229#460—611no—
230#459—611no—
231#449—1221no—
232C8×Q8—21no—
233C4×SD16—1221no—
234C4×D8—1221no—
235#436—22no—
236#436—21no—
237#431—21no—
238#431—611no—
239#431C2×D6111yescharacteristic
240#423—21no—
241#423C2×D6111yescharacteristic
242C2×#104—1221no—
243C2×#104—1222no—
244C4×M16—611no—
245C43—611no—
246#339—1221no—
247D4⋊C8—1221no—
248D4⋊C8—1221no—
249C61#897011yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C2×#41D24411yescharacteristic
252C2×SD16—1221no—
253C2×D8—1221no—
254C2×M16—22no—
255C2×M16—1221no—
256C2×M16C22⋊D6112yes—
257C4⋊Q8—611no—
258#127—611no—
259#124—1221no—
260#121—1221no—
261#121—1221no—
262C4×Q8C4×D6111yescharacteristic
263C4×Q8C22⋊D6111yescharacteristic
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4C4×D6111yescharacteristic
269C4×D4C22⋊D6111yescharacteristic
270#117—21no—
271#117D24411yescharacteristic
272C2×C42—21no—
273C2×C42—1221no—
274C2×C42—1221no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277#107—1221no—
278#106—1221no—
279C4⋊C8—22no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—611no—
283C4⋊C8—611no—
284C4⋊C8C22⋊D6111yescharacteristic
285C4⋊C8C22⋊D6111yescharacteristic
286#104—1224no—
287#104—2444no—
288#103—1221no—
289#102—1221no—
290#102—1221no—
291#102—1221no—
292C22⋊C8—22no—
293C22⋊C8—21no—
294C22⋊C8—1221no—
295#97—1221no—
296C4×C8—22no—
297C4×C8—21no—
298C4×C8—1221no—
299C4×C8—1221no—
300#41—22no—
301#41—41no—
302#41—2442no—
303#41—2442no—
304C2×Q8—1221no—
305C2×Q8#1175819811yescharacteristic
306C2×D4—21no—
307C2×D4—1221no—
308C2×D4—1221no—
309C2×D4—1221no—
310C2×D4—2441no—
311C2×D4#1175819811yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4C2×D24411yescharacteristic
318C22×C4#1175822211yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320SD16—2442no—
321D8—2442no—
322M16—42no—
323M16—82no—
324M16—1222no—
325M16—2441no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—2441no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4#1175819811yescharacteristic
337C4⋊C4#1175819811yescharacteristic
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—1222no—
347C42—1222no—
348C42—1222no—
349C42—2441no—
350C42—2441no—
351C42—2441no—
352C42C2×C4×D6111yescharacteristic
353C42#1175822211yescharacteristic
354C42C2×C22⋊D6111yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C23—21no—
357C23—2441no—
358C23#1451975111yescharacteristic
359Q8—21no—
360Q8—2442no—
361Q8#1451970312yescomplemented
362D4—21no—
363D4—21no—
364D4—42no—
365D4—4882no—
366D4—2442no—
367D4—2441no—
368D4—2441no—
369D4—2441no—
370D4—2441no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4C2×#1175819811yescharacteristic
392C2×C4C4×C22⋊D6111yescharacteristic
393C2×C4C4×C22⋊D6111yescharacteristic
394C2×C4#1451974611yescharacteristic
395C2×C4#1451975111yescharacteristic
396C2×C4#1451976011yescharacteristic
397C8—42no—
398C8—41no—
399C8—41no—
400C8—2442no—
401C8—2441no—
402C8—2441no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1683964111yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—4881no—
420C4—2444no—
421C4C2×#1451970311yescharacteristic
422C4#1683977811yescharacteristic
423C4#1683987711yescharacteristic
424C4#1683988211yescharacteristic
425C4#1683993011yescharacteristic
426C4#1683993511yescharacteristic
427C2—41no—
428C2—81no—
429C2—9761no—
430C2#2144612111yescharacteristic
431C2#2144614611yescharacteristic
432C2#2144619511yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2991C2
socleC22×C61#386
prime radical#21455879C2
cosocleC2×C42×C61C23
hypercenter#2991D61
hypocenterC61#8970
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #21446646C2semidirect
3. C4×#14519704C2semidirect
4. #21444235C2semidirect
5. #21443708C2semidirect
6. #21443651C2semidirect
7. C61×#2991C2semidirect
10. #16840057C4semidirect
11. #16840052C4semidirect
12. #16839962C4semidirect
13. #16839957C4semidirect
143. Q8×C61#104semidirect
212. #2991D61semidirect
249. C61#8970semidirect
361. Q8#14519703semidirect

Maximal subgroups

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

subgroupindex
#214558792
#214466462
C4×#145197042
#214442352
#214437082
#214436512
C61×#29912
#897061