Finite Group Databasebeta

← #30597843

Subgroups — #30597843

Order 15616.

522 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461555C211yesmaximal, characteristic, complemented
2C8×C22⋊#6923C211yesmaximal, characteristic, complemented
3#21461196C211yesmaximal, characteristic
4#21461196C211yesmaximal, characteristic
5#21460701C211yesmaximal, characteristic, complemented
6#21460699C211yesmaximal, characteristic, complemented
7#21454515C211yesmaximal, characteristic
8D61⋊(C22⋊C8)C2211yescharacteristic
9D61⋊(C22⋊C8)C2211yescharacteristic
10C2×#14519831—21no—
11C2×C8×#6923—21no—
12C4×C22⋊#6923C2211yescharacteristic
13C4×C22⋊#6923C2211yescharacteristic
14(C2×#6923)⋊C8—21no—
15(C2×#6923)⋊C8—21no—
16#16840615C412yescomplemented
17#16840615C412yescomplemented
18#16840614C412yescomplemented
19#16840614C412yescomplemented
20C2×#14519783C411yescharacteristic, complemented
21C22×C8×D61C2211yescharacteristic
22#16840274C2211yescharacteristic
23#16839835—21no—
24#16839835—21no—
25C8×D244—21no—
26C8×D244—21no—
27#16839789C411yescharacteristic, complemented
28#16839730C411yescharacteristic, complemented
29#16839725C2211yescharacteristic
30C61×#405C411yescharacteristic, complemented
31C2×D61⋊C8—21no—
32C2×D61⋊C8—21no—
33C2×C22⋊#6923—21no—
34C2×C22⋊#6923—21no—
35C2×C4×#6923—21no—
36C2×C4×#6923—21no—
37#14519834C2×C412yes—
38#14519834C2×C412yes—
39C22⋊#92420—21no—
40C22⋊#92420—21no—
41#14519831—42no—
42C8×#6923—42no—
43(C2×#6923)⋊C4C2×C412yes—
44(C2×#6923)⋊C4C2×C412yes—
45C2×#11758222C2×C411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47#14519783—42no—
48C2×#11758194C2×C411yescharacteristic
49C2×#11758189—21no—
50C22×#92418C2×C411yescharacteristic
51C2×C8×D61—21no—
52C2×C8×D61—42no—
53C2×C8×D61D411yescharacteristic
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56C2×C8×D61D411yescharacteristic
57#14519761—21no—
58#14519757C2×C411yescharacteristic
59#14519757C2×C411yescharacteristic
60#14519752C2×C411yescharacteristic
61#14519752C2×C411yescharacteristic
62#14519747—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4⋊#92418—21no—
66C4⋊#92418—21no—
67#14519689—21no—
68#14519689—21no—
69#14519689—21no—
70#14519689—21no—
71C2×M16×C61C2×C411yescharacteristic
72C22×C8×C61C2×C411yescharacteristic
73#117×C61C2×C411yescharacteristic
74#97×C61—21no—
75C4×C8×C61—21no—
76C23×D61#4111yescharacteristic
77C22×#6923—22no—
78C22×#6923—22no—
79C22×C4×C61C22×C411yescharacteristic
80C2×C22⋊D61—21no—
81C22×Dic61#4111yescharacteristic
82#11758222—41no—
83#11758222—41no—
84C2×D244—21no—
85C2×D244—21no—
86C2×D244C4211yescharacteristic
87C2×C4×D61—21no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61C22×C411yescharacteristic
91C2×C4×D61C22×C411yescharacteristic
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×D411yescharacteristic
94C2×Dic122C4211yescharacteristic
95C22⋊#6923—24no—
96C22⋊#6923—42no—
97C2×#92420—22no—
98C2×#92420—22no—
99C4×#6923—41no—
100C4×#6923—41no—
101C4×#6923—41no—
102C4×#6923—41no—
103D61⋊C8—41no—
104D61⋊C8—41no—
105D61⋊C8—41no—
106D61⋊C8—41no—
107M16×C61—41no—
108C2×C8×C61—22no—
109C2×C8×C61—21no—
110C2×C8×C61C22⋊C411yescharacteristic
111C2×C8×C61C22⋊C411yescharacteristic
112C4⋊C4×C61C4211yescharacteristic
113C4⋊C4×C61C4211yescharacteristic
114C22⋊C4×C61—21no—
115C42×C61—21no—
116#11758198—21no—
117#11758198—21no—
118#11758198—21no—
119#11758198—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122#11758194—41no—
123C2×#92418—21no—
124C2×#92418—41no—
125C2×#92418C22⋊C411yescharacteristic
126C2×#92418C22⋊C411yescharacteristic
127#11758189—42no—
128C8×D61—22no—
129C8×D61—22no—
130C8×D61—41no—
131C8×D61—41no—
132C8×D61—42no—
133C23×C61#11711yescharacteristic
134C22×D61—22no—
135C22×D61—22no—
136C22×D61—21no—
137C22×D61—41no—
138C22×D61C4×D411yescharacteristic
139C22×D61C4×D411yescharacteristic
140C2×#6923—24no—
141C2×#6923—42no—
142C2×#6923—42no—
143C2×#6923—42no—
144C2×#6923—42no—
145C2×#6923—42no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61C2×C4211yescharacteristic
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C22⋊D61—42no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61C4×D411yescharacteristic
156C2×Dic61C4×D411yescharacteristic
157D244—41no—
158D244—42no—
159D244—42no—
160C4×D61—21no—
161C4×D61—21no—
162C4×D61—21no—
163C4×D61—21no—
164C4×D61—41no—
165C4×D61—41no—
166C4×D61—41no—
167C4×D61—41no—
168Dic122—41no—
169#92420—42no—
170#92420—42no—
171C8×C61—21no—
172C8×C61—41no—
173C8×C61#9912yes—
174#92418—21no—
175#92418—21no—
176#92418—41no—
177#8240—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61C4×C22⋊C411yescharacteristic
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—44no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61#44312yes—
191C2×D61#44412yes—
192#6923—44no—
193#6923—82no—
194C4×C61—21no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61C2×#9911yescharacteristic
198C4×C61#41011yescharacteristic
199Dic61—21no—
200Dic61—21no—
201Dic61—41no—
202#2929—611no—
203#1752—611no—
204C8×C22⋊C4—611no—
205#1501—611no—
206#1501—611no—
207#1298—611no—
208#1297—611no—
209C2×C61—41no—
210C2×C61C4×#9911yescharacteristic
211C2×C61#176411yescharacteristic
212C2×C61#176511yescharacteristic
213D61—22no—
214D61—22no—
215D61—81no—
216C2×#131—611no—
217C23×C8—611no—
218#525—611no—
219#435—1221no—
220#435—1221no—
221C8×D4—1221no—
222C8×D4—1221no—
223#424—611no—
224#408—611no—
225#407—611no—
226C2×C22⋊C8—611no—
227C2×C22⋊C8—611no—
228#405#692311yescharacteristic, complemented
229C2×#97—1221no—
230C2×C4×C8—1221no—
231C4×C22⋊C4—611no—
232C4×C22⋊C4—611no—
233#336—1221no—
234#336—1221no—
235#328—612no—
236#328—612no—
237#327—612no—
238#327—612no—
239C61#824011yescharacteristic, complemented, Sylow
240C2×#41—611no—
241C23×C4—611no—
242#131—2442no—
243C2×M16—611no—
244C2×M16—1221no—
245C2×M16C2×#692311yescharacteristic
246C22×C8—611no—
247C22×C8—611no—
248C22×C8—611no—
249C22×C8—611no—
250C22×C8—611no—
251C22×C8—1221no—
252C22×C8—1221no—
253C22×C8—1221no—
254C22×C8—2442no—
255C22×C8C2×#692311yescharacteristic
256#123—1221no—
257C22⋊Q8—611no—
258C22⋊Q8—611no—
259#121—611no—
260#121—611no—
261#120—1221no—
262C4×D4—1221no—
263C4×D4—1221no—
264#117C2×#692311yescharacteristic
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C42—1221no—
268C2×C42—1221no—
269C4⋊C8—1221no—
270C4⋊C8—1221no—
271C22⋊C8—612no—
272C22⋊C8—612no—
273C22⋊C8—1221no—
274C22⋊C8—1221no—
275C22⋊C8—1221no—
276C22⋊C8—1221no—
277C22⋊C8—1221no—
278C22⋊C8—1221no—
279#97—21no—
280#97—2442no—
281C4×C8—21no—
282C4×C8—2442no—
283#95—612no—
284#95—612no—
285C24—611no—
286#41—2441no—
287#41—2441no—
288C2×Q8—611no—
289C2×D4—611no—
290C2×D4—1221no—
291C2×D4—1221no—
292C2×D4—1221no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—611no—
298C22×C4—1222no—
299C22×C4—1221no—
300C22×C4—1222no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4C22×#692311yescharacteristic
304M16—41no—
305M16—2442no—
306M16—2441no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—611no—
310C2×C8—611no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C2×C8—1221no—
316C2×C8—2441no—
317C2×C8—2441no—
318C2×C8—2442no—
319C2×C8—2441no—
320C2×C8—2441no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C2×C8C22⋊#692311yescharacteristic
325C2×C8C22⋊#692311yescharacteristic
326C4⋊C4—1221no—
327C4⋊C4—1221no—
328C4⋊C4C4×#692311yescharacteristic
329C4⋊C4C4×#692311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—1221no—
332C22⋊C4—1221no—
333C22⋊C4—1221no—
334C22⋊C4—1221no—
335C22⋊C4—1224no—
336C22⋊C4—2442no—
337C42—21no—
338C42—2441no—
339C42—2441no—
340C42—2441no—
341C42—2441no—
342C23—611no—
343C23—611no—
344C23—1222no—
345C23—1222no—
346C23—1221no—
347C23—2441no—
348C23#1451985611yescharacteristic
349Q8—2441no—
350D4—2441no—
351D4—2442no—
352D4—2442no—
353D4—2442no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—611no—
359C2×C4—611no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—1221no—
364C2×C4—1224no—
365C2×C4—1221no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4C2×C4×#692311yescharacteristic
377C2×C4C2×C22⋊#692311yescharacteristic
378C8—21no—
379C8—41no—
380C8—1221no—
381C8—1221no—
382C8—2442no—
383C8—2442no—
384C8—2441no—
385C8#1451983612yes—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—612no—
390C22—612no—
391C22—1221no—
392C22—1221no—
393C22—1221no—
394C22—1221no—
395C22—2444no—
396C22—2441no—
397C22—2441no—
398C22—2441no—
399C22C4×C22⋊#692311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—1221no—
404C4—1221no—
405C4—4882no—
406C4—2444no—
407C4—2441no—
408C4C2×#1451983611yescharacteristic
409C4#1684074111yescharacteristic
410C2—41no—
411C2—1222no—
412C2—1222no—
413C2—4881no—
414C2C4×#1451983611yescharacteristic
415C2#2146159311yescharacteristic
416C2#2146159711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#405C4
socleC22×C61C4×C22⋊C4
prime radical#16840274C22
cosocleC22×C4×D61C23
hypercenter#405#6923
hypocenterC61#8240
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #21461555C2semidirect
2. C8×C22⋊#6923C2semidirect
5. #21460701C2semidirect
6. #21460699C2semidirect
16. #16840615C4semidirect
17. #16840615C4semidirect
18. #16840614C4semidirect
19. #16840614C4semidirect
20. C2×#14519783C4semidirect
27. #16839789C4semidirect
28. #16839730C4semidirect
30. C61×#405C4semidirect
228. #405#6923semidirect
239. C61#8240semidirect

Maximal subgroups

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

subgroupindex
#214615552
C8×C22⋊#69232
#214611962
#214607012
#214606992
#214545152
#824061