Finite Group Databasebeta

← #29597006

Subgroups — #29597006

Order 15616.

572 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 117 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#21455401C211yesmaximal, characteristic
2C2×#106×D61C211yesmaximal, characteristic, complemented
3C2×#16839573C211yesmaximal, characteristic, complemented
4C2×#16839571C211yesmaximal, characteristic, complemented
5#21447429C211yesmaximal, characteristic, complemented
6#21443924C211yesmaximal, characteristic, complemented
7C61×#1386C211yesmaximal, characteristic, complemented
8C2×#14519799C2211yescharacteristic
9C22×C8×D61C2211yescharacteristic
10#16840274—21no—
11C2×C4⋊C4×D61—21no—
12C2×#14519717C2211yescharacteristic
13#16840004C412yescomplemented
14#16839999C412yescomplemented
15#16839992—22no—
16C2×#14519709C2211yescharacteristic
17C2×#14519708—21no—
18#106×D61—22no—
19#106×D61—22no—
20#16839756—21no—
21#16839573—22no—
22#16839571—22no—
23#16839541—22no—
24C2×M32×C61C2211yescharacteristic
25C61×#427C2211yescharacteristic
26C2×#106×C61C2211yescharacteristic
27C2×#11758222—21no—
28C22×C4×D61—41no—
29C22×C4×D61D411yescharacteristic
30#14519799—21no—
31#14519799—21no—
32#14519799—21no—
33#14519799—21no—
34C2×Dic244C2×C411yescharacteristic, complemented
35C2×#11758190—21no—
36C2×D488C2×C411yescharacteristic, complemented
37C22×#92418D411yescharacteristic
38C2×C8×D61—21no—
39C2×C8×D61—21no—
40C2×C8×D61—21no—
41C2×C8×D61—21no—
42C2×C8×D61C2×C412yes—
43C2×C8×D61D412yes—
44#14519761—41no—
45#14519757—22no—
46#14519752—22no—
47#14519747—41no—
48C4×D244—42no—
49C2×C4⋊Dic61—21no—
50C2×#11758196—41no—
51C4⋊C4×D61—24no—
52C4⋊C4×D61—42no—
53#14519717—21no—
54#14519717—21no—
55C2×#11758185—21no—
56#14519709—21no—
57#14519709D412yescomplemented
58#14519708—24no—
59#14519700—22no—
60#14519697—22no—
61C22×C8×C61C2311yescharacteristic
62#117×C61—21no—
63C2×C4⋊C4×C61—21no—
64M32×C61—21no—
65M32×C61—21no—
66C2×C16×C61—21no—
67#107×C61C2×C412yescomplemented
68#106×C61—21no—
69#106×C61—21no—
70#106×C61D412yescomplemented
71C23×D61D811yescharacteristic
72C22×C4×C61—41no—
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—41no—
75C22×Dic61—41no—
76C22×Dic61D811yescharacteristic
77#11758222—41no—
78#11758222—41no—
79#11758222—41no—
80#11758222—41no—
81C2×D244—21no—
82C2×D244—42no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—44no—
88C2×C4×D61—41no—
89C2×C4×D61—84no—
90C2×C4×D61C22⋊C412yes—
91C2×C4×D61SD1612yes—
92C2×Dic122—21no—
93C2×C8×C61C22×C411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61#4111yescharacteristic
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104#11758198—42no—
105#11758198—42no—
106C4⋊Dic61—22no—
107C4⋊Dic61—41no—
108C4⋊Dic61—42no—
109#11758196—44no—
110C2×#92418—21no—
111C2×#92418—21no—
112C2×#92418C22⋊C412yes—
113Dic244—21no—
114Dic244—21no—
115D488—21no—
116D488—21no—
117#11758190—41no—
118#11758190—41no—
119C8×D61—22no—
120C8×D61—22no—
121C8×D61—22no—
122C8×D61—22no—
123C16×C61—41no—
124#11758185—41no—
125C23×C61C2×D811yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—24no—
129C22×D61—41no—
130C22×D61#10212yes—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—82no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12311yescharacteristic
137C2×C4×C61C2×SD1611yescharacteristic
138C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C61#13711yescharacteristic
140C22⋊D61—81no—
141C22⋊D61—81no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—41no—
145C2×Dic61—42no—
146C2×Dic61—82no—
147C2×Dic61#10212yes—
148D244—41no—
149D244—41no—
150D244—84no—
151C4×D61—22no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—22no—
155C4×D61—88no—
156C4×D61—81no—
157C4×D61—81no—
158Dic122—41no—
159Dic122—41no—
160C8×C61C4×D411yescharacteristic
161C8×C61C4×D411yescharacteristic
162C8×C61#12111yescharacteristic
163C8×C61#12111yescharacteristic
164#92418—22no—
165#13001—611nomaximal, Sylow
166C22×C61—22no—
167C22×C61C2×#10211yescharacteristic
168C22×C61#44711yescharacteristic
169C22×C61#48011yescharacteristic
170C2×D61—24no—
171C2×D61—24no—
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—81no—
175C2×D61—82no—
176C2×D61#36014yescomplemented
177C4×C61—82no—
178C4×C61—82no—
179C4×C61#38611yescharacteristic
180C4×C61#41711yescharacteristic
181C4×C61#44811yescharacteristic
182C4×C61#48111yescharacteristic
183Dic61—22no—
184Dic61—82no—
185Dic61—81no—
186#3050—611no—
187C22×#106—611no—
188C2×#360—611no—
189C2×#360—611no—
190#2112—611no—
191#1386—611no—
192#1386D6111yescharacteristic, complemented
193C2×C61—21no—
194C2×C61—21no—
195C2×C61#187411yescharacteristic
196C2×C61C2×#36012yes—
197D61—24no—
198D61—161no—
199C2×#135—611no—
200C23×C8—611no—
201#525—1221no—
202C22×C4⋊C4—1221no—
203C2×M32—611no—
204C2×M32C2×D6111yescharacteristic
205#467—612no—
206#466—612no—
207#465—1222no—
208#427C2×D6111yescharacteristic
209C2×#106—611no—
210C2×#106—1222no—
211C2×#106—1221no—
212C2×#106—1222no—
213C2×#106C2×D6111yescharacteristic
214#416—1221no—
215#360—1222no—
216#360—1222no—
217C22⋊C16—1222no—
218C61#1300111yescharacteristic, complemented, Sylow
219C2×#41—1221no—
220C23×C4—611no—
221C23×C4—2441no—
222#135—1221no—
223#135—1221no—
224#135—1221no—
225#135—1221no—
226C2×Q16—611no—
227C2×SD16—1221no—
228C2×D8—611no—
229C22×C8—611no—
230C22×C8—612no—
231C22×C8—612no—
232C22×C8—1221no—
233C22×C8—1221no—
234C22×C8—1221no—
235C22×C8—1221no—
236C22×C8C22×D6111yescharacteristic
237#123—2441no—
238C22⋊Q8—1222no—
239#121—1222no—
240#120—2441no—
241C4×D4—2442no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1224no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—2442no—
247C2×C4⋊C4—2441no—
248M32—21no—
249M32—21no—
250M32—1221no—
251M32—1221no—
252C2×C16—21no—
253C2×C16—1221no—
254#107C4×D6112yescomplemented
255#106—21no—
256#106—21no—
257#106—612no—
258#106—1224no—
259#106—1221no—
260#106D24412yescomplemented
261#103—1222no—
262#102—1222no—
263C24—611no—
264#41—2441no—
265#41—2441no—
266#41—2441no—
267#41—2441no—
268C2×Q8—1221no—
269C2×D4—1221no—
270C2×D4—2442no—
271C2×D4—2441no—
272C22×C4—41no—
273C22×C4—611no—
274C22×C4—612no—
275C22×C4—612no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—4884no—
281C22×C4—2444no—
282C22×C4—2441no—
283C22×C4—2441no—
284C22×C4D4×D6111yescharacteristic
285Q16—1221no—
286Q16—1221no—
287SD16—2441no—
288SD16—2441no—
289D8—1221no—
290D8—1221no—
291C2×C8—612no—
292C2×C8—1221no—
293C2×C8—1221no—
294C2×C8—1222no—
295C2×C8—1222no—
296C2×C8—1222no—
297C2×C8—1222no—
298C2×C8C2×C4×D6111yescharacteristic
299C2×C8C2×D24411yescharacteristic
300C2×C8#1175822211yescharacteristic
301C2×C8D4×D6111yescharacteristic
302C2×C8D4×D6111yescharacteristic
303C2×C8#1175822611yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—1222no—
308C4⋊C4—2442no—
309C4⋊C4—2444no—
310C4⋊C4—2441no—
311C22⋊C4—41no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C42—41no—
315C16—41no—
316C16—2441no—
317C23—612no—
318C23—1224no—
319C23—1221no—
320C23—1221no—
321C23—2441no—
322C23D8×D6111yescharacteristic
323Q8—2441no—
324Q8—2441no—
325D4—4884no—
326D4—4881no—
327D4—4881no—
328D4—2441no—
329D4—2441no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—612no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—1222no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—4888no—
341C2×C4—4882no—
342C2×C4—4881no—
343C2×C4—4881no—
344C2×C4—2441no—
345C2×C4—2442no—
346C2×C4C22⋊C4×D6111yescharacteristic
347C2×C4#1451974711yescharacteristic
348C2×C4#1451976211yescharacteristic
349C2×C4SD16×D6111yescharacteristic
350C2×C4#1451980411yescharacteristic
351C2×C4#1451980911yescharacteristic
352C8—1222no—
353C8#1451973911yescharacteristic
354C8C4×D24411yescharacteristic
355C8#1451975011yescharacteristic
356C8#1451975211yescharacteristic
357C22—22no—
358C22—614no—
359C22—1224no—
360C22—1224no—
361C22—1222no—
362C22—1222no—
363C22—4882no—
364C22—4881no—
365C22#102×D6111yescharacteristic
366C22#1683989711yescharacteristic
367C22#1684007711yescharacteristic
368C4—82no—
369C4—82no—
370C4—1222no—
371C4—4882no—
372C4—4881no—
373C4#1683964011yescharacteristic
374C4#1683976211yescharacteristic
375C4#1683990211yescharacteristic
376C4#1684008211yescharacteristic
377C2—21no—
378C2—21no—
379C2—1224no—
380C2—9761no—
381C2#2144613511yescharacteristic
382C2#360×D6112yes—
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1386C2
socleC22×C61C2×#102
prime radical#21455401C2
cosocleC22×C8×C61C23
hypercenter#1386D61
hypocenterC61#13001
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
2. C2×#106×D61C2semidirect
3. C2×#16839573C2semidirect
4. C2×#16839571C2semidirect
5. #21447429C2semidirect
6. #21443924C2semidirect
7. C61×#1386C2semidirect
13. #16840004C4semidirect
14. #16839999C4semidirect
34. C2×Dic244C2×C4semidirect
36. C2×D488C2×C4semidirect
57. #14519709D4semidirect
67. #107×C61C2×C4semidirect
70. #106×C61D4semidirect
176. C2×D61#360semidirect
192. #1386D61semidirect
218. C61#13001semidirect
254. #107C4×D61semidirect
260. #106D244semidirect

Maximal subgroups

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

subgroupindex
#214554012
C2×#106×D612
C2×#168395732
C2×#168395712
#214474292
#214439242
C61×#13862
#1300161