Finite Group Databasebeta

← #29576293

Subgroups — #29576293

Order 15616.

550 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455828C211yesmaximal, characteristic
2#21448280C211yesmaximal, characteristic
3#21446795C211yesmaximal, characteristic, complemented
4#21444317C211yesmaximal, characteristic, complemented
5#21444243C211yesmaximal, characteristic
6#21443651C211yesmaximal, characteristic, complemented
7C61×#1303C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10C2×#14519734—21no—
11#16840093—21no—
12#16840053C412yescomplemented
13#16839968C412yescomplemented
14#16839915—21no—
15C4×#11758190—22no—
16#16839815C2211yescharacteristic
17#16839663—21no—
18#16839658—21no—
19#16839651—22no—
20C4×#11758198—21no—
21C42×Dic61C2211yescharacteristic
22#16839523—21no—
23#16839523—21no—
24#16839498—22no—
25#16839489—22no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758190—22no—
32#14519777C2×C411yescharacteristic
33#14519777C2×C411yescharacteristic
34#14519769—21no—
35#14519756—21no—
36#14519756—21no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244C2×C412yes—
42C4×D244D412yes—
43#14519736—21no—
44#14519734—42no—
45#14519734—42no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—22no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55#14519709—22no—
56C4⋊#92418—21no—
57#14519702—22no—
58#14519699—22no—
59C22⋊#92418—21no—
60C4×#92418—21no—
61#14519684—22no—
62C2×M16×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C42×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61C2×C412yes—
69C22⋊C8×C61—21no—
70C4×C8×C61—21no—
71#95×C61D412yes—
72C23×D61—21no—
73C22×C4×C61—22no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C2×D244—21no—
81C2×D244—42no—
82C2×D244C22⋊C412yes—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×Dic122—21no—
90C2×Dic122—21no—
91M16×C61—41no—
92M16×C61—41no—
93C2×C8×C61—22no—
94C2×C8×C61—41no—
95C4⋊C4×C61—42no—
96C42×C61—41no—
97C42×C61—41no—
98C42×C61C22×C411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101C42×C61#4111yescharacteristic
102C22⋊Dic61—41no—
103#11758198—22no—
104#11758198—22no—
105#11758198—44no—
106#11758198—44no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61C22⋊C412yes—
113#11758196—44no—
114#11758196—41no—
115#11758196—41no—
116C4×Dic61—21no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—42no—
120C4×Dic61—42no—
121C4×Dic61—41no—
122C4×Dic61SD1612yescomplemented
123C2×#92418—41no—
124#11758190—44no—
125C23×C61#12011yescharacteristic
126C22×D61—22no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—41no—
130C22×D61—41no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×C4×C61C2×SD1611yescharacteristic
143C2×C4×C61#13511yescharacteristic
144C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C61#13711yescharacteristic
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157D244—22no—
158D244—22no—
159D244—42no—
160C4×D61—42no—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—42no—
164Dic122—42no—
165Dic122—81no—
166C8×C61—42no—
167C8×C61—81no—
168#92418—81no—
169#9235—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#38611yescharacteristic
173C22×C61#45011yescharacteristic
174C22×C61#45211yescharacteristic
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61—82no—
179C2×D61—82no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—82no—
184C4×C61#42112yes—
185C4×C61C4×SD1611yescharacteristic
186C4×C61#44011yescharacteristic
187C4×C61#46111yescharacteristic
188C4×C61#46311yescharacteristic
189Dic61—42no—
190Dic61—42no—
191Dic61—42no—
192Dic61—81no—
193Dic61—81no—
194#3101—611no—
195#2293—611no—
196#1994—611no—
197#1503—611no—
198#1486—611no—
199#1303D6111yescharacteristic, complemented
200#1276—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61#187711yescharacteristic
204C2×C61#189711yescharacteristic
205C2×C61#190611yescharacteristic
206D61—82no—
207#533—611no—
208C2×C4×D4—611no—
209C2×#117—1221no—
210#483—1221no—
211#476—612no—
212#461—612no—
213#450—1221no—
214C4×SD16—1222no—
215#431—611no—
216#431C2×D6111yescharacteristic
217#423C2×D6111yescharacteristic
218#390—1221no—
219#389—1221no—
220#388—1222no—
221C4×C4⋊C4C2×D6111yescharacteristic
222C4×C22⋊C4—1221no—
223C43—611no—
224#339—1221no—
225#339—1221no—
226#328—1222no—
227D4⋊C8—1222no—
228C61#923511yescharacteristic, complemented, Sylow
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1222no—
232C2×M16—21no—
233C4⋊Q8—611no—
234C4⋊Q8—611no—
235#125—1221no—
236C22⋊Q8—1221no—
237C22⋊Q8—1221no—
238C4×Q8—1221no—
239C4×Q8—1221no—
240C4×D4—612no—
241C4×D4—612no—
242C4×D4—1221no—
243C4×D4—1221no—
244#117—1221no—
245#117—2442no—
246#117—2442no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1222no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C42—21no—
254C2×C42—611no—
255C2×C42—611no—
256C2×C42—1221no—
257C2×C42—1221no—
258C2×C42C22×D6111yescharacteristic
259#106—1222no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—1221no—
263C4⋊C8C4×D6112yes—
264#103—1222no—
265#102—1222no—
266C22⋊C8—21no—
267C22⋊C8—1221no—
268C4×C8—21no—
269C4×C8—1221no—
270#95—1222no—
271#95D24412yes—
272C24—1221no—
273C2×Q8—1221no—
274C2×Q8—1221no—
275C2×D4—612no—
276C2×D4—1221no—
277C2×D4—2442no—
278C22×C4—22no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—2441no—
285C22×C4—2441no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4C2×D24411yescharacteristic
289C22×C4D4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291SD16—2444no—
292M16—41no—
293M16—41no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—2441no—
297C4⋊C4—42no—
298C4⋊C4—612no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—2444no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C22⋊C4—1222no—
308C22⋊C4—1222no—
309C22⋊C4—2444no—
310C22⋊C4—2444no—
311C22⋊C4—2441no—
312C42—41no—
313C42—41no—
314C42—612no—
315C42—1221no—
316C42—1221no—
317C42—1221no—
318C42—2442no—
319C42—2442no—
320C42—2441no—
321C42C2×C4×D6111yescharacteristic
322C42#1175822211yescharacteristic
323C42D4×D6111yescharacteristic
324C42#1175822611yescharacteristic
325C23—1222no—
326C23—2442no—
327C23—2442no—
328C23—2441no—
329C23—2441no—
330C23#1451974711yescharacteristic
331Q8—4881no—
332Q8—2442no—
333D4—1222no—
334D4—1222no—
335D4—2442no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4C22⋊C4×D6111yescharacteristic
357C2×C4#1451973911yescharacteristic
358C2×C4C4×D24411yescharacteristic
359C2×C4#1451975011yescharacteristic
360C2×C4#1451975211yescharacteristic
361C2×C4#1451976211yescharacteristic
362C2×C4SD16×D6111yescharacteristic
363C2×C4#1451979811yescharacteristic
364C2×C4#1451980111yescharacteristic
365C2×C4#1451980911yescharacteristic
366C8—42no—
367C8—81no—
368C8—4881no—
369C22—21no—
370C22—21no—
371C22—4882no—
372C22—4882no—
373C22—2442no—
374C22—2442no—
375C22—2442no—
376C22#1683964011yescharacteristic
377C22#1683991411yescharacteristic
378C22#1683992711yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—4881no—
384C4—4881no—
385C4—2442no—
386C4—2442no—
387C4—2442no—
388C4#1683977712yes—
389C4#1683984911yescharacteristic
390C4#1683986111yescharacteristic
391C4#1683996611yescharacteristic
392C4#1683997611yescharacteristic
393C2—21no—
394C2—21no—
395C2—4882no—
396C2#2144615411yescharacteristic
397C2#2144626611yescharacteristic
398C2#2144632111yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1303C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1303D61
hypocenterC61#9235
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #21446795C2semidirect
4. #21444317C2semidirect
6. #21443651C2semidirect
7. C61×#1303C2semidirect
12. #16840053C4semidirect
13. #16839968C4semidirect
122. C4×Dic61SD16semidirect
199. #1303D61semidirect
228. C61#9235semidirect

Maximal subgroups

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

subgroupindex
#214558282
#214482802
#214467952
#214443172
#214442432
#214436512
C61×#13032
#923561