Finite Group Databasebeta

← #29576909

Subgroups — #29576909

Order 15616.

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

normal subgroups only

This group has 612 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#21455919C211yesmaximal, characteristic
2#21448181C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic, complemented
4#21444330C211yesmaximal, characteristic, complemented
5#21444263C211yesmaximal, characteristic
6#21443676C211yesmaximal, characteristic, complemented
7C61×#1300C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244—21no—
10C2×C4×D244C2211yescharacteristic
11#16840098—21no—
12#16840043C412yescomplemented
13#16839968C412yescomplemented
14#16839915—21no—
15#16839864—22no—
16#16839819C2211yescharacteristic
17C4×C4⋊Dic61—21no—
18C4×C4⋊Dic61C2211yescharacteristic
19#16839609—22no—
20#16839609—21no—
21#95×D61—21no—
22#16839523—22no—
23#16839498—21no—
24#16839498—21no—
25#16839491—22no—
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—22no—
31C22×C4×D61—21no—
32C2×#11758190—22no—
33#14519777C2×C411yescharacteristic, complemented
34#14519777C2×C411yescharacteristic, complemented
35#14519769—21no—
36#14519756—21no—
37#14519756—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×D244—28no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244C2×C412yes—
44C4×D244D412yes—
45#14519736—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758198—22no—
49C2×#11758198—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61D412yes—
52#14519711—22no—
53C4⋊#92418—21no—
54#14519702—22no—
55#14519699—22no—
56C22⋊#92418—21no—
57#14519688—21no—
58#14519684—22no—
59#14519684—21no—
60C2×M16×C61—21no—
61C2×C42×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61C2×C412yescomplemented
64C22⋊C8×C61—21no—
65#97×C61—21no—
66#97×C61—21no—
67C4×C8×C61—21no—
68#95×C61—21no—
69#95×C61D411yescharacteristic
70#95×C61D411yescharacteristic
71C23×D61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—22no—
79C2×D244—21no—
80C2×D244—24no—
81C2×D244C22⋊C412yes—
82C2×C4×D61—28no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—82no—
86C2×C4×D61—82no—
87C2×C4×D61—82no—
88C2×Dic122—21no—
89C2×Dic122—21no—
90M16×C61—22no—
91M16×C61—41no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C42×C61—24no—
95C42×C61C22×C411yescharacteristic
96C42×C61C2×D411yescharacteristic
97C42×C61#4111yescharacteristic
98C42×C61#4111yescharacteristic
99C22⋊Dic61—41no—
100#11758198—28no—
101#11758198—22no—
102#11758198—22no—
103C4⋊Dic61—24no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61C22⋊C412yes—
110#11758196—41no—
111#11758196—41no—
112C4×Dic61—41no—
113C4×Dic61—42no—
114C4×Dic61—82no—
115C2×#92418—41no—
116#11758190—44no—
117C23×C61#12011yescharacteristic
118C22×D61—24no—
119C22×D61—22no—
120C22×D61—42no—
121C22×D61—41no—
122C22×D61—41no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61C2×C22⋊C411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12311yescharacteristic
139C2×Dic61—24no—
140C2×Dic61—22no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146D244—24no—
147D244—22no—
148D244#10414yes—
149C4×D61—82no—
150C4×D61—82no—
151C4×D61—88no—
152Dic122—42no—
153Dic122—81no—
154C8×C61—42no—
155C8×C61—42no—
156#92418—81no—
157#9323—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45511yescharacteristic
162C22×C61#45611yescharacteristic
163C2×D61—24no—
164C2×D61—44no—
165C2×D61—44no—
166C2×D61—42no—
167C2×D61—42no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—42no—
174C4×C61—42no—
175C4×C61C2×#10412yes—
176Dic61—44no—
177Dic61—42no—
178Dic61—81no—
179Dic61—81no—
180#3110—611no—
181#2275—611no—
182#1986—611no—
183#1506—611no—
184#1490—611no—
185#1300D6111yescharacteristic, complemented
186#1285—611no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61#188211yescharacteristic
190C2×C61#189611yescharacteristic
191C2×C61#190411yescharacteristic
192D61—44no—
193#533—611no—
194C2×C4×D4—611no—
195C2×C4×D4—1221no—
196#484—1221no—
197#474—612no—
198#461—612no—
199#450—1221no—
200#440—1222no—
201#432—611no—
202#432C2×D6111yescharacteristic
203C4×M16C2×D6111yescharacteristic
204C4×C4⋊C4—611no—
205C4×C4⋊C4—1221no—
206C4×C22⋊C4—1222no—
207C4×C22⋊C4—1221no—
208#376C2×D6111yescharacteristic
209C2×#95—1221no—
210#339—1222no—
211#328—1221no—
212#328—1221no—
213D4⋊C8—1222no—
214C61#932311yescharacteristic, complemented, Sylow
215C22×D4—611no—
216C23×C4—1222no—
217C23×C4—1221no—
218C2×SD16—1222no—
219C2×M16—21no—
220C4⋊Q8—611no—
221C4⋊Q8—611no—
222#125—1221no—
223C22⋊Q8—1221no—
224C22⋊Q8—1221no—
225C4×Q8—1221no—
226C4×Q8—1221no—
227C4×D4—612no—
228C4×D4—612no—
229C4×D4—1228no—
230C4×D4—1221no—
231C4×D4—1221no—
232#117—1221no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—1221no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1221no—
237C2×C42—21no—
238C2×C42—612no—
239C2×C42—1222no—
240C2×C42C22×D6111yescharacteristic
241#107—1222no—
242C4⋊C8—1221no—
243C4⋊C8C4×D6112yescomplemented
244#103—1222no—
245#102—1222no—
246C22⋊C8—21no—
247C22⋊C8—1221no—
248#97—21no—
249#97—21no—
250#97—1221no—
251C4×C8—21no—
252#95—21no—
253#95—1222no—
254#95—1221no—
255#95D24411yescharacteristic
256#95D24411yescharacteristic
257C24—1221no—
258C2×Q8—1221no—
259C2×Q8—1221no—
260C2×D4—612no—
261C2×D4—1224no—
262C2×D4—1221no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1228no—
266C22×C4—1222no—
267C22×C4—1221no—
268C22×C4—1222no—
269C22×C4—1222no—
270C22×C4—4882no—
271C22×C4—4882no—
272C22×C4—4882no—
273C22×C4C2×D24411yescharacteristic
274C22×C4D4×D6111yescharacteristic
275C22×C4D4×D6111yescharacteristic
276SD16—2444no—
277M16—22no—
278M16—41no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—2441no—
282C4⋊C4—612no—
283C4⋊C4—1224no—
284C4⋊C4—1221no—
285C4⋊C4—1221no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—1221no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C22⋊C4—1228no—
292C22⋊C4—1222no—
293C22⋊C4—1222no—
294C22⋊C4—2441no—
295C42—24no—
296C42—4882no—
297C42—2442no—
298C42—2441no—
299C42C2×C4×D6111yescharacteristic
300C42#1175822211yescharacteristic
301C42D4×D6111yescharacteristic
302C42#1175822611yescharacteristic
303C23—1224no—
304C23—1222no—
305C23—2442no—
306C23—2441no—
307C23—2441no—
308C23#1451974711yescharacteristic
309Q8—4881no—
310Q8—2442no—
311D4—614no—
312D4—1224no—
313D4—1222no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—1224no—
325C2×C4—1222no—
326C2×C4—4882no—
327C2×C4—4882no—
328C2×C4—4888no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4C22⋊C4×D6111yescharacteristic
335C2×C4#1451973911yescharacteristic
336C2×C4C4×D24411yescharacteristic
337C2×C4#1451975011yescharacteristic
338C2×C4#1451975211yescharacteristic
339C2×C4#1451976211yescharacteristic
340C8—42no—
341C8—42no—
342C8—4881no—
343C22—21no—
344C22—21no—
345C22—1224no—
346C22—2444no—
347C22—2444no—
348C22—2442no—
349C22—2442no—
350C22#1683964011yescharacteristic
351C22#1683994111yescharacteristic
352C22#1683994611yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—4881no—
361C4—4881no—
362C4—2444no—
363C4—2442no—
364C4#104×D6112yes—
365C2—21no—
366C2—21no—
367C2—2444no—
368C2#2144618911yescharacteristic
369C2#2144626111yescharacteristic
370C2#2144630511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1300C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1300D61
hypocenterC61#9323
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #21446763C2semidirect
4. #21444330C2semidirect
6. #21443676C2semidirect
7. C61×#1300C2semidirect
12. #16840043C4semidirect
13. #16839968C4semidirect
33. #14519777C2×C4semidirect
34. #14519777C2×C4semidirect
63. C4⋊C8×C61C2×C4semidirect
185. #1300D61semidirect
214. C61#9323semidirect
243. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214559192
#214481812
#214467632
#214443302
#214442632
#214436762
C61×#13002
#932361