Finite Group Databasebeta

← #29577326

Subgroups — #29577326

Order 15616.

570 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 127 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#21454843C211yesmaximal, characteristic
2#21450995C211yesmaximal, characteristic
3#21444390C212yesmaximal, complemented
4#21443697C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7D4×#92418—24no—
8#16839815—21no—
9C2×C22⋊#92418—21no—
10C2×C22⋊#92418C2212yes—
11#16839701C411yescharacteristic, complemented
12#16839698C411yescharacteristic, complemented
13#16839682C411yescharacteristic, complemented
14#16839674C411yescharacteristic, complemented
15#16839646—21no—
16#16839642—21no—
17#16839635—21no—
18#16839620—21no—
19C2×#14519684C2211yescharacteristic
20#16839498—24no—
21C2×C4×D4×C61C2211yescharacteristic
22C2×#99×C61—21no—
23C61×#380C2212yes—
24C61×#342—21no—
25C22×Dic122C2×C411yescharacteristic
26C23×Dic61—21no—
27C22×#92418—22no—
28C22×#92418—22no—
29#14519756—22no—
30#14519756—22no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C4⋊#92418—22no—
39C22⋊#92418—22no—
40C22⋊#92418—22no—
41C22⋊#92418D414yes—
42C4×#92418—22no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×D4×C61D411yescharacteristic
51C23×C4×C61D411yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61—22no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61D411yescharacteristic
60C2×C42×C61D411yescharacteristic
61#99×C61—42no—
62#95×C61—22no—
63C24×C61C22⋊C411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C2×D4×C61—22no—
66C2×D4×C61—22no—
67C2×D4×C61SD1612yes—
68C2×D4×C61Q1612yes—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—81no—
73C22×C4×C61—81no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D412yes—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4112yes—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61—81no—
88C2×Dic122—22no—
89C2×Dic122—81no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—82no—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—22no—
94C22⋊C4×C61—42no—
95C22⋊C4×C61—42no—
96C42×C61—21no—
97C42×C61—21no—
98C22⋊Dic61—42no—
99C22⋊Dic61—42no—
100C22⋊Dic61—81no—
101C4⋊Dic61—22no—
102#11758196—42no—
103#11758196—41no—
104#11758196—41no—
105#11758196—81no—
106C2×#92418—24no—
107C2×#92418—82no—
108C2×#92418—82no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61#10312yes—
116C23×C61C2×C22⋊C411yescharacteristic
117C23×C61#12112yes—
118C23×C61C2×SD1611yescharacteristic
119C23×C61C2×Q1611yescharacteristic
120C23×C61#13512yes—
121D4×C61—24no—
122D4×C61—24no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—81no—
135C2×C4×C61—82no—
136C2×C4×C61—82no—
137C2×C4×C61—82no—
138C2×C4×C61#10312yes—
139C2×C4×C61C4×D412yes—
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—84no—
155Dic122—82no—
156#92418—44no—
157#9383—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—22no—
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—42no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—81no—
170C22×C61#38611yescharacteristic
171C22×C61C2×#10311yescharacteristic
172C22×C61#41611yescharacteristic
173C22×C61#46512yes—
174C22×C61#46712yes—
175C4×C61—22no—
176C4×C61—21no—
177C4×C61—22no—
178C4×C61—21no—
179C4×C61—41no—
180C4×C61—41no—
181C4×C61—162no—
182Dic61—42no—
183Dic61—81no—
184Dic61—81no—
185Dic61—81no—
186#2972—611no—
187#2564—611no—
188#1799D6111yescharacteristic
189#1520—612no—
190#1294—612no—
191C2×C61—21no—
192C2×C61—22no—
193C2×C61—21no—
194C2×C61—22no—
195C2×C61—42no—
196C2×C61#188411yescharacteristic
197C2×C61#189211yescharacteristic
198C2×C61#191311yescharacteristic
199C2×C22⋊Q8—611no—
200C2×C4×D4C2×D6111yescharacteristic
201C8×D4—1224no—
202#431—1221no—
203C2×#99—21no—
204C2×C22⋊C8—612no—
205C2×C22⋊C8—1221no—
206#400—611no—
207#399—611no—
208#395—611no—
209#393—611no—
210#387—1221no—
211#386—1221no—
212#385—1221no—
213#380—1221no—
214#380C2×D6112yes—
215C2×#95—611no—
216#342—21no—
217#328—1224no—
218C61#938311yescharacteristic, complemented, Sylow
219C22×Q8—611no—
220C22×D4C22⋊D6111yescharacteristic
221C23×C4—1221no—
222C23×C4D24411yescharacteristic
223C23×C4C22×D6111yescharacteristic
224C22×C8—1222no—
225C22×C8—1222no—
226C22⋊Q8—1222no—
227C22⋊Q8—1222no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—611no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4C4×D6111yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4C4×D6111yescharacteristic
241C2×C22⋊C4C22⋊D6111yescharacteristic
242C2×C42D24411yescharacteristic
243C4⋊C8—1222no—
244#99—42no—
245C22⋊C8—614no—
246C22⋊C8—1222no—
247C22⋊C8—1222no—
248C4×C8—1222no—
249#95—22no—
250#95—611no—
251#95—611no—
252#95—1221no—
253#95—1221no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257C24#1175819811yescharacteristic
258C24C2×C22⋊D6111yescharacteristic
259C2×Q8—1222no—
260C2×Q8—4881no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1175820012yes—
264C2×D4#1175820212yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—4881no—
275C22×C4—2442no—
276C22×C4—2442no—
277C22×C4#1175819811yescharacteristic
278C22×C4#1175819811yescharacteristic
279C22×C4#1175819811yescharacteristic
280C22×C4C2×C4×D6111yescharacteristic
281C22×C4C2×D24411yescharacteristic
282C22×C4D4×D6112yes—
283C22×C4#1175822412yes—
284C2×C8—1224no—
285C2×C8—4882no—
286C2×C8—4882no—
287C4⋊C4—22no—
288C4⋊C4—82no—
289C4⋊C4—1222no—
290C4⋊C4—4881no—
291C4⋊C4—2442no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—4881no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C42—21no—
302C42—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1451970012yes—
310C23C2×#1175819811yescharacteristic
311C23#1451974912yes—
312C23C2×#1175820011yescharacteristic
313C23C2×#1175820211yescharacteristic
314C23#1451979712yes—
315Q8—4882no—
316D4—24no—
317D4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—1222no—
334C2×C4—4884no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4#1451970012yes—
347C2×C4#1451973712yes—
348C2×C4#1451974711yescharacteristic
349C2×C4#1451976111yescharacteristic
350C8—2444no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#1683963911yescharacteristic
364C22C2×#1451970011yescharacteristic
365C22#1683975611yescharacteristic
366C22#1683998912yes—
367C22#1684000112yes—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—162no—
375C4—4881no—
376C4—4881no—
377C4—4881no—
378C4—2442no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#2144620011yescharacteristic
385C2#2144624011yescharacteristic
386C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9383
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21444390C2semidirect
4. #21443697C2semidirect
11. #16839701C4semidirect
12. #16839698C4semidirect
13. #16839682C4semidirect
14. #16839674C4semidirect
218. C61#9383semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214509952
#214443902
#214436972
C61×#17992
#938361