Finite Group Databasebeta

← #29584418

Subgroups — #29584418

Order 15616.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#21446011C212yesmaximal
2#21444075C211yesmaximal, characteristic
3#21443697C212yesmaximal
4#21443662C211yesmaximal, characteristic
5C61×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic61C2211yescharacteristic
7C2×#14519708—22no—
8C2×C22⋊#92418C2212yes—
9C2×#14519684—21no—
10C2×#14519684—21no—
11C2×#14519684C2211yescharacteristic
12#16839524—22no—
13C2×C22⋊Q8×C61—21no—
14C61×#398—22no—
15C61×#395—22no—
16C61×#380—21no—
17C61×#380C2212yes—
18C2×#95×C61C2211yescharacteristic
19C23×Dic61—21no—
20C23×Dic61—21no—
21C22×#92418—22no—
22C2×C4⋊Dic61—22no—
23C2×C4⋊Dic61—42no—
24#14519708—44no—
25C22⋊#92418—22no—
26C22⋊#92418Q814yescomplemented
27#14519684—21no—
28#14519684—42no—
29#14519684—42no—
30#14519684Q812yescomplemented
31C22×Q8×C61—21no—
32C23×C4×C61D411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C23×C4×C61C2311yescharacteristic
35C22⋊Q8×C61—42no—
36C22⋊Q8×C61—42no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61D412yes—
42C2×C4⋊C4×C61D412yes—
43C2×C22⋊C4×C61—22no—
44#95×C61—22no—
45#95×C61—22no—
46#95×C61—22no—
47C24×C61C2×D411yescharacteristic
48C2×Q8×C61—42no—
49C2×Q8×C61—81no—
50C2×Q8×C61—81no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—41no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61SD1612yes—
64C22×C4×C61SD1612yes—
65C22×C4×C61C2×D412yes—
66C22×C4×C61C2×Q812yes—
67C22×C4×C61C2×Q811yescharacteristic
68C22×C4×C61#4111yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—41no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—41no—
76C22×Dic61—41no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61—41no—
80C22×Dic61—41no—
81C22×Dic61—41no—
82C22×Dic61—41no—
83C4⋊C4×C61—42no—
84C4⋊C4×C61—44no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—81no—
88C4⋊C4×C61—81no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C4⋊Dic61—24no—
92C4⋊Dic61—42no—
93C2×#92418—24no—
94C2×#92418—42no—
95C2×#92418—42no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61#12011yescharacteristic
101C23×C61C22⋊Q811yescharacteristic
102C23×C61C4⋊Q811yescharacteristic
103C23×C61C2×SD1611yescharacteristic
104C23×C61C2×SD1611yescharacteristic
105C23×C61AGL(1,ℤ/8)11yescharacteristic
106C23×C61#13711yescharacteristic
107Q8×C61—84no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—81no—
128C2×C4×C61—81no—
129C2×C4×C61C22⋊Q812yes—
130C2×C4×C61C4⋊Q812yes—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—81no—
138C2×Dic61—81no—
139C2×Dic61—81no—
140C2×Dic61—81no—
141C2×Dic61—81no—
142C2×Dic61—81no—
143C2×Dic61—81no—
144C2×Dic61—81no—
145#92418—44no—
146#10568—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61#39511yescharacteristic
160C22×C61#44811yescharacteristic
161C22×C61#45011yescharacteristic
162C22×C61#45411yescharacteristic
163C22×C61#45511yescharacteristic
164C22×C61B(4,𝔽2)11yescharacteristic
165C22×C61#45811yescharacteristic
166C22×C61#47511yescharacteristic
167C22×C61#47611yescharacteristic
168C22×C61#49811yescharacteristic
169C22×C61#50111yescharacteristic
170C4×C61—22no—
171C4×C61—22no—
172C4×C61—42no—
173C4×C61—42no—
174C4×C61—82no—
175C4×C61—82no—
176Dic61—82no—
177Dic61—82no—
178#2659D6111yescharacteristic
179#1853—612no—
180#1442—611no—
181#1294—612no—
182#1280—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#159911yescharacteristic
188C2×C61#160111yescharacteristic
189C2×C61#160811yescharacteristic
190C2×C61#161611yescharacteristic
191C2×C61#205811yescharacteristic
192C2×C61#206111yescharacteristic
193C2×C61#206211yescharacteristic
194C2×C22⋊Q8—21no—
195C22×C4⋊C4—611no—
196C2×#106—1222no—
197C2×C22⋊C8—612no—
198#398—22no—
199#395—22no—
200#380—21no—
201#380C2×D6112yes—
202C2×#95—611no—
203C2×#95—1221no—
204C2×#95—1221no—
205C2×#95C2×D6111yescharacteristic
206#340—1222no—
207C61#1056811yescharacteristic, complemented, Sylow
208C22×Q8—21no—
209C23×C4—1221no—
210C23×C4—1221no—
211C23×C4C22⋊D6111yescharacteristic
212C23×C4C22⋊D6111yescharacteristic
213C23×C4C22×D6111yescharacteristic
214C22×C8—1222no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—2442no—
223C2×C4⋊C4C22⋊D6112yes—
224C2×C4⋊C4C22⋊D6112yes—
225C2×C22⋊C4—22no—
226#106—2444no—
227C22⋊C8—614no—
228C22⋊C8—1222no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—612no—
233#95—1221no—
234#95—2442no—
235#95—2442no—
236C24C2×C22⋊D6111yescharacteristic
237C2×Q8—42no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—1222no—
253C22×C4—1222no—
254C22×C4—2441no—
255C22×C4—2441no—
256C22×C4—2441no—
257C22×C4—2441no—
258C22×C4—2441no—
259C22×C4—2441no—
260C22×C4—2441no—
261C22×C4—2441no—
262C22×C4—2441no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4#1175820012yes—
267C22×C4#1175820112yes—
268C22×C4Q8×D6112yes—
269C22×C4Q8×D6111yescharacteristic
270C22×C4#1175822611yescharacteristic
271C22×C4C2×C22⋊D6112yes—
272C2×C8—1224no—
273C2×C8—2442no—
274C2×C8—2442no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—1224no—
282C4⋊C4—2442no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1451974611yescharacteristic
290C23#1451975311yescharacteristic
291C23#1451977511yescharacteristic
292C23C2×#1175820111yescharacteristic
293C23C2×#1175820011yescharacteristic
294C23#1451980311yescharacteristic
295C23#1451980811yescharacteristic
296Q8—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—4881no—
323C2×C4—4881no—
324C2×C4—4881no—
325C2×C4—4881no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4#1451975312yes—
333C2×C4#1451977512yes—
334C8—2444no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1683968111yescharacteristic
348C22#1683990111yescharacteristic
349C22#1683991311yescharacteristic
350C22#1683993511yescharacteristic
351C22#1683994011yescharacteristic
352C22#1683994911yescharacteristic
353C22#1683995211yescharacteristic
354C22#1684004411yescharacteristic
355C22#1684004911yescharacteristic
356C22#1684015811yescharacteristic
357C22#1684016811yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—4882no—
365C4—4882no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2144475911yescharacteristic
371C2#2144477111yescharacteristic
372C2#2144481011yescharacteristic
373C2#2144485611yescharacteristic
374C2#2144714911yescharacteristic
375C2#2144716611yescharacteristic
376C2#2144717111yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2659C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenter#2659D61
hypocenterC61#10568
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
26. C22⋊#92418Q8semidirect
30. #14519684Q8semidirect
207. C61#10568semidirect

Maximal subgroups

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

subgroupindex
#214460112
#214440752
#214436972
#214436622
C61×#26592
#1056861