Finite Group Databasebeta

← #29596271

Subgroups — #29596271

Order 15616.

786 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 99 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#21448401C211yesmaximal, characteristic
2#21445734C211yesmaximal, characteristic
3#21443943C214yesmaximal, complemented
4C61×#1781C211yesmaximal, characteristic
5C2×#14519736C411yescharacteristic, complemented
6C2×#14519736C411yescharacteristic, complemented
7C2×#14519692—21no—
8C2×#14519692C2212yes—
9#16839665—21no—
10#16839665—21no—
11#16839665C412yescomplemented
12#16839665C2212yes—
13#16839618—21no—
14#16839618—21no—
15#16839618—21no—
16C23×D4×C61C2211yescharacteristic
17C2×#99×C61—21no—
18C2×#99×C61C2212yes—
19C61×#379—21no—
20#14519736—42no—
21#14519736—42no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61C2×C412yes—
24C2×C22⋊Dic61—21no—
25C2×C22⋊Dic61—21no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—42no—
32C2×C4×Dic61—21no—
33C2×C4×Dic61C2×C412yes—
34#14519692—24no—
35#14519692—42no—
36C25×C61D411yescharacteristic
37C25×C61D411yescharacteristic
38C22×D4×C61—24no—
39C22×D4×C61—42no—
40C22×D4×C61—42no—
41C22×D4×C61C2×C411yescharacteristic
42C22×D4×C61D411yescharacteristic
43C22×D4×C61D411yescharacteristic
44C22×D4×C61C2311yescharacteristic
45C23×C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—42no—
48#99×C61—24no—
49#99×C61—42no—
50C24×C61—22no—
51C24×C61—22no—
52C24×C61—22no—
53C24×C61—22no—
54C24×C61—41no—
55C24×C61—41no—
56C24×C61—82no—
57C24×C61—82no—
58C24×C61C22⋊C411yescharacteristic
59C24×C61C22⋊C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C24×C61C2×D411yescharacteristic
62C24×C61C2×D411yescharacteristic
63C2×D4×C61—22no—
64C2×D4×C61—22no—
65C2×D4×C61—22no—
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—416no—
75C2×D4×C61#4114yes—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×Dic61—22no—
83C22×Dic61—22no—
84C22×Dic61—42no—
85C22⋊C4×C61—42no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—84no—
88C22⋊Dic61—42no—
89C22⋊Dic61—42no—
90C22⋊Dic61—42no—
91C22⋊Dic61—42no—
92C22⋊Dic61—82no—
93C22⋊Dic61—82no—
94C22⋊Dic61—82no—
95C22⋊Dic61—82no—
96C22⋊Dic61—84no—
97C4⋊Dic61—42no—
98C4⋊Dic61—42no—
99C4×Dic61—42no—
100C4×Dic61—42no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—42no—
114C23×C61—44no—
115C23×C61—44no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—44no—
121C23×C61—44no—
122C23×C61—82no—
123C23×C61—82no—
124C23×C61—84no—
125C23×C61—84no—
126C23×C61—84no—
127C23×C61—84no—
128C23×C61#9912yes—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61#12312yes—
131C23×C61#12312yes—
132D4×C61—44no—
133D4×C61—44no—
134D4×C61—44no—
135D4×C61—44no—
136C2×C4×C61—28no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—82no—
142C2×C4×C61—84no—
143C2×C4×C61#9912yes—
144C2×C4×C61#11712yes—
145C2×Dic61—42no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—82no—
152C2×Dic61—84no—
153#12858—611nomaximal, Sylow
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—42no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—88no—
177C22×C61—88no—
178C22×C61#38111yescharacteristic
179C22×C61C2×#9911yescharacteristic
180C22×C61C2×#9911yescharacteristic
181C4×C61—22no—
182C4×C61—22no—
183C4×C61—162no—
184Dic61—82no—
185Dic61—82no—
186Dic61—162no—
187#2309—611no—
188#1781—611no—
189#1781D6111yescharacteristic
190#1393—614no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—42no—
194C2×C61—42no—
195C2×C61—42no—
196C2×C61—81no—
197C2×C61—81no—
198C2×C61#181911yescharacteristic
199C2×C61#212712yes—
200C23×D4C2×D6111yescharacteristic
201C2×#117—611no—
202C2×#117—611no—
203C2×#99—21no—
204C2×#99—612no—
205C2×#99—1221no—
206C2×#99C2×D6112yes—
207#390—612no—
208#390—612no—
209#390—1221no—
210#390—1221no—
211#379—21no—
212#379—1221no—
213#379—1221no—
214#379—1221no—
215C61#1285811yescharacteristic, complemented, Sylow
216C25D24411yescharacteristic
217C25C22⋊D6111yescharacteristic
218C22×D4—24no—
219C22×D4—42no—
220C22×D4—42no—
221C22×D4C4×D6111yescharacteristic
222C22×D4D24411yescharacteristic
223C22×D4C22⋊D6111yescharacteristic
224C22×D4C22×D6111yescharacteristic
225C23×C4C4×D6111yescharacteristic
226#117—2442no—
227#117—2442no—
228C2×C4⋊C4—612no—
229C2×C4⋊C4—1221no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—1222no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1222no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—2442no—
240C2×C42—612no—
241C2×C42—1221no—
242#99—24no—
243#99—42no—
244#99—1224no—
245#99—2442no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—82no—
253C24—82no—
254C24#1175819811yescharacteristic
255C24#1175819811yescharacteristic
256C24#1175819811yescharacteristic
257C24C2×D24411yescharacteristic
258C24C2×C22⋊D6111yescharacteristic
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—416no—
271C2×D4#1175822414yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—2442no—
279C22×C4#1175819811yescharacteristic
280C22×C4C2×C4×D6111yescharacteristic
281C4⋊C4—2442no—
282C4⋊C4—2442no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—84no—
286C22⋊C4—4884no—
287C22⋊C4—4882no—
288C22⋊C4—4882no—
289C22⋊C4—4882no—
290C22⋊C4—4882no—
291C22⋊C4—2442no—
292C22⋊C4—2442no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C42—2442no—
296C42—2442no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—42no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—44no—
317C23—44no—
318C23—82no—
319C23—82no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23#1451969112yes—
325C23C2×#1175819811yescharacteristic
326C23#1451975812yes—
327C23#1451976112yes—
328D4—44no—
329D4—44no—
330D4—44no—
331D4—44no—
332C2×C4—28no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—82no—
338C2×C4—84no—
339C2×C4—4882no—
340C2×C4—4884no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4#1451969112yes—
348C2×C4#1451973312yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—88no—
372C22—88no—
373C22#1683962311yescharacteristic
374C22C2×#1451969111yescharacteristic
375C22C2×#1451969111yescharacteristic
376C4—22no—
377C4—22no—
378C4—162no—
379C4—9762no—
380C4—4882no—
381C4—4882no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—42no—
386C2—42no—
387C2—81no—
388C2—81no—
389C2#2144589111yescharacteristic
390C2#2144748612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1781C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenter#1781D61
hypocenterC61#12858
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #1781

Complements

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

NH ≅ G/Ntype
3. #21443943C2semidirect
5. C2×#14519736C4semidirect
6. C2×#14519736C4semidirect
11. #16839665C4semidirect
215. C61#12858semidirect

Maximal subgroups

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

subgroupindex
#214484012
#214457342
#214439432
C61×#17812
#1285861